Category Archives: Meta-Analysis

Modeling Heterogeneity in Meta-Analyses

July 10, 2026Meta-Analysisashr, Assumptions, Meta-Analysis, metafor, normal distribution, robustness, Simulation Study, zcurveUlrich Schimmack

Correction (2026-07-14)
The original post claimed that violations of the normality assumption biases standard random effects estimates. That finding was based on a coding error in the simulation program. The revised post shows the results for simulations with normal distributed effect sizes for H1 and mixtures of H0 and H1. In these simulations only estimates of the weight-function selection model (weightr package) are biased when the unimodal normal distribution assumption is violated.

Introduction

The main purpose of meta-analysis is to combine the results of quantitative studies. The simplest form averages the effect-size estimates of studies. The average is a better estimate of the population effect size because it draws on the combined sample size of the individual studies, and larger samples have less sampling error. A more sophisticated version takes each study’s sampling error into account, weighting studies with larger samples more heavily than those with smaller samples. This weighted average approximates the estimate one would obtain by pooling the raw data, under the assumption that all studies estimate the same effect.

These so-called fixed-effect meta-analyses assume that the samples are all drawn from the same population and that studies used interchangeable procedures, so that they all estimate a single population effect size. This assumption is defensible for a few phenomena grounded in basic sensory or cognitive architecture. Weber’s law, for instance — that the just-noticeable difference between two stimuli is a roughly constant proportion of their magnitude — reflects a property of sensory transduction and varies little across procedures and individuals. But such near-constants are the exception. Most effects in psychology vary with situational and personality factors, within and across cultures. This variation adds real differences in the true effect sizes across studies, over and above sampling error.

In recognition of this true variation, statisticians developed random-effects models (DerSimonian & Laird, 1986). To model the variation in population effect sizes, it is necessary to select a model that describes the unobserved variation in population effect sizes. From a statistical perspective, the easiest model assumes that population effect sizes have a normal distribution. The symmetry of this assumption implies that the estimate obtained from a set of studies is an unbiased estimate of the true average because positive and negative deviations from the average population effect size cancel each other out, just like random sampling errors cancel each other out.

Distribution Assumptions

From a theoretical perspective, it is unlikely that population effect sizes are normally distributed, particularly when the mean effect is small relative to the between-study heterogeneity. The reason is that effects are typically coded so that positive values are consistent with a directional theoretical prediction (e.g., conflicting colors produce slower responses in the Stroop task). While the size of the effect may vary across studies, it is often implausible that the true effect in some studies would be reversed (that conflicting colors would genuinely speed responses). A normal distribution, however, has support over the entire range of effect sizes and therefore always assigns some probability to negative true effects — an implication that is difficult to justify for a directionally predicted effect. When the mean is small relative to the heterogeneity, a normal distribution implies that a non-trivial proportion of true effects are negative, which is often theoretically implausible.

In practice, however, when heterogeneity is small, violations of the normality assumption have small and often negligible effects on meta-analytic estimates of the average effect (Hedges & Vevea, 1996). This helps explain why widely used random-effects models share the assumption that population effect sizes are normally distributed, including standard random-effects meta-analysis (DerSimonian & Laird, 1986) as implemented in commonly used software (Viechtbauer, 2010), selection-model approaches (Vevea & Hedges, 1995; Vevea & Woods, 2005), and the random-effects components of Bayesian model-averaging methods (Bartoš et al., 2023; Maier et al., 2023).

Necessary and Unnecessary Assumptions

It is useful to distinguish between necessary and unnecessary assumptions. All statistical methods require some assumptions to reduce the complex information in data to an interpretable statistic. However, not all assumptions of a model are necessary. In general, models with fewer unnecessary assumptions are preferable because they are robust to violations of these unnecessary assumptions by design. For example, the Pearson correlation coefficient assumes normal distribution of the two variables, whereas rank-order correlations do not. This makes rank-order correlations robust to violations of the normal distribution assumption and it is common practice to prefer rank-order correlations when variables’ distributions are not normal.

In meta-analyses, the assumption that population effect sizes are normally distributed is an unnecessary assumption. The idea of estimating the distribution of effects without assuming a parametric form is not new. Laird (1978) and Lindsay (1983) showed that the mixing distribution in a mixture model can be estimated nonparametrically by maximum likelihood, and that the resulting estimate takes the form of a discrete distribution on a finite set of support points. This provided, in principle, a fully flexible way to model heterogeneity without assuming normality. In practice, however, fitting these models was computationally demanding at the time.

Advances in optimization and computing have since removed these hurdles. Efficient algorithms for estimating mixture models — including modern convex-optimization methods (Koenker & Mizera, 2014; Kim et al., 2020) — now make it possible to fit flexible mixtures quickly and reliably. These developments led to the widespread adoption of nonparametric and empirical-Bayes mixture models in fields that analyze large numbers of effects, most notably genomics (Efron, 2010; Stephens, 2017), where the goal is to characterize the distribution of effects across thousands of tests and to identify which are likely to be real. Similar approaches have been applied in astronomy, education, and other areas that deal with many parallel estimates.

Heterogeneity of Population Effect Sizes

Meta-analysis, however, did not adopt these models, and continued to rely on the normal random-effects model. One reason is that the goal of most meta-analyses has been to estimate a single average effect, for which the shape of the effect-size distribution is largely irrelevant. The additional information provided by a flexible mixture — the structure of the heterogeneity itself — was not part of the question being asked.

This has changed with growing recognition of the distinction between direct and conceptual replication (Zwaan et al., 2018). Experimental psychologists rarely repeat a procedure exactly. Instead, they typically vary paradigms across studies in the belief that this is good scientific practice — probing moderators, boundary conditions, and the generalizability of an effect. As a result, the studies combined in a meta-analysis often estimate genuinely different population effects, and meta-analyses of such conceptual replications frequently show high estimates of heterogeneity (van Erp et al., 2017). In these cases the average effect size is of relatively minor interest, because it merely centers a wide distribution of different population effect sizes — an average across paradigm variants that no single study actually instantiates. The scientifically interesting question is not the average, but how and why the effects differ.

Mixture Models

Mixture models have another advantage over random effects models for meta-analysis. In standard meta-analytic models the population distribution of effect sizes is characterized by the mean and standard deviation of a normal distribution. Importantly, this information is removed from the observed data. It is therefore not possible to use this information to make sense of the heterogeneity in the observed effect size estimates in the data. A large effect size in the data may correspond to a small population effect size because it was inflated by sampling error and vice versa. This explains why heterogeneity often plays a minor role in the interpretation of meta-analyses. Heterogeneity exists, but it cannot be explained.

In contrast, mixture models combined with statistical tools that correct for regression to the mean make it possible to obtain estimates of the population effect size of individual studies. Often these effect size estimates for single studies have large uncertainty (wide confidence intervals), but they can be averaged to identify subgroups of studies with large effect sizes. This makes it possible to explore the heterogeneity in observed studies.

Simulation Study

To demonstrate the capabilities and advantages of mixture models over traditional random-effects models, I conducted a simulation study. To isolate the effect of the distributional assumption, the simulation used unbiased data (no publication selection). This removes selection as a confound and allows inclusion of the adaptive-shrinkage mixture model ash (Stephens, 2017), which was developed for genomics and assumes complete, unselected data. The other two models — the weight-function selection model (Vevea & Woods, 2005) and z-curve — model publication bias but reduce to unbiased estimation when no selection is present.

Z-curve is a meta-analytic mixture model (Brunner & Schimmack, 2020; Bartoš & Schimmack, 2022). Version 1 used the noncentrality parameters (ncp) of significant results to estimate the expected replication rate for direct replication studies. Version 2 used the ncp distribution across all studies to estimate the expected discovery rate. Version 3 adds an empirical-Bayes function that estimates population effect sizes by multiplying ncp estimates by their corresponding sampling errors, recovering effects from the latent distribution of noncentrality parameters.

The simulation used fixed sample sizes across studies, which ensures that mixture models fit in the effect-size metric and in the ncp metric produce identical results. Each condition contained k = 1,000 studies; the large k yields small sampling error in the estimates, making systematic biases more visible. The design crossed 4 means and 4 standard deviations of the population effect-size distribution with 4 proportions of true null hypotheses, producing 64 conditions. The population effect sizes when H1 is true were drawn from a normal distribution. However, mixing true H0 and H1 produces non-normal, bimodal distributions. This violates the distribution assumption of standard meta-analytic methods and could produce biased estimates.

zcurve3.0/SimpleMix_All_Normal_ZC_ASHR_VW_RMA.R at main · UlrichSchimmack/zcurve3.0

Results

These results for the mixture models (z-curve, ash) are not surprising. The models make no distribution assumptions and estimates of the mean closely match the simulated true values (z-curve RMSE = .009; ash RMSE = .010). The weight-function selection model had worse fit (RMSE = .138), but the random effects model fit as well as the mixture models (RMA RMSE = .003).

The results for the estimate of heterogeneity (tau) showed that all models estimated heterogeneity well, (z-curve RMSE = .018, ash RMSE = .027, weightr RMSE = .030, RMA RMSE = .013).

The present results show that the random effects model is robust to violations of the normality assumption even when heterogeneity is large and distributions are bimodal. The same cannot be said for the selection model with a normal assumption (weightr). Here the weight parameters to model selection bias can be biased to fit a normal distribution to non-normal observed data. The main implication is that it is possible to improve the performance of selection models by avoiding the normality assumption and modeling the distribution of population effect sizes with a flexible mixture model.

Implications

The present simulation showed that the random-effects model estimates the mean effect well when data are unbiased, even when its distributional assumptions are violated. This is consistent with a substantial prior literature. Simulation studies have repeatedly found that the pooled mean from the normal random-effects model is robust to non-normal random-effects distributions, including skewed and mixture distributions (Yamaguchi et al., 2017; Rubio-Aparicio et al., 2023; Baur et al., 2025). The robustness follows from the estimator itself: the pooled mean is an inverse-variance weighted average, which is consistent for the population mean regardless of the shape of the effect distribution. Large numbers of studies further protect against non-normality (Rubio-Aparicio et al., 2023), and with k = 1,000 the present simulation is in this favorable regime.

Where the normality assumption does matter is not the mean but the characterization of the distribution around it. The same literature finds that non-normality degrades the coverage of confidence and prediction intervals far more than it biases the mean (Baur et al., 2025), and that a symmetric normal model produces a wrongly symmetric prediction interval and a misleading heterogeneity summary when the true distribution is skewed (Yamaguchi et al., 2017). The recognized advantage of flexible mixture and semi-parametric models is therefore not lower bias on the mean but their ability to reveal latent structure — clusters and subgroups of studies — that the mean and a single heterogeneity parameter discard (Baur et al., 2025).

These are real but secondary limitations. The more fundamental problem is one that no distributional flexibility can address: the random-effects model, and the adaptive-shrinkage mixture model (ash) alike, assume that the observed studies are an unbiased sample of the population. In many literatures this assumption is untenable. In psychology, studies with significant results are far more likely to be published; the proportion of significant results often exceeds 90% (Sterling et al., 1995). Such publication bias inflates effect-size estimates, and correcting for it requires modeling the selection process — something neither the random-effects model nor ash attempts. A flexible distribution alone is not enough; what is needed is a model that combines distributional flexibility with a model of selection.

Z-curve occupies this niche: it combines a selection model, which corrects for publication bias, with a flexible mixture model, which captures heterogeneity without a distributional assumption. This is why z-curve outperforms weight-function models when studies are both heterogeneous and affected by publication bias (Schimmack, 2026).

References:

Bartoš, F., Maier, M., Wagenmakers, E.-J., Doucouliagos, H., & Stanley, T. D. (2023). Robust Bayesian meta-analysis: Model-averaging across complementary publication bias adjustment methods. Research Synthesis Methods, 14(1), 99–116.

Böhning, D. (2000). Computer-assisted analysis of mixtures and applications: Meta-analysis, disease mapping and others. Chapman & Hall/CRC.

DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3), 177–188.

Hedges, L. V., & Vevea, J. L. (1996). Estimating effect size under publication bias: Small sample properties and robustness of a random effects selection model. Journal of Educational and Behavioral Statistics, 21(4), 299–332.

Laird, N. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73(364), 805–811.

Lindsay, B. G. (1983). The geometry of mixture likelihoods: A general theory. The Annals of Statistics, 11(1), 86–94.

Maier, M., Bartoš, F., & Wagenmakers, E.-J. (2023). Robust Bayesian meta-analysis: Addressing publication bias with model-averaging. Psychological Methods, 28(1), 107–122.

Stephens, M. (2017). False discovery rates: A new deal. Biostatistics, 18(2), 275–294.

Vevea, J. L., & Hedges, L. V. (1995). A general linear model for estimating effect size in the presence of publication bias. Psychometrika, 60(3), 419–435.

Vevea, J. L., & Woods, C. M. (2005). Publication bias in research synthesis: Sensitivity analysis using a priori weight functions. Psychological Methods, 10(4), 428–443.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48.

Efron, B. (2010). Large-scale inference: Empirical Bayes methods for estimation, testing, and prediction. Cambridge University Press.

Kim, Y., Carbonetto, P., Stephens, M., & Koenker, R. (2020). A fast algorithm for maximum likelihood estimation of mixture proportions using sequential quadratic programming. Journal of Computational and Graphical Statistics, 29(2), 261–273.

Koenker, R., & Mizera, I. (2014). Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. Journal of the American Statistical Association, 109(506), 674–685.

Laird, N. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73(364), 805–811.

Lindsay, B. G. (1983). The geometry of mixture likelihoods: A general theory. The Annals of Statistics, 11(1), 86–94.

Managing the Terror of Meta-Analyses

July 3, 2026Meta-Analysis, Publication Bias, Step-Function Selection Model, Terror Management Theory, Weight-Function Selection Model, Z-CurveUlrich Schimmack

Summary

For method folks, the picture tells the full story: z-curve can estimate the true mean of a set of heterogeneous studies better than the weight-function model because the weight function model makes unrealistic assumptions about the distribution of population effect sizes. Added bonus: z-curves estimates are related to actual studies, whereas the estimates of weightr are population estimates that are not connected to the actual studies.

Try it: TMT analysis with z-curve.3.86

The Long Story

The root cause of the crises in psychology is poor training in scientific thinking and scientific methods. Period! I know because I have been teaching at a top-ranked university in North America for over 25 years now. The most common criticism in student evaluations is that my courses are not psychology courses, but statistics course. The reason: I use numbers when I present research findings. But most students can get a degree in psychology without using numbers. Graduate education does not help because students learn from a mentor, who also never learned to think quantitatively. So, psychology is the worst of both worlds. It is neither qualitative research that pays attention to people’s thoughts, feelings, or actual behaviors, nor is it a quantitative science that use valid quantitative information for the same purpose. It is a pseudo-science that produces meaningless numbers that mainly serve the purpose of claiming scientific support for researchers’ personal beliefs.

The problem that quantitative results in published articles cannot be trusted is now widely recognized and has been called a crisis of confidence a credibility crisis, or the replication crisis. However, the problem also exists at the meta-level when questionable published results are combined into a meta-analysis. Don’t get me wrong. Meta-analysis, like all statistical models, are not wrong. They are only wrong when incompetent researchers use these tools without understand how they work and what assumptions these models make.

Meta-analysis is easy to understand and perform when all data are available. We simply combine summary statistics to reduce sampling error and get a more precise estimate of the population effect size. Instead of running one study with N = 1,000 participants, we combine data from 25 studies with 40 participants. The result is practically the same. However, in psychology, the 25 published study are only a fraction of studies that were conducted and produced a significant result (Sterling et al., 1995). This means the effect sizes in the studies are inflated by publication bias and the same bias leads to an inflated effect size estimate in the meta-analysis. Thus, normal meta-analysis that ignore bias are as useful as a wet tissue paper on a 40°C (104°F) day in the middle of a parking lot at noon.

The solution to this problem is to use fancy statistical models that promise to correct for these biases and reveal the truth hidden in a pile of selected and p-hacked studies. The simple truth is that this goal is as attainable as making gold from base metals. However, as readers also do not understand these models and the problem of using them with uninformative data, the results are now routinely included in meta-analytic articles, if only as a sensitivity analysis that can be dismissed if it shows inconvenient or strange results.

