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What Is the Critical Period Hypothesis?

The critical period hypothesis is a theory in the study of language acquisition which posits that there is a critical period of time in which the human mind can most easily acquire language. This idea is often considered with regard to primary language acquisition, and those who agree with this hypothesis argue that language must be learned in the first few years of life or else the ability to acquire language is greatly hindered. The critical period hypothesis is also used in secondary language acquisition, regarding the idea of a time period in which a secondary language can be most easily acquired.

With regard to primary language acquisition, which refers to the process by which a person learns his or her first language , the critical period hypothesis is quite dramatic. This idea indicates that a person has only a set period of time in which he or she can learn a first language, usually the first three to ten years of development. During this time, language can be learned and acquired through exposure to language; simply hearing others talking on an ongoing and regular basis is sufficient. Once this time period is over, however, those who agree with the critical period hypothesis argue that primary language acquisition may be impossible or greatly impaired.

There is a great deal of research into human brain development that supports this hypothesis, but it is still difficult to prove. One of the only conclusive ways to prove this hypothesis would be to have a person isolated from infancy until about the age of ten, without exposure to human speech. Such upbringing would be unthinkable, however, so this type of experiment cannot be conducted and the hypothesis remains largely unproven.

Unfortunate situations in which a child has been abused and isolated by his or her caregivers have provided opportunities to support the critical period hypothesis. In at least one instance, medical care and study of the child did demonstrate that full language acquisition was nearly impossible. Though this occurrence did support the hypothesis, secondary factors such as possible brain damage make the evidence flawed.

The critical period hypothesis is also frequently applied to secondary language acquisition, though in a somewhat less dramatic way. With regard to secondary language, many linguists and speech therapists agree that a second language can be acquired more easily when someone is young. Studies of the brain indicate that in youth the brain is still developing more quickly and new linguistic information can be processed and incorporated into the brain more easily. Once this period is over, however, secondary language acquisition is still certainly possible, though it can be more difficult.

• https://www.pbs.org/wgbh/nova/transcripts/2112gchild.html

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The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis

Jan vanhove.

Department of Multilingualism, University of Fribourg, Fribourg, Switzerland

Analyzed the data: JV. Wrote the paper: JV.

Associated Data

In second language acquisition research, the critical period hypothesis ( cph ) holds that the function between learners' age and their susceptibility to second language input is non-linear. This paper revisits the indistinctness found in the literature with regard to this hypothesis's scope and predictions. Even when its scope is clearly delineated and its predictions are spelt out, however, empirical studies–with few exceptions–use analytical (statistical) tools that are irrelevant with respect to the predictions made. This paper discusses statistical fallacies common in cph research and illustrates an alternative analytical method (piecewise regression) by means of a reanalysis of two datasets from a 2010 paper purporting to have found cross-linguistic evidence in favour of the cph . This reanalysis reveals that the specific age patterns predicted by the cph are not cross-linguistically robust. Applying the principle of parsimony, it is concluded that age patterns in second language acquisition are not governed by a critical period. To conclude, this paper highlights the role of confirmation bias in the scientific enterprise and appeals to second language acquisition researchers to reanalyse their old datasets using the methods discussed in this paper. The data and R commands that were used for the reanalysis are provided as supplementary materials.

Introduction

In the long term and in immersion contexts, second-language (L2) learners starting acquisition early in life – and staying exposed to input and thus learning over several years or decades – undisputedly tend to outperform later learners. Apart from being misinterpreted as an argument in favour of early foreign language instruction, which takes place in wholly different circumstances, this general age effect is also sometimes taken as evidence for a so-called ‘critical period’ ( cp ) for second-language acquisition ( sla ). Derived from biology, the cp concept was famously introduced into the field of language acquisition by Penfield and Roberts in 1959 [1] and was refined by Lenneberg eight years later [2] . Lenneberg argued that language acquisition needed to take place between age two and puberty – a period which he believed to coincide with the lateralisation process of the brain. (More recent neurological research suggests that different time frames exist for the lateralisation process of different language functions. Most, however, close before puberty [3] .) However, Lenneberg mostly drew on findings pertaining to first language development in deaf children, feral children or children with serious cognitive impairments in order to back up his claims. For him, the critical period concept was concerned with the implicit “automatic acquisition” [2, p. 176] in immersion contexts and does not preclude the possibility of learning a foreign language after puberty, albeit with much conscious effort and typically less success.

sla research adopted the critical period hypothesis ( cph ) and applied it to second and foreign language learning, resulting in a host of studies. In its most general version, the cph for sla states that the ‘susceptibility’ or ‘sensitivity’ to language input varies as a function of age, with adult L2 learners being less susceptible to input than child L2 learners. Importantly, the age–susceptibility function is hypothesised to be non-linear. Moving beyond this general version, we find that the cph is conceptualised in a multitude of ways [4] . This state of affairs requires scholars to make explicit their theoretical stance and assumptions [5] , but has the obvious downside that critical findings risk being mitigated as posing a problem to only one aspect of one particular conceptualisation of the cph , whereas other conceptualisations remain unscathed. This overall vagueness concerns two areas in particular, viz. the delineation of the cph 's scope and the formulation of testable predictions. Delineating the scope and formulating falsifiable predictions are, needless to say, fundamental stages in the scientific evaluation of any hypothesis or theory, but the lack of scholarly consensus on these points seems to be particularly pronounced in the case of the cph . This article therefore first presents a brief overview of differing views on these two stages. Then, once the scope of their cph version has been duly identified and empirical data have been collected using solid methods, it is essential that researchers analyse the data patterns soundly in order to assess the predictions made and that they draw justifiable conclusions from the results. As I will argue in great detail, however, the statistical analysis of data patterns as well as their interpretation in cph research – and this includes both critical and supportive studies and overviews – leaves a great deal to be desired. Reanalysing data from a recent cph -supportive study, I illustrate some common statistical fallacies in cph research and demonstrate how one particular cph prediction can be evaluated.

Delineating the scope of the critical period hypothesis

First, the age span for a putative critical period for language acquisition has been delimited in different ways in the literature [4] . Lenneberg's critical period stretched from two years of age to puberty (which he posits at about 14 years of age) [2] , whereas other scholars have drawn the cutoff point at 12, 15, 16 or 18 years of age [6] . Unlike Lenneberg, most researchers today do not define a starting age for the critical period for language learning. Some, however, consider the possibility of the critical period (or a critical period for a specific language area, e.g. phonology) ending much earlier than puberty (e.g. age 9 years [1] , or as early as 12 months in the case of phonology [7] ).

Second, some vagueness remains as to the setting that is relevant to the cph . Does the critical period constrain implicit learning processes only, i.e. only the untutored language acquisition in immersion contexts or does it also apply to (at least partly) instructed learning? Most researchers agree on the former [8] , but much research has included subjects who have had at least some instruction in the L2.

Third, there is no consensus on what the scope of the cp is as far as the areas of language that are concerned. Most researchers agree that a cp is most likely to constrain the acquisition of pronunciation and grammar and, consequently, these are the areas primarily looked into in studies on the cph [9] . Some researchers have also tried to define distinguishable cp s for the different language areas of phonetics, morphology and syntax and even for lexis (see [10] for an overview).

Fourth and last, research into the cph has focused on ‘ultimate attainment’ ( ua ) or the ‘final’ state of L2 proficiency rather than on the rate of learning. From research into the rate of acquisition (e.g. [11] – [13] ), it has become clear that the cph cannot hold for the rate variable. In fact, it has been observed that adult learners proceed faster than child learners at the beginning stages of L2 acquisition. Though theoretical reasons for excluding the rate can be posited (the initial faster rate of learning in adults may be the result of more conscious cognitive strategies rather than to less conscious implicit learning, for instance), rate of learning might from a different perspective also be considered an indicator of ‘susceptibility’ or ‘sensitivity’ to language input. Nevertheless, contemporary sla scholars generally seem to concur that ua and not rate of learning is the dependent variable of primary interest in cph research. These and further scope delineation problems relevant to cph research are discussed in more detail by, among others, Birdsong [9] , DeKeyser and Larson-Hall [14] , Long [10] and Muñoz and Singleton [6] .

Formulating testable hypotheses

Once the relevant cph 's scope has satisfactorily been identified, clear and testable predictions need to be drawn from it. At this stage, the lack of consensus on what the consequences or the actual observable outcome of a cp would have to look like becomes evident. As touched upon earlier, cph research is interested in the end state or ‘ultimate attainment’ ( ua ) in L2 acquisition because this “determines the upper limits of L2 attainment” [9, p. 10]. The range of possible ultimate attainment states thus helps researchers to explore the potential maximum outcome of L2 proficiency before and after the putative critical period.

One strong prediction made by some cph exponents holds that post- cp learners cannot reach native-like L2 competences. Identifying a single native-like post- cp L2 learner would then suffice to falsify all cph s making this prediction. Assessing this prediction is difficult, however, since it is not clear what exactly constitutes sufficient nativelikeness, as illustrated by the discussion on the actual nativelikeness of highly accomplished L2 speakers [15] , [16] . Indeed, there exists a real danger that, in a quest to vindicate the cph , scholars set the bar for L2 learners to match monolinguals increasingly higher – up to Swiftian extremes. Furthermore, the usefulness of comparing the linguistic performance in mono- and bilinguals has been called into question [6] , [17] , [18] . Put simply, the linguistic repertoires of mono- and bilinguals differ by definition and differences in the behavioural outcome will necessarily be found, if only one digs deep enough.

A second strong prediction made by cph proponents is that the function linking age of acquisition and ultimate attainment will not be linear throughout the whole lifespan. Before discussing how this function would have to look like in order for it to constitute cph -consistent evidence, I point out that the ultimate attainment variable can essentially be considered a cumulative measure dependent on the actual variable of interest in cph research, i.e. susceptibility to language input, as well as on such other factors like duration and intensity of learning (within and outside a putative cp ) and possibly a number of other influencing factors. To elaborate, the behavioural outcome, i.e. ultimate attainment, can be assumed to be integrative to the susceptibility function, as Newport [19] correctly points out. Other things being equal, ultimate attainment will therefore decrease as susceptibility decreases. However, decreasing ultimate attainment levels in and by themselves represent no compelling evidence in favour of a cph . The form of the integrative curve must therefore be predicted clearly from the susceptibility function. Additionally, the age of acquisition–ultimate attainment function can take just about any form when other things are not equal, e.g. duration of learning (Does learning last up until time of testing or only for a more or less constant number of years or is it dependent on age itself?) or intensity of learning (Do learners always learn at their maximum susceptibility level or does this intensity vary as a function of age, duration, present attainment and motivation?). The integral of the susceptibility function could therefore be of virtually unlimited complexity and its parameters could be adjusted to fit any age of acquisition–ultimate attainment pattern. It seems therefore astonishing that the distinction between level of sensitivity to language input and level of ultimate attainment is rarely made in the literature. Implicitly or explicitly [20] , the two are more or less equated and the same mathematical functions are expected to describe the two variables if observed across a range of starting ages of acquisition.

But even when the susceptibility and ultimate attainment variables are equated, there remains controversy as to what function linking age of onset of acquisition and ultimate attainment would actually constitute evidence for a critical period. Most scholars agree that not any kind of age effect constitutes such evidence. More specifically, the age of acquisition–ultimate attainment function would need to be different before and after the end of the cp [9] . According to Birdsong [9] , three basic possible patterns proposed in the literature meet this condition. These patterns are presented in Figure 1 . The first pattern describes a steep decline of the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function up to the end of the cp and a practically non-existent age effect thereafter. Pattern 2 is an “unconventional, although often implicitly invoked” [9, p. 17] notion of the cp function which contains a period of peak attainment (or performance at ceiling), i.e. performance does not vary as a function of age, which is often referred to as a ‘window of opportunity’. This time span is followed by an unbounded decline in ua depending on aoa . Pattern 3 includes characteristics of patterns 1 and 2. At the beginning of the aoa range, performance is at ceiling. The next segment is a downward slope in the age function which ends when performance reaches its floor. Birdsong points out that all of these patterns have been reported in the literature. On closer inspection, however, he concludes that the most convincing function describing these age effects is a simple linear one. Hakuta et al. [21] sketch further theoretically possible predictions of the cph in which the mean performance drops drastically and/or the slope of the aoa – ua proficiency function changes at a certain point.

The graphs are based on based on Figure 2 in [9] .

Although several patterns have been proposed in the literature, it bears pointing out that the most common explicit prediction corresponds to Birdsong's first pattern, as exemplified by the following crystal-clear statement by DeKeyser, one of the foremost cph proponents:

[A] strong negative correlation between age of acquisition and ultimate attainment throughout the lifespan (or even from birth through middle age), the only age effect documented in many earlier studies, is not evidence for a critical period…[T]he critical period concept implies a break in the AoA–proficiency function, i.e., an age (somewhat variable from individual to individual, of course, and therefore an age range in the aggregate) after which the decline of success rate in one or more areas of language is much less pronounced and/or clearly due to different reasons. [22, p. 445].

DeKeyser and before him among others Johnson and Newport [23] thus conceptualise only one possible pattern which would speak in favour of a critical period: a clear negative age effect before the end of the critical period and a much weaker (if any) negative correlation between age and ultimate attainment after it. This ‘flattened slope’ prediction has the virtue of being much more tangible than the ‘potential nativelikeness’ prediction: Testing it does not necessarily require comparing the L2-learners to a native control group and thus effectively comparing apples and oranges. Rather, L2-learners with different aoa s can be compared amongst themselves without the need to categorise them by means of a native-speaker yardstick, the validity of which is inevitably going to be controversial [15] . In what follows, I will concern myself solely with the ‘flattened slope’ prediction, arguing that, despite its clarity of formulation, cph research has generally used analytical methods that are irrelevant for the purposes of actually testing it.

Inferring non-linearities in critical period research: An overview

Group mean or proportion comparisons

[T]he main differences can be found between the native group and all other groups – including the earliest learner group – and between the adolescence group and all other groups. However, neither the difference between the two childhood groups nor the one between the two adulthood groups reached significance, which indicates that the major changes in eventual perceived nativelikeness of L2 learners can be associated with adolescence. [15, p. 270].

Similar group comparisons aimed at investigating the effect of aoa on ua have been carried out by both cph advocates and sceptics (among whom Bialystok and Miller [25, pp. 136–139], Birdsong and Molis [26, p. 240], Flege [27, pp. 120–121], Flege et al. [28, pp. 85–86], Johnson [29, p. 229], Johnson and Newport [23, p. 78], McDonald [30, pp. 408–410] and Patowski [31, pp. 456–458]). To be clear, not all of these authors drew direct conclusions about the aoa – ua function on the basis of these groups comparisons, but their group comparisons have been cited as indicative of a cph -consistent non-continuous age effect, as exemplified by the following quote by DeKeyser [22] :

Where group comparisons are made, younger learners always do significantly better than the older learners. The behavioral evidence, then, suggests a non-continuous age effect with a “bend” in the AoA–proficiency function somewhere between ages 12 and 16. [22, p. 448].

The first problem with group comparisons like these and drawing inferences on the basis thereof is that they require that a continuous variable, aoa , be split up into discrete bins. More often than not, the boundaries between these bins are drawn in an arbitrary fashion, but what is more troublesome is the loss of information and statistical power that such discretisation entails (see [32] for the extreme case of dichotomisation). If we want to find out more about the relationship between aoa and ua , why throw away most of the aoa information and effectively reduce the ua data to group means and the variance in those groups?

Comparison of correlation coefficients

Correlation-based inferences about slope discontinuities have similarly explicitly been made by cph advocates and skeptics alike, e.g. Bialystok and Miller [25, pp. 136 and 140], DeKeyser and colleagues [22] , [44] and Flege et al. [45, pp. 166 and 169]. Others did not explicitly infer the presence or absence of slope differences from the subset correlations they computed (among others Birdsong and Molis [26] , DeKeyser [8] , Flege et al. [28] and Johnson [29] ), but their studies nevertheless featured in overviews discussing discontinuities [14] , [22] . Indeed, the most recent overview draws a strong conclusion about the validity of the cph 's ‘flattened slope’ prediction on the basis of these subset correlations:

In those studies where the two groups are described separately, the correlation is much higher for the younger than for the older group, except in Birdsong and Molis (2001) [ =  [26] , JV], where there was a ceiling effect for the younger group. This global picture from more than a dozen studies provides support for the non-continuity of the decline in the AoA–proficiency function, which all researchers agree is a hallmark of a critical period phenomenon. [22, p. 448].

