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Replicability of Research in Frontiers of Psychology

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Summary

The z-curve analysis of results in this journal shows (a) that many published results are based on studies with low to modest power, (b) selection for significance inflates effect size estimates and the discovery rate of reported results, and (c) there is no evidence that research practices have changed over the past decade. Readers should be careful when they interpret results and recognize that reported effect sizes are likely to overestimate real effect sizes, and that replication studies with the same sample size may fail to produce a significant result again. To avoid misleading inferences, I suggest using alpha = .005 as a criterion for valid rejections of the null-hypothesis. Using this criterion, the risk of a false positive result is below 2%. I also recommend computing a 99% confidence interval rather than the traditional 95% confidence interval for the interpretation of effect size estimates.

Given the low power of many studies, readers also need to avoid the fallacy to report non-significant results as evidence for the absence of an effect. With 50% power, the results can easily switch in a replication study so that a significant result becomes non-significant and a non-significant result becomes significant. However, selection for significance will make it more likely that significant results become non-significant than observing a change in the opposite direction.

The average power of studies in a heterogeneous journal like Frontiers of Psychology provides only circumstantial evidence for the evaluation of results. When other information is available (e.g., z-curve analysis of a discipline, author, or topic, it may be more appropriate to use this information).

Report

Frontiers of Psychology was created in 2010 as a new online-only journal for psychology. It covers many different areas of psychology, although some areas have specialized Frontiers journals like Frontiers in Behavioral Neuroscience.

The business model of Frontiers journals relies on publishing fees of authors, while published articles are freely available to readers.

The number of articles in Frontiers of Psychology has increased quickly from 131 articles in 2010 to 8,072 articles in 2022 (source Web of Science). With over 8,000 published articles Frontiers of Psychology is an important outlet for psychological researchers to publish their work. Many specialized, print-journals publish fewer than 100 articles a year. Thus, Frontiers of Psychology offers a broad and large sample of psychological research that is equivalent to a composite of 80 or more specialized journals.

Another advantage of Frontiers of Psychology is that it has a relatively low rejection rate compared to specialized journals that have limited journal space. While high rejection rates may allow journals to prioritize exceptionally good research, articles published in Frontiers of Psychology are more likely to reflect the common research practices of psychologists.

To examine the replicability of research published in Frontiers of Psychology, I downloaded all published articles as PDF files, converted PDF files to text files, and extracted test-statistics (F, t, and z-tests) from published articles. Although this method does not capture all published results, there is no a priori reason that results reported in this format differ from other results. More importantly, changes in research practices such as higher power due to larger samples would be reflected in all statistical tests.

As Frontiers of Psychology only started shortly before the replication crisis in psychology increased awareness about the problem of low statistical power and selection for significance (publication bias), I was not able to examine replicability before 2011. I also found little evidence of changes in the years from 2010 to 2015. Therefore, I use this time period as the starting point and benchmark for future years.

Figure 1 shows a z-curve plot of results published from 2010 to 2014. All test-statistics are converted into z-scores. Z-scores greater than 1.96 (the solid red line) are statistically significant at alpha = .05 (two-sided) and typically used to claim a discovery (rejection of the null-hypothesis). Sometimes even z-scores between 1.65 (the dotted red line) and 1.96 are used to reject the null-hypothesis either as a one-sided test or as marginal significance. Using alpha = .05, the plot shows 71% significant results, which is called the observed discovery rate (ODR).

Visual inspection of the plot shows a peak of the distribution right at the significance criterion. It also shows that z-scores drop sharply on the left side of the peak when the results do not reach the criterion for significance. This wonky distribution cannot be explained with sampling error. Rather it shows a selective bias to publish significant results by means of questionable practices such as not reporting failed replication studies or inflating effect sizes by means of statistical tricks. To quantify the amount of selection bias, z-curve fits a model to the distribution of significant results and estimates the distribution of non-significant (i.e., the grey curve in the range of non-significant results). The discrepancy between the observed distribution and the expected distribution shows the file-drawer of missing non-significant results. Z-curve estimates that the reported significant results are only 31% of the estimated distribution. This is called the expected discovery rate (EDR). Thus, there are more than twice as many significant results as the statistical power of studies justifies (71% vs. 31%). Confidence intervals around these estimates show that the discrepancy is not just due to chance, but active selection for significance.