Before I show how silly bias-correction of biased literature is, I need to present an example to show that I am not attacking a strawman model of bad meta-analysis. The example comes from a recent meta-analysis of studies that examined the influence of mortality salience on feelings, attitudes, and behaviors (Chen et al., 2025). The meta-analysis is notable for its attempt to deal with publication bias. The title even mentions publication bias in a clever way “Managing the Terror of Publication Bias.” The authors also shared their data. So, the only problem is that they did not consult with experts to make sense of their findings.

Figure 1 shows a simple histogram of the effect size ESTIMATES – these are estimates in small samples with enormous sampling error, not the actual effect sizes without sampling error.

Figure 1.

The most important observation for this blog post is that there are hardly any effect sizes below zero. To understand why this is important, it is important to understand the meaning of the sign of an effect size. In an original study, the sign has no meaning. For example, the height differences between people with XX and XY chromosomes can be positive or negative depending on the coding of XX as 0 or 1 and XY as 1 or 0, respectively.

For a meta-analysis, however, the sign becomes meaningful. In a competently conducted meta-analysis, the sign reflects the substantive hypothesis of a study. If mortality salience is coded as 1 against a control condition coded as 0, and the theory predicted an increase in a dependent variable, a positive sign implies that the result was consistent with the prediction. If the prediction implies a decrease in the DV, a negative sign is consistent with the theory, and the sign has to be reversed. Thus, if researchers mostly make correct predictions about the direction of an effect, we would expect mostly positive signs. Sampling error can still produce negative means in studies, even if the true effect is in the predicted direction, but how often that happens depends on the strength of the effect.

Now we are in the position to make sense of Figure 1. Only 2% of the effect size estimates. There are two possible explanations for this finding. Either TMT studies mostly produce positive results because most studies have true effects or there is selection bias and results that contradict theoretical predictions are not published (a third option would be coding mistakes, where coders code all results as positive and ignore substantive hypotheses).

What happens when these data are analyzed with bias-correction models? It depends on the model. The PET/PEESE model regresses effect sizes on the sampling error under the assumption that all studies have a common effect size and that larger samples are less biased.

The reanalysis that produced Figure 2 reproduced the published estimate of -.114 standard deviations. Thus, even though there are hardly any negative results in the data, the average study is supposed to have made a prediction in the wrong direction because that is what a negative mean means. An analysis that removed the 10% largest effect sizes, produced a positive estimate of .29 standard deviations. It is interesting that removing strong results increases the average. This shows that the results depend on assumptions about the amount of bias for different effect size estimates. Here the largest effect size estimates come also from the smallest studies (N < 10).

The point estimate of .29 should not be confused with the true effect size in each study. After removing sampling error, there is still considerable variability in the effect size estimates that can be quantified with the standard deviation, assuming a normal distribution. The estimate is tau = .40. This also makes it possible to create a prediction interval – a confidence interval for the hypothetical population effect sizes . To get a 95%CI we roughly multiply tau by 2 and get a range of values around the point estimate from .29 – ,80 to .29 + .80. Thus, any particular TMT study could have an effect size anywhere from -.51 to + 1.09. In terms of Cohen’s classification of effect sizes the effect sizes range from a moderate negative effect size to a strong positive effect size. In other words, the data are not telling us anything that we did not know before we ran the analysis. Terror Management effects may sometimes emerge as predicted, sometimes with surprising opposite effects, and sometimes have no notable effects, and we do not know which manipulation produces which effect.

The problem in the published article and many other meta-analysis is that the heterogeneity in effect sizes after taking random sampling error was ignored. The point estimate is only needed to center the prediction interval. The real information is the wide range of possible population effect sizes that are consistent with the model’s assumptions and the data. Every outcome except large negative effect sizes is possible.

Regression models have many limitations and even the developer of this approach has warned against the use of this model for highly heterogenous data (Stanley, 2017). A model that is more suitable for heterogeneous data is the weight-function selection model (Vevea & Woods, 2005). However, this model requires assumptions about selection bias. Chen et al. fitted a model that assumes different selection bias for significant negative results and non-significant results. Importantly, their model assumed the same amount of selection bias for negative non-significant results and positive significant results. This specification is important because Figure 1 shows that there are few negative effect sizes. A better way to see the problem here is to convert effect sizes and sampling error into z-values (z = effect size / standard error) to distinguish between non-significant (z < 1.96) and significant ones.

Figure 3 shows clearly that there is selection against negative results. Sampling error alone cannot explain the drop in effect size estimates from just above zero to just below zero.

The published article reports an estimated average effect size of .36. The model also estimated that only 26% of non-significant results were included in the meta-analysis. In other words, 74% were missing due to to publication bias. Finally, the model estimated that the population effect sizes had high heterogeneity, tau = .71. This leads to a very wide prediction interval around the point estimate of .36 ranging from .36 – 2*.71 to .36 + 2 * .71, which is -1.01 to 1.73. In other words, the model does not even exclude strong negative effect sizes as possible outcomes.

However, this model is misspecified because it ignores that selection against negative non-significant results is stronger than selection against non-significant positive results. I therefore ran the model again with an additional step at p = .5 (one-sided) that separates positive from negative results.. Consistent with the pattern in Figure 3, the model shows stronger selection against negative results (weight = .01, selection 1-weight = 99%) than for nonsignificant positive results (weight = .30, selection bias 70%). This improved model, however, produced a negative estimated average effect size of -.34. It also further increased the estimate of heterogeneity to tau = .96.

To understand this behavior of the model (I am more of a model analyst than a psycho-analyst), we need to understand the model’s assumption about the distribution of the unobserved population effect sizes. The model assumes an unobserved normal distribution, but the data are a truncated distribution at zero with no meaningful negative values. The model therefore fits the positive range of a normal distribution to the observed positive values. If this distribution is very wide, the normal has a large standard deviation and the model extrapolates it into the negative range. This leads to the wide prediction interval ranging from -1.34 to 2.58. Importantly, the negative range is entirely based on distribution assumption of the model . Changing the distribution assumption would change the results.

So, we have to think about the distribution assumption. When studies are more or less identical, there may be some extra variation in population effect sizes aside from sampling error. This variance can be approximated with a normal distribution (Hedges & Vevea, 1996). But when the set of studies has effect sizes ranging from 0 to 2, this is no longer plausible. If the average effect size is small and heterogeneity is large, a normal distribution implies that many substantive hypotheses have the wrong sign, but that is not really plausible. Many studies may have no real effect or really small ones, but it is harder to argue and to believe that researchers often get the sign of an effect wrong, especially when there is a real effect. Reminding people of their death makes them afraid is a reasonable hypothesis, and it would be surprising if studies show the opposite result.

In short, the weight-function model is not wrong, but applying a model that assumes a normal distribution to highly heterogeneous data is wrong. The model predicts many negative results that do not match any observed results. It could be selection bias, but it could also be a false distribution assumption. What to do?

A reasonable approach to make sense of results from the selection model is to focus on the positive side of the distribution. With normal distributions it is easy to get other statistics like the mean of only positive results (or any other subset of studies). We can therefore ignore studies with false substantive hypotheses and focus on studies where researchers made correct predictions about the sign of an effect (H1 is true).

With a mean of -.34 and tau = .956, we get a conditional mean for studies in which H1 is true of .65 standard deviations (a medium to large effect size) with tau of .52. As the lower bound is zero, we only need the upper bound and get .65 + 1.96 * .52 which is 1.67. This would suggest that many studies have strong effect sizes, which seems to contradict the estimated center of the distribution at -.34. This shows how meaningless these point estimates are when heterogeneity is large.

Unfortunately for terror management researchers the truncated moments are not going to rescue their literature because they are hypothetical. The reason is that we are conditioning on an unknown parameter, namely the condition that the hypothesis was true, but for any particular study we do not know whether H1 is true or not. So, the correct way to formulate this result is “if you can identify a study design in which terror management theory makes the right prediction and you can get a fairly precise estimate of the true effect size, you can expect a moderate effect size estimate.

What the weight-function model does not provide is a bias-corrected estimate of the positive effect size estimates in the dataset. The mean of the full distribution includes negative results that were either removed or never obtained. The truncated moment estimate conditions on the unknown status of the null-hypothesis. One is likely too low and the other is likely to high, but neither is conceptually the estimate we want. The average population effect size positive studies that corrects for the selection of nonsignificant results.

To summarize, state of the art meta-analyses in psychology try to deal with the terror of publication bias, but fail to do so. The main reason is that the statistical models that are available do not match the data. They were designed for meta-analysis of close replications with small variation in true effect sizes. They were not intended to be used for meta-analyses of diverse paradigms with large heterogeneity. Other methods that were developed after the replication crisis like p-curve and p-uniform have the same limitation. They work when heterogeneity is small, but they do not work for meta-analyses of diverse studies that are only loosely related by a common hypothesis.

Z-Curve to the Rescue

The quote “Insanity is trying the same thing and expecting a different result” has been attributed to Einstein. Even if that attribution is false, the insight is right. The problem with meta-analytic models is that they try to estimate a single number. This makes sense when the goal is estimation of a single population effect size, but not when every study has a different population effect size.

When we have a heterogenous literature, we need to face heterogeneity head on, and not hide it in some test that is reported and ignored. There is also heterogeneity around the estimate, p < .05. We need to see how much heterogeneity there. But to do that, we first need a model that can deal with heterogeneity without making unrealistic assumptions about the distribution of population effect sizes.

With a fresh look at the problem, we can look to other research areas that have addressed the problem of heterogeneity in effect sizes and large uncertainty about effect sizes of a specific result. Genomics tests millions of DNA segments (SNPs) and tries to find a few segments that show promising results. The goal here is to find the needles in the hey stack rather than averaging across millions of segments that have no relationship with a phenotype. As selection for the strongest observed effects leads to inflated estimates, models are needed to correct for this inflation. However, these corrected estimates are still tight to actual observed results rather than claims about some unobserved distribution of effect sizes. That makes it possible to identify specific segments in the observed data with promising results.

The same logic can be applied to meta-analysis. The goal is no longer to make claims like “the average population effect size is zero” or “the range of plausible effect sizes ranges from -1 to 1.” the goal is now to say “these studies show convincing evidence with meaningful effect sizes.”

One statistical model that can be used to answer this question is zurve (Brunner & Schimmack, 2020; Bartos & Schimmack, 2022). With a few modifications, z-curve can be used for directional meta-analysis where the sign of an effect matters. Rather than fitting z-curve to absolute z-values that ignore the sign and using folded normal components, z-curve can use truncated z-values and truncated normal components. When the model is fitted to only significant results, the difference is minor. More importantly, z-curve estimates of power can also be used to compute bias-corrected effect sizes (Efron, 2005). The reason is that power is a function of effect size and sampling error, so we can use the inverse normal to convert power into a corrected z-value and then multiply it with the sampling error to get a bias corrected effect size. The main challenge is to estimate the sampling error for unobserved non-significant results because their sample sizes are unknown. A simple approach is to use the sampling errors of the just significant results as an approximation. A weighted average of these estimates is the estimate of the true average effect size for the population of studies with positive results before selection for significance. Negative results that are observed are discarded.

Figure 4 shows the results of a simulation study in which the true average power before selection is known. The simulation modeled a beta distribution and graded selection bias. This is important because the weight-function model does well when its assumptions are met. The problem is that the assumptions are are untestable and often questionable. For example, we can simulate a literature with a mean of zero and tau of .4, but this simulation implies that a theories predictions are no better than a coin flip. Once researchers make better predictions, the normal assumption no longer holds.

While z-curve estimates are not perfect, they are conceptually meaningful and closer to the truth than either of the weight-function model’s estimates. We can now apply the model to the TMT data, excluding the few (2%) negative estimates.

The z-curve shows clear evidence of selection bias (the red dotted line is above the light purple bars of the nonsignificant results. However, the EDR estimate of 35% suggests that studies have on average 33% power to produce a significant result. Moreover, an EDR of 33% implies that no more than 10% of the significant results can be false positive results. Even the lower limit of the EDR confidence interval, 20%, allows for only 20% false positive results. This would suggest that many studies, especially significant ones, produced evidence for a true hypotheses with an effect size in the right direction. We can now also quantify the typical effect size. The overall effect size estimate is .63, 95%CI [.46 to .68]. Moreover, we can quantify the average for different ranges of z-values. The average increases from .40 for z-values between 0 and 0.5 to effect sizes greater than 1 for z-values greater than 4.

This finding is surprising, to say the least, because typical effect sizes in psychology are around d = .4 and rarely greater than 1. Before TMT researchers start celebrating, we have to reconcile these findings with the z-curve analysis published in the TMT article.

The z-curve looks notably different in that it does not have a long tail of high z-values. As a result, the EDR estimate is much lower, .08, and the 95% confidence interval includes alpha, 5% to 17%. This implies that there is no t enough evidence to reject the null-hypothesis that all significant results were obtained without a real effect, average effect size: zero, even for z-values greater than 4. So what is it? Is the average effect size close to zero or greater than 1?

To understand the different results, it is important to know that the published z-curve used a different coding of studies than the effect size meta-analysis. I fitted z-curve to these z-values and computed effect size and sampling error estimates from the z-values and degrees of freedom, assuming between-subject designs with equal cell sizes.

The plot is scaled to show the full distribution in the range of non-significant results. The model estimates reproduce the published results. The EDR is 7%, 95%CI = [5%, 17%]. The plot also shows local power for z-values from 0 to 3 stays low. Studies with z-values greater than 4 have acceptable local power but contrary to the previous z-curve, there are hardly any studies. This published z-curve produces dramatically different average effect size estimate, .12 95%CI = .02 to .19. The results also imply much lower heterogeneity because there are hardly any studies with strong evidence (z > 4) and large effect sizes.

Applying the weight-function selection model produces roughly the same results. The average effect size estimate is d = .17, and heterogeneity is small, tau = .17. Now the PET regression result also agrees, intercept = .05, tau = .19.

In conclusion, careful examination of this meta-analysis shows several problems. First, the data were coded inconsistently and different models were given different data. As it turns out, the effect size coding was wrong because F-values were coded as t-values, which dramatically inflates effect size estimates. Second, inconsistent results focused on the point estimate of models, but the point estimate is irrelevant when data are highly heterogenous (due to coding mistakes). Properly interpreted, all models suggest high heterogeneity that allows for large effect sizes among positive results. However, when the data are properly coded, the results show weak evidence that any study produced real effects, a high false positive risk, a small average effect size and small heterogeneity. These results change the final conclusion in the article.

“Given the conflicting findings that emerged across tools and the inherent trade-offs associated with each tool, we caution researchers against drawing firm conclusions about the evidential value of literature through any single analytic tool.”

Correction: The results are consistent and show that most studies provide no evidence for an effect because most effect sizes are small and studies had low power to detect or estimate these effects.

“PET-PEESE can underestimate the effect size when there is publication bias and when p-hacking is present (Carter et al., 2019), which are two conditions likely affecting the literature.”

Here bad research practices are used as an excuse to dismiss the most negative result without mentioning the real problem There is no “effect size.: there is only an average effect size and regression models still allow estimation of heterogeneity that was large in the data the authors used. Even a negative average can be consistent with many true positive effects when heterogeneity is large.

Z-curve can be a powerful tool for inferring the overall composite z-score distribution of a heterogeneous literature. However, unlike the other analyses included in this study, z-curve has not been as thoroughly evaluated by independent researchers so the statistical properties for its power estimates remain under explored. Furthermore, its power estimates are subject to the usual theoretical objections to estimating power from a fixed sample of data (for a recent commentary, see Pek et al., 2022).

This statement ignores that z-curve has been thoroughly evaluated by extensive simulations studies that have been reproduced by the editorial team during an open peer review process. The same cannot be said about the other methods that have not been vetted as rigorously or failed to do well in some conditions (Carter et al., 2019). The reference to Pek is also misleading which has been addressed in several rebuttals to this unfounded claim (Schimmack & Soto, 2026; Soto & Schimmack,2026) with no rejoinder by Pek.

The higher conditional power estimate therefore suggests some evidential value in published studies that yielded significant findings.

The authors are referring to the ERR estimate of 22% [16% , 37%]. Suddenly Pek’s criticism of z-curve is no longer relevant. More importantly, this finding implies that an exact replication of a study with a significant result has a 22% chance of a successful replication outcome. This is abysmal and one of the lowest ever found, not a cause for optimism. Surely reminders of mortality will sometimes have an effect on something, but a research program that uses different designs with an average power of 22% will not be able to identify when a manipulation works or when it is just a chance finding. In fact, the upper limit of the DR estimate is 100%. Thus, these weak studies fail to reject the hypothesis that all studies are pure noise.