In Johnson and Newport's specific case [23] , their correlation-based inference that ua levels off after puberty happened to be largely correct: the gjt scores are more or less randomly distributed around a near-horizontal trend line [26] . Ultimately, however, it rests on the fallacy of confusing correlation coefficients with slopes, which seriously calls into question conclusions such as DeKeyser's (cf. the quote above).

It can then straightforwardly be deduced that, other things equal, the aoa – ua correlation in the older group decreases as the ua variance in the older group increases relative to the ua variance in the younger group (Eq. 3).

Lower correlation coefficients in older aoa groups may therefore be largely due to differences in ua variance, which have been reported in several studies [23] , [26] , [28] , [29] (see [46] for additional references). Greater variability in ua with increasing age is likely due to factors other than age proper [47] , such as the concomitant greater variability in exposure to literacy, degree of education, motivation and opportunity for language use, and by itself represents evidence neither in favour of nor against the cph .

Regression approaches

Having demonstrated that neither group mean or proportion comparisons nor correlation coefficient comparisons can directly address the ‘flattened slope’ prediction, I now turn to the studies in which regression models were computed with aoa as a predictor variable and ua as the outcome variable. Once again, this category of studies is not mutually exclusive with the two categories discussed above.

In a large-scale study using self-reports and approximate aoa s derived from a sample of the 1990 U.S. Census, Stevens found that the probability with which immigrants from various countries stated that they spoke English ‘very well’ decreased curvilinearly as a function of aoa [48] . She noted that this development is similar to the pattern found by Johnson and Newport [23] but that it contains no indication of an “abruptly defined ‘critical’ or sensitive period in L2 learning” [48, p. 569]. However, she modelled the self-ratings using an ordinal logistic regression model in which the aoa variable was logarithmically transformed. Technically, this is perfectly fine, but one should be careful not to read too much into the non-linear curves found. In logistic models, the outcome variable itself is modelled linearly as a function of the predictor variables and is expressed in log-odds. In order to compute the corresponding probabilities, these log-odds are transformed using the logistic function. Consequently, even if the model is specified linearly, the predicted probabilities will not lie on a perfectly straight line when plotted as a function of any one continuous predictor variable. Similarly, when the predictor variable is first logarithmically transformed and then used to linearly predict an outcome variable, the function linking the predicted outcome variables and the untransformed predictor variable is necessarily non-linear. Thus, non-linearities follow naturally from Stevens's model specifications. Moreover, cph -consistent discontinuities in the aoa – ua function cannot be found using her model specifications as they did not contain any parameters allowing for this.

Using data similar to Stevens's, Bialystok and Hakuta found that the link between the self-rated English competences of Chinese- and Spanish-speaking immigrants and their aoa could be described by a straight line [49] . In contrast to Stevens, Bialystok and Hakuta used a regression-based method allowing for changes in the function's slope, viz. locally weighted scatterplot smoothing ( lowess ). Informally, lowess is a non-parametrical method that relies on an algorithm that fits the dependent variable for small parts of the range of the independent variable whilst guaranteeing that the overall curve does not contain sudden jumps (for technical details, see [50] ). Hakuta et al. used an even larger sample from the same 1990 U.S. Census data on Chinese- and Spanish-speaking immigrants (2.3 million observations) [21] . Fitting lowess curves, no discontinuities in the aoa – ua slope could be detected. Moreover, the authors found that piecewise linear regression models, i.e. regression models containing a parameter that allows a sudden drop in the curve or a change of its slope, did not provide a better fit to the data than did an ordinary regression model without such a parameter.

To sum up, I have argued at length that regression approaches are superior to group mean and correlation coefficient comparisons for the purposes of testing the ‘flattened slope’ prediction. Acknowledging the reservations vis-à-vis self-estimated ua s, we still find that while the relationship between aoa and ua is not necessarily perfectly linear in the studies discussed, the data do not lend unequivocal support to this prediction. In the following section, I will reanalyse data from a recent empirical paper on the cph by DeKeyser et al. [44] . The first goal of this reanalysis is to further illustrate some of the statistical fallacies encountered in cph studies. Second, by making the computer code available I hope to demonstrate how the relevant regression models, viz. piecewise regression models, can be fitted and how the aoa representing the optimal breakpoint can be identified. Lastly, the findings of this reanalysis will contribute to our understanding of how aoa affects ua as measured using a gjt .

Summary of DeKeyser et al. (2010)

I chose to reanalyse a recent empirical paper on the cph by DeKeyser et al. [44] (henceforth DK et al.). This paper lends itself well to a reanalysis since it exhibits two highly commendable qualities: the authors spell out their hypotheses lucidly and provide detailed numerical and graphical data descriptions. Moreover, the paper's lead author is very clear on what constitutes a necessary condition for accepting the cph : a non-linearity in the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function, with ua declining less strongly as a function of aoa in older, post- cp arrivals compared to younger arrivals [14] , [22] . Lastly, it claims to have found cross-linguistic evidence from two parallel studies backing the cph and should therefore be an unsuspected source to cph proponents.

The authors set out to test the following hypotheses:

• Hypothesis 1: For both the L2 English and the L2 Hebrew group, the slope of the age of arrival–ultimate attainment function will not be linear throughout the lifespan, but will instead show a marked flattening between adolescence and adulthood.
• Hypothesis 2: The relationship between aptitude and ultimate attainment will differ markedly for the young and older arrivals, with significance only for the latter. (DK et al., p. 417)

Both hypotheses were purportedly confirmed, which in the authors' view provides evidence in favour of cph . The problem with this conclusion, however, is that it is based on a comparison of correlation coefficients. As I have argued above, correlation coefficients are not to be confused with regression coefficients and cannot be used to directly address research hypotheses concerning slopes, such as Hypothesis 1. In what follows, I will reanalyse the relationship between DK et al.'s aoa and gjt data in order to address Hypothesis 1. Additionally, I will lay bare a problem with the way in which Hypothesis 2 was addressed. The extracted data and the computer code used for the reanalysis are provided as supplementary materials, allowing anyone interested to scrutinise and easily reproduce my whole analysis and carry out their own computations (see ‘supporting information’).

Data extraction

In order to verify whether we did in fact extract the data points to a satisfactory degree of accuracy, I computed summary statistics for the extracted aoa and gjt data and checked these against the descriptive statistics provided by DK et al. (pp. 421 and 427). These summary statistics for the extracted data are presented in Table 1 . In addition, I computed the correlation coefficients for the aoa – gjt relationship for the whole aoa range and for aoa -defined subgroups and checked these coefficients against those reported by DK et al. (pp. 423 and 428). The correlation coefficients computed using the extracted data are presented in Table 2 . Both checks strongly suggest the extracted data to be virtually identical to the original data, and Dr DeKeyser confirmed this to be the case in response to an earlier draft of the present paper (personal communication, 6 May 2013).

Results and Discussion

Modelling the link between age of onset of acquisition and ultimate attainment.

I first replotted the aoa and gjt data we extracted from DK et al.'s scatterplots and added non-parametric scatterplot smoothers in order to investigate whether any changes in slope in the aoa – gjt function could be revealed, as per Hypothesis 1. Figures 3 and ​ and4 4 show this not to be the case. Indeed, simple linear regression models that model gjt as a function of aoa provide decent fits for both the North America and the Israel data, explaining 65% and 63% of the variance in gjt scores, respectively. The parameters of these models are given in Table 3 .

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 1.

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 5.

To ensure that both segments are joined at the breakpoint, the predictor variable is first centred at the breakpoint value, i.e. the breakpoint value is subtracted from the original predictor variable values. For a blow-by-blow account of how such models can be fitted in r , I refer to an example analysis by Baayen [55, pp. 214–222].

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

Solid: regression with breakpoint at aoa 16 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

Solid: regression with breakpoint at aoa 6 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

In sum, a regression model that allows for changes in the slope of the the aoa – gjt function to account for putative critical period effects provides a somewhat better fit to the North American data than does an everyday simple regression model. The improvement in model fit is marginal, however, and including a breakpoint does not result in any detectable improvement of model fit to the Israel data whatsoever. Breakpoint models therefore fail to provide solid cross-linguistic support in favour of critical period effects: across both data sets, gjt can satisfactorily be modelled as a linear function of aoa .

On partialling out ‘age at testing’

As I have argued above, correlation coefficients cannot be used to test hypotheses about slopes. When the correct procedure is carried out on DK et al.'s data, no cross-linguistically robust evidence for changes in the aoa – gjt function was found. In addition to comparing the zero-order correlations between aoa and gjt , however, DK et al. computed partial correlations in which the variance in aoa associated with the participants' age at testing ( aat ; a potentially confounding variable) was filtered out. They found that these partial correlations between aoa and gjt , which are given in Table 9 , differed between age groups in that they are stronger for younger than for older participants. This, DK et al. argue, constitutes additional evidence in favour of the cph . At this point, I can no longer provide my own analysis of DK et al.'s data seeing as the pertinent data points were not plotted. Nevertheless, the detailed descriptions by DK et al. strongly suggest that the use of these partial correlations is highly problematic. Most importantly, and to reiterate, correlations (whether zero-order or partial ones) are actually of no use when testing hypotheses concerning slopes. Still, one may wonder why the partial correlations differ across age groups. My surmise is that these differences are at least partly the by-product of an imbalance in the sampling procedure.

The upshot of this brief discussion is that the partial correlation differences reported by DK et al. are at least partly the result of an imbalance in the sampling procedure: aoa and aat were simply less intimately tied for the young arrivals in the North America study than for the older arrivals with L2 English or for all of the L2 Hebrew participants. In an ideal world, we would like to fix aat or ascertain that it at most only weakly correlates with aoa . This, however, would result in a strong correlation between aoa and another potential confound variable, length of residence in the L2 environment, bringing us back to square one. Allowing for only moderate correlations between aoa and aat might improve our predicament somewhat, but even in that case, we should tread lightly when making inferences on the basis of statistical control procedures [61] .

On estimating the role of aptitude

Having shown that Hypothesis 1 could not be confirmed, I now turn to Hypothesis 2, which predicts a differential role of aptitude for ua in sla in different aoa groups. More specifically, it states that the correlation between aptitude and gjt performance will be significant only for older arrivals. The correlation coefficients of the relationship between aptitude and gjt are presented in Table 10 .

The problem with both the wording of Hypothesis 2 and the way in which it is addressed is the following: it is assumed that a variable has a reliably different effect in different groups when the effect reaches significance in one group but not in the other. This logic is fairly widespread within several scientific disciplines (see e.g. [62] for a discussion). Nonetheless, it is demonstrably fallacious [63] . Here we will illustrate the fallacy for the specific case of comparing two correlation coefficients.

Apart from not being replicated in the North America study, does this difference actually show anything? I contend that it does not: what is of interest are not so much the correlation coefficients, but rather the interactions between aoa and aptitude in models predicting gjt . These interactions could be investigated by fitting a multiple regression model in which the postulated cp breakpoint governs the slope of both aoa and aptitude. If such a model provided a substantially better fit to the data than a model without a breakpoint for the aptitude slope and if the aptitude slope changes in the expected direction (i.e. a steeper slope for post- cp than for younger arrivals) for different L1–L2 pairings, only then would this particular prediction of the cph be borne out.

Using data extracted from a paper reporting on two recent studies that purport to provide evidence in favour of the cph and that, according to its authors, represent a major improvement over earlier studies (DK et al., p. 417), it was found that neither of its two hypotheses were actually confirmed when using the proper statistical tools. As a matter of fact, the gjt scores continue to decline at essentially the same rate even beyond the end of the putative critical period. According to the paper's lead author, such a finding represents a serious problem to his conceptualisation of the cph [14] ). Moreover, although modelling a breakpoint representing the end of a cp at aoa 16 may improve the statistical model slightly in study on learners of English in North America, the study on learners of Hebrew in Israel fails to confirm this finding. In fact, even if we were to accept the optimal breakpoint computed for the Israel study, it lies at aoa 6 and is associated with a different geometrical pattern.

Diverging age trends in parallel studies with participants with different L2s have similarly been reported by Birdsong and Molis [26] and are at odds with an L2-independent cph . One parsimonious explanation of such conflicting age trends may be that the overall, cross-linguistic age trend is in fact linear, but that fluctuations in the data (due to factors unaccounted for or randomness) may sometimes give rise to a ‘stretched L’-shaped pattern ( Figure 1, left panel ) and sometimes to a ‘stretched 7’-shaped pattern ( Figure 1 , middle panel; see also [66] for a similar comment).

Importantly, the criticism that DeKeyser and Larsson-Hall levy against two studies reporting findings similar to the present [48] , [49] , viz. that the data consisted of self-ratings of questionable validity [14] , does not apply to the present data set. In addition, DK et al. did not exclude any outliers from their analyses, so I assume that DeKeyser and Larsson-Hall's criticism [14] of Birdsong and Molis's study [26] , i.e. that the findings were due to the influence of outliers, is not applicable to the present data either. For good measure, however, I refitted the regression models with and without breakpoints after excluding one potentially problematic data point per model. The following data points had absolute standardised residuals larger than 2.5 in the original models without breakpoints as well as in those with breakpoints: the participant with aoa 17 and a gjt score of 125 in the North America study and the participant with aoa 12 and a gjt score of 117 in the Israel study. The resultant models were virtually identical to the original models (see Script S1 ). Furthermore, the aoa variable was sufficiently fine-grained and the aoa – gjt curve was not ‘presmoothed’ by the prior aggregation of gjt across parts of the aoa range (see [51] for such a criticism of another study). Lastly, seven of the nine “problems with supposed counter-evidence” to the cph discussed by Long [5] do not apply either, viz. (1) “[c]onfusion of rate and ultimate attainment”, (2) “[i]nappropriate choice of subjects”, (3) “[m]easurement of AO”, (4) “[l]eading instructions to raters”, (6) “[u]se of markedly non-native samples making near-native samples more likely to sound native to raters”, (7) “[u]nreliable or invalid measures”, and (8) “[i]nappropriate L1–L2 pairings”. Problem No. 5 (“Assessments based on limited samples and/or “language-like” behavior”) may be apropos given that only gjt data were used, leaving open the theoretical possibility that other measures might have yielded a different outcome. Finally, problem No. 9 (“Faulty interpretation of statistical patterns”) is, of course, precisely what I have turned the spotlights on.

Conclusions

The critical period hypothesis remains a hotly contested issue in the psycholinguistics of second-language acquisition. Discussions about the impact of empirical findings on the tenability of the cph generally revolve around the reliability of the data gathered (e.g. [5] , [14] , [22] , [52] , [67] , [68] ) and such methodological critiques are of course highly desirable. Furthermore, the debate often centres on the question of exactly what version of the cph is being vindicated or debunked. These versions differ mainly in terms of its scope, specifically with regard to the relevant age span, setting and language area, and the testable predictions they make. But even when the cph 's scope is clearly demarcated and its main prediction is spelt out lucidly, the issue remains to what extent the empirical findings can actually be marshalled in support of the relevant cph version. As I have shown in this paper, empirical data have often been taken to support cph versions predicting that the relationship between age of acquisition and ultimate attainment is not strictly linear, even though the statistical tools most commonly used (notably group mean and correlation coefficient comparisons) were, crudely put, irrelevant to this prediction. Methods that are arguably valid, e.g. piecewise regression and scatterplot smoothing, have been used in some studies [21] , [26] , [49] , but these studies have been criticised on other grounds. To my knowledge, such methods have never been used by scholars who explicitly subscribe to the cph .

