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.
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).
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.
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.
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.
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.
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.
|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.
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.
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