“The selection models provide evidence for a small effect consistent with the MS hypothesis… We suggest that the average effect of the literature may be within the range estimated by the selection models and WAAP-WLS (i.e., r is around .18), although this average may have resulted from a mix of effects, many of which are higher than .18, and many of which are lower than .18.

The average estimate is too high once we correct for the coding mistakes. The real effect size is half of this (d = r / 2), and heterogeneity is small. This is the most conesquences conclusoin. Rather than having evidence of a wide range of positive effect sizes, we have evidence that most effect sizes are small and too small to study with the typical sample sizes of this literature.

We encourage future preregistered replications of the MS hypothesis to use smaller es
timates of effect size (i.e., r = .18).

This inflated effect size estimate will only lead to a replication failure. Given the weak evidence in this literature, it may be better to start a new credible research program about coping with awareness of one’s own mortality than to invest more resources into this failed paradigm with questionable manipulations and dependent variables.

Though on their face the liberal and conservative interpretations feel contradictory, some observations are uncontroversial. The first observation is that the TMT literature consists of highly heterogenous.

Even this conclusion turns out to be false when the proper data are analyzed. Heterogeneity was caused by coding mistakes and practically vanishes when the correct data coded by the authors were analyzed. The authors did not notice that their data were inconsistent, even though a simple comparison of the z-values would have shown the discrepancy. It is natural for humans to make errors, but errors also reveal something about the person who committed the error. In this case, it reveals a lack of understanding of the methods, their assumptions, and why they may produce inconsistent results. Here inconsistency was attributed to properties of the models when the real source were inconsistent data. In the future, meta-analysts should not just report inconsistencies, but also try to explain them. That requires understanding of the tools that they use.

For our entire universe of studies, heterogeneity is estimated at τ = .72 under the selection models, which means that for the estimate of g = 0.36 (r = .18) for the entire literature from the selection models, 95% of the effects underlying studies of MS hypothesis, assuming a normal distribution of g, fall between g = −1.05 and 1.77 (or r = −.47 and .66); an extremely wide range of possible effect sizes arising from differences in study design.

The problem with this wide range of population effect sizes is the assumption of a normal distribution. Even if no negative results are observed, the assumption leads to the conclusion that negative effects were obtained but suppressed. But researchers are flexible and it is more likely that they would change the prediction in the direction of a significant result (Kerr, 1998). Thus, it is highly likely that the predicted negative results are phantom studies that do not exist. These predicted effect sizes surely do not correspond to the positive estimates in the dataset.

With these observations in mind, we conclude that there must be some nonzero underlying effects in the studies we examined.

That sounds more reassuring than it is. We have over 800 results and some of these are not false positives. Great, now what? We do not know which of these results are true or false positives. So, we haven’t really learned anything about mortality awareness from this meta-analysis. Fortunately, the analysis of the data without the coding mistake is more conclusive. Terror management research is an example of a pathological science. Researchers conduct studies but never learn form their data because they find a way to keep their theory alive. A proper analysis shows that we can put this literature to rest. That is ok. The history of science is filled with failures. It is also filled with examples where researchers are unable to learn from their errors. However, science moves on and experimental social psychology with little priming manipulations will be a little footnote in the history books.

P.S. And z-curve works and can now also estimate effect sizes.

Selection Bias in Erik van Zwet’s Concerns about z-curve

May 12, 2026Meta-Analysis, Publication Bias, Z-Curve, ZcurveAssumptions, Bimodal, Publication Bias, Simulations, Step-Function Selection Models, weightr, Z-CurveUlrich Schimmack

In a blog post on Andrew Gelman’s blog, Erik van Zwet voiced serious concerns about the performance of z-curve, a meta-analytic method to detect selection bias. The main concern was that z-curve failed to detect selection bias in a scenario where most observed data come from high-powered studies (noncentrality parameter z = 4) but some come from tests of true H0 (effect size is zero, z = 0). In some cases, there may only be a couple of false positive results and that provides too little information about the file drawer of missing tests of H0).

The first problem that I already addressed is that EvZ’s criticism was invalid because it generalized from a single unrealistic scenario to all other situations and did not mention that z-curve had been validated and performed well in these situations (Schimmack, 2026).

Another selection bias in EvZ’s criticism of z-curve is that he only examined the performance of z-curve and did not compare it to the performance of other models. One advantage of z-curve is that it works even if there is little or no variation in sample sizes, which is a requirement for all regression based methods like Funnel plots, Eggert regression, or PET/PEESE. Thus, the most relevant competitor for z-curve are selection models like Vevea and Wood’s (2005) random-effects, step-function model implemented in the r-package weightr.

I tested the model using the same simulation design that was used to examine the performance of z-curve with identical data. Here I focus on the EvZ scenario where most statistically significant results come tests with high power (d = .6, N = 200, z ~ 4.24, power ~ 98%). Thus, there are few non-significant results that can be suppressed by publication bias.

All simulations had 70% selection bias. That is only 30% of the non-significant results were reported and the distribution was flat. First I examined the performance of weightr with k = 100 significant results. All 100 simulations failed to provide estimates of the selection weight for non-significant results. With k = 300 significant results, bias was detected 92% of the time. With k = 1,000, bias was detected in all 100 simulations.

I then examined performance when 20% of the significant results are false positives – and the other 80% come from the same high-powered distribution as before. Figure 1 shows results for a run with k = 1,000 significant results for z-curve. With k = 1,000 z-curve has no problem detecting the selection bias because the distribution of the significant results is clearly bimodal with the mode for the studies with weak power in the non-significant range. Z-curve shows that there are more observed significant results, observed discover rate ODR = 47% than z-curve predicts based on the distribution of the significant results, expected discovery rate, EDR = 23%. The difference is highly significant, p < .000001.

In contrast, the step-function model falsely interprets the higher percentage of non-significant results as evidence that significant results are missing, w(p-value in .025 to .5 range) = 2.24, 95% 2.05 to 2.43. The reason is that the model does not allow for bimodal distributions and assumes a normal distribution of effect sizes, which also implies a normal distribution of z-values when sample sizes are fixed. This problem with the step-function selection model was already reported by Hedges & Vevea (1995). When the simulated data matched the assumed normal distribution, the model worked well. When the distribution did not match the assumed distribution, the model produced bias estimates. The advantage of z-curve is that it does not make a strong distribution assumption and allows for bimodal distributions like the one in Figure 1.

Conclusion

This blog post shows further evidence that EvZ’ expression of concerns about z-curve are biased and do not provide a balanced account of the strengths and weaknesses of z-curve. It is unreasonable to expect a model to perform well in an edge case that also provides problems for other models. In fact, z-curve handles the problem of bimodal distributions better than other models that assume unimodal distributions. If a heterogeneous literature contains a mixture of studies that tested true and false hypotheses, z-curve is actually the superior method and there are no alternatives because most meta-analytic methods were designed to analyze data where all studies are fairly similar and variation in population effect sizes is small. However, many meta-analyses in psychology show evidence of large heterogeneity and the true distribution of effect sizes across studies is unknown. For these kind of data, z-curve is currently the most appropriate statistical tool.

Men are created equal, p-values are not.

January 15, 2021Expected Discovery Rate, Meta-Analysis, Observed Discovery Rate, Power, Publication Bias, Questionable Research Practices, Z-CurveUlrich Schimmack

Is there still something new to say about p-values? Yes, there is. Most discussions of p-values focus on a scenario where a researcher tests a new hypothesis computes a p-value and now has to interpret the result. The status quo follows Fisher’s – 100 year old – approach to compare the p-value to a value of .05. If the p-value is below .05 (two-sided), the inference is that the population effect size deviates from zero in the same direction as the observed effect in the sample. If the p-value is greater than .05 the results are deemed inconclusive.

This approach to the interpretation of the data assumes that we have no other information about our hypothesis or that we do not trust this information sufficiently to incorporate it in our inference about the population effect size. Over the past decade, Bayesian psychologists have argued that we should replace p-values with Bayes-Factors. The advantage of Bayes-Factors is that they can incorporate prior information to draw inferences from data. However, if no prior information is available, the use of Bayesian statistics may cause more harm than good. To use priors without prior information, Bayes-Factors are computed with generic, default priors that are not based on any information about a research question. Along with other problems of Bayes-Factors, this is not an appealing solution to the problem of p-values.

Here I introduce a new approach to the interpretation of p-values that has been called empirical Bayesian and has been successfully applied in genomics to control the field-wise false positive rate. That is, prior information does not rest on theoretical assumptions or default values, but rather on prior empirical information. The information that is used to interpret a new p-value is the distribution of prior p-values.

P-value distributions

Every study is a new study because it relies on a new sample of participants that produces sampling error that is independent of the previous studies. However, studies are not independent in other characteristics. A researcher who conducted a study with N = 40 participants is likely to have used similar sample sizes in previous studies. And a researcher who used N = 200 is also likely to have used larger sample sizes in previous studies. Researchers are also likely to use similar designs. Social psychologists, for example, prefer between-subject designs to better deceive their participants. Cognitive psychologists care less about deception and study simple behaviors that can be repeated hundreds of times within an hour. Thus, researchers who used a between-subject design are likely to have used a between-subject design in previous studies and researchers who used a within-subject design are likely to have used a within-subject design before. Researchers may also be chasing different effect sizes. Finally, researchers can differ in their willingness to take risks. Some may only test hypotheses that are derived from prior theories that have a high probability of being correct, whereas others may be willing to shoot for the moon. All of these consistent differences between researchers (i.e., sample size, effect size, research design) influence the unconditional statistical power of their studies, which is defined as the long-run probability of obtaining significant results, p < .05.

Over the past decade, in the wake of the replication crisis, interest in the distribution of p-values has increased dramatically. For example, one approach uses the distribution of significant p-values, which is known as p-curve analysis (Simonsohn et al., 2014). If p-values were obtained with questionable research practices when the null-hypothesis is true (p-hacking), the distribution of significant p-values is flat. Thus, if the distribution is monotonically decreasing from 0 to .05, the data have evidential value. Although p-curve analyses has been extended to estimate statistical power, simulation studies show that the p-curve algorithm is systematically biased when power varies across studies (Bartos & Schimmack, 2020; Brunner & Schimmack, 2020).

As shown in simulation studies, a better way to estimate power is z-curve (Bartos & Schimmack, 2020; Brunner & Schimmack, 2020). Here I show how z-curve analyses of prior p-values can be used to demonstrate that p-values from one researcher are not equal to p-values of other researchers when we take their prior research practices into account. By using this prior information, we can adjust the alpha level of individual researchers to take their research practices into account. To illustrate this use of z-curve, I first start with an illustration how different research practices influence p-value distributions.

Scenario 1: P-hacking

In the first scenario, we assume that a researcher only tests false hypotheses (i.e., the null-hypothesis is always true (Bem, 2011; Simonsohn et al., 2011). In theory, it would be easy to spot false positives because replication studies would produce produce 19 non-significant results for every significant one and significant ones would have different signs. However, questionable research practices lead to a pattern of results where only significant results in one direction are reported, which is the norm in psychology (Sterling, 1959, Sterling et al., 1995; Schimmack, 2012).

In a z-curve analysis, p-values are first converted into z-scores, z = -qnorm(p/2) with qnorm being the inverse normal function and p being a two-sided p-value. A z-curve plot shows the histogram of all z-scores, including non-significant ones (Figure 1).

Visual inspection of the z-curve plot shows that all 200 p-values are significant (on the right side of the criterion value z = 1.96). it also shows that the mode of the distribution as at the significance criterion. Most important, visual inspection shows a steep drop from the mode to the range of non-significant values. That is, while z = 1.96 is the most common value, z = 1.95 is never observed. This drop provides direct visual information that questionable research practices were used because normal sampling error cannot produce such dramatic changes in the distribution.

I am skipping the technical details how the z-curve model is fitted to the distribution of z-scores (Bartos & Schimmack, 2020). It is sufficient to know that the model is fitted to the distribution of significant z-scores with a limited number of model parameters that are equally spaced over the range of z-scores from 0 to 6 (7 parameters, z = 0, z = 1, z = 2, …. z = 6). The model gives different weights to these parameters to match the observed distribution. Based on these estimates, z-curve.2.0 computes several statistics that can be used to interpret single p-values that have been published or future p-values by the same researcher, assuming that the same research practices are used.

The most important statistic is the expected discovery rate (EDR), which corresponds to the average power of all studies that were conducted by a researcher. Importantly, the EDR is an estimate that is based on only the significant results, but makes predictions about the number of non-significant results. In this example with N = 200 participants, the EDR is 7%. Of course, we know that it really is only 5% because the expected discovery rate for true hypotheses that are tested with alpha = .05 is 5%. However, sampling error can introduce biases in our estimates. Nevertheless, even with only 200 observations, the estimate of 7% is relatively close to 5%. Thus, z-curve tells us something important about the way these p-values were obtained. They were obtained in studies with very low power that is close to the criterion value for a false positive result.

Z-curve uses bootstrap to compute confidence intervals around the point estimate of the EDR. the 95%CI ranges from 5% to 18%. As the interval includes 5%, we cannot reject the hypothesis that all tests were false positives (which in this scenario is also the correct conclusion). At the upper end we can see that mean power is low, even if some true hypotheses are being tested.

The EDR can be used for two purposes. First, it can be used to examine the extent of selection for significance by comparing the EDR to the observed discovery rate (ODR; Schimmack, 2012). The ODR is simply the percentage of significant results that was observed in the sample of p-values. In this case, this is 200 out of 200 or 100%. The discrepancy between the EDR of 7% and 100% is large and 100% is clearly outside the 95%CI of the EDR. Thus, we have strong evidence that questionable research practices were used, which we know to be true in this simulation because the 200 tests were selected from a much larger sample of 4,000 tests.

Most important for the use of z-curve to interpret p-values is the ability to estimate the maximum False Discovery Rate (Soric, 1989). The false discovery rate is the percentage of significant results that are false positives or type-I errors. The false discovery rate is often confused with alpha, the long-run probability of making a type-I error. The significance criterion ensures that no more than 5% of significant and non-significant results are false positives. When we test 4,000 false hypotheses (i.e., the null-hypothesis is true) were are not going to have more than 5% (4,000 * .05 = 200) false positive results. This is true in general and it is true in this example. However, when only significant results are published, it is easy to make the mistake to assume that no more than 5% of the published 200 results are false positives. This would be wrong because the 200 were selected to be significant and they are all false positives.

The false discovery rate is the percentage of significant results that are false positives. It no longer matters whether non-significant results are published or not. We are only concerned with the population of p-values that are below .05 (z > 1.96). In our example, the question is how many of the 200 significant results could be false positives. Soric (1989 demonstrated that the EDR limits the number of false positive discoveries. The more discoveries there are, the lower is the risk that discoveries are false. Using a simple formula, we can compute the maximum false discovery rate from the EDR.

FDR = (1/(EDR – 1)*(.05/.95), with alpha = .05

With an EDR of 7%, we obtained a maximum FDR of 68%. We know that the true FDR is 100%, thus, the estimate is too low. However, the reason is that sampling error can have dramatic effects on the FDR estimates when the EDR is low. With an EDR of 6%, the FDR estimate goes up to 82% and with an EDR estimate of 5% it is 100%. To take account of this uncertainty, we can use the 95%CI of the EDR to compute a 95%CI for the FDR estimate, 24% to 100%. Now we see that we cannot rule out that the FDR is 100%.

In short, scenario 1 introduced the use of p-value distributions to provide useful information about the risk that the published results are false discoveries. In this extreme example, we can dismiss the published p-values as inconclusive or as lacking in evidential value.

Scenario 2: The Typical Social Psychologist

It is difficult to estimate the typical effect size in a literature. However, a meta-analysis of meta-analyses suggested that the average effect size in social psychology is Cohen’s d = .4 (Richard et al., 2003). A smaller set of replication studies that did not select for significance estimated an effect size of d = .3 for social psychology (d = .2 for JPSP, d = .4 for Psych Science; Open Science Collaboration, 2015). The later estimate may include an unknown number of hypotheses where the null-hypothesis is true and the true effect size is zero. Thus, I used d = .4 as a reasonable effect size for true hypotheses in social psychology (see also LeBel, Campbell, & Loving, 2017).