I suspect that what may be going on is a form of ‘confirmation bias’ [69] , a cognitive bias at play in diverse branches of human knowledge seeking: Findings judged to be consistent with one's own hypothesis are hardly questioned, whereas findings inconsistent with one's own hypothesis are scrutinised much more strongly and criticised on all sorts of points [70] – [73] . My reanalysis of DK et al.'s recent paper may be a case in point. cph exponents used correlation coefficients to address their prediction about the slope of a function, as had been done in a host of earlier studies. Finding a result that squared with their expectations, they did not question the technical validity of their results, or at least they did not report this. (In fact, my reanalysis is actually a case in point in two respects: for an earlier draft of this paper, I had computed the optimal position of the breakpoints incorrectly, resulting in an insignificant improvement of model fit for the North American data rather than a borderline significant one. Finding a result that squared with my expectations, I did not question the technical validity of my results – until this error was kindly pointed out to me by Martijn Wieling (University of Tübingen).) That said, I am keen to point out that the statistical analyses in this particular paper, though suboptimal, are, as far as I could gather, reported correctly, i.e. the confirmation bias does not seem to have resulted in the blatant misreportings found elsewhere (see [74] for empirical evidence and discussion). An additional point to these authors' credit is that, apart from explicitly identifying their cph version's scope and making crystal-clear predictions, they present data descriptions that actually permit quantitative reassessments and have a history of doing so (e.g. the appendix in [8] ). This leads me to believe that they analysed their data all in good conscience and to hope that they, too, will conclude that their own data do not, in fact, support their hypothesis.

I end this paper on an upbeat note. Even though I have argued that the analytical tools employed in cph research generally leave much to be desired, the original data are, so I hope, still available. This provides researchers, cph supporters and sceptics alike, with an exciting opportunity to reanalyse their data sets using the tools outlined in the present paper and publish their findings at minimal cost of time and resources (for instance, as a comment to this paper). I would therefore encourage scholars to engage their old data sets and to communicate their analyses openly, e.g. by voluntarily publishing their data and computer code alongside their articles or comments. Ideally, cph supporters and sceptics would join forces to agree on a protocol for a high-powered study in order to provide a truly convincing answer to a core issue in sla .

Supporting Information

aoa and gjt data extracted from DeKeyser et al.'s North America study.

aoa and gjt data extracted from DeKeyser et al.'s Israel study.

Script with annotated R code used for the reanalysis. All add-on packages used can be installed from within R.

Acknowledgments

I would like to thank Irmtraud Kaiser (University of Fribourg) for helping me to get an overview of the literature on the critical period hypothesis in second language acquisition. Thanks are also due to Martijn Wieling (currently University of Tübingen) for pointing out an error in the R code accompanying an earlier draft of this paper.

Funding Statement

No current external funding sources for this study.

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What are the main arguments for and against the critical period hypothesis, and what are alternative explanations?

Why is the critical period hypothesis so heavily disputed, yet widely accepted; what are its major strengths and weaknesses; what other explanations exist for the perceived "critical period", if it does not exist?

• critical-period

Let us start with a simple, relatively informal statement: “in most cases, those who start learning a language as children become ultimately become more proficient in a language than those who start learning it later”. This is uncontroversial, and something I think the vast majority of second-language acquisition researchers would agree on. However, this is not how the Critical Period Hypothesis (CPH) is understood within the field of Second Language Acquisition (SLA). CPH is a large subject, and your question is hard to answer in a few paragraphs. Therefore, I am reusing large fragments of an assignment I wrote on this very topic for an SLA course a few years ago. Let me know if something is unclear or the style is too terse at some points.

Summary (TL;DR)

There is no universally accepted definition of a critical period within linguistics and some of the controversies are caused by the fact that different researchers use different definitions.

There are a few key findings that are not controversial:

• early learners perform consistently well in all aspects of language use,
• as we move the starting age, they perform statistically worse and worse until puberty,
• however, the decrease in performance is not uniform.

An explanation, provided by Bialystok (1997) as an alternative to CPH, is the different learning style of children, compared to late learners.

Paradoxically, the differences (or lack thereof) between those who learn a foreign language as adults is the key factor in deciding whether CPH is true or not, and a controversial one:

• Some studies found correlations between the age adult learners started learning a language and their ultimate attainment. In other words, these studies suggest that if we compare people who have been learning a language for a very long time, the ultimate attainment of those who started at the age of 20 is statistically higher than the ultimate attainment of those who started at the age of 40. These studies argue that there is no CPH in the childhood, but rather that our abilities in learning a new language consistently decrease throughout our whole lives.
• Other studies found no clear correlations between the starting age and the ultimate attainment among adult language learners. They point out that the correlation between the starting age and ultimate attainment is clear for those who started before puberty. Based on that, they argue that there is something qualitatively different about starting to learn in an early age, and therefore conclude that it is an argument for CPH.

Definitions of the critical period used by those who argue against CPH

Controversies with the Critical Period Hypothesis (CPH) are related to the issue of ultimate attainment of early and late language learners, that is, the highest language proficiency level they can attain. The patterns in ultimate attainment may be explained by CPH, but they may also have different explanations. Some researchers support some form of the Critical Period Hypothesis (Johnson and Newport 1989, DeKeyser and Larson-Hall 2005, Patkowsky 1994, Scovel 1988), while others argue against it (Bialystok 1997).

A major problem with the Critical Period Hypothesis is that there appears to be no universally accepted definition of a critical period within linguistics . Bialystok (1997) bases her discussion of the critical/sensitive period (which she takes to be synonymous 1 ) on a specific technical definition used in ethology, which includes 14 essential structural characteristics that describe such a period (Bornstein 1989). She argues that one of these characteristics is especially problematic – the system: “structure or function altered in the sensitive period” (Bornstein 1989:184). In other words, she argues that there is no period where a structure in the brain is modified in a way that makes subsequent language learning harder or impossible. Bialystok does, however, agree that there is an optimal period for language learning – something that can be characterised by the statement “ On average, children are more successful than adults when faced with the task of learning a second language ” (Bialystok 1997:117). Despite the controversy around other issues, this fact is uncontested and has been verified by numerous studies .

Bialystok (1997) rejects the existence of a critical period, because of lack of postulated structure that is modified when the period is over. She postulates that an important factor that causes differences in ultimate attainment between early and late starters is learning style: children prefer accommodation (creating new concepts) over assimilation (extending existing concepts). The question remains: why do they prefer accommodation? She suggests that “[t]his may be because children are in the process of creating new categories all the time as they are learning new information” (Bialystok 1997:132).

Definitions of the critical period used by supporters of CPH

The researchers who support some form of the Critical Period Hypothesis (Johnson and Newport 1989, DeKeyser and Larson-Hall 2005), formulate it in a form that is much weaker than Bialystok's (1997) formulation. What they postulate often resembles what Bialystok calls the optimal age.

Johnson and Newport (1989) reformulated CPH into two alternative hypotheses, in order to fit second language acquisition into the picture:

The exercise hypothesis : “Early in life, humans have a superior capacity for acquiring languages. If the capacity is not exercised during this time, it will disappear or decline with maturation. If the capacity is exercised, however, further language learning abilities will remain intact throughout life.” (Johnson and Newport 1989:64)

The maturational state hypothesis : “Early in life, humans have a superior capacity for acquiring languages. This capacity disappears or declines with maturation.” (Johnson and Newport 1989:64)

We can see that if a critical period was found for second language acquisition, we could be almost sure that it exists for L1 acquisition as well (the maturational state hypothesis). However, we cannot deduce in this way in case of the exercise hypothesis – non-existence of a critical period for L2 acquisition does not exclude in any way a possibility of such period for the first language (Bialystok 1997).

DeKeyser and Larson-Hall (2005) formulate the hypothesis as: “language acquisition from mere exposure (i.e. implicit learning) […] is severely limited in older adolescents and adults”. Their formulation is quite vague, as is the constatation that there is a “qualitative change in language learning capacities somewhere between 4 and 18 years”.

There are also definitions that restrict the Critical Period Hypothesis to specific subareas of the language faculty. The most commonly mentioned area is phonology, see e.g. Patkowsky (1994, cited in Bialystok 1997), Scovel (1988, cited in Bialystok 1997).

Age effects before and after puberty

The current consensus is that early learners perform consistently well in all aspects of language use. As we move the starting age, they perform statistically worse and worse until puberty. The decrease in performance is not uniform, and in some areas (such as phonology) it is particularly visible. Performance on the same level as early bilinguals is possible, but rare.

Probably the most controversial aspect is the performance of adult learners. After puberty there is much bigger variance in the performance, so data are more prone to different interpretations. The results obtained by Derwing and Munro (2013) suggest that comprehensibility and good accent are negatively correlated with the age of arrival, that is, the age when English language immersion started. Johnson and Newport (1989) found no correlation of starting age after puberty with ultimate language proficiency, while Bialystok (1997) re-analysed these data and found some negative correlation. A meta-analysis by DeKeyser and Larson-Hall (2005) downplays the role of post-adolescent correlations. As we can see, the jury is still out on this debate.

1 In neuroscience critical period and sensitive period are two separate concepts, see Knudsen (2004).

Bibliography

• Bialystok, E. 1997. The structure of age: in search of barriers to second language acquisition. Second Language Research 13(2): 116-137.
• Bornstein, M.H. 1989. Sensitive periods in development: structural characteristics and causal interpretations. Psychological Bulletin 105,179–97.
• DeKeyser, R. and J. Larson-Hall. 2005. What does the critical period really mean? In J. F. Kroll and A. M. B. de Groot. 2005. Handbook of bilingualism: Psycholinguistic approaches . Cary, NC: Oxford University Press. Pp. 109–27.
• Derwing, T. M., & Munro, M. J. 2013. The development of L2 oral language skills in two L1 groups: A 7-year study. Language Learning 63, 163-185.
• Johnson, J.S., & Newport, E.L. 1989. Critical period effects in second language learning: The influence of maturational state on the acquisition of English as a second language. Cognitive Psychology , 21, 60-99.
• Knudsen, E. I. 2004. Sensitive periods in the development of the brain and behavior. Journal of Cognitive Neuroscience , 16, 1412-25
• Newport, E. L., & Supalla, T. 1987. A critical period effect in the acquisition of a primary language .
• Patkowsky, M. 1980. The sensitive period for the acquisition of syntax in a second language. Language Learning 30, 449–72
• Scovel, T. 1988. A time to speak: a psycholinguistic inquiry into the critical period for human speech . New York: Newbury House

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In This Article Expand or collapse the "in this article" section Critical Periods

Introduction, general overviews.

• Critical Periods in Language: Background Readings
• First-Language Acquisition (L1A)
• Age and Second-Language Acquisition (L2A)
• Controversy around the Critical Period Hypothesis for Second-Language Acquisition (CPH/L2A)
• Critical Period Geometry and Timing in L2A
• Brain-Based Studies of Critical Periods in L2A
• Bilingualism
• First-Language Attrition
• Sign Language
• Foreign Language Education
• Animal Models of Critical Periods
• Absolute Pitch

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Critical Periods by David P. Birdsong LAST REVIEWED: 12 January 2023 LAST MODIFIED: 12 January 2023 DOI: 10.1093/obo/9780199772810-0139

A critical period is a bounded maturational span during which experiential factors interact with biological mechanisms to determine neurocognitive and behavioral outcomes. In humans, the construct of critical period (CP) is commonly applied to first-language (L1) and second-language (L2) development. Some language researchers hold that during a CP, various mechanisms are at work that result in successful language acquisition and language processing. Outside of the period, other factors and mechanisms are involved, resulting in deficits in acquisition and processing. Many researchers believe that L1 development is constrained by maturationally based CPs. However, this notion is more controversial in L2 acquisition research, where the Critical Period Hypothesis for L2 acquisition (CPH/L2A) is debated on empirical, theoretical, and methodological grounds. Bilingualism researchers study the possibility that CPs may govern the likelihood and degree of loss (attrition) of the L1 among bilinguals as they age. Studies of CPs in L1 acquisition and L2 acquisition have been conducted with learners of spoken languages, signed languages, and artificial languages. CP research is considered in educational policy, particularly in the context of foreign language instruction. On a terminological note, a distinction is sometimes drawn between “critical” and “sensitive” periods, the latter term denoting receptivity of the organism to shaping by experience (or, in certain studies, suggesting relatively mild effects). Some researchers use these terms interchangeably, while others use one but not the other. Here, “critical period” will be used as a cover term unless specific reference is being made to sensitive period.

Lillard and Erisir 2011 describes juvenile CPs in language, imprinting, and vision. The article includes an informative table covering seven levels of neural changes in the brain in juveniles versus adults, with notes on the time course of changes and affected brain areas in animal and human models. The authors observe that changes in neural architecture triggered by early versus late experiences differ in degree more than type, and that the variety of triggering experiences is reduced with age. A second table summarizes neuroanatomical, electrophysiological, and neuroimaging techniques for observing specific types of neuroplasticity. Knudsen 2004 is exceptionally informative with respect to: prerequisites for CPs; the properties, mechanisms, and timing of plasticity; reopening of critical periods; the roles of presence and absence of relevant stimulation; and sensitive periods versus critical periods. Knudsen points out that complex behaviors (which include language use) may be regulated by multiple CPs. Takesian and Hensch 2013 emphasizes the individual-level plasticity of the timing of CP onset, peak, and offset, which may vary according to excitatory/inhibitory circuit balance that is sensitive to drugs, sleep, trauma, and genetic perturbation (see also Werker and Hensch 2015 [cited under First-Language Acquisition (L1A) ). Reh, et al. 2020 examines how plasticity is regulated at multiple timescales during development and provides examples from language processing, mental illness, and recovery from brain injury. Gabard-Durnam and McLaughlin 2020 outlines a set of current approaches to the study of sensitive periods in humans. These approaches include environmental manipulations (deprivation, enrichment, substitution), plasticity manipulations via pharmacological intervention, and computational modeling. Frankenhuis and Walasek 2020 develops an evolutionary model that accounts for sensitive periods that occur beyond the early stages of ontogeny.

Frankenhuis, Willem E., and Nicole Walasek. 2020. Modeling the evolution of sensitive periods . Developmental Cognitive Neuroscience 41:100715.

DOI: 10.1016/j.dcn.2019.100715

Sensitive periods in mid-ontogeny are favored by natural selection as a function of the reliability of relevant environmental cues.

Gabard-Durnam, Laurel, and Katie A. McLaughlin. 2020. Sensitive periods in human development: Charting a course for the future. Current Opinion in Behavioral Sciences 36:120–128.

DOI: 10.1016/j.cobeha.2020.09.003

Figures 1, 2 and 3 and their captions are particularly informative.

Knudsen, Eric I. 2004. Sensitive periods in the development of brain and behavior. Journal of Cognitive Neuroscience 16.8: 1412–1425.

DOI: 10.1162/0898929042304796

Focuses on the role of experience in modifying neural circuits during periods of plasticity, leading to connectivity patterns that become stable and less energy intensive, and making up what Knudsen calls the “stability landscape.”

Lillard, Angeline S., and Alev Erisir. 2011. Old dogs learning new tricks: Neuroplasticity beyond the juvenile period. Developmental Review 31:207–239.

DOI: 10.1016/j.dr.2011.07.008

A largely uncritical review and synthesis of well-known studies.

Reh, Rebecca K., Brian G. Dias, Charles A. Nelson III, et al. 2020. Critical period regulation across multiple timescales . Proceedings of the National Academy of Sciences of the United States of America 117.38: 23242–23251.

DOI: 10.1073/pnas.1820836117

A diverse group of specialists’ account of neurobiological CP mechanisms in animals and humans. Notes that “cortical plasticity is not only influenced by an animal’s life experiences but may also be modified by that of the parents. This occurs via parental behavior during the offspring’s early postnatal life, the in utero environment during gestation, or modification of the parental or fetal germ cells” (p. 23246).

Takesian, Anne E., and Takao K. Hensch. 2013. Balancing plasticity/stability across brain development. Progress in Brain Research 207:3–34.

DOI: 10.1016/B978-0-444-63327-9.00001-1

Illuminates the dynamic between the intrinsic plasticity of CP and the stabilization of neural networks, which limits maladaptive proliferation of circuit rewiring past the CP.

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Multilingual Pedagogy and World Englishes

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Critical Period Hypothesis (CPH)

Tom Scovel writes, “The CPH [critical period hypothesis] is conceivably the most contentious issue in SLA because there is disagreement over its exact age span; people disagree strenuously over which facets of language are affected; there are competing explanations for its existence; and, to top it off, many people don’t believe it exists at all” (113). Proposed by Wilder Penfield and Lamar Roberts in 1959, the Critical Period Hypothesis (CPH) argues that there is a specific period of time in which people can learn a language without traces of the L1 (a so-called “foreign” accent or even L1 syntactical features) manifesting in L2 production (Scovel 48). If a learner’s goal is to sound “native,” there may be age-related limitations or “maturational constraints” as Kenneth Hyltenstam and Niclas Abrahamsson call them, on how “native” they can sound. Reducing the impression left by the L1 is certainly possible after puberty, but eliminating that impression entirely may not be possible.

