114 Graduate Student Journal of Psychology 2022, Vol. 19 Copyright 2022 by the Department of Counseling and Clinical Psychology Teachers College, Columbia University Evaluating the Sampling Precision of Social Identity Related Published Research �ȅLjǓ�vǩǹșȅǿॹ���ƺΚǩǏ�eȖƺЙǾȅΛॹ��ƺǿǏ��eȅǿΡ�vƺǿǠॹ New Mexico State University Cong Wang, University of Nebraska Omaha Social identity theory states that a person’s sense of who they are is based largely on their group membership(s). We categorize ourselves, identify with groups, and compare our groups with others, in the hopes that our self-esteem is ǾƺǩǿȠƺǩǿǓǏ�ȅȖ�LjȅȅșȠǓǏ�ǟȖȅǾ�ȠǦǩș�ljȅǾȒƺȖǩșȅǿঀ���ȒȖǓȖǓȕȣǩșǩȠǓ�ȅǟ�șljǩǓǿȠǩЙlj�ȖǓșǓƺȖljǦॹ�ǓΚǓǿ�ȖǓǠƺȖǏǩǿǠ�șȅljǩƺǹ�ǩǏǓǿȠǩȠΡॹ�ǩș� ȠǦƺȠ�ȖǓșǓƺȖljǦǓȖș�ǿǓǓǏ�Ƞȅ�LjǓ�ljȅǿЙǏǓǿȠ�ȠǦƺȠ�ȠǦǓ�ǓǾȒǩȖǩljƺǹ�ǟƺljȠș�ȖǓƺǹǹΡ�ƺȖǓ�ǟƺljȠȣƺǹআ�ȠǦƺȠ�ǩșॹ�ȠǦƺȠ�ȠǦǓ�șƺǾȒǹǓ�șȠƺȠǩșȠǩljș�ȖǓȒȅȖȠǓǏ� accurately estimate corresponding population parameters; this is known as sampling precision, or how precisely our sample corresponds to our population. By employing the recently invented a priori procedure, the present research assesses the sampling precision with which published experimental and correlational social identity research statistics, across three time periods, estimate corresponding population parameters. We hypothesized 1: The precision of re- search in the social identity area should be hopefully below the 0.10 level or at least the 0.20 level for true experimental designs and 2: Precision in the social identity area should be improving, with recent social identity research enjoying ƺ�ȒȖǓljǩșǩȅǿ�ƺǏΚƺǿȠƺǠǓ�ȅΚǓȖ� ǹǓșș� ȖǓljǓǿȠ� șȅljǩƺǹ� ǩǏǓǿȠǩȠΡ� ȖǓșǓƺȖljǦঀ���șƺǾȒǹǓ�ȅǟࢶࢸ��ƺljƺǏǓǾǩlj�ȒƺȒǓȖșॹ� ƺljȖȅșșࢷࢵ��ǏǩАǓȖǓǿȠ� journals was collected for analysis. For experimental studies, the mean precision level was 0.51 and the median pre- cision level was 0.50 (n = 39). For correlational studies, the mean precision level was 0.24 and the median precision ǹǓΚǓǹ�Λƺșࢱ�ঀࢱࢳ�শǿ�઀ࢶࢶ�ষঀ�eǦǓ�Ǿƺǩǿ�ЙǿǏǩǿǠș�ƺȖǓ�ȒǓșșǩǾǩșȠǩljॹ�LjȣȠ�ΛǩȠǦ�ȠǦǓ�ǠǹǩǾǾǓȖ�ȅǟ�ǹǩǠǦȠ�ȠǦƺȠ�ȒȖǓljǩșǩȅǿ�ǩș�ǩǾȒȖȅΚǩǿǠঀ� ?Dzͧ͡ȤȵǮȸ࣒�ȱȵDzǨȈȸȈȤȞ࣓�ǨȤȞϫǮDzȞǨDz࣓�ȸȤǨȈǙȘ�ȈǮDzȞȿȈȿͧ�ȿȅDzȤȵ࣓ͧ�Ǚ�ȱȵȈȤȵȈ�ȱȵȤǨDzǮɂȵDz࣓�DzȸȿȈȝǙȿȈȤȞ The overarching idea of social identity theo- ry is that the individual and the group are inter- twined. People are born into social societies, they develop ties to social groups, they classify them- selves as members of those social groups, and the ȣǿǩȕȣǓ� ljȅǾLjǩǿƺȠǩȅǿș� ȅǟ� ȠǦȅșǓ� șȅljǩƺǹ� ljǹƺșșǩЙljƺȠǩȅǿș� create unique individuals. According to Stets and �ȣȖǷǓ� শࢱࢱࢱࢳষॹ� ǩǏǓǿȠǩȠΡ� ǩș� ȖǓМǓΠǩΚǓ� ǩǿ� ȠǦƺȠ� ǩȠ�ǾƺǷǓș� ǩȠ- șǓǹǟ� ƺǿ� ȅLjǴǓljȠ� ΛǦǩljǦ� ljƺǿ� LjǓ� ljƺȠǓǠȅȖǩΦǓǏॹ� ljǹƺșșǩЙǓǏॹ� ƺǿǏ� ǿƺǾǓǏ� ǩǿ� ȖǓǹƺȠǩȅǿ� Ƞȅ� ȅȠǦǓȖ� șȅljǩƺǹ� ljǹƺșșǩЙljƺȠǩȅǿșঀ� Social identity is a person’s knowledge that they belong to a social group or category (Hogg & Abrams, 1988). A social group consists of individuals who hold ƺ�șǓȠ�ȅǟ�ljȅǾǾȅǿ�șȅljǩƺǹ� ǩǏǓǿȠǩЙljƺȠǩȅǿșঀ�eǦǓșǓ�ǠȖȅȣȒș� are further categorized as the ingroup, which consists of individuals who are like the self, and the outgroup, which consists of individuals who are unlike the self. This process of accentuating the similarities with the ǩǿǠȖȅȣȒ�ƺǿǏ�ǏǩАǓȖǓǿljǓș�ΛǩȠǦ�ȠǦǓ�ȅȣȠǠȖȅȣȒ�ǹǓƺǏș�Ƞȅ�ƺ� second process called social comparison. The process of identifying and categorizing yourself is known as self-categorization. Social identity researchers com- monly assume that by classifying social identities through self-categorization (and by extension accen- tuating similarities with the ingroup, etc.), individ- uals will enhance their own positive self-esteem by promoting the ingroup and denigrating the outgroup. No matter what the theoretical criticisms of so- cial identity happen to be, the theoretical basis of identity and social identity research is not the current ȕȣǓșȠǩȅǿঀ�eȖƺЙǾȅΛ� ƺǿǏ�EΡȧΦ� শࢺࢲࢱࢳষ� ǏǓǾȅǿșȠȖƺȠǓǏ� ǩǾȒȖǓljǩșǩȅǿ� ΛǩȠǦ� ȖǓșȒǓljȠ� Ƞȅ� ЙΚǓ� ƺȖǓƺș� ȅǟ� ȒșΡljǦȅǹȅǠΡ� (social, developmental, clinical, cognitive, and neuro) ƺǿǏ� eȖƺЙǾȅΛॹ� /ΡǾƺǿॹ� ƺǿǏ� ?