It is also known that a rule of thumb in experimental social psychology was to allocate n = 20 participants to a condition, resulting in a sample size of N = 40 in studies with two groups. In a 2 x 2 design, the main effect would be tested with N = 80. However, to keep this scenario simple, I used d = .4 and N = 40 for true effects. This affords 23% power to obtain a significant result.

Finkel, Eastwick, and Reis (2017) argued that power of 25% is optimal if 75% of the hypotheses that are being tested are true. However, the assumption that 75% of hypotheses are true may be on the optimistic side. Wilson and Wixted (2018) suggested that the false discovery risk is closer to 50%. With 23% power for true hypotheses, this implies a false discovery rate of Given uncertainty about the actual false discovery rate in social psychology, I used a scenario with 50% true and 50% false hypotheses.

I kept the number of significant results at 200. To obtain 200 significant results with an equal number of true and false hypotheses, we need 1,428 tests. The 714 true hypotheses contribute 714*.23 = 164 true positives and the 714 false hypotheses produce 714*.05 = 36 false positive results; 164 + 36 = 200. This implies a false discovery rate of 36/200 = 18%. The true EDR is (714*.23+714*.05)/(714+714) = 14%.

The z-curve plot looks very similar to the previous plot, but they are not identical. Although the EDR estimate is higher, it still includes zero. The maximum FDR is well above the actual FDR of 18%, but the 95%CI includes the actual value of 18%.

A notable difference between Figure 1 and Figure 2 is the expected replication rate (ERR), which corresponds to the average power of significant p-values. It is called the estimated replication rate (ERR) because it predicts the percentage of significant results if the studies that were selected for significance were replicated exactly (Brunner & Schimmack, 2020). When power is heterogeneous, power of the studies with significant results is higher than power of studies with non-significant results (Brunner & Schimmack, 2020). In this case, with only two power values, the reason is that false positives have a much lower chance to be significant (5%) than true positives (23%). As a result, the average power of significant studies is higher than the average power of all studies. In this simulation, the true average power of significant studies is the weighted average of true and false positives with significant results, (164*.23 +36*.05)/(164+36) = 20%. Z-curve perfectly estimated this value.

Importantly, the 95% CI of the ERR, 11% to 34%, does not include zero. Thus, we can reject the null-hypotheses that all of the significant results are false positives based on the ERR. In other words, the significant results have evidential value. However, we do not know the composition of this average. It could be a large percentage of false positives and a few true hypotheses with high power or it could be many true positives with low power. We also do not know which of the 200 significant results is a true positive or a false positive. Thus, we would need to conduct replication studies to distinguish between true and false hypotheses. And given the low power, we would only have a 23% chance of successfully replicating a true positive result. This is exactly what happened with the reproducibility project. And the inconsistent results lead to debates and require further replications. Thus, we have real-world evidence how uninformative p-values are when they are obtained this way.

Social psychologists might argue that the use of small samples is justified because most hypotheses in psychology are true. Thus, we can use prior information to assume that significant results are true positives. However, this logic fails when social psychologists test false hypotheses. In this case, the observed distribution of p-values (Figure 1) is not that different from the distribution that is observed when most significant results are true positives that were obtained with low power (Figure 2). Thus, it is doubtful that this is really an optimal use of resources (Finkel et al., 2015). However, until recently this was the way experimental social psychologists conducted their research.

Scenario 3: Cohen’s Way

In 1962 (!), Cohen conducted a meta-analysis of statistical power in social psychology. The main finding was that studies had only a 50% chance to get significant results with a median effect size of d = .5. Cohen (1988) also recommended that researchers should plan studies to have 80% power. However, this recommendation was ignored.

To achieve 80% power with d = .4, researchers need N = 200 participants. Thus, the number of studies is reduced from 5 studies with N = 40 to one study with N = 200. As Finkel et al. (2017) point out, we can make more discoveries with many small studies than a few large ones. However, this ignores that the results of the small studies are difficult to replicate. This was not a concern when social psychologists did not bother to test whether their discoveries are false discoveries or whether they can be replicated. The replication crisis shows the problems of this approach. Now we have results from decades of research that produced significant p-values without providing any information whether these significant results are true or false discoveries.

Scenario 3 examines what social psychology would look like today, if social psychologists had listened to Cohen. The scenario is the same as in the second scenario, including publication bias. There are 50% false hypotheses and 50% true hypotheses with an effect size of d = .4. The only difference is that researchers used N = 200 to test their hypotheses to achieve 80% power.

With 80% power, we need 470 tests (compared to 1,428 in Scenario 2) to produce 200 significant results, 235*.80 + 235*.05 = 188 + 12 = 200. Thus, the EDR is 200/470 = 43%. The true false discovery rate is 6%. The expected replication rate is 188*.80 + 12*.05 = 76%. Thus, we see that higher power increases replicability from 20% to 76% and lowers the false discovery rate from 18% to 6%.

Figure 3 shows the z-curve plot. Visual inspection shows that Figure 3 looks very different from Figures 1 and 2. The estimates are also different. In this example, sampling error inflated the EDR to be 58%, but the 95%CI includes the true value of 46%. The 95%CI does not include the ODR. Thus, there is evidence for publication bias, which is also visible by the steep drop in the distribution at 1.96.

Even with a low EDR of 20%, the maximum FDR is only 21%. Thus, we can conclude with confidence that at least 79% of the significant results are true positives. Remember, in the previous scenario, we could not rule out that most results are false positives. Moreover, the estimated replication rate is 73%, which underestimates the true replication rate of 76%, but the 95%CI includes the true value, 95%CI = 61% – 84%. Thus, if these studies were replicated, we would have a high success rate for actual replication studies.

Just imagine for a moment what social psychology might look like in a parallel universe where social psychologists followed Cohen’s advice. Why didn’t they? The reason is that they did not have z-curve. All they had was p < .05, and using p < .05, all three scenarios are identical. All three scenarios produced 200 significant results. Moreover, as Finkel et al. (2015) pointed out, smaller samples produce 200 significant results quicker than large samples. An additional advantage of small samples is that they inflate point estimates of the population effect size. Thus, the social psychologists with the smallest samples could brag about the biggest (illusory) effect sizes as long as nobody was able to publish replication studies with larger samples that deflated effect sizes of d = .8 to d = .08 (Joy-Gaba & Nosek, 2010).

This game is over, but social psychology – and other social sciences – have published thousands of significant p-values, and nobody knows whether they were obtained using scenario 1, 2, or 3, or probably a combination of these. This is where z-curve can make a difference. P-values are no longer equal when they are considered as a data point from a p-value distribution. In scenario 1, a p-value of .01 and even a p-value of .001 has no meaning. In contrast, in scenario 3 even a p-value of .02 is meaningful and more likely to reflect a true positive than a false positive result. This means that we can use z-curve analyses of published p-values to distinguish between probably false and probably true positives.

I illustrate this with three concrete examples from a project that examined the p-value distributions of over 200 social psychologists (Schimmack, in preparation). The first example has the lowest EDR in the sample. The EDR is 11% and because there are only 210 tests, the 95%CI is wide and includes 5%.

The maximum EDR estimate is high with 41% and the 95%CI includes 100%. This suggests that we cannot rule out the hypothesis that most significant results are false positives. However, the replication rate is 57% and the 95%CI, 45% to 69%, does not include 5%. Thus, some tests tested true hypotheses, but we do not know which ones.

Visual inspection of the plot shows a different distribution than Figure 2. There are more just significant p-values, z = 2.0 to 2.2 and more large z-scores (z > 4). This shows more heterogeneity in power. A comparison of the ODR with the EDR shows that the ODR falls outside the 95%CI of the EDR. This is evidence of publication bias or the use of questionable research practices. One solution to the presence of publication bias is to lower the criterion for statistical significance. As a result, the large number of just significant results is no longer significant and the ODR decreases. This is a post-hoc correction for publication bias. For example, we can lower alpha to .005.

As expected, the ODR decreases considerably from 70% to 39%. In contrast, the EDR increases. The reason is that many questionable research practices produce a pile of just significant p-values. As these values are no longer used to fit the z-curve, it predicts a lot fewer non-significant p-values. The model now underestimates p-values between 2 and 2.2. However, these values do not seem to come from a sampling distribution. Rather they stick out like a tower. By excluding them, the p-values that are still significant with alpha = .005 look more credible. Thus, we can correct for the use of QRPs by lowering alpha and by examining whether these p-values produced interesting discoveries. At the same time, we can ignore the p-values between .05 and .005 and await replication studies to provide empirical evidence whether these hypotheses receive empirical support.

The second example was picked because it was close to the median EDR (33) and ERR (66) in the sample of 200 social psychologists.

The larger sample of tests (k = 1,529) helps to obtain more precise estimates. A comparison of the ODR, 76%, and the 95%CI of the EDR, 12% to 48%, shows that publication bias is present. However, with an EDR of 33%, the maximum FDR is only 11% and the upper limit of the 95%CI is 39%. Thus, we can conclude with confidence that fewer than 50% of the significant results are false positives, however numerous findings might be false positives. Only replication studies can provide this information.

In this example, lowering alpha to .005 did not align the ODR and the EDR. This suggests that these values come from a sampling distribution where non-significant results were not published. Thus, adjusting the there is no simple fix to adjust the significance criterion. In this situation, we can conclude that the published p-values are unlikely to be false positives, but that replication studies are needed to ensure that published significant results are not false positives.

The third example is the social psychologists with the highest EDR. In this case, the EDR is actually a little bit lower than the ODR, suggesting that there is no publication bias. The high EDR also means that the maximum FDR is very small and even the upper limit of the 95%CI is only 7%.

Another advantage of data without publication bias is that it is not necessary to exclude non-significant results from the analysis. Fitting the model to all p-values produces much tighter estimates of the EDR and the maximum FDR.

The upper limit of the 95%CI for the FDR is now 4%. Thus, we conclude that no more than 5% of the p-values less than .05 are false positives. Even p = .02 is unlikely to be a false positive. Finally, the estimated replication rate is 84% with a tight confidence interval ranging from 78% to 90%. Thus, most of the published p-values are expected to replicate in an exact replication study.

I hope these examples make it clear how useful it can be to evaluate single p-values with prior information about the p-values distribution of a lab. As labs differ in their research practices, significant p-values are also different. Only if we ignore the research context and focus on a single result p = .02 equals p = .02. But once we see the broader distribution, p-values of .02 can provide stronger evidence against the null-hypothesis than p-values of .002.

Implications

Cohen tried and failed to change the research culture of social psychologists. Meta-psychological articles have puzzled why meta-analyses of power failed to increase power (Maxwell, 2004; Schimmack, 2012; Sedelmeier & Gigerenzer, 1989). Finkel et al. (2015) provided an explanation. In a game where the winner publishes as many significant results as possible, the optimal strategy is to conduct as many studies as possible with low power. This strategy continues to be rewarded in psychology, where jobs, promotions, grants, and pay raises are based on the number of publications. Cohen (1990) said less is more, but that is not true in a science that does not self-correct and treats every p-value less than .05 as a discovery.

To improve psychology as a science, we need to change the incentive structure and author-wise z-curve analyses can do this. Rather than using p < .05 (or p < .005) as a general rule to claim discoveries, claims of discoveries can be adjusted to the research practices of a researchers. As demonstrated here, this will reward researchers who follow Cohen’s rules and punish those who use questionable practices to produce p-values less than .05 (or Bayes-Factors > 3) without evidential value. And maybe, there is a badge for credible p-values one day.

(incomplete) References

Richard, F. D., Bond, C. F., Jr., & Stokes-Zoota, J. J. (2003). One hundred years of social psychology quantitatively described. Review of General Psychology, 7, 331–363. http://dx.doi.org/10.1037/1089-2680.7.4.331

The Replicability Index Is the Most Powerful Tool to Detect Publication Bias in Meta-Analyses

January 1, 2020Meta-Analysis, Post-Hoc Power, Power, Publication Bias, Statistical PowerUlrich Schimmack

Abstract

Methods for the detection of publication bias in meta-analyses were first introduced in the 1980s (Light & Pillemer, 1984). However, existing methods tend to have low statistical power to detect bias, especially when population effect sizes are heterogeneous (Renkewitz & Keiner, 2019). Here I show that the Replicability Index (RI) is a powerful method to detect selection for significance while controlling the type-I error risk better than the Test of Excessive Significance (TES). Unlike funnel plots and other regression methods, RI can be used without variation in sampling error across studies. Thus, it should be a default method to examine whether effect size estimates in a meta-analysis are inflated by selection for significance. However, the RI should not be used to correct effect size estimates. A significant results merely indicates that traditional effect size estimates are inflated by selection for significance or other questionable research practices that inflate the percentage of significant results.

Evaluating the Power and Type-I Error Rate of Bias Detection Methods

Just before the end of the year, and decade, Frank Renkewitz and Melanie Keiner published an important article that evaluated the performance of six bias detection methods in meta-analyses (Renkewitz & Keiner, 2019).

The article makes several important points.

1. Bias can distort effect size estimates in meta-analyses, but the amount of bias is sometimes trivial. Thus, bias detection is most important in conditions where effect sizes are inflated to a notable degree (say more than one-tenth of a standard deviation, e.g., from d = .2 to d = .3).

2. Several bias detection tools work well when studies are homogeneous (i.e. ,the population effect sizes are very similar). However, bias detection is more difficult when effect sizes are heterogeneous.

3. The most promising tool for heterogeneous data was the Test of Excessive Significance (Francis, 2013; Ioannidis, & Trikalinos, 2013). However, simulations without bias showed that the higher power of TES was achieved by a higher false-positive rate that exceeded the nominal level. The reason is that TES relies on the assumption that all studies have the same population effect size and this assumption is violated when population effect sizes are heterogeneous.

This blog post examines two new methods to detect publication bias and compares them to the TES and the Test of Insufficient Variance (TIVA) that performed well when effect sizes were homogeneous (Renkewitz & Keiner , 2019). These methods are not entirely new. One method is the Incredibility Index, which is similar to TES (Schimmack, 2012). The second method is the Replicability Index, which corrects estimates of observed power for inflation when bias is present.

The Basic Logic of Power-Based Bias Tests

The mathematical foundations for bias tests based on statistical power were introduced by Sterling et al. (1995). Statistical power is defined as the conditional probability of obtaining a significant result when the null-hypothesis is false. When the null-hypothesis is true, the probability of obtaining a significant result is set by the criterion for a type-I error, alpha. To simplify, we can treat cases where the null-hypothesis is true as the boundary value for power (Brunner & Schimmack, 2019). I call this unconditional power. Sterling et al. (1995) pointed out that for studies with heterogeneity in sample sizes, effect sizes or both, the discoery rate; that is the percentage of significant results, is predicted by the mean unconditional power of studies. This insight makes it possible to detect bias by comparing the observed discovery rate (the percentage of significant results) to the expected discovery rate based on the unconditional power of studies. The empirical challenge is to obtain useful estimates of unconditional mean power, which depends on the unknown population effect sizes.

Ioannidis and Trialinos (2007) were the first to propose a bias test that relied on a comparison of expected and observed discovery rates. The method is called Test of Excessive Significance (TES). They proposed a conventional meta-analysis of effect sizes to obtain an estimate of the population effect size, and then to use this effect size and information about sample sizes to compute power of individual studies. The final step was to compare the expected discovery rate (e.g., 5 out of 10 studies) with the observed discovery rate (8 out of 10 studies) with a chi-square test and to test the null-hypothesis of no bias with alpha = .10. They did point out that TES is biased when effect sizes are heterogeneous (see Renkewitz & Keiner, 2019, for a detailed discussion).

Schimmack (2012) proposed an alternative approach that does not assume a fixed effect sizes across studies, called the incredibility index. The first step is to compute observed-power for each study. The second step is to compute the average of these observed power estimates. This average effect size is then used as an estimate of the mean unconditional power. The final step is to compute the binomial probability of obtaining as many or more significant results that were observed for the estimated unconditional power. Schimmack (2012) showed that this approach avoids some of the problems of TES when effect sizes are heterogeneous. Thus, it is likely that the Incredibility Index produces fewer false positives than TES.

