Article Refinements to Effect Sizes for Tests of Categorical Moderation and Differential Prediction Organizational Research Methods 1-9 ª The Author(s) 2017 Reprints and permission: sagepub.com/journalsPermissions.nav DOI: 10.1177/1094428117736591 journals.sagepub.com/home/orm Jeffrey A. Dahlke1 and Paul R. Sackett1 Abstract We provide a follow-up treatment of Nye and Sackett’s (2017) recently proposed dMod standardized effect-size measures for categorical-moderation analyses. We offer several refinements to Nye and Sackett’s effect-size equations that increase the precision of dMod estimates by accounting for asymmetries in predictor distributions, facilitate the interpretation of moderated effects by separately quantifying positive and negative differences in prediction, and permit the computation of nonparametric effect sizes. To aid in the implementation of our refinements to dMod, we provide software written in the R programming language that computes Nye and Sackett’s effect sizes with all of our refinements and that includes options for easily computing bootstrapped standard errors and bootstrapped confidence intervals. Keywords categorical moderation, multiple regression, differential prediction, bias, effect size Nye and Sackett (2017) recently derived a class of effect-size measures to quantify categorically moderated effects. These new dMod effect sizes summarize interactions in a consistent metric across studies, are easier to interpret than R2 values from regression models (see Nye & Sackett, 2017, for a discussion of the difficulties of using R2 as an effect size for interactions), and provide an intuitive way to detect categorical moderation without significance testing. We build on Nye and Sackett’s equations and introduce several refinements that increase dMod’s versatility and ease of interpretation. We begin with a brief summary of Nye and Sackett’s (2017) methods as context for the present work, followed by a discussion of updates to dMod effect sizes. These updates include (a) adjustment factors to offset bias that results from violating distributional assumptions, (b) directional effect sizes to separately quantify negative and positive differences in prediction between two groups’ 1 Department of Psychology, University of Minnesota, Minneapolis, MN, USA Corresponding Author: Jeffrey A. Dahlke, Department of Psychology, University of Minnesota, 75 E. River Rd., Minneapolis, MN 55455, USA. Email: [email protected] 2 Organizational Research Methods XX(X) regression lines, and (c) nonparametric versions of the dMod equations for use with observed distributions of predictor scores. Our review of Nye and Sackett’s dMod equations is meant to provide minimally sufficient context for the presently proposed updates; we encourage readers to consult Nye and Sackett (2017) for detailed information about dMod. Nye and Sackett’s (2017) dMod Effect-Size Measures Nye and Sackett’s (2017) dMod effect-size measures facilitate the comparison of two regression models: one model regressing Y on X for each of two categorically different groups.1 The interpretation of dMod effect sizes is similar to the interpretation of Cohen’s d, except that dMod summarizes differences between distributions of predicted scores rather than distributions of observed scores. A dMod effect size indicates the weighted average difference in prediction between a referent regression model (e.g., a model summarizing data from a majority or control group) and a focal regression model (e.g., a model summarizing data from a minority or experimental group), scaled in terms of the referent group’s criterion standard deviation (SDY 1 ). Table 1 arrays all of the equations that will be discussed in this article, including Nye and Sackett’s (2017) formulations of the dMod equations (lightly modified to be in slope-intercept form). Nye and Sackett’s (2017) dMod_Signed effect-size measure (see Equation 1a) represents the weighted average net difference in prediction between two models across an operational range of predictor scores. A positive (negative) sign for dMod_Signed means that focal-group criterion scores predicted from the focal-group regression model are, on average, lower (higher) than focal-group criterion scores predicted from the referent-group regression model. When subgroup regression lines cross within an operational range of predictor scores and one computes dMod_Signed, positive and negative differences in prediction will cancel out because the signed effect size summarizes the net difference in predicted criterion scores. For example, if two groups’ regression lines cross at the focal group’s mean predictor score when the predictor is normally distributed, positive and negative differences would completely cancel out and dMod_Signed would be zero. In settings such as this, dMod_Signed would fail to suggest the existence of a moderated effect. To overcome this, Nye and Sackett (2017) created an unsigned effect size, dMod_Unsigned, that quantifies differences