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Title: Dealing with Artificially Dichotomized Variables in Meta-Analytic Structural Equation Modeling
Authors: de Jonge, Hannelies
Jak, Suzanne
Kan, Kees-Jan
Issue Date: 29-May-2019
Publisher: ZPID (Leibniz Institute for Psychology Information)
Abstract: Background: Meta-analysis (Glass, 1976) is a commonly used statistical technique to aggregate sample effect sizes of different independent primary studies in order to draw inferences concerning population effects. To extend the range of research questions that can be answered, new meta-analytic models have been developed, such as meta-analytic structural equation modeling (MASEM) (Becker, 1992, 1995; Cheung, 2014, 2015a; Cheung & Chang, 2005; Jak, 2015; Viswesvaran & Ones, 1995). In primary studies, an effect size may represent the strength and direction of the association between any two variables of interest. Such an effect size can be expressed in different ways, for example as Pearson product-moment correlation, Cohens’ d, biserial correlation, and point-biserial correlation. How an effect size is expressed depends on the nature of the variables (e.g., continuous or dichotomous), but also on the way the variables are measured or analyzed. If one of the two continuous variables is artificially dichotomized, one may express the effect size as a point-biserial correlation. However, this typically provides a negatively biased estimate of the true underlying Pearson product-moment correlation (e.g., Cohen, 1983; MacCallum, Zhang, Preacher, & Rucker, 2002). The biserial correlation on the other hand should generally provide an unbiased estimate (Soper, 1914; Tate, 1955). Bias in the effect size of any primary study may affect meta-analytic results in the same direction (Jacobs & Viechtbauer, 2017). Therefore, we may expect that the use of the point-biserial correlation for the relationship between an artificially dichotomized and continuous variable also biases MASEM-parameters. In the current study we will evaluate how using point-biserial correlations versus biserial correlations from primary studies may affects path coefficients, their standard errors, and model fit in MASEM. Based on the results, we expect to be able to inform researchers about which of the two investigated effect sizes is the most appropriate to use in MASEM-applications and under which conditions. Aim: Our aim is to investigate the effects of using (1) the point-biserial correlation and (2) the biserial correlation for the relationship between an artificially dichotomized variable and a continuous variable on MASEM-parameters and model fit. Specifically, our interest lies in path coefficients, standard errors of these coefficients, and model fit. Method: We simulated meta-analytic data according to a full mediation (hence overidentified) population model (see Figure 1), with a continuous predictor variable X, continuous mediator M, and a continuous variable Y as outcome. Depending on the condition, the predictor variable X is artificially dichotomized in all or a given percentage of the primary studies. We chose this population model because in educational research the median number of variables in a ‘typical’ meta-analysis is three (de Jonge & Jak, 2018) and because mediation is a popular research topic. Figure 1. Population model with fixed parameter values. Under this population model, random meta-analytic datasets were generated under different conditions. We systematically varied the following: (1) the size of the (standardized) path coefficient between X and M (.16, .23, .33), (2) the percentage of primary studies in which X was artificially dichotomized (25%, 75%, 100%), and (3) the cut-off point at which X was artificially dichotomized (at the median value, so a proportion of .05, or when groups become unbalance, at a proportion of .01). These choices were mainly based on typical situations in educational research. The size of the path coefficient, reflect the minimum, mean/median, and maximum pooled Pearson product-moment correlations in a ‘typical’ meta-analysis in educational research (de Jonge & Jak, 2018). The 75% primary studies that artificially dichotomize the variable X, is based on a comparable example of a meta-analysis in educational research (Jansen, Elffers, & Jak, 2019). We used between-study variances of .01. The number of primary studies in a meta-analysis was fixed at the median number of a ‘typical’ meta-analysis, which is 44 (de Jonge & Jak, 2018). Because we use a random-effects MASEM-method, the assumption is thus that the population comprises 44 subpopulations from which the 44 samples are drawn, and that the weighted mean of the subpopulation parameters equals the population parameter. Given a specific condition and the fixed number of 44 primary studies, we randomly sampled the within primary study sample sizes from a positively skewed distribution as used in Hafdahl (2007) with a mean of 421.75, yielding ‘typical’ sample sizes (de Jonge & Jak, 2018) for every iteration. We imposed 39% missing correlations (Sheng, Kong, Cortina, & Hou, 2016) by (pseudo) randomly deleting either variable M or Y from 26 of the 44 studies. In each condition, we generated 2000 meta-analytic datasets drawn from the 44 subpopulations, which we analyzed using (1) the point-biserial and (2) the biserial correlation as effect size between the artificially dichotomized predictor X and continuous mediator M. The full mediation model was fitted using random-effects two stage structural equation modeling (TSSEM) (Cheung, 2014) within the R-package ‘metaSEM’ (Cheung, 2015b). As recommended (Becker, 2009; Hafdahl, 2007), we used the weighted mean correlation across the included primary studies to estimate the sampling variances and covariances of the correlation coefficients in the primary studies. Next, over the converged simulated datasets, we (1) estimated the relative percentage bias in both path coefficients (less than 5% bias was considered negligible; Hoogland & Boomsma, 1998), (2) calculated the relative percentage bias of the standard errors of these path coefficients (less than 10% bias was considered acceptable; Hoogland & Boomsma, 1998), (3) calculated the rejection rates of the chi-square statistic of the model of Stage 2 (df = 1,  = .05) and tested whether the rejection rate significantly differed from the nominal -level with the proportion test, and (4) compared the theoretical chi-square distribution (df = 1) with the empirical chi-square distribution (by means of QQplots and the Kolmogorov-Smirnov test). Main Results: When the point-biserial correlation for the relation between an artificially dichotomized predictor and a continuous mediator was used, the path coefficient of this relationship in the population (βMX) seems systematically underestimated. When the biserial correlation was used instead of the point-biserial correlation, this path coefficient could be considered unbiased in each condition. The estimated path coefficient between the two continuous variables (βYM) could also be considered unbiased in all conditions, no matter if the biserial or point-biserial correlation was used. The relative percentage bias in the standard errors of all path coefficients could be considered as not substantial according to the criteria that were applied. However, we noticed that the relative percentage bias in the standard error of the path coefficient between the predictor and mediator (βMX) seems systematically negatively biased when the biserial correlation was used. We also found that the relative percentage bias in the standard error of the path coefficient between the continuous variables Y and M (βYM) seems systematically negative, regardless if the point-biserial or biserial correlation was used. In most conditions, the rejection rate of the chi-square test of model fit at Stage 2 of the random-effects TSSEM was slightly above the nominal -level, no matter if the point-biserial or biserial correlation was used. The results of the Kolmogorov-Smirnov test and QQplots show that when the biserial correlation was used, there was a statistically significant difference between the empirical chi-square distribution and the theoretical chi-square distribution in five of the 18 conditions. When the point-biserial correlation was used, there was a significant difference in the same five conditions plus in three other conditions. There seems to be no clear pattern in which conditions the distributions differed significantly or not. 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Citation: De Jonge, H., Jak, S., & Kan, K.-J. (2019). Dealing with Artificially Dichotomized Variables in Meta-Analytic Structural Equation Modeling. ZPID (Leibniz Institute for Psychology Information).
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