Like TES, the incredibility index has low power to detect bias because bias inflates observed power. Thus, the expected discovery rate is inflated, which makes it a conservative test of bias. Schimmack (2016) proposed a solution to this problem. As the inflation in the expected discovery rate is correlated with the amount of bias, the discrepancy between the observed and expected discovery rate indexes inflation. Thus, it is possible to correct the estimated discovery rate by the amount of observed inflation. For example, if the expected discovery rate is 70% and the observed discovery rate is 90%, the inflation is 20 percentage points. This inflation can be deducted from the expected discovery rate to get a less biased estimate of the unconditional mean power. In this example, this would be 70% – 20% = 50%. This inflation-adjusted estimate is called the Replicability Index. Although the Replicability Index risks a higher type-I error rate than the Incredibility Index, it may be more powerful and have a better type-I error control than TES.

To test these hypotheses, I conducted some simulation studies that compared the performance of four bias detection methods. The Test of Insufficient Variance (TIVA; Schimmack, 2015) was included because it has good power with homogeneous data (Renkewitz & Keiner, 2019). The other three tests were TES, ICI, and RI.

Selection bias was simulated with probabilities of 0, .1, .2, and 1. A selection probability of 0 implies that non-significant results are never published. A selection probability of .1 implies that there is a 10% chance that a non-significant result is published when it is observed. Finally, a selection probability of 1 implies that there is no bias and all non-significant results are published.

Effect sizes varied from 0 to .6. Heterogeneity was simulated with a normal distribution with SDs ranging from 0 to .6. Sample sizes were simulated by drawing from a uniform distribution with values between 20 and 40, 100, and 200 as maximum. The number of studies in a meta-analysis were 5, 10, 20, and 30. The focus was on small sets of studies because power to detect bias increases with the number of studies and power was often close to 100% with k = 30.

Each condition was simulated 100 times and the percentage of significant results with alpha = .10 (one-tailed) was used to compute power and type-I error rates.

RESULTS

Bias

Figure 1 shows a plot of the mean observed d-scores as a function of the mean population d-scores. In situations without heterogeneity, mean population d-scores corresponded to the simulated values of d = 0 to d = .6. However, with heterogeneity, mean population d-scores varied due to sampling from the normal distribution of population effect sizes.

The figure shows that bias could be negative or positive, but that overestimation is much more common than underestimation. Underestimation was most likely when the population effect size was 0, there was no variability (SD = 0), and there was no selection for significance. With complete selection for significance, bias always overestimated population effect sizes, because selection was simulated to be one-sided. The reason is that meta-analysis rarely show many significant results in both directions.

An Analysis of Variance (ANOVA) with number of studies (k), mean population effect size (mpd), heterogeneity of population effect sizes (SD), range of sample sizes (Nmax) and selection bias (sel.bias) showed a four-way interaction, t = 3.70. This four-way interaction qualified main effects that showed bias decreases with effect sizes (d), heterogeneity (SD), range of sample sizes (N), and increased with severity of selection bias (sel.bias).

The effect of selection bias is obvious in that effect size estimates are unbiased when there is no selection bias and increases with severity of selection bias. Figure 2 illustrates the three way interaction for the remaining factors with the most extreme selection bias; that is, all non-significant results are suppressed.

The most dramatic inflation of effect sizes occurs when sample sizes are small (N = 20-40), the mean population effect size is zero, and there is no heterogeneity (light blue bars). This condition simulates a meta-analysis where the null-hypothesis is true. Inflation is reduced, but still considerable (d = .42), when the population effect is large (d = .6). Heterogeneity reduces bias because it increases the mean population effect size. However, even with d = .6 and heterogeneity, small samples continue to produce inflated estimates by d = .25 (dark red). Increasing sample sizes (N = 20 to 200) reduces inflation considerably. With d = 0 and SD = 0, inflation is still considerable, d = .52, but all other conditions have negligible amounts of inflation, d < .10.

As sample sizes are known, they provide some valuable information about the presence of bias in a meta-analysis. If studies with large samples are available, it is reasonable to limit a meta-analysis to the larger and more trustworthy studies (Stanley, Jarrell, & Doucouliagos, 2010).

Discovery Rates

If all results are published, there is no selection bias and effect size estimates are unbiased. When studies are selected for significance, the amount of bias is a function of the amount of studies with non-significant results that are suppressed. When all non-significant results are suppressed, the amount of selection bias depends on the mean power of the studies before selection for significance which is reflected in the discovery rate (i.e., the percentage of studies with significant results). Figure 3 shows the discovery rates for the same conditions that were used in Figure 2. The lowest discovery rate exists when the null-hypothesis is true. In this case, only 2.5% of studies produce significant results that are published. The percentage is 2.5% and not 5% because selection also takes the direction of the effect into account. Smaller sample sizes (left side) have lower discovery rates than larger sample sizes (right side) because larger samples have more power to produce significant results. In addition, studies with larger effect sizes have higher discovery rates than studies with small effect sizes because larger effect sizes increase power. In addition, more variability in effect sizes increases power because variability increases the mean population effect sizes, which also increases power.

In conclusion, the amount of selection bias and the amount of inflation of effect sizes varies across conditions as a function of effect sizes, sample sizes, heterogeneity, and the severity of selection bias. The factorial design covers a wide range of conditions. A good bias detection method should have high power to detect bias across all conditions with selection bias and low type-I error rates across conditions without selection bias.

Overall Performance of Bias Detection Methods

Figure 4 shows the overall results for 235,200 simulations across a wide range of conditions. The results replicate Renkewitz and Keiner’s finding that TES produces more type-I errors than the other methods, although the average rate of type-I errors is below the nominal level of alpha = .10. The error rate of the incredibility index is practically zero, indicating that it is much more conservative than TES. The improvement for type-I errors does not come at the cost of lower power. TES and ICI have the same level of power. This finding shows that computing observed power for each individual study is superior than assuming a fixed effect size across studies. More important, the best performing method is the Replicability Index (RI), which has considerably more power because it corrects for inflation in observed power that is introduced by selection for significance. This is a promising results because one of the limitation of the bias tests examined by Renkewitz and Keiner was the low power to detect selection bias across a wide range of realistic scenarios.

Logistic regression analyses for power showed significant five-way interactions for TES, IC, and RI. For TIVA, two four-way interactions were significant. For type-I error rates no four-way interactions were significant, but at least one three-way interaction was significant. These results show that results systematic vary in a rather complex manner across the simulated conditions. The following results show the performance of the four methods in specific conditions.

Number of Studies (k)

Detection of bias is a function of the amount of bias and the number of studies. With small sets of studies (k = 5), it is difficult to detect power. In addition, low power can suppress false-positive rates because significant results without selection bias are even less likely than significant results with selection bias. Thus, it is important to examine the influence of the number of studies on power and false positive rates.

Figure 5 shows the results for power. TIVA does not gain much power with increasing sample sizes. The other three methods clearly become more powerful as sample sizes increase. However, only the R-Index shows good power with twenty studies and still acceptable studies with just 10 studies. The R-Index with 10 studies is as powerful as TES and ICI with 10 studies.

Figure 6 shows the results for the type-I error rates. Most important, the high power of the R-Index is not achieved by inflating type-I error rates, which are still well-below the nominal level of .10. A comparison of TES and ICI shows that ICI controls type-I error much better than TES. TES even exceeds the nominal level of .10 with 30 studies and this problem is going to increase as the number of studies gets larger.

Selection Rate

Renkewitz and Keiner noticed that power decreases when there is a small probability that non-significant results are published. To simplify the results for the amount of selection bias, I focused on the condition with n = 30 studies, which gives all methods the maximum power to detect selection bias. Figure 7 confirms that power to detect bias deteriorates when non-significant results are published. However, the influence of selection rate varies across methods. TIVA is only useful when only significant results are selected, but even TES and ICI have only modest power even if the probability of a non-significant result to be published is only 10%. Only the R-Index still has good power, and power is still higher with a 20% chance to select a non-significant result than with a 10% selection rate for TES and ICI.

Population Mean Effect Size

With complete selection bias (no significant results), power had ceiling effects. Thus, I used k = 10 to illustrate the effect of population effect sizes on power and type-I error rates. (Figure 8)

In general, power decreased as the population mean effect sizes increased. The reason is that there is less selection because the discovery rates are higher. Power decreased quickly to unacceptable levels (< 50%) for all methods except the R-Index. The R-Index maintained good power even with the maximum effect size of d = .6.

Figure 9 shows that the good power of the R-Index is not achieved by inflating type-I error rates. The type-I error rate is well below the nominal level of .10. In contrast, TES exceeds the nominal level with d = .6.

Variability in Population Effect Sizes

I next examined the influence of heterogeneity in population effect sizes on power and type-I error rates. The results in Figure 10 show that hetergeneity decreases power for all methods. However, the effect is much less sever for the RI than for the other methods. Even with maximum heterogeneity, it has good power to detect publication bias.

Figure 11 shows that the high power of RI is not achieved by inflating type-I error rates. The only method with a high error-rate is TES with high heterogeneity.

Variability in Sample Sizes

With a wider range of sample sizes, average power increases. And with higher power, the discovery rate increases and there is less selection for significance. This reduces power to detect selection for significance. This trend is visible in Figure 12. Even with sample sizes ranging from 20 to 100, TIVA, TES, and IC have modest power to detect bias. However, RI maintains good levels of power even when sample sizes range from 20 to 200.

Once more, only TES shows problems with the type-I error rate when heterogeneity is high (Figure 13). Thus, the high power of RI is not achieved by inflating type-I error rates.

Stress Test

The following analyses examined RI’s performance more closely. The effect of selection bias is self-evident. As more non-significant results are available, power to detect bias decreases. However, bias also decreases. Thus, I focus on the unfortunately still realistic scenario that only significant results are published. I focus on the scenario with the most heterogeneity in sample sizes (N = 20 to 200) because it has the lowest power to detect bias. I picked the lowest and highest levels of population effect sizes and variability to illustrate the effect of these factors on power and type-I error rates. I present results for all four set sizes.

The results for power show that with only 5 studies, bias can only be detected with good power if the null-hypothesis is true. Heterogeneity or large effect sizes produce unacceptably low power. This means that the use of bias tests for small sets of studies is lopsided. Positive results strongly indicate severe bias, but negative results are inconclusive. With 10 studies, power is acceptable for homogeneous and high effect sizes as well as for heterogeneous and low effect sizes, but not for high effect sizes and high heterogeneity. With 20 or more studies, power is good for all scenarios.

The results for the type-I error rates reveal one scenario with dramatically inflated type-I error rates, namely meta-analysis with a large population effect size and no heterogeneity in population effect sizes.

Solutions

The high type-I error rate is limited to cases with high power. In this case, the inflation correction over-corrects. A solution to this problem is found by considering the fact that inflation is a non-linear function of power. With unconditional power of .05, selection for significance inflates observed power to .50, a 10 fold increase. However, power of .50 is inflated to .75, which is only a 50% increase. Thus, I modified the R-Index formula and made inflation contingent on the observed discovery rate.

RI2 = Mean.Observed.Power – (Observed Discovery Rate – Mean.Observed.Power)*(1-Observed.Discovery.Rate). This version of the R-Index reduces power, although power is still superior to the IC.

It also fixed the type-I error problem at least with sample sizes up to N = 30.

Example 1: Bem (2011)

Bem’s (2011) sensational and deeply flawed article triggered the replication crisis and the search for bias-detection tools (Francis, 2012; Schimmack, 2012). Table 1 shows that all tests indicate that Bem used questionable research practices to produce significant results in 9 out of 10 tests. This is confirmed by examination of his original data (Schimmack, 2018). For example, for one study, Bem combined results from four smaller samples with non-significant results into one sample with a significant result. The results also show that both versions of the Replicability Index are more powerful than the other tests.

Test	p	1/p
TIVA	0.008	125
TES	0.018	56
IC	0.031	32
RI	0.00002	45754
RI2	0.00013	7255

Example 2: Francis (2014) Audit of Psychological Science

Francis audited multiple-study articles in the journal Psychological Science from 2009-2012. The main problem with the focus on single articles is that they often contain relatively few studies and the simulation studies showed that bias tests tend to have low power if 5 or fewer studies are available (Renkewitz & Keiner, 2019). Nevertheless, Francis found that 82% of the investigated articles showed signs of bias, p < .10. This finding seems very high given the low power of TES in the simulation studies. It would mean that selection bias in these articles was very high and power of the studies was extremely low and homogeneous, which provides the ideal conditions to detect bias. However, the high type-I error rates of TES under some conditions may have produced more false positive results than the nominal level of .10 suggests. Moreover, Francis (2014) modified TES in ways that may have further increased the risk of false positives. Thus, it is interesting to reexamine the 44 studies with other bias tests. Unlike Francis, I coded one focal hypothesis test per study.

I then applied the bias detection methods. Table 2 shows the p-values.

Year	Author	Francis	TIVA	TES	IC	RI1	RI2
2012	Anderson, Kraus, Galinsky, & Keltner	0.167	0.388	0.122	0.387	0.111	0.307
2012	Bauer, Wilkie, Kim, & Bodenhausen	0.062	0.004	0.022	0.088	0.000	0.013
2012	Birtel & Crisp	0.133	0.070	0.076	0.193	0.004	0.064
2012	Converse & Fishbach	0.110	0.130	0.161	0.319	0.049	0.199
2012	Converse, Risen, & Carter Karmic	0.043	0.000	0.022	0.065	0.000	0.010
2012	Keysar, Hayakawa, &	0.091	0.115	0.067	0.119	0.003	0.043
2012	Leung et al.	0.076	0.047	0.063	0.119	0.003	0.043
2012	Rounding, Lee, Jacobson, & Ji	0.036	0.158	0.075	0.152	0.004	0.054
2012	Savani & Rattan	0.064	0.003	0.028	0.067	0.000	0.017
2012	van Boxtel & Koch	0.071	0.496	0.718	0.498	0.200	0.421
2011	Evans, Horowitz, & Wolfe	0.426	0.938	0.986	0.628	0.379	0.606
2011	Inesi, Botti, Dubois, Rucker, & Galinsky	0.026	0.043	0.061	0.122	0.003	0.045
2011	Nordgren, Morris McDonnell, & Loewenstein	0.090	0.026	0.114	0.196	0.012	0.094
2011	Savani, Stephens, & Markus	0.063	0.027	0.030	0.080	0.000	0.018
2011	Todd, Hanko, Galinsky, & Mussweiler	0.043	0.000	0.024	0.051	0.000	0.005
2011	Tuk, Trampe, & Warlop	0.092	0.000	0.028	0.097	0.000	0.017
2010	Balcetis & Dunning	0.076	0.113	0.092	0.126	0.003	0.048
2010	Bowles & Gelfand	0.057	0.594	0.208	0.281	0.043	0.183
2010	Damisch, Stoberock, & Mussweiler	0.057	0.000	0.017	0.073	0.000	0.007
2010	de Hevia & Spelke	0.070	0.351	0.210	0.341	0.062	0.224
2010	Ersner-Hershfield, Galinsky, Kray, & King	0.073	0.004	0.005	0.089	0.000	0.013
2010	Gao, McCarthy, & Scholl	0.115	0.141	0.189	0.361	0.041	0.195
2010	Lammers, Stapel, & Galinsky	0.024	0.022	0.113	0.061	0.001	0.021
2010	Li, Wei, & Soman	0.079	0.030	0.137	0.231	0.022	0.129
2010	Maddux et al.	0.014	0.344	0.100	0.189	0.010	0.087
2010	McGraw & Warren	0.081	0.993	0.302	0.148	0.006	0.066
2010	Sackett, Meyvis, Nelson, Converse, & Sackett	0.033	0.002	0.025	0.048	0.000	0.011
2010	Savani, Markus, Naidu, Kumar, & Berlia	0.058	0.011	0.009	0.062	0.000	0.014
2010	Senay, Albarracín, & Noguchi	0.090	0.000	0.017	0.081	0.000	0.010
2010	West, Anderson, Bedwell, & Pratt	0.157	0.223	0.226	0.287	0.032	0.160
2009	Alter & Oppenheimer	0.071	0.000	0.041	0.053	0.000	0.006
2009	Ashton-James, Maddux, Galinsky, & Chartrand	0.035	0.175	0.133	0.270	0.025	0.142
2009	Fast & Chen	0.072	0.006	0.036	0.073	0.000	0.014
2009	Fast, Gruenfeld, Sivanathan, & Galinsky	0.069	0.008	0.042	0.118	0.001	0.030
2009	Garcia & Tor	0.089	1.000	0.422	0.190	0.019	0.117
2009	González & McLennan	0.139	0.080	0.194	0.303	0.055	0.208
2009	Hahn, Close, & Graf	0.348	0.068	0.286	0.474	0.175	0.390
2009	Hart & Albarracín	0.035	0.001	0.048	0.093	0.000	0.015
2009	Janssen & Caramazza	0.083	0.051	0.310	0.392	0.115	0.313
2009	Jostmann, Lakens, & Schubert	0.090	0.000	0.026	0.098	0.000	0.018
2009	Labroo, Lambotte, & Zhang	0.008	0.054	0.071	0.148	0.003	0.051
2009	Nordgren, van Harreveld, & van der Pligt	0.100	0.014	0.051	0.135	0.002	0.041
2009	Wakslak & Trope	0.061	0.008	0.029	0.065	0.000	0.010
2009	Zhou, Vohs, & Baumeister	0.041	0.009	0.043	0.097	0.002	0.036

The Figure shows the percentage of significant results for the various methods. The results confirm that despite the small number of studies, the majority of multiple-study articles show significant evidence of bias. Although statistical significance does not speak directly to effect sizes, the fact that these tests were significant with a small set of studies implies that the amount of bias is large. This is also confirmed by a z-curve analysis that provides an estimate of the average bias across all studies (Schimmack, 2019).