between subgroup regression lines without allowing signed differences to cancel out (see Equation 2a). As an unsigned index of an effect, dMod_Unsigned is useful when one wishes to quantify the overall magnitude of a moderated effect and the net direction of predicted differences between groups is not relevant to one’s research question. Nye and Sackett (2017) offered two supplementary effect sizes that facilitate the interpretation of dMod_Signed and dMod_Unsigned. The dMin (see Equation 3) and dMax (see Equation 4) effect sizes indicate the smallest and largest absolute-value differences, respectively, between groups’ regression lines. By computing dMin and dMax, one can easily communicate the range of differences that were used to compute dMod_Signed and dMod_Unsigned and identify whether subgroup regression lines cross within the operational range of predictor scores (an occurrence signaled by a dMin equal to zero that is paired with a nonzero dMax). We refer readers to Nye and Sackett (2017) for more information about dMod_Signed, dMod_Unsigned, dMin, and dMax. The remainder of this article will describe our refinements to dMod. Refinements to Nye and Sackett’s (2017) dMod Effect-Size Measures Our suggested refinements include (a) adjustments to dMod to reduce bias from modest violations of distributional assumptions, (b) special cases of dMod_Signed that quantify directional differences in predicted criterion scores, and (c) nonparametric methods for computing dMod effect sizes. We introduce each of these refinements in the sections that follow. 3 (2a) dMod Unsigned Nye & Sackett (2017) formulation dMod Unsigned revised formulation Over dMod dMod Signed dMod Unsigned Under f2 ðXÞ½Xðb11 b12 Þ þ b01 b02 dX f2 ðXÞdX Under Under j jdMod (8) X: Y^1 >Y^2 R Over f2 ðXÞ½Xðb11 b12 Þ þ b01 b02 dX X: Y^1 <Y^2 Over þ dMod þ dMod X: Y^1 >Y^2 dMod SDY1 R f2 ðXÞdX f2 ðXInf ÞdXInf X: Y^1 <Y^2 XInf :Y^1 >Y^2 R SDY1 XInf : Max½jXðb11 b12 Þ þ b01 b02 j f ðXInf ÞdXInf R ^ ^ 2 R Y 1 <Y 2 f2 ðXÞ½Xðb11 b12 Þ þ b01 b02 dX f2 ðXÞjXðb11 b12 Þ þ b01 b02 jdX 1 RSDY1 R Min½jXðb11 b12 Þ þ b01 b02 j f2 ðXÞdX 1 SDY1 R (7) (6) (5) (4) dMax dMod (3) dMin R R qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ f2 ðXÞ½Xðb11 b12 Þ þ b01 b02 dX f2 ðXÞdX f2 ðXInf ÞdXInf SDY1 R 1 SDY1 R f2 ðXInf ÞdXInf SDY1 R 1 Formula R 1 f2 ðXÞ½Xðb11 b12 Þ þ b01 b02 dX SDY Parametric Version (14) (13) (12) (11) (10) (9) Eq. No. Max½jxðb11 b12 Þ þ b01 b02 j Min½jxðb11 b12 Þ þ b01 b02 j nTOver ½xOver ðb11 b12 Þþb01 b02 SDY1 nT 1 nTUnder ½xUnder ðb11 b12 Þþb01 b02 SDY1 nT 1 1 SDY1 1 SDY1 nT jxðb11 b12 Þþb01 b02 j SDY1 nT 1 nT ½xðb11 b12 Þþb01 b02 SDY1 nT 1 Formula Nonparametric Version Note: X represents values within the range of observed focal-group predictor scores; X is bounded by the minimum and maximum possible scores in the focal group. XInf represents values from an infinite distribution of predictor scores; the distribution of XInf is integrated to account for the finite range of X. Eq. No. is the formula’s equation number for in-text reference. SDY1 is the referent group’s observed criterion standard deviation. f2 is the normal-density function for the focal group’s unrestricted predictor scores. b11 and b12 are the referent- and focal-group slopes, respectively and b01 and b02 are the referent- and focal-group intercepts, respectively. The subscripts of the integrals for dMod Under and dMod Over specify the values of X to include in the integral; X values are included in the integral as a function of the predicted values of Y associated with X for the referent and focal regression models, where Y^1 ¼ b01 þ Xb11 and Y^2 ¼ b02 þ Xb12 . x is a column vector of observed focal-group predictor scores. nT is a row vector of focal-group frequencies with elements that correspond in an ordered fashion to the predictor scores arrayed in x. xUnder and xOver are column vectors of focal-group predictor scores that contain the values from x that satisfy Y^1 < Y^2 and Y^1 > Y^2 , respectively. nTUnder and nTOver are row vectors of focal-group frequencies with elements that correspond in an ordered fashion to the predictor scores arrayed in xUnder and xOver , respectively. 1 is a column vector of 1 s with as many elements as are in nT . Overall (additive computation; recommended over direct computation) Directional Extrema (1b) dMod Signed revised formulation (2b) (1a) dMod Signed Nye & Sackett (2017) formulation Overall (direct computation) Eq. No. Effect-Size Measure Type of Effect Table 1. Compendium of Parametric and Nonparametric dMod Formulas. 