Using a formula developed by Soric (1989), it is possible to estimate the false discovery risk (FDR). That is, the probability that a significant result was obtained without a real effect (a type-I error). The estimated FDR is 12%. This may not be alarming, but the risk varies as a function of the strength of evidence (the magnitude of the z-score). Z-scores that correspond to p-values close to p =.05 have a higher false positive risk and large z-scores have a smaller false positive risk. Moreover, even true results are unlikely to replicate when significance was obtained with inflated effect sizes. The most optimistic estimate of replicability is the expected replication rate (ERR) of 69%. This estimate, however, assumes that a study can be replicated exactly, including the same sample size. Actual replication rates are often lower than the ERR and tend to fall between the EDR and ERR. Thus, the predicted replication rate is around 50%. This is slightly higher than the replication rate in the Open Science Collaboration replication of 100 studies which was 37%.

Figure 2 examines how things have changed in the next five years.

The observed discovery rate decreased slightly, but statistically significantly, from 71% to 66%. This shows that researchers reported more non-significant results. The expected discovery rate increased from 31% to 40%, but the overlapping confidence intervals imply that this is not a statistically significant increase at the alpha = .01 level. (if two 95%CI do not overlap, the difference is significant at around alpha = .01). Although smaller, the difference between the ODR of 60% and the EDR of 40% is statistically significant and shows that selection for significance continues. The ERR estimate did not change, indicating that significant results are not obtained with more power. Overall, these results show only modest improvements, suggesting that most researchers who publish in Frontiers in Psychology continue to conduct research in the same way as they did before, despite ample discussions about the need for methodological reforms such as a priori power analysis and reporting of non-significant results.

The results for 2020 show that the increase in the EDR was a statistical fluke rather than a trend. The EDR returned to the level of 2010-2015 (29% vs. 31), but the ODR remained lower than in the beginning, showing slightly more reporting of non-significant results. The size of the file drawer remains large with an ODR of 66% and an EDR of 72%.

The EDR results for 2021 look again better, but the difference to 2020 is not statistically significant. Moreover, the results in 2022 show a lower EDR that matches the EDR in the beginning.

Overall, these results show that results published in Frontiers in Psychology are selected for significance. While the observed discovery rate is in the upper 60%s, the expected discovery rate is around 35%. Thus, the ODR is nearly twice the rate of the power of studies to produce these results. Most concerning is that a decade of meta-psychological discussions about research practices has not produced any notable changes in the amount of selection bias or the power of studies to produce replicable results.

How should readers of Frontiers in Psychology articles deal with this evidence that some published results were obtained with low power and inflated effect sizes that will not replicate? One solution is to retrospectively change the significance criterion. Comparisons of the evidence in original studies and replication outcomes suggest that studies with a p-value below .005 tend to replicate at a rate of 80%, whereas studies with just significant p-values (.050 to .005) replicate at a much lower rate (Schimmack, 2022). Demanding stronger evidence also reduces the false positive risk. This is illustrated in the last figure that uses results from all years, given the lack of any time trend.

In the Figure the red solid line moved to z = 2.8; the value that corresponds to p = .005, two-sided. Using this more stringent criterion for significance, only 45% of the z-scores are significant. Another 25% were significant with alpha = .05, but are no longer significant with alpha = .005. As power decreases when alpha is set to more stringent, lower, levels, the EDR is also reduced to only 21%. Thus, there is still selection for significance. However, the more effective significance filter also selects for more studies with high power and the ERR remains at 72%, even with alpha = .005 for the replication study. If the replication study used the traditional alpha level of .05, the ERR would be even higher, which explains the finding that the actual replication rate for studies with p < .005 is about 80%.