A comparison of the methods shows with real data that the R-Index (RI1) is the most powerful method and even more powerful than Francis’s method that used multiple studies from a single study. The good performance of TIVA shows that population effect sizes are rather homogeneous as TIVA has low power with heterogeneous data. The Incredibility Index has the worst performance because it has an ultra-conservative type-I error rate. The most important finding is that the R-Index can be used with small sets of studies to demonstrate moderate to large bias.

Discussion

In 2012, I introduced the Incredibility Index as a statistical tool to reveal selection bias; that is, the published results were selected for significance from a larger number of results. I compared the IC with TES and pointed out some advantages of averaging power rather than effect sizes. However, I did not present extensive simulation studies to compare the performance of the two tests. In 2014, I introduced the replicability index to predict the outcome of replication studies. The replicability index corrects for the inflation of observed power when selection for significance is present. I did not think about RI as a bias test. However, Renkewitz and Keiner (2019) demonstrated that TES has low power and inflated type-I error rates. Here I examined whether IC performed better than TES and I found it did. Most important, it has much more conservative type-I error rates even with extreme heterogeneity. The reason is that selection for significance inflates observed power which is used to compute the expected percentage of significant results. This led me to see whether the bias correction that is used to compute the Replicability Index can boost power, while maintaining acceptable type-I error rates. The present results shows that this is the case for a wide range of scenarios. The only exception are meta-analysis of studies with a high population effect size and low heterogeneity in effect sizes. To avoid this problem, I created an alternative R-Index that reduces the inflation adjustment as a function of the percentage of non-significant results that are reported. I showed that the R-Index is a powerful tool that detects bias in Bem’s (2011) article and in a large number of multiple-study articles published in Psychological Science. In conclusion, the replicability index is the most powerful test for the presence of selection bias and it should be routinely used in meta-analyses to ensure that effect sizes estimates are not inflated by selective publishing of significant results. As the use of questionable practices is no longer acceptable, the R-Index can be used by editors to triage manuscripts with questionable results or to ask for a new, pre-registered, well-powered additional study. The R-Index can also be used in tenure and promotion evaluations to reward researchers that publish credible results that are likely to replicate.

References

Francis, G. (2013). Replication, statistical consistency, and publication bias. Journal of Mathematical Psychology, 57, 153–169. https://doi.org/10.1016/j.jmp.2013.02.003

Ioannidis, J. P. A., & Trikalinos, T. A. (2007). An exploratory test for an excess of significant findings. Clinical Trials: Journal of the Society for Clinical Trials, 4, 245–253. https://doi.org/10.1177/1740774507079441

R. J. Light; D. B. Pillemer (1984). Summing up: The Science of Reviewing Research. Cambridge, Massachusetts: Harvard University Press.

Renkewitz, F., & Keiner, M. (2019). How to Detect Publication Bias in Psychological Research
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Baby Einstein: The Numbers Do Not Add Up

December 31, 2019Incredibility Index, Meta-Analysis, Publication BiasUlrich Schimmack

A small literature suggests that babies can add and subtract. Wynn (1992) showed 5-month olds a Mickey Mouse doll, covered this toy, and placed another doll behind the cover to imply addition (1 + 1 = 2). A second group of infants saw two Mickey Mouse dolls, that were covered and then one Mickey Mouse was removed (2 – 1 = 1). When the cover was removed, either 1 or 2 Mickeys were visible. Infants looked longer at the incongruent display, suggesting that they expected 2 Mickeys in the addition scenario and one Mickey in the subtraction scenario.

Both studies produced just significant results; Study 1, t(30) = 2.078, p = .046 (two-tailed), Study 2 , t(14) = 1.795, p = .047 (one-tailed). Post-2011, these just significant results raise a red flag about the replicability of these results.

This study produced a small literature that was meta-analyzed by Christodoulou, Lac, and Moore (2017). The headline finding was that a random-effects meta-analysis showed a significant effect, d = .34, “suggesting that the phenomenon Wynn originally reported is reliable.”

The problem with effect-size meta-analysis is that effect sizes are inflated when published results are selected for significance. Christodoulou et al. (2017) examined the presence of publication bias using a variety of statistical tests that produced inconsistent results. The Incredibility Index showed that there were just as many significant results (k = 12) as one would predict based on median observed power (k = 11). Trim-and-fill suggested some bias, but the corrected effect size estimate would still be significant, d = .24. However, PEESE showed significant evidence of publication bias, and no significant effect after correcting for bias.

Christodoulou et al. (2017) dismiss the results obtained with PEESE that would suggest the findings are not robust.

For instance, the PET-PEESE has been criticized on grounds that it severely penalizes samples with a small N (Cunningham & Baumeister, 2016), is inappropriate for syntheses involving a limited number of studies (Cunningham & Baumeister, 2016), is sometimes inferior in performance compared to estimation methods that do not correct for publication bias (Reed, Florax, & Poot, 2015), and is premised on acceptance of the assumption that large sample sizes confer unbiased effect size estimates (Inzlicht, Gervais, & Berkman, 2015). Each of the other four tests used have been criticized on various grounds as well (e.g., Cunningham & Baumeister, 2016)

These arguments are not very convincing. Studies with larger samples produce more robust results than studies with smaller samples. Thus, placing a greater emphasizes on larger samples is justified by the smaller sampling error in these studies. In fact, random effects meta-analysis gives too much weight to small samples. It is also noteworthy that Baumeister and Inzlicht are not unbiased statisticians. Their work has been criticized as unreliable using PEESE and their responses are at least partially motivated by defending their work.

I will demonstrate that the PEETSE results are credible and that the other methods failed to reveal publication bias because effect-size meta-analyses fail to reveal the selection bias in original articles. For example, Wynn’s (1992) seminal finding was only significant with a one-sided test. However, the meta-analysis used a two-sided p-value of .055, which was coded as a non-significant result. This is a coding mistake because the result was used to reject the null-hypothesis with a different alpha level. A follow-up study by McCrink and Wynn (2004) reported a significant interaction effect with opposite effects for addition and subtraction, p = .016. However, the meta-analysis coded addition and subtraction separately, which produced one significant, p = .01, and one non-significant result, p = .504. The coding by subgroups is common in meta-analysis to conduct moderator analyses. However, this practices mutes the selection bias, which makes it more difficult to detect selection bias. Thus, bias tests need to be applied to the focal tests that supported authors’ main conclusions.

I recoded all 12 articles that reported 14 independent tests of the hypothesis that babies can add and subtract. I found only two articles that reported a failure to reject the null-hypothesis. Wakeley,Rivera, and Langer’s (2000) article is a rare example of an article in a major journal that reported a series of failed replication studies before 2011. “Unlike Wynn, we found no systematic evidence of either imprecise or precise adding and subtracting in young infants” (p. 1525). Moore and Cocas (2006) published two studies. Study 2 reported a non-significant result with an effect in the opposite direction. They clearly stated that this result failed to replicate Wynn’s results. “This test failed to reveal a reliable difference between the two groups’ fixation preferences, t(87) = -1.31, p = .09” However, they continued to examine the data with an Analysis of Variance that produced a significant four-way interaction, F(1, 85) = 4.80, p = .031. If this result had been used as the focal test, there would be only 2 non-significant results. However, I coded the study as reporting a non-significant result. Thus, the success rate across 14 studies in 12 articles is 11/14 = 78.6%. Without Wakeley et al.’s exceptional report of replication failures, the success rate would have been 93%, which is the norm in psychology publications (Sterling, 1959; Sterling et al., 1995).

The mean observed power of the 14 studies was MOP = 57%. The binomial probabilty of obtaining 11 or more significant results in 14 studies with 57% power is p = .080. This shows significant bias with the typical alpha level of .10 for bias tests due to the low power of these tests in small samples.

I also developed a more powerful bias tests that corrects for the inflation in the estimate of observed mean power that is based on the replicability index (Schimmack, 2016). Simulation studies show that this method has higher power, while maintaining good type-I error rates. To correct for inflation, I subtract the difference between the success rate and observed mean power from the observed mean power (simulation studies show that the mean is superior to the median that was used in the 2016 manuscript). This yields a value of .57 – (.79 – .57) = .35. The binomial probability of obtaining 11 out of 14 significant results with just 35% power is p = .001. These results confirm the results obtained with PEESE that publication bias contributes to the evidence in favor of babies’ math abilities.

To examine the credibilty of the published literature, I submitted the 11 significant results to a z-curve analysis (Brunner & Schimmack, 2019). The z-curve analysis also confirms the presence of publication bias. Whereas the observed discovery rate is 79%, 95%CI = 57% to 100%, the expected discovery rate is only 6%, 95%CI = 5% to 31%. As the confidence intervals do not overlap, the difference is statistically significant. The expected replication rate is 15%. Thus, if the 11 studies could be replicated exactly only 2 rather than 11 are expected to be significant again. The 95%CI included a value of 5% which means that all studies could be false positives. This shows that the published studies do not provide empirical evidence to reject the null-hypothesis that babies cannot add or subtract.

Meta-analyses also have another drawback. They focus on results that are common across studies. However, subsequent studies are not mere replication studies. Several studies in this literature examined whether the effect is an artifact of the experimental procedure and showed that performance is altered by changing the experimental setup. These studies first replicate the original finding and then show that the effect can be attributed to other factors. Given the low power to replicate the effect, it is not clear how credible this evidence is. However, it does show that even if the effect were robust, it does not warrant the conclusion that infants can do math.

Conclusion

The problems with bias tests in standard meta-analysis are by no means unique to this article. It is well known that original articles publish nearly exclusively confirmatory evidence with success rates over 90%. However, meta-analyses often include a much larger number of non-significant results. This paradox is explained by the coding of original studies that produces non-significant results that were either not published or not the focus of an original article. This coding practices mutes the signal and makes it difficult to detect publication bias. This does not mean that the bias has disappeared. Thus, most published meta-analysis are useless because effect sizes are inflated to an unknown degree by selection for significance in the primary literature.

The Black Box of Meta-Analysis: Personality Change

August 11, 2019Meta-Analysis, Personality, Personality Change, personality stabilityUlrich Schimmack

Psychologists treat meta-analyses as the gold standard to answer empirical questions. The idea is that meta-analyses combine all of the relevant information into a single number that reveals the answer to an empirical question. The problem with this naive interpretation of meta-analyses is that meta-analyses cannot provide more information than the original studies contained. If original studies have major limitations, a meta-analytic integration does not make these limitations disappear. Meta-analyses can only reduce random sampling error, but they cannot fix problems of original studies. However, once a meta-analysis is published, the problems are often ignored and the preliminary conclusion is treated as an ultimate truth.

In this regard meta-analyses are like collateralized debt obligations that were popular until problems with CDOs triggered the financial crisis in 2008. A collateralized debt obligation (CDO) pools together cash flow-generating assets and repackages this asset pool into discrete tranches that can be sold to investors. The problem is when a CDO is considered to be less risky than the actual debt in the CDO actually is and investors believe they get high returns with low risks, when the actual debt is much more risky than investors believe.

In psychology, the review process and publication in a top journal give the appeal that information is trustworthy and can be cited as solid evidence. However, a closer inspection of the original studies might reveal that the results of a meta-analysis rest on shaky foundations.

Roberts et al. (2006) published a highly-cited meta-analysis in the prestigious journal Psychological Bulletin. The key finding of this meta-analysis was that personality levels change with age in longitudinal studies of personality.

The strongest change was observed for conscientiousness. According to the figure, conscientiousness doesn’t change much during adolescence, when the prefrontal cortex is still developing, but increases from d ~ .4 to d ~ .9 from age 30 to age 70 by about half a standard deviation.

Like many other consumers, I bought the main finding and used the results in my Introduction to Personality lectures without carefully checking the meta-anlysis. However, when I analyzed new data from longitudinal studies with large national representative samples, I could not find the predicted pattern (Schimmack, 2019a, 2019b, 2019c). Thus, I decided to take a closer look at the meta-analysis.

Roberts and colleagues list all the studies that were used with information about sample sizes, personality dimensions, and the ages that were studied. Thus, it is easy to find the studies that examined conscientiousness with participants who were 30 years or older at the start of the study.

Study	N	Weight	Start1	Max.Interval		ES
Costa et al. (2000)	2274	0.44	41	9	9	0.00
Costa et al. (1980)	433	0.08	36	6	44	0.00
Costa & McCrae (1988)	398	0.08	35	6	46	NA
Labouvie-Vief & Jain (2002)	300	0.06	39	6	39	NA
Branje et al. (2004)	285	0.06	42	2	4	NA
Small et al. (2003)	223	0.04	68	6	6	NA
P. Martin (2002)	179	0.03	65	5	46	0.10
Costa & McCrae (1992)	175	0.03	53	7	7	0.06
Cramer (2003)	155	0.03	33	14	14	NA
Haan, Millsap, & Hartka (1986)	118	0.02	33	10	10	NA
Helson & Kwan (2000)	106	0.02	33	42	47	NA
Helson & Wink (1992)	101	0.02	43	9	9	0.20
Grigoriadis & Fekken (1992)	89	0.02	30	3	3
Roberts et al. (2002)	78	0.02	43	9	9
Dudek & Hall (1991)	70	0.01	49	25	25
Mclamed et al. (1974)	62	0.01	36	3	3
Cartwright & Wink (1994)	40	0.01	31	15	15
Weinryb et al. (1992)	37	0.01	39	2	2
Wink & Helson (1993)	21	0.00	31	25	25
Total N / Average	5144	1.00	41	11	19

There are 19 studies with a total sample size of N = 5,144 participants. However, sample sizes vary dramatically across studies from a low of N = 21 to a high of N = 2,274. Table 1 shows the proportion of participants that would be used to weight effect sizes according to sample sizes. By far the largest study found no significant increase in conscientiousness. I tried to find information about effect sizes from the other studies, but the published articles didn’t contain means or the information was from an unpublished source. I did not bother to obtain information from samples with less than 100 participants, because they contribute only 8% to the total sample size. Even big effects would be washed out by the larger samples.

The main conclusion that can be drawn from this information is that there is no reliable information to make claims about personality change throughout adulthood. If we assume that conscientiousness changes by half a standard deviation over a 40 year period, the average effect size for a decade is d = .12. For studies with even shorter retest intervals, the predicted effect size is even weaker. It is therefore highly speculative to extrapolate from this patchwork of data and make claims about personality change during adulthood.

Fortunately, much better information is now available from longitudinal panels with over thousand participants who have been followed for 12 (SOEP) or 20 (MIDUS) years with three or four retests. Theories of personality stability and change need to be revisited in the light of this new evidence. Updating theories in the face of new data is at the basis of science. Citing an outdated meta-analysis as if it provided a timeless answer to a question is not.

The Hierarchy of Consistency Revisited

July 25, 2019Hierarchy of Consistency, https://doi.org/10.1016/0191-8869(84)90133-8, James J. Conley, Meta-Analysis, personality stabilityUlrich Schimmack

In 1984, James J. Conley published one of the most interesting studies of personality stability. However, this important article was published in Personality and Individual Differences and has been ignored. Even today, the article has only 184 citations in WebofScience. In contrast, the more recent meta-analyses of personality stability by Roberts and DelVeccio (2001) has 1,446 citations.