Kenji Hakuta et al. explains that the relationship between age and L1 interference in L2 production is really not up for debate:

“The diminished average achievement of older learners is supported by personal anecdote and documented by empirical evidence….What is controversial, though, is whether this pattern meets the conditions for concluding that a critical period constrains learning in a way predicted by the theory” (31).

Some learners manage to overcome the “constraints” that Scovel believes are “probably accounted for by neurological factors that are genetically specified in our species” (114), but these learners are exceptional rather than the rule. It may be biology; it may be due to something else. The debate will continue, but evidence seems to indicate that the older learners become, the more difficult complete acquisition can be.

“David Birdsong, Looking Inside and Beyond the Critical Period Hypothesis.”  YouTube,  uploaded by IWL Channel, 09 May 2016, https://www.youtube.com/watch?v=9Bo0C4dj7Mw.

Application

Instructors should consider taking the CPH into account when assessing their students’ oral communication in the target language. When “maturational constraints” are a potential concern, it seems more fair for instructors to weight comprehension more heavily than nativeness. A thorough understanding of the CPH can also help instructors to counteract adult learners’ “self-handicapping” by helping the learners understand that, in spite of constraints due to aging, they are still capable of acquiring many–if not most–aspects of the target language.

Bibliography

Hakuta, Kenji, et al. “Critical Evidence: A Test of the Critical-Period Hypothesis for Second-Language Acquisition.”  Psychological Science , vol. 14, no. 1, 2003, pp. 31–38.  JSTOR , www.jstor.org/stable/40063748.

Hyltenstam, Kenneth, and Niclas Abrahamsson. “Comments on Stefka H. Marinova-Todd, D. Bradford Marshall, and Catherine E. Snow’s ‘Three Misconceptions about Age and L2 Learning’: Age and L2 Learning: The Hazards of Matching Practical ‘Implications’ with Theoretical ‘Facts.’”  TESOL Quarterly , vol. 35, no. 1, 2001, pp. 151–170.  JSTOR , www.jstor.org/stable/3587863.

Nemer, Randa. “Critical Period Hypothesis.”  Prezi,  04 Dec. 2013, https://prezi.com/zzuch40ibrlq/critical-period-hypothesis-sla/#.

Scovel, Tom.  Learning New Languages . Heinle & Heinle, 2001.

Critical Period In Brain Development and Childhood Learning

Charlotte Nickerson

Research Assistant at Harvard University

Undergraduate at Harvard University

Charlotte Nickerson is a student at Harvard University obsessed with the intersection of mental health, productivity, and design.

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Saul McLeod, PhD

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BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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BSc (Hons) Psychology, MSc Psychology of Education

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Key Takeaways

• Critical period is an ethological term that refers to a fixed and crucial time during the early development of an organism when it can learn things that are essential to survival. These influences impact the development of processes such as hearing and vision, social bonding, and language learning.
• The term is most often experienced in the study of imprinting, where it is thought that young birds could only develop an attachment to the mother during a fixed time soon after hatching.
• Neurologically, critical periods are marked by high levels of plasticity in the brain before neural connections become more solidified and stable. In particular, critical periods tend to end when synapses that inhibit the neurotransmitter GABA mature.
• In contrast to critical periods, sensitive periods, otherwise known as “weak critical periods,” happen when an organism is more sensitive than usual to outside factors influencing behavior, but this influence is not necessarily restricted to the sensitive period.
• Scholars have debated the extent to which older organisms can develop certain skills, such as natively-accented foreign languages, after the critical period.

The critical period is a biologically determined stage of development where an organism is optimally ready to acquire some pattern of behavior that is part of typical development. This period, by definition, will not recur at a later stage.

If an organism does not receive exposure to the appropriate stimulus needed to learn a skill during a critical period, it may be difficult or even impossible for that organism to develop certain functions associated with that skill later in life.

This happens because a range of functional and structural elements prevent passive experiences from eliciting significant changes in the brain (Cisneros-Franco et al., 2020).

The first strong proponent of the theory of critical periods was Charles Stockhard (1921), a biologist who attempted to experiment with the effects of various chemicals on the development of fish embryos, though he gave credit to Dareste for originating the idea 30 years earlier (Scott, 1962).

Stockhard’s experiments showed that applying almost any chemical to fish embryos at a certain stage of development would result in one-eyed fish.

These experiments established that the most rapidly growing tissues in an embryo are the most sensitive to any change in conditions, leading to effects later in development (Scott, 1962).

Meanwhile, psychologist Sigmund Freud attempted to explain the origins of neurosis in human patients as the result of early experiences, implying that infants are particularly sensitive to influences at certain points in their lives.

Lorenz (1935) later emphasized the importance of critical periods in the formation of primary social bonds (otherwise known as imprinting) in birds, remarking that this psychological imprinting was similar to critical periods in the development of the embryo.

Soon thereafter, McGraw (1946) pointed out the existence of critical periods for the optimal learning of motor skills in human infants (Scott, 1962).

Example: Infant-Parent Attachment

The concept of critical or sensitive periods can also be found in the domain of social development, for example, in the formation of the infant-parent attachment relationship (Salkind, 2005).

Attachment describes the strong emotional ties between the infant and caregiver, a reciprocal relationship developing over the first year of the child’s life and particularly during the second six months of the first year.

During this attachment period , the infant’s social behavior becomes increasingly focused on the principal caregivers (Salkind, 2005).

The 20th-century English psychiatrist John Bowlby formulated and presented a comprehensive theory of attachment influenced by evolutionary theory.

Bowlby argued that the infant-parent attachment relationship develops because it is important to the survival of the infant and that the period from six to twenty-four months of age is a critical period of attachment.

This coincides with an infant’s increasing tendency to approach familiar caregivers and to be wary of unfamiliar adults. After this critical period, it is still possible for a first attachment relationship to develop, albeit with greater difficulty (Salkind, 2005).

This has brought into question, in a similar vein to language development, whether there is actually a critical development period for infant-caregiver attachment.

Sources debating this issue typically include cases of infants who did not experience consistent caregiving due to being raised in institutions prior to adoption (Salkind, 2005).

Early research into the critical period of attachment, published in the 1940s, reports consistently that children raised in orphanages subsequently showed unusual and maladaptive patterns of social behavior, difficulty in forming close relationships, and being indiscriminately friendly toward unfamiliar adults (Salkind, 2005).

Later, research from the 1990s indicated that adoptees were actually still able to form attachment relationships after the first year of life and also made developmental progress following adoption.

Nonetheless, these children had an overall increased risk of insecure or maladaptive attachment relationships with their adoptive parents. This evidence supports the notion of a sensitive period, but not a critical period, in the development of first attachment relationships (Salkind, 2005).

Mechanisms for Critical Periods

Both genetics and sensory experiences from outside the body shape the brain as it develops (Knudsen, 2004). However, the developmental stage that an organism is in significantly impacts how much the brain can change based on these experiences.

In scientific terms, the brain’s plasticity changes over the course of a lifespan. The brain is very plastic in the early stages of life before many key connections take root, but less so later.

This is why researchers have shown that early experience is crucial for the development of, say, language and musical abilities, and these skills are more challenging to take up in adulthood (Skoe and Kraus, 2013; White et al., 2013; Hartshorne et al., 2018).

As brains mature, the connections in them become more fixed. The brain’s transitions from a more plastic to a more fixed state advantageously allow it to retain new and complex processes, such as perceptual, motor, and cognitive functions (Piaget, 1962).

Children’s gestures, for example, pride and predict how they will acquire oral language skills (Colonnesi et al., 2010), which in turn are important for developing executive functions (Marcovitch and Zelazo, 2009).

However, this formation of stable connections in the brain can limit how the brain’s neural circuitry can be revised in the future. For example, if a young organism has abnormal sensory experiences during the critical period – such as auditory or visual deprivation – the brain may not wire itself in a way that processes future sensory inputs properly (Gallagher et al., 2020).

One illustration of this is the timing of cochlear implants – a prosthesis that restores hearing in some deaf people. Children who receive cochlear implants before two years of age are more likely to benefit from them than those who are implanted later in life (Kral and Eggermont, 2007; Gallagher et al., 2020).

Similarly, the visual deprivation caused by cataracts in infants can cause similar consequences. When cataracts are removed during early infancy, individuals can develop relatively normal vision; however, when the cataracts are not removed until adulthood, this results in substantially poorer vision (Martins Rosa et al., 2013).

After the critical period closes, abnormal sensory experiences have a less drastic effect on the brain and lead to – barring direct damage to the central nervous system – reversible changes (Gallagher et al., 2020). Much of what scientists know about critical periods derives from animal studies , as these allow researchers greater control over the variables that they are testing.

This research has found that different sensory systems, such as vision, auditory processing, and spatial hearing, have different critical periods (Gallagher et al., 2020).

The brain regulates when critical periods open and close by regulating how much the brain’s synapses take up neurotransmitters , which are chemical substances that affect the transmission of electrical signals between neurons.

In particular, over time, synapses decrease their uptake of gamma-aminobutyric acid, better known as GABA. At the beginning of the critical period, outside sources become more effective at influencing changes and growth in the brain.

Meanwhile, as the inhibitory circuits of the brain mature, the mature brain becomes less sensitive to sensory experiences (Gallagher et al., 2020).

Critical Periods vs Sensitive Periods

Critical periods are similar to sensitive periods, and scholars have, at times, used them interchangeably. However, they describe distinct but overlapping developmental processes.

A sensitive period is a developmental stage where sensory experiences have a greater impact on behavioral and brain development than usual; however, this influence is not exclusive to this time period (Knudsen, 2004; Gallagher, 2020). These sensitive periods are important for skills such as learning a language or instrument.

In contrast, A critical period is a special type of sensitive period – a window where sensory experience is necessary to shape the neural circuits involved in basic sensory processing, and when this window opens and closes is well-defined (Gallagher, 2020).

Researchers also refer to sensitive periods as weak critical periods. Some examples of strong critical periods include the development of vision and hearing, while weak critical periods include phenome tuning – how children learn how to organize sounds in a language, grammar processing, vocabulary acquisition, musical training, and sports training (Gallagher et al., 2020).

Critical Period Hypothesis

One of the most notable applications of the concept of a critical period is in linguistics. Scholars usually trace the origins of the debate around age in language acquisition to Penfield and Robert’s (2014) book Speech and Brain Mechanisms.

In the 1950s and 1960s, Penfield was a staunch advocate of early immersion education (Kroll and De Groot, 2009). Nonetheless, it was Lenneberg, in his book Biological Foundations of Language, who coined the term critical period (1967) in describing the language period.

Lennenberg (1967) described a critical period as a period of automatic acquisition from mere exposure” that “seems to disappear after this age.” Scovel (1969) later summarized and narrowed Penfield’s and Lenneberg’s view on the critical period hypothesis into three main claims:

• Adult native speakers can identify non-natives by their accents immediately and accurately.
• The loss of brain plasticity at about the age of puberty accounts for the emergence of foreign accents./li>
• The critical period hypothesis only holds for speech (whether or not someone has a native accent) and does not affect other areas of linguistic competence.

Linguists have since attempted to find evidence for whether or not scientific evidence actually supports the critical period hypothesis, if there is a critical period for acquiring accentless speech, for “morphosyntactic” competence, and if these are true, how age-related differences can be explained on the neurological level (Scovel, 2000).

The critical period hypothesis applies to both first and second-language learning. Until recently, research around the critical period’s role in first language acquisition revolved around findings about so-called “feral” children who had failed to acquire language at an older age after having been deprived of normal input during the critical period.

However, these case studies did not account for the extent to which social deprivation, and possibly food deprivation or sensory deprivation, may have confounded with language input deprivation (Kroll and De Groot, 2009).

More recently, researchers have focused more systematically on deaf children born to hearing parents who are therefore deprived of language input until at least elementary school.

These studies have found the effects of lack of language input without extreme social deprivation: the older the age of exposure to sign language is, the worse its ultimate attainment (Emmorey, Bellugi, Friederici, and Horn, 1995; Kroll and De Groot, 2009).

However, Kroll and De Groot argue that the critical period hypothesis does not apply to the rate of acquisition of language. Adults and adolescents can learn languages at the same rate or even faster than children in their initial stage of acquisition (Slavoff and Johnson, 1995).

However, adults tend to have a more limited ultimate attainment of language ability (Kroll and De Groot, 2009).

There has been a long lineage of empirical findings around the age of acquisition. The most fundamental of this research comes from a series of studies since the late 1970s documenting a negative correlation between age of acquisition and ultimate language mastery (Kroll and De Grott, 2009).

Nonetheless, different periods correspond to sensitivity to different aspects of language. For example, shortly after birth, infants can perceive and discriminate speech sounds from any language, including ones they have not been exposed to (Eimas et al., 1971; Gallagher et al., 2020).

Around six months of age, exposure to the primary language in the infant’s environment guides phonetic representations of language and, subsequently, the neural representations of speech sounds of the native language while weakening those of unused sounds (McClelland et al., 1999; Gallagher et al., 2020).

Vocabulary learning experiences rapid growth at about 18 months of age (Kuhl, 2010).

Critical Evaluation

More than any other area of applied linguistics, the critical period hypothesis has impacted how teachers teach languages. Consequently, researchers have critiqued how important the critical period is to language learning.

For example, several studies in early language acquisition research showed that children were not necessarily superior to older learners in acquiring a second language, even in the area of pronunciation (Olson and Samuels, 1973; Snow and Hoefnagel-Hohle, 1978; Scovel, 2000).

In fact, the majority of researchers at the time appeared to be skeptical about the existence of a critical period, with some explicitly denying its existence.

Counter to one of the primary tenets of Scovel’s (1969) critical period hypothesis, there have been several cases of people who have acquired a second language in adulthood speaking with native accents.

For example, Moyer’s study of highly proficient English-speaking learners of German suggested that at least one of the participants was judged to have native-like pronunciation in his second language (1999), and several participants in Bongaerts (1999) study of highly proficient Dutch speakers of French spoke with accents judged to be native (Scovel, 2000).

Bongaerts, T. (1999). Ultimate attainment in L2 pronunciation: The case of very advanced late L2 learners. Second language acquisition and the critical period hypothesis, 133-159.

Cisneros-Franco, J. M., Voss, P., Thomas, M. E., & de Villers-Sidani, E. (2020). Critical periods of brain development. In Handbook of Clinical Neurolog y (Vol. 173, pp. 75-88). Elsevier.

Colonnesi, C., Stams, G. J. J., Koster, I., & Noom, M. J. (2010). The relation between pointing and language development: A meta-analysis. Developmental Review, 30 (4), 352-366.

Eimas, P. D., Siqueland, E. R., Jusczyk, P., & Vigorito, J. (1971). Speech perception in infants. Science, 171 (3968), 303-306.

Emmorey, K., Bellugi, U., Friederici, A., & Horn, P. (1995). Effects of age of acquisition on grammatical sensitivity: Evidence from on-line and off-line tasks. Applied Psycholinguistics, 16 (1), 1-23.

Knudsen, E. I. (2004). Sensitive periods in the development of the brain and behavior. Journal of cognitive neuroscience, 16 (8), 1412-1425.

Hartshorne, J. K., Tenenbaum, J. B., & Pinker, S. (2018). A critical period for second language acquisition: Evidence from 2/3 million English speakers. Cognition, 177 , 263-277.

Kral, A., & Eggermont, J. J. (2007). What’s to lose and what’s to learn: development under auditory deprivation, cochlear implants and limits of cortical plasticity. Brain Research Reviews, 56(1), 259-269.

Kroll, J. F., & De Groot, A. M. (Eds.). (2009). Handbook of bilingualism: Psycholinguistic approaches . Oxford University Press.

Kuhl, P. K. (2010). Brain mechanisms in early language acquisition. Neuron, 67 (5), 713-727.

Lenneberg, E. H. (1967). The biological foundations of language. Hospital Practice, 2( 12), 59-67.

Lorenz, K. (1935). Der kumpan in der umwelt des vogels. Journal für Ornithologie, 83 (2), 137-213.

Marcovitch, S., & Zelazo, P. D. (2009). A hierarchical competing systems model of the emergence and early development of executive function. Developmental science, 12 (1), 1-18.

McClelland, J. L., Thomas, A. G., McCandliss, B. D., & Fiez, J. A. (1999). Understanding failures of learning: Hebbian learning, competition for representational space, and some preliminary experimental data. Progress in brain research, 121, 75-80.