4 Organizational Research Methods XX(X) Modifications to Reduce Bias From Modest Violations of Normality A subtlety of the dMod effect-size measures that makes them particularly well-suited for use in operational settings is that one can integrate differences in prediction within a finite range of scores (i.e., one need not integrate the infinite normal distribution if one is studying predictor scores that span a bounded operational range). However, when dMod requires integration involving a finite minimum value and/or a finite maximum value, the cumulative density within that range may fall short of unity by a nontrivial amount due to departures from normality. A cumulative density smaller than 1 indicates that the dMod effect size one has computed may differ from what one would have obtained if one’s operational distribution of scores were indeed normally distributed. We propose a simple adjustment to dMod_Signed and dMod_Unsigned to offset this bias; we describe an even more effective adjustment later in the article, but the logic of the initial refinement described below is necessary to lay the groundwork for the more advanced method. Integrating over a finite distribution can result in a biased dMod effect size unless one rescales the effect size by the sum of the weights generated by the normal-density function (i.e., the cumulative density). The problem of integrating overP a finite distribution of scores is analogous to algebraically n wx computing a weighted average (i.e., x ¼ Pi n i i , where w is a weight and x is a score) in which the i wi sum of weights in the denominator exceeds the sum of the weights used in the numerator. Of course, cumulative densities appreciably smaller than unity may indicate that the assumption of a normal distribution is inappropriate. When normality cannot be reasonably assumed of one’s data, one should use the nonparametric versions of dMod described later in this article. For our present discussion, we assume that departures from normality are modest and that using a normal-density function to compute dMod is a reasonable choice. When integrating a dMod function over a finite range of predictor scores, an option for addressing a cumulative density smaller than one is to simply divide the dMod effect size by the cumulative density of scores within the operational range. For example, if the operational range of scores includes 95% of the theoretical normal distribution, dividing the dMod effect size by .95 will rescale the weights to account for the fact that scores outside of the operational range are impossible to achieve and that 95% of the theoretical distribution represents 100% of the distribution of possible scores. An adjustment factor that implements this correction is incorporated in Equations 1b and 2b and we recommend using these updated equations over Equations 1a and 2a. The gains in precision from using our corrections are likely to be small in most settings, but the corrections will consistently attenuate the bias from minor violations of normality. Unless otherwise noted, all subsequent mentions of dMod_Signed and dMod_Unsigned in this article will refer to Equations 1b and 2b, respectively. In formulating our updated version of dMod_Unsigned in Equation 2b, we have further reduced bias by modifying the way in which the sign is removed from differences in prediction. Although Nye and Sackett’s dMod_Unsigned formula (Equation 2a) accomplishes the advertised goal of removing the sign from differences in prediction, it does so in a way that slightly alters the meaning of the resulting dMod_Unsigned effect size relative to the meaning of the dMod_Signed effect size (Equation 1a). Under the radical in Equation 2a, one only squares the differences between regression lines and does not square the density weights. Due to the fact that only the differences between regression predictions are squared (rather than squaring the products of the differences and densities), the square root of the product under the radical does not result in an absolute difference like one might expect. Over the range of predictor scores, taking the square root of the unsquared densities in Equation 2a alters the proportional weight given to each predictor score: Scores toward the middle of the distribution receive too little weight and extreme scores receive too much weight. See Figure 1 Dahlke and Sackett 5 Figure 1. Comparisons of the distributions of densities and square roots of densities for the normal distributions used in Equations 2a and 2b, respectively. Distributions of proportional weights were computed by dividing the density associated with each predictor score by the corresponding cumulative density. Comparison of the figure panels reveals that taking the square roots of densities distorts the weight given to each predictor score, whereas using raw densities does not. for comparisons of the distributions of raw and square-root densities, as well as the corresponding distributions of proportional weights. As absolute differences are the intuitive metric for unsigned differences, dMod_Unsigned should be computed using Equation 2b rather than Equation 2a. A limitation of Equations 1b and 2b is that they assume that violations of normality are symmetric, with similar effects on the low and high ends of a score distribution, but this not likely to be the case in all settings. We present methods that account for asymmetric violations of assumptions after first