ȅșȠΡǷ� শࢱࢳࢱࢳষ� ǏǓǾȅǿ- strated imprecision with respect to marketing research șȒǓljǩЙljƺǹǹΡঀ� KȣȖ� Ǡȅƺǹ� ǿȅΛ� ǩș� Ƞȅ� ǓΠȒƺǿǏ� ȠǦƺȠ� ǹǩǿǓ� ȅǟ� inquiry and address a potential criticism that, to our ǷǿȅΛǹǓǏǠǓॹ�Ǧƺș�ǿȅȠ�LjǓǓǿ�ǾƺǏǓঀ�^ȒǓljǩЙljƺǹǹΡॹ�ȅȣȖ�Ǡȅƺǹ� is to address the issue of the sampling precision of so- cial identity research. Sampling precision is a measure of how close a sample’s descriptive statistics are to the corresponding population parameters. Usually, and in the present case, this is measured as a fraction of a stan- dard deviation. If the sampling precision of that re- search is impressive, it would imply that, whatever the ȅȠǦǓȖ� ǏǓЙljǩǓǿljǩǓș� ȅǟ� șȅljǩƺǹ� ǩǏǓǿȠǩȠΡ� ȖǓșǓƺȖljǦॹ� ƺȠ� ǹǓƺșȠ� there would be good reason to trust that the sample statistics are a good estimate of the corresponding pop- ulation parameters. In contrast, if the sampling preci- sion of social identity research is poor, it would imply that the sample statistics are not good estimates of the corresponding population parameters. A further im- ȒǹǩljƺȠǩȅǿ�ΛȅȣǹǏ�LjǓ�ȠǦƺȠ�ȠǦǓ�ЙǿǏǩǿǠș�ljƺǿǿȅȠ�LjǓ�ȠȖȣșȠǓǏॹ� which would be a preliminary problem that future re- 115 SOCIAL IDENTITY PRECISION search would have to solve before addressing the more ljȅǿljǓȒȠȣƺǹ�ǩșșȣǓș�ǩǏǓǿȠǩЙǓǏ�ǩǿ�ȠǦǓ�ǟȅȖǓǠȅǩǿǠ�ȒƺȖƺǠȖƺȒǦঀ To summarize the present thinking, we believe that getting the empirical facts straight is vital both for theorizing and for evaluating theories. If a theory is unable to explain human behavior reliably and val- idly, then it is not a strong psychological theory. By ensuring that sample statistics are good estimators of population parameters, the reliability and validity of theoretical conclusions related to social identity theory ljƺǿ�ǩǿșȒǩȖǓ�ǠȖǓƺȠǓȖ�ljȅǿЙǏǓǿljǓঀ�'ȣȖȠǦǓȖॹ�ȠǦǓ�ǠǓǿǓȖƺǹǩΦƺ- tion of these theories would also be improved. Thus, for the purpose of this article, the actual content of social identity theory is unimportant. What is most important, is the sampling practices of that research. To examine this issue, we use the a priori procedure. The A Priori Procedure The a priori procedure (APP) assumes the cru- ciality of obtaining sample statistics that are good estimators of corresponding population parameters শeȖƺЙǾȅΛ� ǓȠ� ƺǹঀॹ� �ষঀࢱࢳࢱࢳ @ǓșȠ� ȠǦǓ� ȖǓƺǏǓȖ� ǏȅȣLjȠș� ȠǦǩșॹ� imagine a fanciful scenario where Laplace’s omni- scient demon appears and informs us that there is no relationship between sample statistics and correspond- ing population parameters. In that case, no empirical reports would be trusted, nor would sample statis- tics be taken to provide good tests of hypotheses. Of course, there is no demon, but considering the fanciful scenario renders salient the importance of estimation. Given that estimation is crucial, there are two re- lated questions. First, there is the precision question: How close do we desire that sample statistics be to their corresponding population parameters? Second, ȠǦǓȖǓ� ǩș� ȠǦǓ� ljȅǿЙǏǓǿljǓ� ȕȣǓșȠǩȅǿॸ� vǦƺȠ� ȒȖȅLjƺLjǩǹǩȠΡ� do we insist on of being that close? The basic idea of ȠǦǓ��XX�ǩș�ȠǦƺȠ�ȠǦǓ�ȖǓșǓƺȖljǦǓȖ�ǾƺǷǓș�șȒǓljǩЙljƺȠǩȅǿș�ǟȅȖ� ȒȖǓljǩșǩȅǿ�ƺǿǏ�ljȅǿЙǏǓǿljǓॹ�ƺǿǏ�ȠǦǓǿ�ƺǿ��XX�ǓȕȣƺȠǩȅǿ� ȒȖȅΚǩǏǓș�ȠǦǓ�ǿǓljǓșșƺȖΡ�șƺǾȒǹǓ�șǩΦǓ�Ƞȅ�ǾǓǓȠ�ȠǦǓ�șȒǓljǩЙ- cations. If the researcher collects the necessary sample size, or a larger one, then the researcher can be assured ȅǟ�ǾǓǓȠǩǿǠ�ȠǦǓ�șȒǓljǩЙljƺȠǩȅǿș�শeȖƺЙǾȅΛ�ǓȠ�ƺǹঀॹࢱࢳࢱࢳ�ষঀ Consider an example. Suppose that a researcher intends to collect a single sample and is interested in ЙǿǏǩǿǠ� ȅȣȠ� ȠǦǓ� șƺǾȒǹǓ� șǩΦǓ� ǿǓǓǏǓǏ� Ƞȅ� LjǓ� �ઔࢶࢺ ljȅǿЙ- dent of obtaining a sample mean within one-tenth of a standard deviation of the population mean. Us- ǩǿǠ�ƺǿ�ǓȕȣƺȠǩȅǿ�LjΡ�eȖƺЙǾȅΛ�শࢸࢲࢱࢳআࢺࢲࢱࢳ�ষॹ�� � � � � � � � � � � � ॹ where n is the sample size, f is the desired precision, and z(1-c)/2 is the z-score that corresponds to the de- șǩȖǓǏ� ǏǓǠȖǓǓ� ȅǟ� ljȅǿЙǏǓǿljǓঀ� 2ǿșȠƺǿȠǩƺȠǩǿǠ� ΚƺǹȣǓș� ǩǿȠȅ� the equation indicates the following: n=(1.96/0.1)2 ઀ࢵࢹࢴঀࢷࢲઇࢶࢹࢴঀ�eǦȣșॹ�ȠǦǓ�ȖǓșǓƺȖljǦǓȖ�ΛȅȣǹǏ�ǦƺΚǓ�Ƞȅ�ljȅǹ- ǹǓljȠ� ��ȒƺȖȠǩljǩȒƺǿȠșࢶࢹࢴ Ƞȅ�ǾǓǓȠ� șȒǓljǩЙljƺȠǩȅǿșঀ� 2ǿșǩșȠǩǿǠ� on 385 participants may seem extreme compared to the much smaller sample sizes in most research, however, this provides a much greater level of precision than in ȠΡȒǩljƺǹ�ȒșΡljǦȅǹȅǠΡ�ȅȖ�ǾƺȖǷǓȠǩǿǠ�ȖǓșǓƺȖljǦ�শeȖƺЙǾȅΛ�૭� EΡȧΦॹࢺࢲࢱࢳ�আ�eȖƺЙǾȅΛॹ�/ΡǾƺǿॹ�૭�?