McGraw, M. B. (1946). Maturation of behavior. In Manual of child psychology. (pp. 332-369). John Wiley & Sons Inc.

Moyer, A. (1999). Ultimate attainment in L2 phonology: The critical factors of age, motivation, and instruction. Studies in second language acquisition, 21 (1), 81-108.

Gallagher, A., Bulteau, C., Cohen, D., & Michaud, J. L. (2019). Neurocognitive Development: Normative Development. Elsevier.

Olson, L. L., & Jay Samuels, S. (1973). The relationship between age and accuracy of foreign language pronunciation. The Journal of Educational Research, 66 (6), 263-268.

Penfield, W., & Roberts, L. (2014). Speech and brain mechanisms. Princeton University Press.

Piaget, J. (1962). The stages of the intellectual development of the child. Bulletin of the Menninger Clinic, 26 (3), 120.

Rosa, A. M., Silva, M. F., Ferreira, S., Murta, J., & Castelo-Branco, M. (2013). Plasticity in the human visual cortex: an ophthalmology-based perspective. BioMed research international, 2013.

Salkind, N. J. (Ed.). (2005). Encyclopedia of human development . Sage Publications.

Scott, J. P. (1962). Critical periods in behavioral development. Science, 138 (3544), 949-958.

Scovel, T. (1969). Foreign accents, language acquisition, and cerebral dominance 1. Language learning, 19 (3‐4), 245-253.

Scovel, T. (2000). A critical review of the critical period research. Annual review of applied linguistics, 20 , 213-223.

Skoe, E., & Kraus, N. (2013). Musical training heightens auditory brainstem function during sensitive periods in development. Frontiers in Psychology, 4, 622.

Slavoff, G. R., & Johnson, J. S. (1995). The effects of age on the rate of learning a second language. Studies in Second Language Acquisition, 17 (1), 1-16.

Snow, C. E., & Hoefnagel-Höhle, M. (1978). The critical period for language acquisition: Evidence from second language learning. Child development, 1114-1128.

Stockard, C. R. (1921). Developmental rate and structural expression: an experimental study of twins,‘double monsters’ and single deformities, and the interaction among embryonic organs during their origin and development. American Journal of Anatomy, 28 (2), 115-277.

White, E. J., Hutka, S. A., Williams, L. J., & Moreno, S. (2013). Learning, neural plasticity and sensitive periods: implications for language acquisition, music training and transfer across the lifespan. Frontiers in systems neuroscience, 7, 90.

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Statistics By Jim

Making statistics intuitive

Critical Value: Definition, Finding & Calculator

By Jim Frost 2 Comments

What is a Critical Value?

A critical value defines regions in the sampling distribution of a test statistic. These values play a role in both hypothesis tests and confidence intervals. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits.

In both cases, critical values account for uncertainty in sample data you’re using to make inferences about a population . They answer the following questions:

• How different does the sample estimate need to be from the null hypothesis to be statistically significant?
• What is the margin of error (confidence interval) around the sample estimate of the population parameter ?

In this post, I’ll show you how to find critical values, use them to determine statistical significance, and use them to construct confidence intervals. I also include a critical value calculator at the end of this article so you can apply what you learn.

Because most people start learning with the z-test and its test statistic, the z-score, I’ll use them for the examples throughout this post. However, I provide links with detailed information for other types of tests and sampling distributions.

Related posts : Sampling Distributions and Test Statistics

Using a Critical Value to Determine Statistical Significance

In this context, the sampling distribution of a test statistic defines the probability for ranges of values. The significance level (α) specifies the probability that corresponds with the critical value within the distribution. Let’s work through an example for a z-test.

The z-test uses the z test statistic. For this test, the z-distribution finds probabilities for ranges of z-scores under the assumption that the null hypothesis is true. For a z-test, the null z-score is zero, which is at the central peak of the sampling distribution. This sampling distribution centers on the null hypothesis value, and the critical values mark the minimum distance from the null hypothesis required for statistical significance.

Critical values depend on your significance level and whether you’re performing a one- or two-sided hypothesis. For these examples, I’ll use a significance level of 0.05. This value defines how improbable the test statistic must be to be significant.

Related posts : Significance Levels and P-values and Z-scores

Two-Sided Tests

Two-sided hypothesis tests have two rejection regions. Consequently, you’ll need two critical values that define them. Because there are two rejection regions, we must split our significance level in half. Each rejection region has a probability of α / 2, making the total likelihood for both areas equal the significance level.

The probability plot below displays the critical values and the rejection regions for a two-sided z-test with a significance level of 0.05. When the z-score is ≤ -1.96 or ≥ 1.96, it exceeds the cutoff, and your results are statistically significant.

One-Sided Tests

One-tailed tests have one rejection region and, hence, only one critical value. The total α probability goes into that one side. The probability plots below display these values for right- and left-sided z-tests. These tests can detect effects in only one direction.

Related post : Understanding One-Tailed and Two-Tailed Hypothesis Tests and Effects in Statistics

Using a Critical Value to Construct Confidence Intervals

Confidence intervals use the same critical values (CVs) as the corresponding hypothesis test. The confidence level equals 1 – the significance level. Consequently, the CVs for a significance level of 0.05 produce a confidence level of 1 – 0.05 = 0.95 or 95%.

For example, to calculate the 95% confidence interval for our two-tailed z-test with a significance level of 0.05, use the CVs of -1.96 and 1.96 that we found above.

To calculate the upper and lower limits of the interval, take the positive critical value and multiply it by the standard error of the mean. Then take the sample mean and add and subtract that product from it.

• Lower Limit = Sample Mean – (CV * Standard Error of the Mean)
• Upper Limit = Sample Mean + (CV * Standard Error of the Mean)

To learn more about confidence intervals and how to construct them, read my posts about Confidence Intervals and How Confidence Intervals Work .

Related post : Standard Error of the Mean

How to Find a Critical Value

Unfortunately, the formulas for finding critical values are very complex. Typically, you don’t calculate them by hand. For the examples in this article, I’ve used statistical software to find them. However, you can also use statistical tables.

To learn how to use these critical value tables, read my articles that contain the tables and information about using them. The process for finding them is similar for the various tests. Using these tables requires knowing the correct test statistic, the significance level, the number of tails, and, in most cases, the degrees of freedom.

The following articles provide the statistical tables, explain how to use them, and visually illustrate the results.

• T distribution table
• Chi-square table

Related post : Degrees of Freedom

Critical Value Calculator

Another method for finding CVs is to use a critical value calculator, such as the one below. These calculators are handy for finding the answer, but they don’t provide the context for the results.

This calculator finds critical values for the sampling distributions of common test statistics.

For example, choose the following in the calculator:

• Z (standard normal)
• Significance level = 0.05

The calculator will display the same ±1.96 values we found earlier in this article.

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Reader Interactions

January 16, 2024 at 5:26 pm

Hello, I am currently taking statistics and am reviewing confidence intervals. I would like to know what is the equation for calculating a two-tailed test for upper and lower limits? I would like to know is there a way to calculate one and two-tailed tests without using a confidence interval calculator and can you explain further?

January 16, 2024 at 6:43 pm

If you’re talking about calculating the critical values values for a test statistic for two-tailed test, the calculations are fairly complex. Consequently, you’ll either use statistical software, an online calculator, or a statistical table to find those limits.

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S.3.1 hypothesis testing (critical value approach).

The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the " critical value ." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. To conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error — which is denoted \(\alpha\) (greek letter "alpha") and is called the " significance level of the test " — is small (typically 0.01, 0.05, or 0.10).
• Compare the test statistic to the critical value. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less extreme than the critical value, do not reject the null hypothesis.

Example S.3.1.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose we take a random sample of n = 15 students majoring in mathematics. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right-Tailed

The critical value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the t -value, denoted t \(\alpha\) , n - 1 , such that the probability to the right of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value t 0.05,14 is 1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3 if the test statistic t * is greater than 1.7613. Visually, the rejection region is shaded red in the graph.

Left-Tailed

The critical value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the t -value, denoted -t ( \(\alpha\) , n - 1) , such that the probability to the left of it is \(\alpha\). It can be shown using either statistical software or a t -table that the critical value -t 0.05,14 is -1.7613. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3 if the test statistic t * is less than -1.7613. Visually, the rejection region is shaded red in the graph.

There are two critical values for the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 — one for the left-tail denoted -t ( \(\alpha\) / 2, n - 1) and one for the right-tail denoted t ( \(\alpha\) / 2, n - 1) . The value - t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the left of it is \(\alpha\)/2, and the value t ( \(\alpha\) /2, n - 1) is the t -value such that the probability to the right of it is \(\alpha\)/2. It can be shown using either statistical software or a t -table that the critical value -t 0.025,14 is -2.1448 and the critical value t 0.025,14 is 2.1448. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3 if the test statistic t * is less than -2.1448 or greater than 2.1448. Visually, the rejection region is shaded red in the graph.

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Critical Period in Brain Development: Definition, Importance

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• When Does the Critical Period Begin and End?
• The Critical Period Hypothesis—What It States
• What Happens to the Brain in the Critical Period?
• What Kind of Events Impact the Brain During the Critical Period?
• How Do Adverse Events Impact the Brain?
• What's the Difference Between a Critical Period and a Sensitive Period?
• What Happens to the Brain When the Critical Period Ends?

The critical period in brain development is an immensely significant and specific time frame during which the brain is especially receptive to environmental stimuli and undergoes a series of rapid changes. 

These changes have lifelong effects as essential neural connections and pathways are established, playing a vital role in cognitive, emotional, and social development. 

This article will explore the timeline, impacting events, and subsequent consequences of the critical period on brain development. It also explores the distinction between critical periods and sensitive periods and what happens to the brain once the critical period ends.

When Does the Critical Period Begin and End? 

The starting point of the critical period is at conception. The brain starts to form and develop from the moment you are conceived. During pregnancy, a baby's brain is already beginning to shape itself for the world outside. The brain is gearing up and getting ready to absorb a massive amount of information.

The Early Years of a Child's Life

Once the baby is born, the brain kicks into high gear. The early years of a child's life, from birth to around the age of five, are generally considered the core of the critical period. The brain is incredibly absorbent during these years, taking in information rapidly. Everything from language to motor skills to social cues is being learned and processed extensively.

Different aspects of learning and development have different critical periods. For instance, the critical period for language acquisition extends into early adolescence. This means that while the brain is still very good at learning languages during early childhood, it continues to be relatively efficient at it until the teenage years.

The brain is incredibly absorbent during these years, taking in information rapidly. Everything from language to motor skills to social cues is being learned and processed extensively.

Vision Develops During This Period

On the other hand, for certain sensory abilities like vision, the critical period might end much earlier. This means that the brain is most receptive to developing visual abilities in the first few years of life, and after that, it becomes significantly harder to change or improve these abilities.

The Critical Period Hypothesis—What It States 

The brain has a certain time window when it's exceptionally good at learning new things, especially languages. This window of time is what is referred to as the "critical period."

Younger People Learn Languages Faster Than Older People

Eric Lenneberg, a neuropsychologist, introduced the Critical Period Hypothesis. He was very interested in how people learn languages . Through his observations and research, Lenneberg noticed that younger people were much more adept at learning languages than older people. This observation led him to the idea that there is a specific period during which the brain is highly efficient and capable of absorbing languages.

As You Age, It Becomes More Difficult to Absorb New Information

If the critical period is a wide open window in the early years of life, allowing the brain to take in an abundance of information quickly and efficiently, as time progresses, this window begins to close gradually. As it closes, the brain becomes less capable of easily absorbing languages.

This doesn't mean that learning becomes impossible as you age; it merely indicates that the ease and efficiency with which the brain learns start to decline.

What Happens to the Brain in the Critical Period? 

During the critical period, the brain experiences explosive growth. Let's take a look at some of the changes that happen in the brain during the critical period.

Neurons Form Connections

In the early stages, neurons in the brain start to form connections. These connections are called synapses.

Synapses are bridges that help different parts of the brain communicate with each other. In the critical period, the brain is building these bridges at an incredible pace.

Neuroplasticity Strengthens Brain Connections

As a baby interacts with the world, certain connections strengthen while others weaken. For instance, if a baby hears a lot of music, the parts of the brain associated with sounds and music will become stronger. This process of strengthening certain connections is known as brain plasticity because the brain molds itself like plastic.

Attachment to Primary Caregivers

An essential aspect of the critical period is the development of attachment to caregivers. During the early months and years, babies and toddlers form strong bonds with the people caring for them .

These attachments are critical for emotional development. When a caregiver responds to a baby's needs with warmth and care, the baby learns to form secure attachments . This lays the foundation for healthy relationships later in life.

What Happens When Children Are Not Given Attention?

What if a child is not given the attention and care they need during the critical period? This is a significant concern. Without proper attention and stimulation, the brain doesn't develop as effectively. The bridges or connections that should be built might not form properly. This can lead to various issues, including difficulty forming relationships, emotional problems, and learning difficulties.

When a child is given proper attention, stimulation, and care during the critical period, their brain thrives. The connections form rapidly and robustly. This sets the stage for better learning, emotional regulation, and relationship-building throughout life.

What Kind of Events Impact the Brain During the Critical Period? 

When a child is exposed to a rich, stimulating environment where they can play, explore, and learn, it tremendously impacts the brain. Engaging in interactive learning, being read to, and having supportive relationships with caregivers can significantly contribute to a well-developed brain.

Events such as abuse, neglect, head trauma , or extreme stress—collectively known as adverse childhood experiences (ACEs)—can be detrimental to brain development. These adverse events can impede the formation of neural connections and lead to behavioral, emotional, and cognitive difficulties later in life. "Unfortunately, disruptions to normal brain development due to environmental influences such as poverty, neglect, or exposure to toxins can cause lasting damage. This is why it is so important for children to receive adequate nutrition, stimulation, and parental care during these first few years of life; without it, they may suffer developmental delays and other issues that could potentially be avoided with proper attention,"  Harold Hong, MD , a board-certified psychiatrist says.

How Do Adverse Events Impact the Brain? 

When a child is neglected or abused, stress can impact how their brain develops. The parts of the brain involved in emotions and handling stress might not develop properly. This can make it hard for the person to manage their emotions later in life.

The hippocampus, involved in learning and memory, and the amygdala, which plays a role in emotion processing, are especially vulnerable. 

Similarly, if a child does not have enough food to eat or a safe place to live, the chronic stress of these conditions can impact brain development. The brain might focus on survival instead of other important areas of development, like learning and building relationships.

Even accidents that cause head injuries can impact the brain during the critical period. If a child experiences head trauma, it can affect the brain's development depending on the injury's severity and location.

What's the Difference Between a Critical Period and a Sensitive Period?

It is imperative to distinguish between critical periods and sensitive periods.

• Critical periods are specific windows of time during development when the brain is exceptionally receptive to certain types of learning and experiences. Once this period is over, acquiring those skills or attributes becomes significantly more challenging.
• Sensitive periods are phases in which the brain is more responsive to certain experiences. It's easier to learn or be influenced by specific experiences during sensitive periods, but unlike critical periods, missing this timeframe doesn't make it impossible to acquire those skills or traits later.

For example, while there is a critical period for acquiring native-like pronunciation and grammar, there is also a sensitive period for language learning. Children are more adept at learning new languages when they are young, but even if someone misses this window, they can still learn languages later in life.

One way to visualize the difference is to think of critical periods as a tightly defined window of time with a clear beginning and end, during which certain development must occur. In contrast, sensitive periods are more like a gradual slope, where learning at the beginning is optimal, but the ability doesn't disappear entirely over time.

What Happens to the Brain When the Critical Period Ends? 

It's essential to recognize that the end of the critical period does not mean the end of learning or brain development. Instead, it signifies a shift in how the brain learns and adapts. 

During the critical period, the brain is highly plastic, meaning it can change and form new connections rapidly. As this period ends, the brain doesn't lose this plasticity entirely, but the rate at which it can make new connections slows down. 

According to Hong, although some of these connections can still be altered by experiences later in life, such as learning a new language or practicing a skill, it is much harder to make significant changes after the critical period has ended. This highlights just how important it is for parents to provide proper care and nurture during those first few years.