introducing our directional dMod effect sizes. Separate Effect Sizes for Positive and Negative Differences in Predicted Criterion Scores The dMod_Signed and dMod_Unsigned effect sizes are useful for quantifying effects that are moderated by a categorical variable. However, some categorically moderated effects occur in domains in which the regions of negative and positive differences in predicted criterion values have substantive importance. An example of this is predictive bias from the industrial and organizational psychology literature. Predictive bias occurs when assessment scores predict performance differently as a function of one’s protected-class status (e.g., one’s sex or race). For example, a biased test might predict lower performance for Black job applicants when the White (referent group) regression model is used to make predictions than when the Black (focal group) regression model is used. If predicted criterion scores are Y^1 ¼ b01 þ Xb11 using the referent model (where X is a vector of focal predictor scores) and Y^2 ¼ b01 þ Xb11 using the focal model, “underprediction” for the focal group occurs when Y^1 < Y^2 and “overprediction” occurs when Y^1 > Y^2 . We suggest that quantifying these directional differences can have interpretive value. The dMod_Signed and dMod_Unsigned effect sizes both combine directional differences in prediction into a single effect size, which can occasionally make it difficult to interpret the precise form of an interaction. Nye and Sackett (2017) suggested that dMod effect sizes could be computed over any meaningful range of predictor scores, which means that one could compute separate dMod effect sizes within any segments of a distribution that are of interest to one’s research question. In our use of dMod effect sizes to quantify predictive bias, we have found it informative to break dMod_Signed into two directional effect sizes. We propose an effect size 6 Organizational Research Methods XX(X) called dMod_Under that only quantifies differences in prediction in the score range where negative differences in prediction occur (dMod_Under is the standardized average of differences in prediction for all Y^1 < Y^2 ; see Equation 5) and an effect size called dMod_Over that only quantifies differences in prediction in the score range where positive differences in prediction occur (dMod_Over is the standardized average of differences in prediction for all Y^1 > Y^2 ; see Equation 6). These directional effect sizes facilitate the interpretation of moderated effects, especially when groups’ regression lines cross within an operational score range. If subgroup regression lines do not cross and there are no Y^1 < Y^2 differences in prediction within an operational score range, dMod_Under will be zero and dMod_Over will be equal to dMod_Signed. Similarly, if subgroup regression lines do not cross and there are only Y^1 < Y^2 differences in prediction within an operational score range, dMod_Over will be zero and dMod_Under will be equal to dMod_Signed. The dMod_Under and dMod_Over directional effect sizes are useful for isolating the magnitudes of negative (i.e., Y^1 < Y^2 ) and positive (i.e., Y^1 > Y^2 ) differences in prediction, respectively. However, provided that the cumulative density of the operational predictor distribution equals unity, dMod_Under and dMod_Over can also be used to compute dMod_Signed and dMod_Unsigned via addition. When dMod_Under and dMod_Over effect sizes computed from a complete normal distribution are added together, they equal dMod_Signed and, when their absolute values are added together, they equal dMod_Unsigned. The two directional effect sizes are therefore quite versatile and, because of this, we view dMod_Under and dMod_Over as elemental equations for quantifying moderated effects. The results from dMod_Under and dMod_Over capture all of the information necessary to make sense of the magnitude of a moderated effect: Separately, they communicate information about negative and positive differences in prediction and, by addition, they can result in dMod_Signed and dMod_Unsigned. The information gained from dMod_Under and dMod_Over is complementary to the information found by comparing dMin and dMax and these directional effect sizes can be helpful for making sense of dMod_Signed and dMod_Unsigned. Based on our recommendations for rescaling dMod_Signed and dMod_Unsigned by the cumulative density of predictor scores, Equations 5 and 6 include adjustment factors that rescale the effect sizes by the ratio of the cumulative density from integrating the infinite normal distribution to the cumulative density from integrating over operational scores (i.e., the ratio of the sum of the theoretical weights to the sum of the actual weights). In general, the adjustment to the dMod_Under and dMod_Over effect sizes is more appropriate than making a global correction to the dMod_Signed and dMod_Unsigned effect sizes as we did in Equations 1b and 2b. This is because the separate corrections made to