The lower alpha also reduces the risk of false positive results, even though the EDR is reduced. The FDR is only 2%. Thus, the null-hypothesis is unlikely to be true. The caveat is that the standard null-hypothesis in psychology is the nil-hypothesis and that the population effect size might be too small to be of practical significance. Thus, readers who interpret results with p-values below .005 should also evaluate the confidence interval around the reported effect size, using the more conservative 99% confidence interval that correspondence to alpha = .005 rather than the traditional 95% confidence interval. In many cases, this confidence interval is likely to be wide and provide insufficient information about the strength of an effect.

2021 Replicability Report for the Psychology Department at New York University

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Introduction

Since 2011, it is an open secret that many published results in psychology journals do not replicate. The replicability of published results is particularly low in social psychology (Open Science Collaboration, 2015).

A key reason for low replicability is that researchers are rewarded for publishing as many articles as possible without concerns about the replicability of the published findings. This incentive structure is maintained by journal editors, review panels of granting agencies, and hiring and promotion committees at universities.

To change the incentive structure, I developed the Replicability Index, a blog that critically examined the replicability, credibility, and integrity of psychological science. In 2016, I created the first replicability rankings of psychology departments (Schimmack, 2016). Based on scientific criticisms of these methods, I have improved the selection process of articles to be used in departmental reviews.

1. I am using Web of Science to obtain lists of published articles from individual authors (Schimmack, 2022). This method minimizes the chance that articles that do not belong to an author are included in a replicability analysis. It also allows me to classify researchers into areas based on the frequency of publications in specialized journals. Currently, I cannot evaluate neuroscience research. So, the rankings are limited to cognitive, social, developmental, clinical, and applied psychologists.

2. I am using department’s websites to identify researchers that belong to the psychology department. This eliminates articles that are from other departments.

3. I am only using tenured, active professors. This eliminates emeritus professors from the evaluation of departments. I am not including assistant professors because the published results might negatively impact their chances to get tenure. Another reason is that they often do not have enough publications at their current university to produce meaningful results.

Like all empirical research, the present results rely on a number of assumptions and have some limitations. The main limitations are that
(a) only results that were found in an automatic search are included
(b) only results published in 120 journals are included (see list of journals)
(c) published significant results (p < .05) may not be a representative sample of all significant results
(d) point estimates are imprecise and can vary based on sampling error alone.

These limitations do not invalidate the results. Large difference in replicability estimates are likely to predict real differences in success rates of actual replication studies (Schimmack, 2022).

New York University

I used the department website to find core members of the psychology department. I found 13 professors and 6 associate professors. Figure 1 shows the z-curve for all 12,365 tests statistics in articles published by these 19 faculty members. I use the Figure to explain how a z-curve analysis provides information about replicability and other useful meta-statistics.

1. All test-statistics are converted into absolute z-scores as a common metric of the strength of evidence (effect size over sampling error) against the null-hypothesis (typically H0 = no effect). A z-curve plot is a histogram of absolute z-scores in the range from 0 to 6. The 1,239 (~ 10%) of z-scores greater than 6 are not shown because z-scores of this magnitude are extremely unlikely to occur when the null-hypothesis is true (particle physics uses z > 5 for significance). Although they are not shown, they are included in the meta-statistics.

2. Visual inspection of the histogram shows a steep drop in frequencies at z = 1.96 (dashed blue/red line) that corresponds to the standard criterion for statistical significance, p = .05 (two-tailed). This shows that published results are selected for significance. The dashed red/white line shows significance for p < .10, which is often used for marginal significance. There is another drop around this level of significance.

3. To quantify the amount of selection bias, z-curve fits a statistical model to the distribution of statistically significant results (z > 1.96). The grey curve shows the predicted values for the observed significant results and the unobserved non-significant results. The full grey curve is not shown to present a clear picture of the observed distribution. The statistically significant results (including z > 6) make up 20% of the total area under the grey curve. This is called the expected discovery rate because the results provide an estimate of the percentage of significant results that researchers actually obtain in their statistical analyses. In comparison, the percentage of significant results (including z > 6) includes 70% of the published results. This percentage is called the observed discovery rate, which is the rate of significant results in published journal articles. The difference between a 70% ODR and a 20% EDR provides an estimate of the extent of selection for significance. The difference of 50 percentage points is large. The upper level of the 95% confidence interval for the EDR is 28%. Thus, the discrepancy is not just random. To put this result in context, it is possible to compare it to the average for 120 psychology journals in 2010 (Schimmack, 2022). The ODR is similar (70 vs. 72%), but the EDR is a bit lower (20% vs. 28%), although the difference might be largely due to chance.