The Brain Becomes More Specialized Via Adult Plasticity

The brain also becomes more specialized in the skills and information it has acquired as this period ends. During the critical period, the brain forms numerous connections, and as it ends, it starts to use these connections more efficiently for specialized tasks.

Even though the critical period ends, the brain still possesses a degree of plasticity and continues to learn throughout life. This is called adult plasticity.

Adult plasticity is not as robust during the critical period, but it allows for the continuous adaptation and learning necessary for us to navigate the ever-changing demands of life.

The conventional view is that critical periods close relatively tightly. However, research has started to challenge this rigid view. It's more accurate to say that the doors of critical periods close but do not necessarily lock.

While the brain's plasticity decreases after these periods, learning and adaptation can still take place, albeit with more effort and over a longer time. This phenomenon of 'metaplasticity'—the brain's ability to change its plasticity levels—remains an exciting area of ongoing research,  Dr. Ryan Sultan , a neuroscientist, child psychiatrist, and professor of psychiatry at Columbia University, says. 

What This Means For You

The critical period represents an invaluable window during which the foundations for cognitive, emotional, and social abilities are established. The environment, experiences, and attachments formed during this period have far-reaching consequences on a person's life.  Understanding the nuances of the critical period is essential for educators, parents, and policymakers to create nurturing environments that support healthy brain development. Providing support and early interventions for children exposed to adverse experiences is vital for ensuring their potential is not hindered by the circumstances of their early life.

Siahaan F. The critical period hypothesis of sla eric lenneberg’s . Journal of Applied Linguistics . 2022;2(1):40-45.

Nelson CA, Gabard-Durnam LJ. Early adversity and critical periods: neurodevelopmental consequences of violating the expectable environment. Trends in Neurosciences. 2020;43(3):133-143.

Colombo J, Gustafson KM, Carlson SE. Critical and sensitive periods in development and nutrition. Ann Nutr Metab . 2019;75(Suppl. 1):34-42.

Patton MH, Blundon JA, Zakharenko SS. Rejuvenation of plasticity in the brain: opening the critical period. Current Opinion in Neurobiology . 2019;54:83-89.

By Toketemu Ohwovoriole Toketemu has been multimedia storyteller for the last four years. Her expertise focuses primarily on mental wellness and women’s health topics.

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Research Article

The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis

* E-mail: [email protected]

Affiliation Department of Multilingualism, University of Fribourg, Fribourg, Switzerland

• Jan Vanhove

• Published: July 25, 2013
• https://doi.org/10.1371/journal.pone.0069172
• Reader Comments

17 Jul 2014: The PLOS ONE Staff (2014) Correction: The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis. PLOS ONE 9(7): e102922. https://doi.org/10.1371/journal.pone.0102922 View correction

In second language acquisition research, the critical period hypothesis ( cph ) holds that the function between learners' age and their susceptibility to second language input is non-linear. This paper revisits the indistinctness found in the literature with regard to this hypothesis's scope and predictions. Even when its scope is clearly delineated and its predictions are spelt out, however, empirical studies–with few exceptions–use analytical (statistical) tools that are irrelevant with respect to the predictions made. This paper discusses statistical fallacies common in cph research and illustrates an alternative analytical method (piecewise regression) by means of a reanalysis of two datasets from a 2010 paper purporting to have found cross-linguistic evidence in favour of the cph . This reanalysis reveals that the specific age patterns predicted by the cph are not cross-linguistically robust. Applying the principle of parsimony, it is concluded that age patterns in second language acquisition are not governed by a critical period. To conclude, this paper highlights the role of confirmation bias in the scientific enterprise and appeals to second language acquisition researchers to reanalyse their old datasets using the methods discussed in this paper. The data and R commands that were used for the reanalysis are provided as supplementary materials.

Citation: Vanhove J (2013) The Critical Period Hypothesis in Second Language Acquisition: A Statistical Critique and a Reanalysis. PLoS ONE 8(7): e69172. https://doi.org/10.1371/journal.pone.0069172

Editor: Stephanie Ann White, UCLA, United States of America

Received: May 7, 2013; Accepted: June 7, 2013; Published: July 25, 2013

Copyright: © 2013 Jan Vanhove. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: No current external funding sources for this study.

Competing interests: The author has declared that no competing interests exist.

Introduction

In the long term and in immersion contexts, second-language (L2) learners starting acquisition early in life – and staying exposed to input and thus learning over several years or decades – undisputedly tend to outperform later learners. Apart from being misinterpreted as an argument in favour of early foreign language instruction, which takes place in wholly different circumstances, this general age effect is also sometimes taken as evidence for a so-called ‘critical period’ ( cp ) for second-language acquisition ( sla ). Derived from biology, the cp concept was famously introduced into the field of language acquisition by Penfield and Roberts in 1959 [1] and was refined by Lenneberg eight years later [2] . Lenneberg argued that language acquisition needed to take place between age two and puberty – a period which he believed to coincide with the lateralisation process of the brain. (More recent neurological research suggests that different time frames exist for the lateralisation process of different language functions. Most, however, close before puberty [3] .) However, Lenneberg mostly drew on findings pertaining to first language development in deaf children, feral children or children with serious cognitive impairments in order to back up his claims. For him, the critical period concept was concerned with the implicit “automatic acquisition” [2, p. 176] in immersion contexts and does not preclude the possibility of learning a foreign language after puberty, albeit with much conscious effort and typically less success.

sla research adopted the critical period hypothesis ( cph ) and applied it to second and foreign language learning, resulting in a host of studies. In its most general version, the cph for sla states that the ‘susceptibility’ or ‘sensitivity’ to language input varies as a function of age, with adult L2 learners being less susceptible to input than child L2 learners. Importantly, the age–susceptibility function is hypothesised to be non-linear. Moving beyond this general version, we find that the cph is conceptualised in a multitude of ways [4] . This state of affairs requires scholars to make explicit their theoretical stance and assumptions [5] , but has the obvious downside that critical findings risk being mitigated as posing a problem to only one aspect of one particular conceptualisation of the cph , whereas other conceptualisations remain unscathed. This overall vagueness concerns two areas in particular, viz. the delineation of the cph 's scope and the formulation of testable predictions. Delineating the scope and formulating falsifiable predictions are, needless to say, fundamental stages in the scientific evaluation of any hypothesis or theory, but the lack of scholarly consensus on these points seems to be particularly pronounced in the case of the cph . This article therefore first presents a brief overview of differing views on these two stages. Then, once the scope of their cph version has been duly identified and empirical data have been collected using solid methods, it is essential that researchers analyse the data patterns soundly in order to assess the predictions made and that they draw justifiable conclusions from the results. As I will argue in great detail, however, the statistical analysis of data patterns as well as their interpretation in cph research – and this includes both critical and supportive studies and overviews – leaves a great deal to be desired. Reanalysing data from a recent cph -supportive study, I illustrate some common statistical fallacies in cph research and demonstrate how one particular cph prediction can be evaluated.

Delineating the scope of the critical period hypothesis

First, the age span for a putative critical period for language acquisition has been delimited in different ways in the literature [4] . Lenneberg's critical period stretched from two years of age to puberty (which he posits at about 14 years of age) [2] , whereas other scholars have drawn the cutoff point at 12, 15, 16 or 18 years of age [6] . Unlike Lenneberg, most researchers today do not define a starting age for the critical period for language learning. Some, however, consider the possibility of the critical period (or a critical period for a specific language area, e.g. phonology) ending much earlier than puberty (e.g. age 9 years [1] , or as early as 12 months in the case of phonology [7] ).

Second, some vagueness remains as to the setting that is relevant to the cph . Does the critical period constrain implicit learning processes only, i.e. only the untutored language acquisition in immersion contexts or does it also apply to (at least partly) instructed learning? Most researchers agree on the former [8] , but much research has included subjects who have had at least some instruction in the L2.

Third, there is no consensus on what the scope of the cp is as far as the areas of language that are concerned. Most researchers agree that a cp is most likely to constrain the acquisition of pronunciation and grammar and, consequently, these are the areas primarily looked into in studies on the cph [9] . Some researchers have also tried to define distinguishable cp s for the different language areas of phonetics, morphology and syntax and even for lexis (see [10] for an overview).

Fourth and last, research into the cph has focused on ‘ultimate attainment’ ( ua ) or the ‘final’ state of L2 proficiency rather than on the rate of learning. From research into the rate of acquisition (e.g. [11] – [13] ), it has become clear that the cph cannot hold for the rate variable. In fact, it has been observed that adult learners proceed faster than child learners at the beginning stages of L2 acquisition. Though theoretical reasons for excluding the rate can be posited (the initial faster rate of learning in adults may be the result of more conscious cognitive strategies rather than to less conscious implicit learning, for instance), rate of learning might from a different perspective also be considered an indicator of ‘susceptibility’ or ‘sensitivity’ to language input. Nevertheless, contemporary sla scholars generally seem to concur that ua and not rate of learning is the dependent variable of primary interest in cph research. These and further scope delineation problems relevant to cph research are discussed in more detail by, among others, Birdsong [9] , DeKeyser and Larson-Hall [14] , Long [10] and Muñoz and Singleton [6] .

Formulating testable hypotheses

Once the relevant cph 's scope has satisfactorily been identified, clear and testable predictions need to be drawn from it. At this stage, the lack of consensus on what the consequences or the actual observable outcome of a cp would have to look like becomes evident. As touched upon earlier, cph research is interested in the end state or ‘ultimate attainment’ ( ua ) in L2 acquisition because this “determines the upper limits of L2 attainment” [9, p. 10]. The range of possible ultimate attainment states thus helps researchers to explore the potential maximum outcome of L2 proficiency before and after the putative critical period.

One strong prediction made by some cph exponents holds that post- cp learners cannot reach native-like L2 competences. Identifying a single native-like post- cp L2 learner would then suffice to falsify all cph s making this prediction. Assessing this prediction is difficult, however, since it is not clear what exactly constitutes sufficient nativelikeness, as illustrated by the discussion on the actual nativelikeness of highly accomplished L2 speakers [15] , [16] . Indeed, there exists a real danger that, in a quest to vindicate the cph , scholars set the bar for L2 learners to match monolinguals increasingly higher – up to Swiftian extremes. Furthermore, the usefulness of comparing the linguistic performance in mono- and bilinguals has been called into question [6] , [17] , [18] . Put simply, the linguistic repertoires of mono- and bilinguals differ by definition and differences in the behavioural outcome will necessarily be found, if only one digs deep enough.

A second strong prediction made by cph proponents is that the function linking age of acquisition and ultimate attainment will not be linear throughout the whole lifespan. Before discussing how this function would have to look like in order for it to constitute cph -consistent evidence, I point out that the ultimate attainment variable can essentially be considered a cumulative measure dependent on the actual variable of interest in cph research, i.e. susceptibility to language input, as well as on such other factors like duration and intensity of learning (within and outside a putative cp ) and possibly a number of other influencing factors. To elaborate, the behavioural outcome, i.e. ultimate attainment, can be assumed to be integrative to the susceptibility function, as Newport [19] correctly points out. Other things being equal, ultimate attainment will therefore decrease as susceptibility decreases. However, decreasing ultimate attainment levels in and by themselves represent no compelling evidence in favour of a cph . The form of the integrative curve must therefore be predicted clearly from the susceptibility function. Additionally, the age of acquisition–ultimate attainment function can take just about any form when other things are not equal, e.g. duration of learning (Does learning last up until time of testing or only for a more or less constant number of years or is it dependent on age itself?) or intensity of learning (Do learners always learn at their maximum susceptibility level or does this intensity vary as a function of age, duration, present attainment and motivation?). The integral of the susceptibility function could therefore be of virtually unlimited complexity and its parameters could be adjusted to fit any age of acquisition–ultimate attainment pattern. It seems therefore astonishing that the distinction between level of sensitivity to language input and level of ultimate attainment is rarely made in the literature. Implicitly or explicitly [20] , the two are more or less equated and the same mathematical functions are expected to describe the two variables if observed across a range of starting ages of acquisition.

But even when the susceptibility and ultimate attainment variables are equated, there remains controversy as to what function linking age of onset of acquisition and ultimate attainment would actually constitute evidence for a critical period. Most scholars agree that not any kind of age effect constitutes such evidence. More specifically, the age of acquisition–ultimate attainment function would need to be different before and after the end of the cp [9] . According to Birdsong [9] , three basic possible patterns proposed in the literature meet this condition. These patterns are presented in Figure 1 . The first pattern describes a steep decline of the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function up to the end of the cp and a practically non-existent age effect thereafter. Pattern 2 is an “unconventional, although often implicitly invoked” [9, p. 17] notion of the cp function which contains a period of peak attainment (or performance at ceiling), i.e. performance does not vary as a function of age, which is often referred to as a ‘window of opportunity’. This time span is followed by an unbounded decline in ua depending on aoa . Pattern 3 includes characteristics of patterns 1 and 2. At the beginning of the aoa range, performance is at ceiling. The next segment is a downward slope in the age function which ends when performance reaches its floor. Birdsong points out that all of these patterns have been reported in the literature. On closer inspection, however, he concludes that the most convincing function describing these age effects is a simple linear one. Hakuta et al. [21] sketch further theoretically possible predictions of the cph in which the mean performance drops drastically and/or the slope of the aoa – ua proficiency function changes at a certain point.

• PPT PowerPoint slide
• PNG larger image
• TIFF original image

The graphs are based on based on Figure 2 in [9] .

https://doi.org/10.1371/journal.pone.0069172.g001

Although several patterns have been proposed in the literature, it bears pointing out that the most common explicit prediction corresponds to Birdsong's first pattern, as exemplified by the following crystal-clear statement by DeKeyser, one of the foremost cph proponents:

[A] strong negative correlation between age of acquisition and ultimate attainment throughout the lifespan (or even from birth through middle age), the only age effect documented in many earlier studies, is not evidence for a critical period…[T]he critical period concept implies a break in the AoA–proficiency function, i.e., an age (somewhat variable from individual to individual, of course, and therefore an age range in the aggregate) after which the decline of success rate in one or more areas of language is much less pronounced and/or clearly due to different reasons. [22, p. 445].

DeKeyser and before him among others Johnson and Newport [23] thus conceptualise only one possible pattern which would speak in favour of a critical period: a clear negative age effect before the end of the critical period and a much weaker (if any) negative correlation between age and ultimate attainment after it. This ‘flattened slope’ prediction has the virtue of being much more tangible than the ‘potential nativelikeness’ prediction: Testing it does not necessarily require comparing the L2-learners to a native control group and thus effectively comparing apples and oranges. Rather, L2-learners with different aoa s can be compared amongst themselves without the need to categorise them by means of a native-speaker yardstick, the validity of which is inevitably going to be controversial [15] . In what follows, I will concern myself solely with the ‘flattened slope’ prediction, arguing that, despite its clarity of formulation, cph research has generally used analytical methods that are irrelevant for the purposes of actually testing it.

Inferring non-linearities in critical period research: An overview

Group mean or proportion comparisons.

[T]he main differences can be found between the native group and all other groups – including the earliest learner group – and between the adolescence group and all other groups. However, neither the difference between the two childhood groups nor the one between the two adulthood groups reached significance, which indicates that the major changes in eventual perceived nativelikeness of L2 learners can be associated with adolescence. [15, p. 270].

Similar group comparisons aimed at investigating the effect of aoa on ua have been carried out by both cph advocates and sceptics (among whom Bialystok and Miller [25, pp. 136–139], Birdsong and Molis [26, p. 240], Flege [27, pp. 120–121], Flege et al. [28, pp. 85–86], Johnson [29, p. 229], Johnson and Newport [23, p. 78], McDonald [30, pp. 408–410] and Patowski [31, pp. 456–458]). To be clear, not all of these authors drew direct conclusions about the aoa – ua function on the basis of these groups comparisons, but their group comparisons have been cited as indicative of a cph -consistent non-continuous age effect, as exemplified by the following quote by DeKeyser [22] :

Where group comparisons are made, younger learners always do significantly better than the older learners. The behavioral evidence, then, suggests a non-continuous age effect with a “bend” in the AoA–proficiency function somewhere between ages 12 and 16. [22, p. 448].

The first problem with group comparisons like these and drawing inferences on the basis thereof is that they require that a continuous variable, aoa , be split up into discrete bins. More often than not, the boundaries between these bins are drawn in an arbitrary fashion, but what is more troublesome is the loss of information and statistical power that such discretisation entails (see [32] for the extreme case of dichotomisation). If we want to find out more about the relationship between aoa and ua , why throw away most of the aoa information and effectively reduce the ua data to group means and the variance in those groups?