ȅșȠΡǷॹࢱࢳࢱࢳ�ষঀ�2ǟ� the researcher is willing to settle for less precision, such as precision at the 0.3 or 0.4 level typical in much social psychology research, the necessary sample size would drop dramatically. This precision can be treated much ǹǩǷǓ�ljȅǿЙǏǓǿljǓ�ǹǓΚǓǹॹ�ΛǦǓȖǓ�ΛǓ�ΛƺǿȠ�Ƞȅ�șǦȅȅȠ�ǟȅȖ�ƺࢶࢺ�ઔ� ljȅǿЙǏǓǿljǓ�ǹǓΚǓǹ�LjȣȠ�ǾƺΡ�ǦƺΚǓ�Ƞȅ�șǓȠȠǹǓ�ǟȅȖࢱࢺ�ઔॹ�ǟȅȖ�ȒȖǓ- cision we can shoot for .1 but may have to settle for .3 or.4, depending on the circumstances of the research. Although the APP was designed to be used pre-data, it can be used post-data too, which is a neces- sary condition for the present work. Remaining with the foregoing example, suppose the researcher wanted to collect 385 participants but only succeeded in ob- ȠƺǩǿǩǿǠࢱࢱࢲ��ȅǟ�ȠǦǓǾঀ�hșǩǿǠࢶࢺ�ઔ�ljȅǿЙǏǓǿljǓ�ƺș�ƺ�ljȅǿ- ventional value, what is the precision entailed by 100 participants? To answer, the foregoing equation can be algebraically manipulated to yield f as opposed to Thus, if the sample size is 100, rather than 385, the actual precision would be 0.196 rather than ȠǦǓ� ǏǓșǩȖǓǏ� ΚƺǹȣǓ� ȅǟ� �ࢱࢲঀࢱ শeȖƺЙǾȅΛ� ǓȠ� ƺǹঀॹ� ষঀࢹࢲࢱࢳ Researchers are usually interested in more com- ȒǹǓΠ� ljƺșǓș� șȣljǦ� ƺș� ǏǩАǓȖǓǿljǓș� LjǓȠΛǓǓǿ� ǠȖȅȣȒșॹ� ljȅȖ- ȖǓǹƺȠǩȅǿ� ljȅǓГljǩǓǿȠș� ȅȖ� ǟȣǿljȠǩȅǿș� ȅǟ� ȠǦǓǾ� শǓঀǠঀॹ� ȖǓ- ǠȖǓșșǩȅǿ� ΛǓǩǠǦȠșষॹ� ƺǿǏ� șȅ� ȅǿ� শeȖƺЙǾȅΛॹ� �ষঀࢺࢲࢱࢳ 2ǿ� more complex cases, the mathematics can become extremely complex; but it is not necessary to address that complexity here because Hui et al. (2020) pub- lished links to programs for rendering the computa- tions and these are free and user-friendly. However, ȠǦȅșǓ�ǩǿȠǓȖǓșȠǓǏ�ǩǿ�ȠǦǓ�ǾƺȠǦǓǾƺȠǩljș�ljƺǿ�ljȅǿșȣǹȠ�eȖƺЙ- mow and MacDonald (2017) for multiple groups, eȖƺЙǾȅΛॹ�vƺǿǠॹ�ƺǿǏ�vƺǿǠ�শࢱࢳࢱࢳষ�ǟȅȖ�ǏǩАǓȖǓǿljǓș� ǩǿ� means for independent samples or dependent samples, ƺǿǏ� vƺǿǠ� ǓȠ� ƺǹঀ� শࢲࢳࢱࢳষ� ǟȅȖ� ljȅȖȖǓǹƺȠǩȅǿ� ljȅǓГljǩǓǿȠșঀ To place into perspective the importance of preci- 116 WILSON, TRAFIMOW, WANG, WANG șǩȅǿ�ΚƺǹȣǓșॹ�ΛǓ�ȠȣȖǿ�Ƞȅ�eȖƺЙǾȅΛ�ǓȠ�ƺǹ�শࢹࢲࢱࢳষঀ�vǩȠǦǩǿ� șljǩǓǿȠǩЙlj�ǩǿȕȣǩȖΡॹ�ȖǓȒǹǩljƺȠǩȅǿ�ǩș�ƺ�ΚǩȠƺǹ�Ƞȅȅǹ�ǟȅȖ�ȠǦǓ�Κƺ- lidity and reliability of a theory. If the results based on a theory cannot be replicated, then that theory lacks șȣГljǩǓǿȠ�șȣȒȒȅȖȠঀ�2ǿ�ƺ�șǩǿǠǹǓেșƺǾȒǹǓ�șȠȣǏΡॹ� ǩǟ�ΛǓ�ǏǓ- sire a precision value of .1, and we only have a sample size of 10, then the probability of replicating would only be .06! However, with 111 participants, that probability becomes .5, and so on as the sample size in- creases. The bottom line, then, is that the APP can be used, in a post-data fashion, to determine the sampling precision of varied published social identity research, where the sample sizes are reported. And this in turn can be used to draw conclusions about the possible replicability, validity, etc. of that published research. One question that often comes up is why sam- ȒǹǩǿǠ�ȒȖǓljǩșǩȅǿ�ǾƺȠȠǓȖș�ƺȠ�ƺǹǹঁ�2ǟ�ΛǓ�ǦƺΚǓ�ƺ�ǠȅȅǏ�ǓАǓljȠ� size, then what is the point of the APP? Well, think ƺLjȅȣȠ�ǓАǓljȠ�șǩΦǓঀ�2ǟ�ΛǓ�ƺȖǓ�ǹȅȅǷǩǿǠ�ǟȅȖ�ƺ�șȒǓljǩЙlj�ǓΠȒǓȖ- ǩǾǓǿȠƺǹ�ǓАǓljȠॹ�ƺǿǏ�ȅȣȖ�ǓАǓljȠ�șǩΦǓ� ǩș� ǹƺȖǠǓ�ǓǿȅȣǠǦॹ�ǿȅ� matter how small a sample we have, we will be able to șǓǓ�ȠǦǓ�ǓАǓljȠঀ�/ȅΛǓΚǓȖॹ�ǏȅǓș�ȠǦƺȠ�ǾǓƺǿ�ȠǦƺȠ�ȠǦǓ�ΛǦȅǹǓ� population of the world will also undergo that same ǓАǓljȠঁ�[ǓƺǹǩșȠǩljƺǹǹΡ�ǿȅȠঀ�eǦƺȠ� ǩșॹ� ǿȅ�ǾƺȠȠǓȖ�ΛǦƺȠ� ȠǦǓ� ȅLjȠƺǩǿǓǏ� ǓАǓljȠ� șǩΦǓ� ǟȅȖ� ȅȣȖ� șƺǾȒǹǓॹ�ΛǓ� ǦƺΚǓ� ǿȅ� ǩǏǓƺ� ǩǟ� ȠǦƺȠ�ǓАǓljȠ�șǩΦǓ� ǩș�ljǹȅșǓ�Ƞȅ�ȠǦǓ�ȒȅȒȣǹƺȠǩȅǿ�ǓАǓljȠ�șǩΦǓ� unless there has been some sort of APP calculation. Therefore, there is no way to