Sometimes more recent and more citations doesn’t mean better. The biggest problem in studies of stability is that random and occasion specific measurement error attenuates observed retest correlations. Thus, observed retest correlations are prone to underestimate the true stability of personality traits. With a single retest-correlation it is impossible to separate measurement error from real change. However, when more then two repeated measurements are observed, it is possible to separate random measurement error from true change, using a statistical approach that was developed by Heise (1969).

The basic idea of Heise’s model is that change accumulates over time. Thus, if traits change from T1 to T2 and from T2 to T3, the trait changed even more from T1 to T3.

Without going into mathematical details, the observed retest correlation from T1 to T3 should match the product of the retest correlations from T1 to T2 and T2 to T3.

For example, if r12 = .8 and r 23 = .8, r13 should be .8 * .8 = .64.

The same is also true if the retest correlations are not identical. Maybe more change occurred from T1 to T2 than from T2 to T3. The total stability is still a function of the product of the two partial stabilities. For example, r12 = .8 and r23 = .5 yields r13 = .8 * .5 = .4.

However, if there is random measurement error, the r13 correlation will be larger than the product of the r12 and r23 correlations. For example, using the above example and a reliability of .8, we get r12 = .8 * .8 = .64, r23 = .4 * .8 = .32 and the product is .64 * .32 = .20, while the actual r13 correlation is .32 * .8 = .256. Assuming that reliability is constant, we have three equations with three unknowns and it is possible to solve the equations to estimate reliability.

(1) r12 = rel*s1; s1 = r12/rel
(2) r23 = rel*s2; s2 = r23/rel
(3) r13 = rel*s1*s2, rel = r13/(s1*s2)

r = (r12*r23)/r13

with r12 = .64, r23 = .32, and r13 = .256, we get (.64*.32)/.256 = .8.

Heise’s model is called an autoregressive model which implies that over time, retest correlations will become smaller and smaller until they approach zero. However, if stability is high, this can take a long time. For example, Conley (1984) estimated that the annual stability of IQ tests is r = .99. With this high stability, the retest correlation over 40 years is still r = .67. Consistent with Conley’s prediction a study found that the retest correlation from age 11 to age 70 of r = .67 (ref), which is even higher than predicted by Conley.

The Figure below shows Conley’s estimate for personality traits like extraversion and neuroticism. The figure shows that reliability varies across studies and instruments from as low as .4 to as high as .9. After correcting for unreliability, the estimated annual stability of personality traits is s = .98.

The figure also shows that most studies in this meta-analysis of retest correlations covered short time-intervals from a few month up to 10 years. Studies with 10 or more years are rare. As a result, Conley’s estimates are not very precise.

To test Conley’s predictions, I used the three waves of the Midlife in the US study (MIDUS). Each wave was approximately 10 years apart with a total time span of 20 years. To analyze the data, I fitted a measurement model to the personality items in the MIDUS. The fit of the measurement model has been examined elsewhere (Schimmack, 2019). The measurement model was constrained for all three waves (see OSF for syntax). The model had acceptable overall fit, CFI = .963, RMSEA = .018, SRMR = .035 (see OSF for output).

The key finding are the retest correlations r12, r23, and r13 for the Big Five and two method factors; a factor for evaluative bias (halo) and acquiescence bias.

	r12	r23	r13
Neuroticism	0.78	0.80	0.72
Extraversion	0.87	0.90	0.85
Openness	0.80	0.84	0.75
Agreeableness	0.79	0.81	0.75
Conscientiousness	0.78	0.83	0.77
Halo	0.72	0.76	0.68
Aquiescence	0.53	0.53	0.54

For all traits except acquiescence bias, the r13 correlation is lower than the r12 or r23 correlation, indicating some real change. However, for all traits, the r13 correlation is higher than the product of r12 and r23, indicating the presence of random measurement error or occasion specific variance.

The next table shows the decomposition of the retest-correlations into a reliability component and a stability component.

	Reliable	20Y Stability	1Y Stability
Neuroticism	0.87	0.83	0.99
Extraversion	0.92	0.92	1.00
Openness	0.90	0.84	0.99
Agreeableness	0.85	0.88	0.99
Conscientiousness	0.84	0.92	1.00
Halo	0.80	0.85	0.99
Aquiescence	0.52	1.04	1.00

The reliability estimates range from .84 to .92 for the Big Five scales. Reliability of the method factor is estimated to be lower. After correcting for unreliability, 20-year stability estimates increase from observed levels of .72 to .85 to estimated levels of .83 to .1. The implied annual stability estimates are above .99, which is higher than Conley’s estimate of .98.

Unfortunately, three time points are not enough to test the assumptions of Heise’s model. Maybe reliability increases over time. Another possibility is that some of the variance in personality is influenced by stable factors that never change (e.g., genetic variance). In this case, retest correlations do not approach zero, but to a level that is set by the influence of stable factors.

Anusic and Schimmack’s meta-analysis suggested that for the oldest age group, the amount of stable variance is 80, and that this asymptote is reached very quickly (see picture). However, this model predicts that 10-year retest correlations are equivalent to 20-year retest correlations, which is not consistent with the results in Table 1. Thus, the MIDUS data suggest that the model in Figure 1 overestimates the amount of stable trait variance in personality. More data are needed to model the contribution of stable factors to stability of personality traits. However, both models predict high stability of personality over a long period of 20 years.

Conclusion

Science can be hard. Astronomy required telescopes to study the universe. Psychologists need longitudinal studies to examine stability of personality and personality development. The first telescopes were imperfect and led to false beliefs about canals and life on Mars. Similarly, longitudinal data are messy and provide imperfect glimpses into the stability of personality. However, the accumulating evidence shows impressive stability in personality differences. Many psychologists are dismayed by this finding because they have a fixation on disorders and negative traits. However, the Big Five traits are not disorders or undesirable traits. They are part of human diversity. When it comes to normal diversity, stability is actually desirable. Imagine you train for a job and after ten years of training you don’t like it anymore. Imagine you marry a quiet introvert and five year later, he is a wild party animal. Imagine, you never know who you are because your personality is constantly changing. The grass on the other side of the fence is often greener, but self-acceptance and building on one’s true strength may be a better way to live a happy life than to try to change your personality to fit cultural norms or parental expectations. Maybe stability and predictability aren’t so bad after all.

The results also have implications for research on personality change and development. If natural variation in factors that influence personality produces only very small changes over periods of a few years, it will be difficult to study personality change. Moreover, small real changes will be contaminated with relatively large amounts of random measurement error. Good measurement models that can separate real change from noise are needed to do so.

References

Conley, J. J. (1984). The hierarchy of consistency: A review and model of longitudinal findings on adult individual differences in intelligence, personality and self-opinion. Personality and Individual Differences, 84, 11-25.

Heise D. R. (1969) Separating reliability and stability in test-retest correlation. Am. social. Rev. 34, 93-101.

Roberts, B. W., & DelVecchio, W. F. (2000). The rank-order consistency of personality traits from childhood to old age: A quantitative review of longitudinal studies. Psychological Bulletin, 126, 3–25.

Where Do Non-Significant Results in Meta-Analysis Come From?

June 4, 2019Meta-Analysis, Publication BiasUlrich Schimmack

It is well known that focal hypothesis tests in psychology journals nearly always reject the null-hypothesis (Sterling, 1959; Sterling et al., 1995). However, meta-analyses often contain a fairly large number of non-significant results. To my knowledge, the emergence of non-significant results in meta-analysis has not been examined systematically (happy to be proven wrong). Here I used the extremely well-done meta-analysis of money priming studies to explore this issue (Lodder, Ong, Grasman, & Wicherts, 2019).

I downloaded their data and computed z-scores by (1) dividing Cohen’s d by sampling errror (2/sqrt(N)) to compute t-values, (2) convert the absolute t-values into two-sided p-values, and (3) converting the p-values into absolute z-scores. The z-scores were submitted to a z-curve analysis (Brunner & Schimmack, 2019).

The first figure shows the z-curve for all test-statistics. Out of 282 tests, only 116 (41%) are significant. This finding is surprising, given the typical discovery rates over 90% in psychology journals. The figure also shows that the observed discovery rate of 41% is higher than the expected discovery rate of 29%, although the difference is relatively small and the confidence intervals overlap. This might suggest that publication bias in the money priming literature is not a serious problem. On the other hand, meta-analysis may mask the presence of publication bias in the published literature for a number of reasons.

Published vs. Unpublished Studies

Publication bias implies that studies with non-significant results end up in the proverbial file-drawer. Meta-analysts try to correct for publication bias by soliciting unpublished studies. The money-priming meta-analysis included 113 unpublished studies.

Figure 2 shows the z-curve for these studies. The observed discovery rate is slightly lower than for the full set of studies, 29%, and more consistent with the expected discovery rate, 25%. Thus, there this set of studies appears to be unbiased.

The complementary finding for published studies (Figure 3) is that the observed discovery rate increases, 49%, while the expected discovery rate remains low, 31%. Thus, published articles report a higher percentage of significant results without more statistical power to produce significant results.

A New Type of Publications: Independent Replication Studies

In response to concerns about publication bias and questionable research practices, psychology journals have become more willing to publish null-results. An emerging format are pre-registered replication studies with the explicit aim of probing the credibility of published results. The money priming meta-analysis included 47 independent replication studies.

Figure 4 shows that independent replication studies had a very low observed discovery rate, 4%, that is matched by a very low expected discovery rate, 5%. It is remarkable that the discovery rate for replication studies is lower than the discovery rate for unpublished studies. One reason for this discrepancy is that significance alone is not sufficient to get published and authors may be selective in the sharing of unpublished results.

Removing independent replication studies from the set of published studies further increases the observed discovery rate, 66%. Given the low power of replication studies, the expected discovery rate also increases somewhat, but it is notably lower than the observed discovery rate, 35%. The difference is now large enough to be statistically significant, despite the rather wide confidence interval around the expected discovery rate estimate.

Coding of Interaction Effects

After a (true or false) effect has been established in the literature, follow up studies often examine boundary conditions and moderators of an effect. Evidence for moderation is typically demonstrated with interaction effects that are sometimes followed by contrast analysis for different groups. One way to code these studies would be to focus on the main effect and to ignore the moderator analysis. However, meta-analysts often split the sample and treat different subgroups as independent samples. This can produce a large number of non-significant results because a moderator analysis allows for the fact that the effect emerged only in one group. The resulting non-significant results may provide false evidence of honest reporting of results because bias tests rely on the focal moderator effect to examine publication bias.

The next figure is based on studies that involved an interaction hypothesis. The observed discovery rate, 42%, is slightly higher than the expected discovery rate, 25%, but bias is relatively mild and interaction effects contribute 34 non-significant results to the meta-analysis.

The analysis of the published main effect shows a dramatically different pattern. The observed discovery rate increased to 56/67 = 84%, while the expected discovery rate remained low with 27%. The 95%CI do not overlap, demonstrating that the large file-drawer of missing studies is not just a chance finding.

I also examined more closely the 7 non-significant results in this set of studies.

Gino and Mogliner (2014) reported results of a money priming study with cheating as the dependent variable. There were 98 participants in 3 conditions. Results were analyzed with percentage of cheating participants and extent of cheating. The percentage of cheating participants produced a significant contrast of the money priming and control condition, chi2(1, N = 65) = 3.97. However, the meta-analysis used the extent of cheating dependent variable, which should only a marginally significant effect with a one-tailed p-value of .07. “Simple contrasts revealed that participants cheated more in the money condition (M = 4.41, SD = 4.25) than in both the control condition (M = 2.76, SD = 3.96; p = .07) and the time condition (M = 1.55, SD = 2.41; p = .002).” Thus, this non-significant results was presented as supporting evidence in the original article.
Jin, Z., Shiomura, K., & Jiang, L. (2015) conducted a priming studies with reaction times as dependent variables. This design is different from social priming studies in the meta-analysis. Moreover, money priming effects were examined within-participants, and the study produced several significant complex interaction effects. Thus, this study also does not count as a published failure to replicate money priming effects.
Mukherjee, S., Nargundkar, M., & Manjaly, J. A. (2014) examined the influence of money primes on various satisfaction judgments. Study 1 used a small sample of N = 48 participants with three dependent variables. Two achieved significance, but the meta-analysis aggregated across DVs, which resulted in a non-significant outcome. Study 2 used a larger sample and replicated significance for two outcomes. It was not included in the meta-analysis. In this case, aggregation of DVs explains a non-significant result in the meta-analysis, while the original article reported significant results.
I was unable to retrieve this article, but the abstract suggests that the article reports a significant interaction. ” We found that although money-primed reactance in control trials in which the majority provided correct responses, this effect vanished in critical trials in which the majority provided incorrect answers.”
[https://www.sbp-journal.com/index.php/sbp/article/view/3227]
Wierzbicki, J., & Zawadzka, A. (2014) published two studies. Study 1 reported a significant result. Study 2 added a non-significant result to the meta-analysis. Although the effect for money priming was not significant, this study reported a significant effect for credit-card priming and a money priming x morality interaction effect. Thus, the article also did not report a money-priming failure as the key finding.
Gasiorowska, A. (2013) is an article in Polish.
is a duplication of article 5

In conclusion, none of the 7 studies with non-significant results in the meta-analysis that were published in a journal reported that money priming had no effect on a dependent variable. All articles reported some significant results as the key finding. This further confirms how dramatically publication bias distorts the evidence reported in psychology journals.

Conclusion

In this blog post, I examined the discrepancy between null-results in journal articles and in meta-analysis, using a meta-analysis of money priming. While the meta-analysis suggested that publication bias is relatively modest, published articles showed clear evidence of publication bias with an observed discovery rate of 89%, while the expected discovery rate was only 27%.

Three factors contributed to this discrepancy: (a) the inclusion of unpublished studies, (b) independent replication studies, and (c) the coding of interaction effects as separate effects for subgroups rather than coding the main effect.

After correcting for publication bias, expected discovery rates are consistently low with estimates around 30%. The main exception are the independent replication studies that found no evidence at all. Overall, these results confirm that published money priming studies and other social priming studies cannot be trusted because the published studies overestimate replicability and effect sizes.

It is not the aim of this blog post to examine whether some money priming paradigms can produce replicable effects. The main goal was to explain why publication bias in meta-analysis is often small, when publication bias in the published literature is large. The results show that several factors contribute to this discrepancy and that the inclusion of unpublished studies, independent replication studies, and coding of effects explain most of these discrepancies.

The (lacking) predictive validity of the race IAT

February 6, 2019Discriminant Validity, Implicit Association Test, Meta-Analysis, Predictive Validity, race IAT, ValidityUlrich Schimmack

Good science requires valid measures. This statement is hardly controversial. Not surprisingly, all authors of some psychological measure claim that their measure is valid. However, validation research is expensive and difficult to publish in prestigious journals. As a result, psychological science has a validity crisis. Many measures are used in hundreds of articles without clear definitions of constructs and without quantitative information about their validity (Schimmack, 2010).

The Implicit Association Test (AT) is no exception. The IAT was introduced in 1998 with strong and highly replicable evidence that average attitudes towards objects pairs (e.g., flowers vs. spiders) can be measured with reaction times in a classification task (Greenwald et al., 1998). Although the title of the article promised a measure of individual differences, the main evidence in the article were mean differences between groups. Thus, the original article provided little evidence that the IAT is a valid measure of individual differences.

The use of the IAT as a measure of individual differences in attitudes requires scientific evidence that tests scores are linked to variation in attitudes. Key evidence for the validity of a test are reliability, convergent validity, discriminant validity, and incremental predictive validity (Campbell & Fiske, 1959).

The validity of the IAT as a measure of attitudes has to be examined on a case by case basis because the link between associations and attitudes can vary depending on the attitude object. For attitude objects like pop drinks, Coke vs. Pepsi, associations may be strongly related to attitudes. In fact, the IAT has good predictive validity for choices between two pop drinks (Hofmann, Gawronski, Gschwendner, & Schmitt, 2005). However, it lacks convergent validity when it is used to measure self-esteem (Bosson & Swan, & Pennebaker, 2000).