dMod_Under and dMod_Over capture the possibility that violations of normality are asymmetric and differentially impact regions where Y^1 < Y^2 and where Y^1 > Y^2 . We recommend computing dMod_Under and dMod_Over effect sizes separately (ensuring that the raw directional effect sizes are appropriately rescaled, as shown in Equations 5 and 6) and then adding them together to obtain dMod Signed and dMod Unsigned (see Equations 7 and 8). Although the adjustment factors included in Equations 1b and 2b account for the overall cumulative density’s deviation from unity, we recom mend using dMod Signed and dMod Unsigned instead of dMod_Signed and dMod_Unsigned when possible because the asterisked additive effect sizes are more robust to asymmetric violations of normality than are the directly computed effect sizes. We note that our corrections for nonunity cumulative densities do not perfectly correct for abnormalities in the distribution of actual predictor scores. However, they offer a relatively simple way to approximate the effect size of interest. If deviations from the assumed normal distribution are substantial, use of normal-density weights will result in effect sizes of questionable validity. When deviations from normality are great and one has access to the actual distribution of predictor scores, we recommend computing dMod using nonparametric equations. Dahlke and Sackett 7 Nonparametric Methods for Computing dMod Effect Sizes The dMod equations discussed thus far have assumed that there is an underlying parametric function that describes the distribution of predictor scores. This assumption is not always reasonable and there may be settings in which researchers wish to use observed frequencies as weights when they compute dMod. To compute nonparametric versions of all of the effect-size measures described above, we recommend using Equations 9 through 14 in Table 1. These equations provide standardized weighted averages of the differences in predicted criterion scores between referent and focal models using frequencies of observed focal-group scores as weights. These equations are direct algebraic analogs of their parametric counterparts listed in Table 1; if the observed data are normally distributed, the results from corresponding parametric and nonparametric procedures should agree within a reasonable margin of error. Our notation for the nonparametric formulas implies that one is using a frequency distribution as weights, but it is also possible to perform this procedure with raw data directly. If X is a vector of the focal group’s observed scores in which each entry represents a case, one can simply compute the average of X ðb11 b12 Þ þ b01 b02 and divide that average by the referent-group criterion standard deviation to obtain a nonparametric version of dMod_Signed. The same logic applies to computing the other nonparametric dMod effect sizes from observed scores. These nonparametric versions of dMod will be appropriate only if one can reasonably assume that the population to which one seeks to generalize will follow approximately the same distribution as the observed frequencies. If the sample used to derive the weights is atypical of the population of interest, these nonparametric equations will only be suitable for describing differences in prediction in the sample and will not be useful for making generalizations. Empirical Examples To illustrate the impact of our refinements, we analyzed a data set from the GATB validation project that substantially overlaps with the data set analyzed by Nye and Sackett (2017) in terms of the occupational groups represented. We computed unstandardized regression models for all of the occupational groups reported by Nye and Sackett that were represented in our version of the data set (see Nye & Sackett, 2017, for information on how these occupational groups were identified). Table 2 presents comparisons of the effect sizes computed using Nye and Sackett’s equations and using our revised equations. Some results with large proportional differences in Table 2 are associated with very small raw differences, but other differences (e.g., raw differences of .07 and .08 for dMod_Unsigned, associated with proportional differences of 24.14% and 18.18%, respectively) are of magnitudes that could impact interpretations of effect sizes and complicate comparisons of effect sizes across contexts. The majority of the differences from using the revised equations came from our use of absolute values in computing dMod_Unsigned, which illustrates that the differences between the weight distributions depicted in Figure 1 can have noticeable, practical impacts on effect-size estimates. Discussion In this article, we have outlined several refinements to Nye and Sackett’s (2017) dMod effect-size measures that we discovered during our use of these effect sizes. Our goal has been to update readers on methods for computing dMod effect sizes and to share the progress that has been made since the concept of dMod was first