know how well the ob- ȠƺǩǿǓǏ�ǓАǓljȠ�șȣȒȒȅȖȠș�ȅȖ�ǏǩșljȅǿЙȖǾș�ȠǦǓ�ǦΡȒȅȠǦǓșǩș�ȅȖ� theory from which the hypothesis was derived. This is the strength of the APP and sampling precision. Sampling precision values, and the APP, can be used Ƞȅ�ȒȖȅΚǩǏǓ� ǩǿljȖǓƺșǓǏ�ljȅǿЙǏǓǿljǓ� ǩǿ�ȅȣȖ� șƺǾȒǹǓ� șȠƺȠǩș- Ƞǩljșॹ�ƺǿǏ�ȠǦȣș� ǩǿljȖǓƺșǓǏ�ljȅǿЙǏǓǿljǓ�ȠǦƺȠ� ȠǦǓ�ǓАǓljȠ�ΛǓ� are seeing is representative of the overall population, ǿȅȠ�ǴȣșȠ�ȠǦƺȠ�ȠǦǓ�ǓАǓljȠ�ǩș�ȒȖǓșǓǿȠ�ǩǿ�ȅȣȖ�șƺǾȒǹǓঀ�KΚǓȖ- all, the APP sampling precision values can be viewed ƺș� ƺ� ljǹȅșǓǿǓșș� șȠƺȠǩșȠǩlj� șǩǾǩǹƺȖ� Ƞȅ� ljȅǿЙǏǓǿljǓ� ǹǓΚǓǹș� ǩǿ� that the more stringent criteria we use, the more con- ЙǏǓǿȠ� ΛǓ� ljƺǿ� LjǓ� ǩǿ� ȅȣȖ� ǩǿǟǓȖǓǿljǓș� ƺǿǏ� ljȅǿljǹȣșǩȅǿșঀ In this study, we tested competing hypotheses from optimistic versus pessimistic viewpoints. From an opti- mistic viewpoint, the expectations would be as follows. H1: The precision of research in the social identi- ty area should be hopefully below the 0.10 level or at ǹǓƺșȠ�ȠǦǓࢱ�ঀࢱࢳ�ǹǓΚǓǹ�ǟȅȖ�ȠȖȣǓ�ǓΠȒǓȖǩǾǓǿȠƺǹ�ǏǓșǩǠǿș�শeȖƺЙ- mow, 2018). H2: Precision in the social identity area should be improving, with recent social identity research enjoy- ing a precision advantage over less recent social identi- ty research. Of course, from a pessimistic perspective, the foregoing hypotheses should not be supported. If the precision of social identity research is worse than 0.30 শȠǦƺȠ� ǩș� ǹƺȖǠǓȖষॹ� ȠǦƺȠ�ΛȅȣǹǏ�șȣȒȒȅȖȠ�ȒǓșșǩǾǩșǾ�শeȖƺЙ- ǾȅΛॹࢹࢲࢱࢳ�ষঀ�'ȣȖȠǦǓȖॹ�ǩȠ�ΛȅȣǹǏ�LjǓ�ȣǿǏǓșǩȖƺLjǹǓ�Ƞȅ�ЙǿǏ� that precision is not improving, as an increase in preci- sion is related to an increase in replicability and stron- ger generalization of theory, and as research methods have supposedly improved as time passes, we should hope to see an improvement in sampling precision as well. Finally, because the APP has never been used systematically to analyze correlational research, the relevant correlational analyses to be presented can be ljȅǿșǩǏǓȖǓǏ�ǓΠȒǹȅȖƺȠȅȖΡ�ƺș� ƺ�ЙȖșȠ� șȠǓȒ� ȠȅΛƺȖǏș�ƺ�ǾȅȖǓ� integral use of the APP in sampling analysis. Method The goal of this research was to investigate the pre- cision (how close a sample statistic would be to its pop- ulation parameter) of a sample of published social iden- tity research. To investigate the question of whether șȅljǩƺǹ�ǩǏǓǿȠǩȠΡ�ȖǓșǓƺȖljǦ�ЙǿǏǩǿǠș�ƺȖǓ�ȒȖǓljǩșǓ�ǓǿȅȣǠǦ�Ƞȅ� be trusted, we applied the APP programs presented by Hui et al. (2020) to test the precision of the studies in- cluded in a sample of published social identity research. The present a posteriori use of the APP has been sys- ȠǓǾƺȠǩljƺǹǹΡ�ǓǾȒǹȅΡǓǏ�LjΡ�eȖƺЙǾȅΛ�ƺǿǏ�EΡȧΦ�শࢺࢲࢱࢳষ� ƺǿǏ�eȖƺЙǾȅΛॹ�/ΡǾƺǿॹ�ƺǿǏ�?ȅșȠΡǷ�শࢱࢳࢱࢳষ�ǟȅȖ�ǓΠȒǓȖ- imental research but not for correlational research. Procedure A total sample of 75 academic papers (see Table 1) was collected, prior to data analysis, broken down into 25 papers per time period across three time peri- ods: 2014 to 2021, 1995 to 2001, and 1975 to 1981. 2Ƞ�Λƺș� ǩǾȒȅȖȠƺǿȠ� Ƞȅ� ǩǿljǹȣǏǓ� ƺ� șƺǾȒǹǓ� șȣГljǩǓǿȠǹΡ� Ǐǩ- verse to be representative of the existing literature, while also ensuring that the sample broke into equal portions. The sample included articles from 46 dif- ferent journals, randomly selected from the set of ar- ticles that met our criteria, from multiple academic disciplines (i.e., psychology, advertising, business, so- ciology, etc.). The primary criteria for selection were: 1. Must include social identity as a primary tar- get of interest for investigation. This may be as 117 a theoretical concept, an applied concept, etc. This was checked via the mention of social iden- tity in subject terms, titles, or the paper itself. 2. Must directly specify the methodology (cor- relational, between-subjects, etc.). This is im- portant, as determining the method of anal- ysis is vital to understanding the precision of particular samples using the APP equations. 