Comparison of correlation coefficients.

Correlation-based inferences about slope discontinuities have similarly explicitly been made by cph advocates and skeptics alike, e.g. Bialystok and Miller [25, pp. 136 and 140], DeKeyser and colleagues [22] , [44] and Flege et al. [45, pp. 166 and 169]. Others did not explicitly infer the presence or absence of slope differences from the subset correlations they computed (among others Birdsong and Molis [26] , DeKeyser [8] , Flege et al. [28] and Johnson [29] ), but their studies nevertheless featured in overviews discussing discontinuities [14] , [22] . Indeed, the most recent overview draws a strong conclusion about the validity of the cph 's ‘flattened slope’ prediction on the basis of these subset correlations:

In those studies where the two groups are described separately, the correlation is much higher for the younger than for the older group, except in Birdsong and Molis (2001) [ =  [26] , JV], where there was a ceiling effect for the younger group. This global picture from more than a dozen studies provides support for the non-continuity of the decline in the AoA–proficiency function, which all researchers agree is a hallmark of a critical period phenomenon. [22, p. 448].

In Johnson and Newport's specific case [23] , their correlation-based inference that ua levels off after puberty happened to be largely correct: the gjt scores are more or less randomly distributed around a near-horizontal trend line [26] . Ultimately, however, it rests on the fallacy of confusing correlation coefficients with slopes, which seriously calls into question conclusions such as DeKeyser's (cf. the quote above).

https://doi.org/10.1371/journal.pone.0069172.g002

Lower correlation coefficients in older aoa groups may therefore be largely due to differences in ua variance, which have been reported in several studies [23] , [26] , [28] , [29] (see [46] for additional references). Greater variability in ua with increasing age is likely due to factors other than age proper [47] , such as the concomitant greater variability in exposure to literacy, degree of education, motivation and opportunity for language use, and by itself represents evidence neither in favour of nor against the cph .

Regression approaches.

Having demonstrated that neither group mean or proportion comparisons nor correlation coefficient comparisons can directly address the ‘flattened slope’ prediction, I now turn to the studies in which regression models were computed with aoa as a predictor variable and ua as the outcome variable. Once again, this category of studies is not mutually exclusive with the two categories discussed above.

In a large-scale study using self-reports and approximate aoa s derived from a sample of the 1990 U.S. Census, Stevens found that the probability with which immigrants from various countries stated that they spoke English ‘very well’ decreased curvilinearly as a function of aoa [48] . She noted that this development is similar to the pattern found by Johnson and Newport [23] but that it contains no indication of an “abruptly defined ‘critical’ or sensitive period in L2 learning” [48, p. 569]. However, she modelled the self-ratings using an ordinal logistic regression model in which the aoa variable was logarithmically transformed. Technically, this is perfectly fine, but one should be careful not to read too much into the non-linear curves found. In logistic models, the outcome variable itself is modelled linearly as a function of the predictor variables and is expressed in log-odds. In order to compute the corresponding probabilities, these log-odds are transformed using the logistic function. Consequently, even if the model is specified linearly, the predicted probabilities will not lie on a perfectly straight line when plotted as a function of any one continuous predictor variable. Similarly, when the predictor variable is first logarithmically transformed and then used to linearly predict an outcome variable, the function linking the predicted outcome variables and the untransformed predictor variable is necessarily non-linear. Thus, non-linearities follow naturally from Stevens's model specifications. Moreover, cph -consistent discontinuities in the aoa – ua function cannot be found using her model specifications as they did not contain any parameters allowing for this.

Using data similar to Stevens's, Bialystok and Hakuta found that the link between the self-rated English competences of Chinese- and Spanish-speaking immigrants and their aoa could be described by a straight line [49] . In contrast to Stevens, Bialystok and Hakuta used a regression-based method allowing for changes in the function's slope, viz. locally weighted scatterplot smoothing ( lowess ). Informally, lowess is a non-parametrical method that relies on an algorithm that fits the dependent variable for small parts of the range of the independent variable whilst guaranteeing that the overall curve does not contain sudden jumps (for technical details, see [50] ). Hakuta et al. used an even larger sample from the same 1990 U.S. Census data on Chinese- and Spanish-speaking immigrants (2.3 million observations) [21] . Fitting lowess curves, no discontinuities in the aoa – ua slope could be detected. Moreover, the authors found that piecewise linear regression models, i.e. regression models containing a parameter that allows a sudden drop in the curve or a change of its slope, did not provide a better fit to the data than did an ordinary regression model without such a parameter.

To sum up, I have argued at length that regression approaches are superior to group mean and correlation coefficient comparisons for the purposes of testing the ‘flattened slope’ prediction. Acknowledging the reservations vis-à-vis self-estimated ua s, we still find that while the relationship between aoa and ua is not necessarily perfectly linear in the studies discussed, the data do not lend unequivocal support to this prediction. In the following section, I will reanalyse data from a recent empirical paper on the cph by DeKeyser et al. [44] . The first goal of this reanalysis is to further illustrate some of the statistical fallacies encountered in cph studies. Second, by making the computer code available I hope to demonstrate how the relevant regression models, viz. piecewise regression models, can be fitted and how the aoa representing the optimal breakpoint can be identified. Lastly, the findings of this reanalysis will contribute to our understanding of how aoa affects ua as measured using a gjt .

Summary of DeKeyser et al. (2010)

I chose to reanalyse a recent empirical paper on the cph by DeKeyser et al. [44] (henceforth DK et al.). This paper lends itself well to a reanalysis since it exhibits two highly commendable qualities: the authors spell out their hypotheses lucidly and provide detailed numerical and graphical data descriptions. Moreover, the paper's lead author is very clear on what constitutes a necessary condition for accepting the cph : a non-linearity in the age of onset of acquisition ( aoa )–ultimate attainment ( ua ) function, with ua declining less strongly as a function of aoa in older, post- cp arrivals compared to younger arrivals [14] , [22] . Lastly, it claims to have found cross-linguistic evidence from two parallel studies backing the cph and should therefore be an unsuspected source to cph proponents.

The authors set out to test the following hypotheses:

• Hypothesis 1: For both the L2 English and the L2 Hebrew group, the slope of the age of arrival–ultimate attainment function will not be linear throughout the lifespan, but will instead show a marked flattening between adolescence and adulthood.
• Hypothesis 2: The relationship between aptitude and ultimate attainment will differ markedly for the young and older arrivals, with significance only for the latter. (DK et al., p. 417)

Both hypotheses were purportedly confirmed, which in the authors' view provides evidence in favour of cph . The problem with this conclusion, however, is that it is based on a comparison of correlation coefficients. As I have argued above, correlation coefficients are not to be confused with regression coefficients and cannot be used to directly address research hypotheses concerning slopes, such as Hypothesis 1. In what follows, I will reanalyse the relationship between DK et al.'s aoa and gjt data in order to address Hypothesis 1. Additionally, I will lay bare a problem with the way in which Hypothesis 2 was addressed. The extracted data and the computer code used for the reanalysis are provided as supplementary materials, allowing anyone interested to scrutinise and easily reproduce my whole analysis and carry out their own computations (see ‘supporting information’).

Data extraction

In order to verify whether we did in fact extract the data points to a satisfactory degree of accuracy, I computed summary statistics for the extracted aoa and gjt data and checked these against the descriptive statistics provided by DK et al. (pp. 421 and 427). These summary statistics for the extracted data are presented in Table 1 . In addition, I computed the correlation coefficients for the aoa – gjt relationship for the whole aoa range and for aoa -defined subgroups and checked these coefficients against those reported by DK et al. (pp. 423 and 428). The correlation coefficients computed using the extracted data are presented in Table 2 . Both checks strongly suggest the extracted data to be virtually identical to the original data, and Dr DeKeyser confirmed this to be the case in response to an earlier draft of the present paper (personal communication, 6 May 2013).

https://doi.org/10.1371/journal.pone.0069172.t001

https://doi.org/10.1371/journal.pone.0069172.t002

Results and Discussion

Modelling the link between age of onset of acquisition and ultimate attainment.

I first replotted the aoa and gjt data we extracted from DK et al.'s scatterplots and added non-parametric scatterplot smoothers in order to investigate whether any changes in slope in the aoa – gjt function could be revealed, as per Hypothesis 1. Figures 3 and 4 show this not to be the case. Indeed, simple linear regression models that model gjt as a function of aoa provide decent fits for both the North America and the Israel data, explaining 65% and 63% of the variance in gjt scores, respectively. The parameters of these models are given in Table 3 .

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 1.

https://doi.org/10.1371/journal.pone.0069172.g003

The trend line is a non-parametric scatterplot smoother. The scatterplot itself is a near-perfect replication of DK et al.'s Fig. 5.

https://doi.org/10.1371/journal.pone.0069172.g004

https://doi.org/10.1371/journal.pone.0069172.t003

To ensure that both segments are joined at the breakpoint, the predictor variable is first centred at the breakpoint value, i.e. the breakpoint value is subtracted from the original predictor variable values. For a blow-by-blow account of how such models can be fitted in r , I refer to an example analysis by Baayen [55, pp. 214–222].

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g005

Solid: regression with breakpoint at aoa 18 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g006

https://doi.org/10.1371/journal.pone.0069172.t004

https://doi.org/10.1371/journal.pone.0069172.g007

Solid: regression with breakpoint at aoa 16 (dashed lines represent its 95% confidence interval); dot-dash: regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g008

Solid: regression with breakpoint at aoa 6 (dashed lines represent its 95% confidence interval); dot-dash (hardly visible due to near-complete overlap): regression without breakpoint.

https://doi.org/10.1371/journal.pone.0069172.g009

https://doi.org/10.1371/journal.pone.0069172.t005

https://doi.org/10.1371/journal.pone.0069172.t006

https://doi.org/10.1371/journal.pone.0069172.t007

https://doi.org/10.1371/journal.pone.0069172.t008

In sum, a regression model that allows for changes in the slope of the the aoa – gjt function to account for putative critical period effects provides a somewhat better fit to the North American data than does an everyday simple regression model. The improvement in model fit is marginal, however, and including a breakpoint does not result in any detectable improvement of model fit to the Israel data whatsoever. Breakpoint models therefore fail to provide solid cross-linguistic support in favour of critical period effects: across both data sets, gjt can satisfactorily be modelled as a linear function of aoa .

On partialling out ‘age at testing’

As I have argued above, correlation coefficients cannot be used to test hypotheses about slopes. When the correct procedure is carried out on DK et al.'s data, no cross-linguistically robust evidence for changes in the aoa – gjt function was found. In addition to comparing the zero-order correlations between aoa and gjt , however, DK et al. computed partial correlations in which the variance in aoa associated with the participants' age at testing ( aat ; a potentially confounding variable) was filtered out. They found that these partial correlations between aoa and gjt , which are given in Table 9 , differed between age groups in that they are stronger for younger than for older participants. This, DK et al. argue, constitutes additional evidence in favour of the cph . At this point, I can no longer provide my own analysis of DK et al.'s data seeing as the pertinent data points were not plotted. Nevertheless, the detailed descriptions by DK et al. strongly suggest that the use of these partial correlations is highly problematic. Most importantly, and to reiterate, correlations (whether zero-order or partial ones) are actually of no use when testing hypotheses concerning slopes. Still, one may wonder why the partial correlations differ across age groups. My surmise is that these differences are at least partly the by-product of an imbalance in the sampling procedure.

https://doi.org/10.1371/journal.pone.0069172.t009

The upshot of this brief discussion is that the partial correlation differences reported by DK et al. are at least partly the result of an imbalance in the sampling procedure: aoa and aat were simply less intimately tied for the young arrivals in the North America study than for the older arrivals with L2 English or for all of the L2 Hebrew participants. In an ideal world, we would like to fix aat or ascertain that it at most only weakly correlates with aoa . This, however, would result in a strong correlation between aoa and another potential confound variable, length of residence in the L2 environment, bringing us back to square one. Allowing for only moderate correlations between aoa and aat might improve our predicament somewhat, but even in that case, we should tread lightly when making inferences on the basis of statistical control procedures [61] .

On estimating the role of aptitude

Having shown that Hypothesis 1 could not be confirmed, I now turn to Hypothesis 2, which predicts a differential role of aptitude for ua in sla in different aoa groups. More specifically, it states that the correlation between aptitude and gjt performance will be significant only for older arrivals. The correlation coefficients of the relationship between aptitude and gjt are presented in Table 10 .

https://doi.org/10.1371/journal.pone.0069172.t010

The problem with both the wording of Hypothesis 2 and the way in which it is addressed is the following: it is assumed that a variable has a reliably different effect in different groups when the effect reaches significance in one group but not in the other. This logic is fairly widespread within several scientific disciplines (see e.g. [62] for a discussion). Nonetheless, it is demonstrably fallacious [63] . Here we will illustrate the fallacy for the specific case of comparing two correlation coefficients.

Apart from not being replicated in the North America study, does this difference actually show anything? I contend that it does not: what is of interest are not so much the correlation coefficients, but rather the interactions between aoa and aptitude in models predicting gjt . These interactions could be investigated by fitting a multiple regression model in which the postulated cp breakpoint governs the slope of both aoa and aptitude. If such a model provided a substantially better fit to the data than a model without a breakpoint for the aptitude slope and if the aptitude slope changes in the expected direction (i.e. a steeper slope for post- cp than for younger arrivals) for different L1–L2 pairings, only then would this particular prediction of the cph be borne out.

Using data extracted from a paper reporting on two recent studies that purport to provide evidence in favour of the cph and that, according to its authors, represent a major improvement over earlier studies (DK et al., p. 417), it was found that neither of its two hypotheses were actually confirmed when using the proper statistical tools. As a matter of fact, the gjt scores continue to decline at essentially the same rate even beyond the end of the putative critical period. According to the paper's lead author, such a finding represents a serious problem to his conceptualisation of the cph [14] ). Moreover, although modelling a breakpoint representing the end of a cp at aoa 16 may improve the statistical model slightly in study on learners of English in North America, the study on learners of Hebrew in Israel fails to confirm this finding. In fact, even if we were to accept the optimal breakpoint computed for the Israel study, it lies at aoa 6 and is associated with a different geometrical pattern.

Diverging age trends in parallel studies with participants with different L2s have similarly been reported by Birdsong and Molis [26] and are at odds with an L2-independent cph . One parsimonious explanation of such conflicting age trends may be that the overall, cross-linguistic age trend is in fact linear, but that fluctuations in the data (due to factors unaccounted for or randomness) may sometimes give rise to a ‘stretched L’-shaped pattern ( Figure 1, left panel ) and sometimes to a ‘stretched 7’-shaped pattern ( Figure 1 , middle panel; see also [66] for a similar comment).

Importantly, the criticism that DeKeyser and Larsson-Hall levy against two studies reporting findings similar to the present [48] , [49] , viz. that the data consisted of self-ratings of questionable validity [14] , does not apply to the present data set. In addition, DK et al. did not exclude any outliers from their analyses, so I assume that DeKeyser and Larsson-Hall's criticism [14] of Birdsong and Molis's study [26] , i.e. that the findings were due to the influence of outliers, is not applicable to the present data either. For good measure, however, I refitted the regression models with and without breakpoints after excluding one potentially problematic data point per model. The following data points had absolute standardised residuals larger than 2.5 in the original models without breakpoints as well as in those with breakpoints: the participant with aoa 17 and a gjt score of 125 in the North America study and the participant with aoa 12 and a gjt score of 117 in the Israel study. The resultant models were virtually identical to the original models (see Script S1 ). Furthermore, the aoa variable was sufficiently fine-grained and the aoa – gjt curve was not ‘presmoothed’ by the prior aggregation of gjt across parts of the aoa range (see [51] for such a criticism of another study). Lastly, seven of the nine “problems with supposed counter-evidence” to the cph discussed by Long [5] do not apply either, viz. (1) “[c]onfusion of rate and ultimate attainment”, (2) “[i]nappropriate choice of subjects”, (3) “[m]easurement of AO”, (4) “[l]eading instructions to raters”, (6) “[u]se of markedly non-native samples making near-native samples more likely to sound native to raters”, (7) “[u]nreliable or invalid measures”, and (8) “[i]nappropriate L1–L2 pairings”. Problem No. 5 (“Assessments based on limited samples and/or “language-like” behavior”) may be apropos given that only gjt data were used, leaving open the theoretical possibility that other measures might have yielded a different outcome. Finally, problem No. 9 (“Faulty interpretation of statistical patterns”) is, of course, precisely what I have turned the spotlights on.