introduced. As a supplement to this article, we have produced software to compute dMod effect sizes. Our software is written in the R programming language (R Core Team, 2017) and is part of the 8 Directional Effects Percentage Difference Between Nye and Sackett’s (2017) Equations and Revised Equations 268 97 150 57 67 195 522 348 151 276 131 112 118 130 118 51 75 96 91 52 84 244 64 59 53 185 56 65 .06 .38 .28 –.18 .22 .67 .56 –.01 .10 .43 .46 –.04 –.02 –.39 .18 .40 .28 .18 .36 .70 .57 .07 .18 .45 .56 .04 .02 .52 .06 .40 .28 –.18 .22 .68 .56 –.01 .11 .44 .47 –.03 –.02 –.39 .15 .40 .28 .18 .29 .68 .56 .05 .15 .44 .49 .03 .02 .44 –.04 .00 .00 –.18 –.04 .00 .00 –.03 –.02 .00 –.01 –.03 –.02 –.41 .10 .40 .28 .00 .26 .68 .56 .02 .13 .44 .48 .00 .00 .02 .00 –.02 .00 .00 .00 –.01 .00 .00 –.01 –.01 –.01 –.01 .00 .00 .03 .00 .00 .00 .07 .02 .01 .02 .03 .01 .07 .01 .00 .08 0.00 –5.00 0.00 0.00 0.00 –1.47 0.00 0.00 –9.09 –2.27 –2.13 33.33 0.00 0.00 20.00 0.00 0.00 0.00 24.14 2.94 1.79 40.00 20.00 2.27 14.29 33.33 0.00 18.18 n White n Black dMod_Signed dMod_Unsigned dMod_Signed dMod_Unsigned dMod_Under dMod_Over dMod_Signed dMod_Unsigned dMod_Signed dMod_Unsigned Results From Revised Equations Raw Difference Between Nye and Sackett’s (2017) Equations and Revised Equations Note: Equations used to compute all tabled effect-size estimates are displayed in Table 1. The occupations listed in this table are those for which we could match the occupational groups analyzed by Nye and Sackett (2017) to cases in our version of the GATB data set. Sample sizes for millwrights and typists differ from the sample sizes reported by Nye and Sackett (2017) because we included all individuals in the data set whose Dictionary of Occupational Titles (DOT) codes matched with these occupations’ DOT codes. All effect sizes were computed using unstandardized regression models for congruence with how the effect sizes are computed in practice; thus, the effect sizes computed here using Nye and Sackett’s equations may differ from the effect sizes computed from standardized data reported by Nye and Sackett (2017). Raw difference ¼ Nye and Sackett equation – revised equation. Percentage difference ¼ (Nye and Sackett equation – revised equation) / revised equation 100. Clerks Bindery workers Ship fitters Child care workers Bench assemblers Forming machine operators Millwrights Typists Salespersons Steel workers Computer operators Psychiatric aides Electrical assemblers (employer 1) Etchers Occupational Group Sample Sizes Results From Nye and Sackett’s (2017) Equations Table 2. dMod Effect Sizes Computed for Selected Occupational Groups in the GATB Validation Database. Dahlke and Sackett 9 “psychmeta” R package (Dahlke & Wiernik, 2017). The general-purpose “compute_dmod” function computes parametric and nonparametric dMod effect sizes from a raw data set and can compute corresponding bootstrapped uncertainty statistics. We also offer functions for computing dMod from descriptive statistics and regression coefficients without raw data. We have found dMod effect-size measures to be informative for interpreting moderated effects and we hope that our modifications to Nye and Sackett’s (2017) equations will encourage more researchers to use dMod. Our open-source software’s compatibility with all common operating systems will support the use of these effect-size measures in future research. Acknowledgments The authors would like to thank Christopher D. Nye for his helpful comments on an early version of this article. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. Note 1. dMod can be used to compare regressions from more than two groups, but this requires computing multiple effect sizes because dMod can compare only two groups at a time. References Dahlke, J. A., & Wiernik, B. M. (2017). psychmeta: Psychometric meta-analysis toolkit (Version 0.1.1) [Computer software]. Retrieved from https://CRAN.R-project.org/package=psychmeta. Nye, C. D., & Sackett, P. R. (2017). New effect sizes for tests of categorical moderation and differential prediction. Organizational Research Methods, 20, 639-664. doi:10.1177/1094428116644505 R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from https://www.R-project.org/ Author Biographies Jeffrey A. Dahlke is a PhD student in industrial-organizational psychology at the University of Minnesota and earned his MA degree in industrial-organizational psychology from Minnesota State University, Mankato. His research is generally within the domains of personnel assessment and selection, occupational health, research synthesis, and quantitative methods. Paul R. Sackett has research interests that involve various aspects of testing and assessment in workplace, educational, and military settings. He has served as editor of two journals, Industrial and Organizational Psychology: Perspectives on Science and Practice and Personnel Psychology, and as president of the Society for Industrial and Organizational Psychology.

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