3. Must have a publication date within one of the three date ranges (1975-1981, 1995-2001, 2014-2021). These date ranges were chosen as an exploratory (not exhaustive) representation of research across time. The articles were picked using the institution- ƺǹ� ǹǩLjȖƺȖΡ� șǓƺȖljǦ� Ƞȅȅǹঀ�?ǓΡΛȅȖǏș�ȣșǓǏ� Ƞȅ�ЙǹȠǓȖ� ȖǓșȣǹȠș� were as follows: social identity, social identity theory, SIT, experimental, correlational, and research. Arti- cles were picked using a random method of 2 articles per results page starting from the top. If the article did not specify social identity, the research method, or was outside of the appropriate date ranges, then it was skipped, and the next article was picked. Once two articles on the page were selected, we moved to the next results page and began selection again. All studies picked in this sample were quantita- tive leaning (i.e., analysis was done quantitatively). Once the articles were collected, the relevant in- formation was cataloged via an Excel spreadsheet শǏƺȠƺ� ƺΚƺǩǹƺLjǹǓ� ǟȖȅǾ� vǩǹșȅǿॹ� eȖƺЙǾȅΛॹ� vƺǿǠॹ� ૭� Wang, 2021 via the OSF open-access database). The information cataloged included the citation for the paper, journal, publication year, number of studies included in the paper, sample sizes, and number of conditions. While some of this information is not pertinent for the actual precision analysis, examin- ǩǿǠ� ǏǩАǓȖǓǿljǓș� ǩǿ� ȠǦǓ� ǾǓƺǿș� ƺǿǏ� ǾǓǏǩƺǿș� LjΡ� ȒȣLj- lication year and methodology type is of interest for potential broader investigations and conclusions. To determine the precision of each study, we used the APP programs listed in Hui et al. (2020). Given sam- ple sizes reported in the articles, and assuming a conven- Ƞǩȅǿƺǹࢶࢺ�ઔ�ljȅǿЙǏǓǿljǓ�ǹǓΚǓǹॹ�ȠǦǓ�ȒȖȅǠȖƺǾș�ȒȖȅΚǩǏǓ�ȠǦǓ� ȒȖǓljǩșǩȅǿ� ǹǓΚǓǹॹ� LjȣȠ�ΛǩȠǦ� ƺ� ljȅǾȒǹǩljƺȠǩȅǿঀ� ^ȒǓljǩЙljƺǹǹΡॹ� because some studies were experimental whereas oth- ers were correlational, there are mathematical reasons why the precision of the correlational studies cannot be compared directly to the precision of the experimental studies (Wang et al., 2021)1. Consequently, results per- taining to experimental studies and results pertaining to correlational studies will be presented separately. Results There were too few mixed designs for analysis and so we focused on between-participants experi- mental studies and correlational studies. For experi- mental studies, the mean precision level was 0.51 (SD = .25) and the median precision level was 0.50 (n = �ষঀ�eǦǓșǓ�ЙǿǏǩǿǠșࢺࢴ șȣȒȒȅȖȠ�ƺ�ȒǓșșǩǾǩșȠǩlj�ΚǩǓΛ�ȅǟ�ǓΠ- perimental social identity research. For correlational studies, the mean precision level was 0.24 (SD = .16) and the median precision level was 0.20 (n = 55). How- ever, we reiterate that the precision of correlational research cannot be compared to the precision of ex- perimental research. As correlations range from -1 to +1, and the precision value pertains to the fraction of a variance (not a fraction of a standard deviation), the degree to which this level of precision in correlation- al research is pessimistic or optimistic is a judgment call. Certain theories and paradigms may require more șȠȖǩǿǠǓǿȠ�ljȣȠȅАș�ǟȅȖ�ΛǦƺȠ� ǩș�ljȅǿșǩǏǓȖǓǏ�ȅȒȠǩǾǩșȠǩlj�ȅȖ� pessimistic, and since correlational APP equations use variance instead of standard deviation in analy- șǩșॹ� ȠǦǓ� ljȣȠȅАș� ƺȖǓ� ƺ� ǴȣǏǠǾǓǿȠ� ljƺǹǹ� ȅǟ� ȠǦǓ� ȖǓșǓƺȖljǦǓȖঀ � eȅ�ǓΚƺǹȣƺȠǓ�ȠǦǓ�ǓАǓljȠ�ȅǟ�ȠǦǓ�ΡǓƺȖ�ȅǟ�ȒȣLjǹǩljƺȠǩȅǿ�ȅǿ� precision, we performed multiple correlational anal- yses. First, we obtained mean and median precision levels across the three time periods for the experimen- tal research. The mean precision levels were 0.59 (n = 17, SD = .20), 0.64 (n = 10, SD = .24), and 0.29 (n = 12, SD = .16) for the most distant to most recent time periods, respectively. The corresponding median precision levels were 0.59 (n = 17), 0.71 (n = 10), and 0.23 (n = 12). With respect to correlational research, the mean precision levels were 0.23 (n = 11, SD = .09), 0.27 (n = 24, SD = .19), and 0.21 (n = 20, SD = .15) for