Conclusions

The critical period hypothesis remains a hotly contested issue in the psycholinguistics of second-language acquisition. Discussions about the impact of empirical findings on the tenability of the cph generally revolve around the reliability of the data gathered (e.g. [5] , [14] , [22] , [52] , [67] , [68] ) and such methodological critiques are of course highly desirable. Furthermore, the debate often centres on the question of exactly what version of the cph is being vindicated or debunked. These versions differ mainly in terms of its scope, specifically with regard to the relevant age span, setting and language area, and the testable predictions they make. But even when the cph 's scope is clearly demarcated and its main prediction is spelt out lucidly, the issue remains to what extent the empirical findings can actually be marshalled in support of the relevant cph version. As I have shown in this paper, empirical data have often been taken to support cph versions predicting that the relationship between age of acquisition and ultimate attainment is not strictly linear, even though the statistical tools most commonly used (notably group mean and correlation coefficient comparisons) were, crudely put, irrelevant to this prediction. Methods that are arguably valid, e.g. piecewise regression and scatterplot smoothing, have been used in some studies [21] , [26] , [49] , but these studies have been criticised on other grounds. To my knowledge, such methods have never been used by scholars who explicitly subscribe to the cph .

I suspect that what may be going on is a form of ‘confirmation bias’ [69] , a cognitive bias at play in diverse branches of human knowledge seeking: Findings judged to be consistent with one's own hypothesis are hardly questioned, whereas findings inconsistent with one's own hypothesis are scrutinised much more strongly and criticised on all sorts of points [70] – [73] . My reanalysis of DK et al.'s recent paper may be a case in point. cph exponents used correlation coefficients to address their prediction about the slope of a function, as had been done in a host of earlier studies. Finding a result that squared with their expectations, they did not question the technical validity of their results, or at least they did not report this. (In fact, my reanalysis is actually a case in point in two respects: for an earlier draft of this paper, I had computed the optimal position of the breakpoints incorrectly, resulting in an insignificant improvement of model fit for the North American data rather than a borderline significant one. Finding a result that squared with my expectations, I did not question the technical validity of my results – until this error was kindly pointed out to me by Martijn Wieling (University of Tübingen).) That said, I am keen to point out that the statistical analyses in this particular paper, though suboptimal, are, as far as I could gather, reported correctly, i.e. the confirmation bias does not seem to have resulted in the blatant misreportings found elsewhere (see [74] for empirical evidence and discussion). An additional point to these authors' credit is that, apart from explicitly identifying their cph version's scope and making crystal-clear predictions, they present data descriptions that actually permit quantitative reassessments and have a history of doing so (e.g. the appendix in [8] ). This leads me to believe that they analysed their data all in good conscience and to hope that they, too, will conclude that their own data do not, in fact, support their hypothesis.

I end this paper on an upbeat note. Even though I have argued that the analytical tools employed in cph research generally leave much to be desired, the original data are, so I hope, still available. This provides researchers, cph supporters and sceptics alike, with an exciting opportunity to reanalyse their data sets using the tools outlined in the present paper and publish their findings at minimal cost of time and resources (for instance, as a comment to this paper). I would therefore encourage scholars to engage their old data sets and to communicate their analyses openly, e.g. by voluntarily publishing their data and computer code alongside their articles or comments. Ideally, cph supporters and sceptics would join forces to agree on a protocol for a high-powered study in order to provide a truly convincing answer to a core issue in sla .

Supporting Information

Dataset s1..

aoa and gjt data extracted from DeKeyser et al.'s North America study.

https://doi.org/10.1371/journal.pone.0069172.s001

Dataset S2.

aoa and gjt data extracted from DeKeyser et al.'s Israel study.

https://doi.org/10.1371/journal.pone.0069172.s002

Script with annotated R code used for the reanalysis. All add-on packages used can be installed from within R.

https://doi.org/10.1371/journal.pone.0069172.s003

Acknowledgments

I would like to thank Irmtraud Kaiser (University of Fribourg) for helping me to get an overview of the literature on the critical period hypothesis in second language acquisition. Thanks are also due to Martijn Wieling (currently University of Tübingen) for pointing out an error in the R code accompanying an earlier draft of this paper.

Author Contributions

Analyzed the data: JV. Wrote the paper: JV.

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Chapter 9: Hypothesis Testing

Back to chapter, critical region, critical values and significance level, previous video 9.2: null and alternative hypotheses, next video 9.4: p -value.

Hypothesis testing requires the sample statistics—such as proportion, mean, or standard deviation—to be converted into a value or score known as the test statistics.

Assuming that the null hypothesis is true, the test statistic for each sample statistic is calculated using the following equations.

As samples assume a particular distribution, a given test statistic value would fall into a specific area under the curve with some probability.

Such an area, which includes all the values of a test statistic that indicates that the null hypothesis must be rejected, is termed the rejection region or critical region.

The value that separates a critical region from the rest is termed the critical value. The critical values are the  z, t,  or chi-square values calculated at the desired confidence level.

The probability that the test statistic will fall in the critical region when the null hypothesis is actually true is called the significance level.

In the example of testing the proportion of healthy and scabbed apples, if the sample proportion is 0.9, the hypothesis can be tested as follows.

The critical region, critical value, and significance level are interdependent concepts crucial in hypothesis testing.

In hypothesis testing, a sample statistic is converted to a test statistic using z , t , or chi-square distribution. A critical region is an area under the curve in  probability distributions demarcated by the critical value. When the test statistic falls in this region, it suggests that the null hypothesis must be rejected. As this region contains all those values of the test statistic (calculated using the sample data) that suggest rejecting the null hypothesis, it is also known as the rejection region or region of rejection. The critical region may fall at the right, left, or both tails of the distribution based on the direction indicated in the alternative hypothesis and the calculated critical value.

A critical value is calculated using the z , t, or chi-square distribution table at a specific significance level. It is a fixed value for the given sample size and the significance level. The critical value creates a demarcation between all those values that suggest rejection of the null hypothesis and all those other values that indicate the opposite. A critical value is  based on a pre-decided significance level.

A significance level or level of significance or statistical significance is defined as the probability that the calculated test statistic will fall in the critical region. In other words, it is a statistical measure that indicates that the evidence for rejecting a true null hypothesis is strong enough. The significance level is indicated by α, and it is commonly 0.05 or 0.01.

What is a critical value?

A critical value is a point on the distribution of the test statistic under the null hypothesis that defines a set of values that call for rejecting the null hypothesis. This set is called critical or rejection region. Usually, one-sided tests have one critical value and two-sided test have two critical values. The critical values are determined so that the probability that the test statistic has a value in the rejection region of the test when the null hypothesis is true equals the significance level (denoted as α or alpha).

Critical values on the standard normal distribution for α = 0.05

Figure A shows that results of a one-tailed Z-test are significant if the value of the test statistic is equal to or greater than 1.64, the critical value in this case. The shaded area represents the probability of a type I error (α = 5% in this example) of the area under the curve. Figure B shows that results of a two-tailed Z-test are significant if the absolute value of the test statistic is equal to or greater than 1.96, the critical value in this case. The two shaded areas sum to 5% (α) of the area under the curve.

Examples of calculating critical values

In hypothesis testing, there are two ways to determine whether there is enough evidence from the sample to reject H 0 or to fail to reject H 0 . The most common way is to compare the p-value with a pre-specified value of α, where α is the probability of rejecting H 0 when H 0 is true. However, an equivalent approach is to compare the calculated value of the test statistic based on your data with the critical value. The following are examples of how to calculate the critical value for a 1-sample t-test and a One-Way ANOVA.

Calculating a critical value for a 1-sample t-test

• Select Calc > Probability Distributions > t .
• Select Inverse cumulative probability .
• In Degrees of freedom , enter 9 (the number of observations minus one).
• In Input constant , enter 0.95 (one minus one-half alpha).

This gives you an inverse cumulative probability, which equals the critical value, of 1.83311. If the absolute value of the t-statistic value is greater than this critical value, then you can reject the null hypothesis, H 0 , at the 0.10 level of significance.

Calculating a critical value for an analysis of variance (ANOVA)

• Choose Calc > Probability Distributions > F .
• In Numerator degrees of freedom , enter 2 (the number of factor levels minus one).
• In Denominator degrees of freedom , enter 9 (the degrees of freedom for error).
• In Input constant , enter 0.95 (one minus alpha).

This gives you an inverse cumulative probability (critical value) of 4.25649. If the F-statistic is greater than this critical value, then you can reject the null hypothesis, H 0 , at the 0.05 level of significance.

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What Is a Hypothesis? (Science)

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A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject.

In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.

In the study of logic, a hypothesis is an if-then proposition, typically written in the form, "If X , then Y ."

In common usage, a hypothesis is simply a proposed explanation or prediction, which may or may not be tested.

Writing a Hypothesis

Most scientific hypotheses are proposed in the if-then format because it's easy to design an experiment to see whether or not a cause and effect relationship exists between the independent variable and the dependent variable . The hypothesis is written as a prediction of the outcome of the experiment.

Null Hypothesis and Alternative Hypothesis

Statistically, it's easier to show there is no relationship between two variables than to support their connection. So, scientists often propose the null hypothesis . The null hypothesis assumes changing the independent variable will have no effect on the dependent variable.

In contrast, the alternative hypothesis suggests changing the independent variable will have an effect on the dependent variable. Designing an experiment to test this hypothesis can be trickier because there are many ways to state an alternative hypothesis.

For example, consider a possible relationship between getting a good night's sleep and getting good grades. The null hypothesis might be stated: "The number of hours of sleep students get is unrelated to their grades" or "There is no correlation between hours of sleep and grades."

An experiment to test this hypothesis might involve collecting data, recording average hours of sleep for each student and grades. If a student who gets eight hours of sleep generally does better than students who get four hours of sleep or 10 hours of sleep, the hypothesis might be rejected.

But the alternative hypothesis is harder to propose and test. The most general statement would be: "The amount of sleep students get affects their grades." The hypothesis might also be stated as "If you get more sleep, your grades will improve" or "Students who get nine hours of sleep have better grades than those who get more or less sleep."

In an experiment, you can collect the same data, but the statistical analysis is less likely to give you a high confidence limit.

Usually, a scientist starts out with the null hypothesis. From there, it may be possible to propose and test an alternative hypothesis, to narrow down the relationship between the variables.

Example of a Hypothesis

Examples of a hypothesis include:

• If you drop a rock and a feather, (then) they will fall at the same rate.
• Plants need sunlight in order to live. (if sunlight, then life)
• Eating sugar gives you energy. (if sugar, then energy)
• White, Jay D.  Research in Public Administration . Conn., 1998.
• Schick, Theodore, and Lewis Vaughn.  How to Think about Weird Things: Critical Thinking for a New Age . McGraw-Hill Higher Education, 2002.
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Understanding Hypothesis Testing

Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

To test the validity of the claim or assumption about the population parameter:

• A sample is drawn from the population and analyzed.
• The results of the analysis are used to decide whether the claim is true or not.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

• Null hypothesis (H 0 ): In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured cases or no relationship among groups. In other words, it is a basic assumption or made based on the problem knowledge. Example : A company’s mean production is 50 units/per da H 0 : [Tex]\mu [/Tex] = 50.
• Alternative hypothesis (H 1 ): The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis.  Example: A company’s production is not equal to 50 units/per day i.e. H 1 : [Tex]\mu [/Tex] [Tex]\ne [/Tex] 50.

Key Terms of Hypothesis Testing

• Level of significance : It refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible for accepting a hypothesis, so we, therefore, select a level of significance that is usually 5%. This is normally denoted with  [Tex]\alpha[/Tex] and generally, it is 0.05 or 5%, which means your output should be 95% confident to give a similar kind of result in each sample.
• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

• Left-Tailed (Left-Sided) Test: The alternative hypothesis asserts that the true parameter value is less than the null hypothesis. Example: H 0 ​: [Tex]\mu \geq 50 [/Tex] and H 1 : [Tex]\mu < 50 [/Tex]
• Right-Tailed (Right-Sided) Test : The alternative hypothesis asserts that the true parameter value is greater than the null hypothesis. Example: H 0 : [Tex]\mu \leq50 [/Tex] and H 1 : [Tex]\mu > 50 [/Tex]

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]

To delve deeper into differences into both types of test: Refer to link

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

• Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha( [Tex]\alpha [/Tex] ).
• Type II errors : When we accept the null hypothesis, but it is false. Type II errors are denoted by beta( [Tex]\beta [/Tex] ).

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ​), suggesting an effect or difference.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

• If the p-value is less than or equal to the significance level i.e. ( [Tex]p\leq\alpha [/Tex] ), you reject the null hypothesis. This indicates that the observed results are unlikely to have occurred by chance alone, providing evidence in favor of the alternative hypothesis.
• If the p-value is greater than the significance level i.e. ( [Tex]p\geq \alpha[/Tex] ), you fail to reject the null hypothesis. This suggests that the observed results are consistent with what would be expected under the null hypothesis.

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]

• [Tex]\bar{x} [/Tex] is the sample mean,
• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]

• [Tex]O_{ij}[/Tex] is the observed frequency in cell [Tex]{ij} [/Tex]
• i,j are the rows and columns index respectively.
• [Tex]E_{ij}[/Tex] is the expected frequency in cell [Tex]{ij}[/Tex] , calculated as : [Tex]\frac{{\text{{Row total}} \times \text{{Column total}}}}{{\text{{Total observations}}}}[/Tex]

Real life Examples of Hypothesis Testing

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Case A

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )

T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] ​ and we get accordingly , Z =2.039999999999992.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Python Implementation of Case B

import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )

Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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Conceptual Theory Art as a Method for Interdisciplinary Research, Hypothesis, and Critical Analysis Both Within and Beyond Traditional Western Social Sciences

10 Pages Posted: 6 Sep 2024

Adam Daley Wilson

Stanford Law School

Date Written: August 01, 2024

This paper researches whether and how Theory Art, an emergent conceptual art form across visual art, music, literature, and other artistic disciplines, might serve as a method for integrating existing methodologies of interdisciplinary research, hypothesis-making, and critical analysis, both in and beyond the traditional social sciences of the West, by creatively blending elements of artistic and philosophical traditions across cultures, thought systems, and time periods, to observe, propose, critique, and preserve human theories, arguments, and narratives, including theories conveyed through spoken-word storytelling. The paper first provides an academic research survey as a foundation to measure the intellectual rigor of Theory Art's emphasis on construction of meaning by the audience-viewer-listener-reader, rather than by the artist-creator alone. The paper then evaluates Theory Art relative to other artistic movements to date. The paper then measures Theory Art in relation to traditional Western social sciences, including economics and other quantitative social sciences, to measure the potential degree and scope of Theory Art's interdisciplinary nature and theoretical foundations. The paper then proposes several observable standards and elements that appear to indicate that Theory Art is a new form of Conceptual Art, including in relation to, and distinct from, traditional Conceptual Art as well as Post-conceptual Art, Post-minimalism, Postmodernism, and Neo-conceptual Art. By integrating a wide survey of relevant cross-cultural researchers and theorists, from multiple societies and time periods, this research paper is a first attempt to analyze Theory Art's substantive elements as an artistic movement. This paper is also a first attempt to measure Theory Art's intellectual and methodological rigor, in order to assess Theory Art as a method for interdisciplinary research, hypothesis-making, and critical analysis of human theories both within and beyond traditional Western social sciences. The paper concludes that Theory Art offers new theory-communication methodologies that have new implications for art, and cultural discourse, not only in Western cultures but all cultures worldwide. Finally, this research paper is intended as a foundation allowing future researchers and scholars to develop hypotheses by which to test not only the rigor of Theory Art itself as an art movement related to Conceptual Art, but also the rigor of any given work that may be made by an artist-creator.

Keywords: Theory Art, Conceptual Art, Art Theory, Art History, Social Sciences, Post-Theory Art, Post-Contemporary Art

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Adam Daley Wilson (Contact Author)

Stanford law school ( email ), do you have a job opening that you would like to promote on ssrn, paper statistics, related ejournals, aesthetics & philosophy of art ejournal.

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