The IAT is best known as a measure of prejudice, racial bias, or attitudes of White Americans towards African Americans. On the one hand, the inventor of the IAT, Greenwald, argues that the race IAT has predictive validity (Greenwald et al., 2009). Others take issue with the evidence: “Implicit Association Test scores did not permit prediction of individual-level behaviors” (Blanton et al., 2009, p. 567); “the IAT provides little insight into who will discriminate against whom, and provides no more insight than explicit measures of bias” (Oswald et al., 2013).

Nine years later, Greenwald and colleagues present a new meta-analysis of predictive validity of the IAT (Kurdi et al., 2018) based on 217 research reports and a total sample size of N = 36,071 participants. The results of this meta-analysis are reported in the abstract.

We found significant implicit– criterion correlations (ICCs) and explicit– criterion correlations (ECCs), with unique contributions of implicit (beta = .14) and explicit measures (beta = .11) revealed by structural equation modeling.

The problem with meta-analyses is that they aggregate information with diverse methods, measures, and criterion variables, and the meta-analysis showed high variability in predictive validity. Thus, the headline finding does not provide information about the predictive validity of the race IAT. As noted by the authors, “Statistically, the high degree of heterogeneity suggests that any single point estimate of the implicit– criterion relationship would be misleading” (p. 7).

Another problem of meta-analysis is that it is difficult to find reliable moderator variables if original studies have small samples and large sampling error. As a result, a non-significant moderator effect cannot be interpreted as evidence that results are homogeneous. Thus, a better way to examine the predictive validity of the race IAT is to limit the meta-analysis to studies that used the race IAT.

Another problem of small studies is that they introduce a lot of noise because point estimates are biased by sampling error. Stanley, Jarrell, and Doucouliagos (2010) made the ingenious suggestion to limit meta-analysis to the top 10% of studies with the largest sample sizes. As these studies have small sampling error to begin with, aggregating them will produce estimates with even smaller sampling error and inclusion of many small studies with high heterogeneity is not necessary. A smaller number of studies also makes it easier to evaluate the quality of studies and to examine sources of heterogeneity across studies. I used this approach to examine the predictive validity of the race IAT using the studies included in Kurdi et al.’s (2018) meta-analysis (data).

Description of the Data

The datafile contained the variable groupStemCat2 that coded the groups compared in the IAT. Only studies classified as groupStemCat2 == “African American and Africans” were selected, leaving 1328 entries (rows). Next, I selected only studies with an IAT-criterion correlation, leaving 1004 entries. Next, I selected only entries with a minimum sample size of N = 100, leaving 235 entries (more than 10%).

The 235 entries were based on 21 studies, indicating that the meta-analysis coded, on average, more than 10 different effects for each study.

The median IAT-criterion correlation across all 235 studies was r = .070. In comparison, the median r for the 769 studies with N < 100 was r = .044. Thus, selecting for studies with large N did not reduce the effect size estimate.

When I first computed the median for each study and then the median across studies, I obtained a similar median correlation of r = .065. There was no significant correlation between sample size and median ICC-criterion correlation across the 21 studies, r = .12. Thus, there is no evidence of publication bias.

I now review the 21 studies in decreasing order of the median IAT-criterion correlation. I evaluate the quality of the studies with 1 to 5 stars ranging from lowest to highest quality. As some studies were not intended to be validation studies, this evaluation does not reflect the quality of a study per se. The evaluation is based on the ability of a study to validate the IAT as a measure of racial bias.

1. * Ma et al. (Study 2), N = 303, r = .34

Ma et al. (2012) used several IATs to predict voting intentions in the 2012 US presidential election. Importantly, Study 2 did not include the race IAT that was used in Study 1 (#15, median r = .03). Instead, the race IAT was modified to include pictures of the two candidates Obama and Romney. Although it is interesting that an IAT that requires race classifications of candidates predicted voting intentions, this study cannot be used to claim that the race IAT as a measure of racial bias has predictive validity because the IAT measures specific attitudes towards candidates rather than attitudes towards African Americans in general.

2. *** Knowles et al., N = 285, r = .26

This study used the race IAT to predict voting intentions and endorsement of Obama’s health care reforms. The main finding was that the race IAT was a significant predictor of voting intentions (Odds Ratio = .61; r = .20) and that this relationship remained significant after including the Modern Racism scale as predictor (Odds Ratio = .67, effect size r = .15). The correlation is similar to the result obtained in the next study with a larger sample.

3. ***** Greenwald et al. (2009), N = 1,057, r = .17

The most conclusive results come from Greenwald et al.’s (2009) study with the largest sample size of all studies. In a sample of N = 1,057 participants, the race IAT predicted voting intentions in the 2008 US election (Obama vs. McCain), r = .17. However, in a model that included political orientation as predictor of voting intentions, only explicit attitude measures added incremental predictive validity, b = .10, SE = .03, t = 3.98, but the IAT did not, b = .00, SE = .02, t = 0.18.

4. * Cooper et al., N = 178, r = .12

The sample size in the meta-analysis does not match the sample size of the original study. Although 269 patients were involved, the race IAT was administered to 40 primary care clinicians. Thus, predictive validity can only be assessed on a small sample of N = 40 physicians who provided independent IAT scores. Table 3 lists seven dependent variables and shows two significant results (p = .02, p = .02) for Black patients.

5. * Biernat et al. (Study 1), N = 136, r = .10

Study 1 included the race IAT and donations to a Black vs. other student organizations as the criterion variable. The negative relationship was not significant (effect size r = .05). The meta-analysis also included the shifting standard variable (effect size r = .14). Shifting standards refers to the extent to which participants shifted standards in their judgments of Black versus White targets’ academic ability. The main point of the article was that shifting standards rather than implicit attitude measures predict racial bias in actual behavior. “In three studies, the tendency to shift standards was uncorrelated with other measures of prejudice but predicted reduced allocation of funds to a Black student organization.” Thus, it seems debatable to use shifting standards as a validation criterion for the race IAT because the key criterion variable were the donations, while shifting standards were a competing indirect measure of prejudice.

6. ** Zhang et al. (Study 2), N = 196, r = .10

This study examined thought listings after participants watched a crime committed by a Black offender on Law and Order. “Across two programs, no statistically significant relations between the nature of the thoughts and the scores on IAT were found, F(2, 85) = 2.4, p < .11 for program 1, and F(2, 84) = 1.98, p < .53 for program 2.” The main limitation of this study is that thought listings are not a real social behavior. As the effect size for this study is close to the median, excluding it has no notable effect on the final result.

7. * Ashburn et al., N = 300, r = .09

The title of this article is “Race and the psychological health of African Americans.” The sample consists of 300 African American participants. Although it is interesting to examine racial attitudes of African Americans, this study does not address the question whether the race IAT is a valid measure of prejudice against African Americans.

8. *** Eno et al. (Study 1), N = 105, r = .09

This article examines responses to a movie set during the Civil Rights Era; “Remember the Titans.” After watching the movie, participants made several ratings about interpretations of events. Only one event, attributing Emma’s actions to an accident, showed a significant correlation with the IAT, r = .20, but attributions to racism also showed a correlation in the same direction, r = .10. For the other events, attributions had the same non-significant effect size, Girls interests r = .12, Girls race, r = .07; Brick racism, r = -.10, Brick Black coach’s actions, r = -.10.

9. *** Aberson & Haag, N = 153, r = .07

Abserson and Haag administered the race IAT to 153 participants and asked questions about quantity and quality of contact with African Americans. They found non-significant correlations with quantity, r = -.12 and quality, r = -.10, and a significant positive correlation with the interaction, r = .17. The positive interaction effect suggests that individuals with low contact, which implies low quality contact as well, are not different from individuals with frequent high quality contact.

10. *Hagiwara et al., N = 106, r = .07

This study is another study of Black patients and non-Black physician. The main limitation is that there were only 14 physicians and only 2 were White.

11. **** Bar-Anan & Nosek, N = 397, r = .06

This study used contact as a validation criterion. The race IAT showed a correlation of r = -.14 with group contact. , N in the range from 492-647. The Brief IAT showed practically the same relationship, r = -.13. The appendix reports that contact was more strongly correlated with the explicit measures; thermometer r = .27, preference r = .31. Using structural equation modeling, as recommended by Greenwald and colleagues, I found no evidence that the IAT has unique predictive validity in the prediction of contact when explicit measures were included as predictors, b = .03, SE = .07, t = 0.37.

12. *** Aberson & Gaffney, N = 386, median r = .05

This study related the race IAT to measures of positive and negative contact, r = .10, r = -.01, respectively. Correlations with an explicit measure were considerably stronger, r = .38, r = -.35, respectively. These results mirror the results presented above.

13. * Orey et al., N = 386, median r = .04

This study examined racial attitudes among Black respondents. Although this is an interesting question, the data cannot be used to examine the predictive validity of the race IAT as a measure of prejudice.

14. * Krieger et al., N = 708, median r = .04

This study used the race IAT with 442 Black participants and criterion measures of perceived discrimination and health. Although this is a worthwhile research topic, the results cannot be used to evaluate the validity of the race IAT as a measure of prejudice.

15. *** Ma et al. (Study 1), N = 335, median r = .03

This study used the race IAT to predict voter intentions in the 2012 presidential election. The study found no significant relationship. “However, neither category-level measures were related to intention to vote for Obama (rs ≤ .06, ps ≥ .26)” (p. 31). The meta-analysis recorded a correlation of r = .045, based on email correspondence with the authors. It is not clear why the race IAT would not predict voting intentions in 2012, when it did predict voting intentions in 2008. One possibility is that Obama was now seen as a an individual rather than as a member of a particular group so that general attitudes towards African Americans no longer influenced voting intentions. No matter what the reason is, this study does not provide evidence for the predictive validity of the race IAT.

16. **** Oliver et al., N = 105, median r = .02

This study was on online study of 543 family and internal medicine physicians. They completed the race IAT and gave treatment recommendations for a hypothetical case. Race of the patient was experimentally manipulated. The abstract states that “physicians possessed explicit and implicit racial biases, but those biases did not predict
treatment recommendations” (p. 177). The sample size in the meta-analysis is smaller because the total sample was broken down into smaller subgroups.

17. * Nosek & Hansen, N = 207, median r = .01

This study did not include a clear validation criterion. The aim was to examine the relationship between the race IAT and cultural knowledge about stereoetypes. “In seven studies (158 samples, N = 107,709), the IAT was reliably and variably related to explicit attitudes, and explicit attitudes accounted for the relationship between the IAT and cultural knowledge.” The cultural knowledge measures were used as criterion variables. A positive relation, r = .10, was obtained for the item “If given the choice, who would most employers choose to hire, a Black American or a White American? (1 definitely White to 7 definitely Black).” A negative relation, r = -.09, was obtained for the item “Who is more likely to be a target of discrimination, a Black American or a White American? (1 definitely White to 7 definitely Black).”

18. *Plant et al., N = 229, median r = .00

This article examined voting intentions in a sample of 229 students. The results are not reported in the article. The meta-analysis reported a positive r = .04 and a negative r = -.04 for two separate entries with different explicit measures, which must be a coding mistake. As voting behavior has been examined in larger and more representative samples (#3, #15), these results can be ignored.

19. *Krieger et al. (2011), N = 503, r = .00

This study recruited 504 African Americans and 501 White Americans. All participants completed the race IAT. However, the study did not include clear validation criteria. The meta-analysis used self-reported experiences of discrimination as validation criterion. However, the important question is whether the race IAT predicts behaviors of people who discriminate, not the experience of victims of discrimination.

20. *Fiedorowicz, N = 257, r = -.01

This study is a dissertation and the validation criterion was religious fundamentalism.

21. *Heider & Skowronski, N = 140, r = -.02

This study separated the measurement of prejudice with the race IAT and the measurement of the criterion variables by several weeks. The criterion was cooperative behavior in a prisoner dilemma game. The results showed that “both the IAT (b = -.21, t = -2.51, p = .013) and the Pro-Black subscore (b = .17, t = 2.10, p = .037) were significant predictors of more cooperation with the Black confederate. However, these results were false and have been corrected (see Carlsson et al., 2018, for a detailed discussion).

Heider, J. D., & Skowronski, J.J. (2011). Addendum to Heider and Skowronski (2007): Improving the predictive validity of the Implicit Association Test. North American Journal of Psychology, 13, 17-20

Discussion

In summary, a detailed examination of the race IAT studies included in the meta-analysis shows considerable heterogeneity in the quality of the studies and their ability to examine the predictive validity of the race IAT. The best study is Greenwald et al.’s (2009) study with a large sample and voting in the Obama vs. McCain race as the criterion variable. However, another voting study failed to replicate these findings in 2012. The second best study was BarAnan and Nosek’s study with intergroup contact as a validation criterion, but it failed to show incremental predictive validity of the IAT.

Studies with physicians show no clear evidence of racial bias. This could be due to the professionalism of physicians and the results should not be generalized to the general population. The remaining studies were considered unsuitable to examine predictive validity. For example, some studies with African American participants did not use the IAT to measure prejudice.

Based on this limited evidence it is impossible to draw strong conclusions about the predictive validity of the race IAT. My assessment of the evidence is rather consistent with the authors of the meta-analysis, who found that “out of the 2,240 ICCs included in this metaanalysis, there were only 24 effect sizes from 13 studies that (a) had the relationship between implicit cognition and behavior as their primary focus” (p. 13).

This confirms my observation in the introduction that psychological science has a validation crisis because researchers rarely conduct validation studies. In fact, despite all the concerns about replicability, the lack of replication studies are much more numerous than validation studies. The consequences of the validation crisis is that psychologists routinely make theoretical claims based on measures with unknown validity. As shown here, this is also true for the IAT. At present, it is impossible to make evidence-based claims about the validity of the IAT because it is unknown what the IAT measures and how well it measures what it measures.

Theoretical Confusion about Implicit Measures

The lack of theoretical understanding of the IAT is evident in Greenwald and Banaji’s (2017) recent article, where they suggest that “implicit cognition influences explicit cognition that, in turn, drives behavior” (Kurdi et al., p. 13). This model would imply that implicit measures like the IAT do not have a direct link to behavior because conscious processes ultimately determine actions. This speculative model is illustrated with Bar-Anan and Nosek’s (#11) data that showed no incremental predictive validity on contact. The model can be transformed into a causal chain by changing the bidiretional path into an assumed causal relationship between implicit and explicit attitudes.

However, it is also possible to change the model into a single factor model, that considers unique variance in implicit and explicit measures as mere method variance.

Thus, any claims about implicit bias and explicit bias is premature because the existing data are consistent with various theoretical models. To make scientific claims about implicit forms of racial bias, it would be necessary to obtain data that can distinguish empirically between single construct and dual-construct models.

Conclusion

The race IAT is 20 years old. It has been used in hundreds of articles to make empirical claims about prejudice. The confusion between measures and constructs has created a public discourse about implicit racial bias that may occur outside of awareness. However, this discourse is removed from the empirical facts. The most important finding of the recent meta-analysis is that a careful search of the literature uncovered only a handful of serious validation studies and that the results of these studies are suggestive at best. Even if future studies would provide more conclusive evidence of incremental predictive validity, this finding would be insufficient to claim that the IAT is a valid measure of implicit bias. The IAT could have incremental predictive validity even if it were just a complementary measure of consciously accessible prejudice that does not share method variance with explicit measures. A multi-method approach is needed to examine the construct validity of the IAT as a measure of implicit race bias. Such evidence simply does not exist. Greenwald and colleagues had 20 years and ample funding to conduct such validation studies, but they failed to do so. In contrast, their articles consistently confuse measures and constructs and give the impression that the IAT measures unconscious processes that are hidden from introspection (“conscious experience provides only a small window into how the mind works”, “click here to discover your hidden thoughts”).

Greenwald and Banaji are well aware that their claims matter. “Research on implicit social cognition has witnessed higher levels of attention both from the general public and from governmental and commercial entities, making regular reporting of what is known an added responsibility” (Kurdi et al., 2018, p. 3). I concur. However, I do not believe that their meta-analysis fulfills this promise. An unbiased assessment of the evidence shows no compelling evidence that the race IAT is a valid measure of implicit racial bias; and without a valid measure of implicit racial bias it is impossible to make scientific statements about implicit racial bias. I think the general public deserves to know this. Unfortunately, there is no need for scientific evidence that prejudice and discrimination still exists. Ideally, psychologists will spend more effort in developing valid measures of racism that can provide trustworthy information about variation across individuals, geographic regions, groups, and time. Many people believe that psychologists are already doing it, but this review of the literature shows that this is not the case. It is high time to actually do what the general public expects from us.