the most distant to most recent time periods, re- spectively. The corresponding median precision levels for correlational studies were 0.25 (n = 11), 0.23 (n = 24), and 0.18 (n = 20). Overall, both correlational and experimental precisions have shown improve- ment over time, with experimental studies showing much more improvement than correlational studies. Second, we correlated precision with the year of 1 1 These include that correlations are bounded (-1, +1) and that the precision level refers to a squared standard deviation (variance) as opposed to a stan- These include that correlations are bounded (-1, +1) and that the precision level refers to a squared standard deviation (variance) as opposed to a stan- dard deviation. dard deviation. SOCIAL IDENTITY PRECISION 118118 publication. Because smaller values indicate better precision, an optimistic perspective suggests a negative correlation whereas a pessimistic perspective suggests no correlation or a positive correlation. The correla- tion was -0.445 (n = 39, p = .004) for experimental research and -0.056 (n = 55, p = .687) for correlational research. Figure 1 contains a scatterplot pertaining to experimental research and Figure 2 contains a scatter- plot pertaining to correlational research, both with the LjǓșȠ�ЙȠ�ȠȖǓǿǏǹǩǿǓ�Ƞȅ�ǓΠƺǾǩǿǓ�ȠǦǓ�ȒȖȅǠȖǓșșǩȅǿ�ȅǟ�ȠǦǓ�ǏƺȠƺঀ From the data presented in Figure 1, we can see a clear negative trend across time. As the publica- tion year becomes more recent, the precision level Figure 1. Scatterplot representing precision of experimental social identity research along the vertical axis as a function of year of publication along the horizontal axis. Figure 1. Scatterplot representing precision of correlational social identity research along the vertical axis as a function of year of publication along the horizontal axis. moves further towards zero, indicating a promising improvement in experimental precision. In contrast, 'ǩǠȣȖǓࢳ��ǏȅǓș�ǿȅȠ�șǦȅΛ�ǾȣljǦ�ȅǟ�ƺǿ�ǓАǓljȠ�ȅǟ�ȠǩǾǓ�ȅǿ� correlational precision. These time-period correla- tions should be taken as exploratory however due to the small sample sizes within each time period. Discussion Social identity research composes an important domain within social psychology that is relevant to oth- ǓȖ�ЙǓǹǏș�Ƞȅȅ�șȣljǦ�ƺș�ǾƺȖǷǓȠǩǿǠॹ�șȅljǩȅǹȅǠΡॹ�ƺǿǏ�ȅȠǦǓȖșঀ� Hence, it is unsurprising that there is a large literature and many arguments both praising and criticizing so- cial identity theory. The present work is agnostic about ȠǦǓ�ǹƺȖǠǓȖ�șȅljǩƺǹ�ǩǏǓǿȠǩȠΡ�ǩșșȣǓș�LjȖǩǓМΡ�ǾǓǿȠǩȅǿǓǏ�ǓƺȖǹǩ- er. This is not to say that these issues are unimportant; in contrast, we believe they are very important. How- ever, we also believe that if empirical facts are going to be used either to test theories or to provide the founda- tion for theory formation, it is crucial to be clear that the empirical facts really are factual. That is, it is a pre- requisite that researchers have good reason to believe that the sample statistics researchers report are rea- sonably good estimates of corresponding population parameters. Our goal was to test whether this prereq- uisite has been met in a sample of published research. The picture is very clear with respect to experi- ǾǓǿȠƺǹ� ȖǓșǓƺȖljǦঀ� ^ƺǾȒǹǓ� șǩΦǓș� ƺȖǓ� șǩǾȒǹΡ� ǩǿșȣГljǩǓǿȠॹ� thereby resulting in precision levels that one would consider untrustworthy. It is true that there are many ǟƺljȠȅȖș�ȠǦƺȠ�ljƺǿ�ljȖǓƺȠǓ�ȠǦǓșǓ�ǩǿșȣГljǩǓǿȠ�ǿȣǾLjǓȖș�șȣljǦ� as funding constraints, time, feasibility, and more. However, this does not disqualify the conclusion of ǩǿșȣГljǩǓǿȠ� șƺǾȒǹǓ� șǩΦǓșॹ� LjȣȠ� ȅǿǹΡ� ǩǿljȖǓƺșǓș� ȠǦǓ�ǿǓǓǏ� for more detailed and stringent research practices. It is important to note that quality research with smaller sample sizes is still very much alive and is in no way dis- ȕȣƺǹǩЙǓǏ�ǏȣǓ�Ƞȅ�șƺǾȒǹǩǿǠ�ȒȖǓljǩșǩȅǿঀ�2ǿșȠǓƺǏॹ�șƺǾȒǹǩǿǠ� precision should be taken as an additional tool to assist researchers, in an a priori fashion, to determine a sam- ple size needed for sample statistics that meet their cor- responding population parameters, in essence giving researchers a goal for their eventual work to shoot for. There is some good news, which is that precision is improving, with research in the most recent period of research exhibiting more precision than research in previous periods. But even with the improvement, ǩȠ�ΛȅȣǹǏ�LjǓ� ǏǩГljȣǹȠ� Ƞȅ� ƺȖǠȣǓ� ȠǦƺȠ� ȠǦǓ� ǹǓΚǓǹ� ȅǟ� ȒȖǓljǩ- WILSON, TRAFIMOW, WANG, WANG 119 SOCIAL IDENTITY PRECISION șǩȅǿ� ǩș� șȣГljǩǓǿȠঀ�/ȅΛǓΚǓȖॹ� �eȖƺЙǾȅΛ�ƺǿǏ�EΡȧΦ� শࢺࢲࢱࢳষ� ǏǓǾȅǿșȠȖƺȠǓǏ� ǩǾȒȖǓljǩșǩȅǿ� ΛǩȠǦ� ȖǓșȒǓljȠ� Ƞȅ� ЙΚǓ� ƺȖǓƺș� ȅǟ� psychology (social, developmental, clinical, cognitive, ƺǿǏ� ǿǓȣȖȅষॹ� ƺǿǏ� eȖƺЙǾȅΛॹ�/ΡǾƺǿॹ� ƺǿǏ� ?ȅșȠΡǷ� শࢱࢳࢱࢳষ� demonstrated imprecision with respect to marketing, there is no reason to be more pessimistic with respect to so- cial identity research than other social science research. It is perhaps better to view social identity research as anoth- er domain in which researchers should devote increased ȖǓșǓƺȖljǦ� ǓАȅȖȠș� ǩǿ� ȠǦǓ� ǏǩȖǓljȠǩȅǿ� ȅǟ� ǩǾȒȖȅΚǓǏ� ȒȖǓljǩșǩȅǿঀ � �ș� ȠǦǓ� ȒȖǓșǓǿȠ� ΛȅȖǷ� ljȅǿșȠǩȠȣȠǓș� ȠǦǓ� ЙȖșȠ� șΡșȠǓǾƺȠ- ic application of the APP to correlational research, the correlational analyses are better considered exploratory ȠǦƺǿ�ǏǓЙǿǩȠǩΚǓঀ��ȅȖȖǓǹƺȠǩȅǿƺǹ� ȖǓșǓƺȖljǦ�ȅАǓȖș�LjȅȠǦ�ƺǏΚƺǿ- tages and disadvantages relative to experimental research. From an APP perspective, an advantage of correlation- al research is that there is only one group, thereby ren- dering precision easier to obtain. Going beyond an APP perspective, correlational research can be argued to be more representative of reality because of a lack of po- ȠǓǿȠǩƺǹǹΡ� ƺȖȠǩЙljǩƺǹ� ǹƺLjȅȖƺȠȅȖΡেǩǿǏȣljǓǏ� ǾƺǿǩȒȣǹƺȠǩȅǿঀ� 2ǿ� addition, it is easier to obtain diverse samples in cor- relational contexts, though the recent proliferation of ΛǓLjেȒǓȖǟȅȖǾǓǏ� ǓΠȒǓȖǩǾǓǿȠș� ǩș� ȖǓǏȣljǩǿǠ� ȠǦǩș� ǏǩАǓȖǓǿljǓঀ But there are disadvantages too. From an APP perspec- tive, a disadvantage is the lack of previous systematic APP ƺȒȒǹǩljƺȠǩȅǿș�Ƞȅ�ljȅȖȖǓǹƺȠǩȅǿ�ljȅǓГljǩǓǿȠșॹ�ȠǦǓȖǓLjΡ�ȖǓǿǏǓȖǩǿǠ� ljȅǾȒƺȖǩșȅǿ�ǏǩГljȣǹȠঀ�eǦǩș�ǩș�ǿȅȠ�ƺ�ǟƺȣǹȠ�ȅǟ�ȠǦǓ�ljȅȖȖǓǹƺȠǩȅǿƺǹ� research itself, but rather due to the historical fact that APP equations applicable to experimental data were developed prior to APP equations applicable to correlational data, and thus the basic assumptions underlying the equations cannot be directly compared on an “A = B” comparison. More generally, it is well-known that correlational research provides a less convincing case for causal mechanisms than experimental research. On the other hand, however, given the present demonstration of the lack of precision of ex- perimental social identity research, it is not clear that even ȠǦǓ�ǓΠȒǓȖǩǾǓǿȠƺǹ�ЙǿǏǩǿǠș�ȒȖȅΚǩǏǓ�șȠȖȅǿǠ�ljƺȣșƺǹ�ǓΚǩǏǓǿljǓঀ� 2ǟ�ƺ�șƺǾȒǹǓ�ǾǓƺǿ�ǏǩАǓȖǓǿljǓ�ljƺǿǿȅȠ�LjǓ�ȠȖȣșȠǓǏ�Ƞȅ�ȖǓƺșȅǿƺLjǹΡ� ǓșȠǩǾƺȠǓ� ȠǦǓ�ȒȅȒȣǹƺȠǩȅǿ�ǾǓƺǿ�ǏǩАǓȖǓǿljǓॹ� ƺ� șȠȖȅǿǠ� ljƺȣșƺǹ� conclusion is contraindicated. Therefore, the disadvan- tage of correlational research paradigms relative to exper- imental research paradigms might be considered decreased in the context of small sample experimental research. Major implications can be discussed surrounding șƺǾȒǹǩǿǠ� ȒȖǓljǩșǩȅǿ� ƺǿǏ� ǦȅΛ� ǩȠ� ǾǩǠǦȠ� ƺАǓljȠ� șȅljǩƺǹ� ǩǏǓǿ- ȠǩȠΡ� ȖǓșǓƺȖljǦ� ƺǿǏ� ȠǦǓȅȖǓȠǩljƺǹ� ЙǿǏǩǿǠș� ȅǟ� șȅljǩƺǹ� ǩǏǓǿȠǩȠΡ� theory. It is important, as mentioned previously, to have sample statistics that are good estimates of corresponding population parameters. Without this, basic assumptions of the applicability of the- ȅȖǩǓș� ƺǿǏ� șƺǾȒǹǓ�ЙǿǏǩǿǠș� ƺȖǓ�ǿȅȠ�ǾǓȠঀ� 2ǟ� șƺǾȒǹǩǿǠ� precision for research in social identity work is con- sistently low across the board, with no real attempt Ƞȅ� ǩǾȒȖȅΚǓॹ� ȠǦǓǿ� ȠǦǓ� ȖǓȒǹǩljƺLjǩǹǩȠΡ�ȅǟ� ȠǦȅșǓ�ЙǿǏǩǿǠș� ǩș� ǩǿ� ǴǓȅȒƺȖǏΡঀ� (ǩΚǓǿ� ȠǦǓ� ǾƺǴȅȖ� ȒȣșǦ� LjΡ� șljǩǓǿȠǩЙlj� ƺǏΚȅljƺȠǓș� ǟȅȖ� ǠȖǓƺȠǓȖ� ȖǓȒǹǩljƺLjǩǹǩȠΡ� ǩǿ� șljǩǓǿȠǩЙlj� ȖǓ- search, especially in social sciences, this means that sampling precision is a vital step towards a more ȅȒǓǿ� ƺǿǏ� ȠȖȣȠǦǟȣǹ� șljǩǓǿȠǩЙlj�ǏǩșljǩȒǹǩǿǓঀ� 2ǟ� șƺǾȒǹǩǿǠ� ȒȖǓljǩșǩȅǿ�ΛǓȖǓ�Ƞȅ�ȖǓǾƺǩǿ�ǹȅΛॹ�ȠǦǓ�ЙǿǏǩǿǠș�ȅǟ�șȅljǩƺǹ� research and the applicability of theories developed ǟȖȅǾ�ȠǦȅșǓ�ЙǿǏǩǿǠș�ΛȅȣǹǏ�LjǓ�ȕȣǓșȠǩȅǿƺLjǹǓ�ƺȠ�LjǓșȠঀ � �ș� ƺǹΛƺΡș� ΛǩȠǦ� șljǩǓǿȠǩЙlj� ȖǓșǓƺȖljǦॹ� ȠǦǓȖǓ� ƺȖǓ� some limitations. One limitation is that we only tested three time periods. This of course was done due to time constraints as well as the exploratory শȖƺȠǦǓȖ�ȠǦƺǿ�Йǿƺǹষ�ǿƺȠȣȖǓ�ȅǟ�ȠǦǓ�ȖǓșǓƺȖljǦঀ���șǓljȅǿǏ� limitation is that the sample size of the studies in- cluded was limited (again due to time constraints). Thus, conclusions may be clear, but a more com- prehensive analysis with a much larger sample size would provide a more detailed and stringent review of sampling precision in a posteriori fashion. Unfor- tunately, no APP techniques have yet been devel- oped to estimate the number of studies that should be included in an analysis such as that conducted here, and so traditional APP techniques are invalid to determine the sample size for a meta-style analysis of this kind. A third limitation is that, even within social identity research, there are research catego- ries not addressed here. For example, there is social identity research with a basic or applied focus, a fo- cus on integrating other literature, and many others. Of course, one potential avenue for future re- search is to address the foregoing limitations. A sec- ond potential avenue is to expand to domains that are not precisely about social identity but are related. These could include work in attachment, aggression, or stereotyping. A third avenue is to pursue non nor- mal distributions. Because the researchers in the ex- perimental papers all performed statistics based on the assumption of normality, we used that assump- tion too in the present analyses for the sake of con- sistency. However, this assumption is likely wrong, 120 WILSON, TRAFIMOW, WANG, WANG as most distributions are skewed (Blanca et al., 2013; Ho & Yu, 2015; Micceri, 1989). The usual counter to skewness arguments is that the Central Limit Theorem renders deviations from normality unimportant, but that depends on the goal of the research. For example, ǟȅȖ�ȠǦǓ�ǟƺǾǩǹΡ�ȅǟ�șǷǓΛ�ǿȅȖǾƺǹ�ǏǩșȠȖǩLjȣȠǩȅǿșॹ�eȖƺЙǾȅΛॹ� Wang, and Wang (2019; 2021) have demonstrated that there are important precision gains to be had for analy- ses analogous to those conducted here, even under low levels of skewness, provided that the researcher focuses on locations as opposed to means. Because locations are a parameter of skew normal distributions, whereas means are not, it makes sense to use locations rather than means anyway as locations are the more generally applicable parameter. And because the location equals the mean when there is normality, nothing is lost by including locations in APP analyses, even in the rare cases where the normality assumption is true. It is also important to note again that this research is not an examination of the accuracy of social identity theory itself and is not limited to “social identity researchers’’ in the traditional sense (that is those who test the the- ory of SIT). The goal of this research was to examine a broad and eclectic mix of social identity applications. In conclusion, the notion that theory should be checked against reality is a staple of science, including the social sciences. But the reality in the social sciences tends to be characterized by summary statistics such ƺș�ǾǓƺǿșॹ�șȠƺǿǏƺȖǏ�ǏǓΚǩƺȠǩȅǿșॹ�ljȅȖȖǓǹƺȠǩȅǿ�ljȅǓГljǩǓǿȠșॹ� etcetera. Social scientists do not obtain such sample summary statistics as ends in themselves. Rather, social scientists obtain sample summary statistics because of the faith they have that these provide good estimates of corresponding population parameters. 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