4. The EDR can be used to estimate the risk that published results are false positives (i.e., a statistically significant result when H0 is true), using Soric’s (1989) formula for the maximum false discovery rate. An EDR of 20% implies that no more than 20% of the significant results are false positives, however the upper limit of the 95%CI of the EDR, 28%, allows for 36% false positive results. Most readers are likely to agree that this is an unacceptably high risk that published results are false positives. One solution to this problem is to lower the conventional criterion for statistical significance (Benjamin et al., 2017). Figure 2 shows that alpha = .005 reduces the point estimate of the FDR to 3% with an upper limit of the 95% confidence interval of XX%. Thus, without any further information readers could use this criterion to interpret results published by NYU faculty members.

5. The estimated replication rate is based on the mean power of significant studies (Brunner & Schimmack, 2020). Under ideal condition, mean power is a predictor of the success rate in exact replication studies with the same sample sizes as the original studies. However, as NYU professor van Bavel pointed out in an article, replication studies are never exact, especially in social psychology (van Bavel et al., 2016). This implies that actual replication studies have a lower probability of producing a significant result, especially if selection for significance is large. In the worst case scenario, replication studies are not more powerful than original studies before selection for significance. Thus, the EDR provides an estimate of the worst possible success rate in actual replication studies. In the absence of further information, I have proposed to use the average of the EDR and ERR as a predictor of actual replication outcomes. With an ERR of 62% and an EDR of 20%, this implies an actual replication prediction of 41%. This is close to the actual replication rate in the Open Science Reproducibility Project (Open Science Collaboration, 2015). The prediction for results published in 120 journals in 2010 was (ERR = 67% + ERR = 28%)/ 2 = 48%. This suggests that results published by NYU faculty are slightly less replicable than the average result published in psychology journals, but the difference is relatively small and might be mostly due to chance.

6. There are two reasons for low replication rates in psychology. One possibility is that psychologists test many false hypotheses (i.e., H0 is true) and many false positive results are published. False positive results have a very low chance of replicating in actual replication studies (i.e. 5% when .05 is used to reject H0), and will lower the rate of actual replications a lot. Alternative, it is possible that psychologists tests true hypotheses (H0 is false), but with low statistical power (Cohen, 1961). It is difficult to distinguish between these two explanations because the actual rate of false positive results is unknown. However, it is possible to estimate the typical power of true hypotheses tests using Soric’s FDR. If 20% of the significant results are false positives, the power of the 80% true positives has to be (.62 – .2*.05)/.8 = 76%. This would be close to Cohen’s recommended level of 80%, but with a high level of false positive results. Alternatively, the null-hypothesis may never be really true. In this case, the ERR is an estimate of the average power to get a significant result for a true hypothesis. Thus, power is estimated to be between 62% and 76%. The main problem is that this is an average and that many studies have less power. This can be seen in Figure 1 by examining the local power estimates for different levels of z-scores. For z-scores between 2 and 2.5, the ERR is only 47%. Thus, many studies are underpowered and have a low probability of a successful replication with the same sample size even if they showed a true effect.

Area

The results in Figure 1 provide highly aggregated information about replicability of research published by NYU faculty. The following analyses examine potential moderators. First, I examined social and cognitive research. Other areas were too small to be analyzed individually.

The z-curve for the 11 social psychologists was similar to the z-curve in Figure 1 because they provided more test statistics and had a stronger influence on the overall result.

The z-curve for the 6 cognitive psychologists looks different. The EDR and ERR are higher for cognitive psychology, and the 95%CI for social and cognitive psychology do not overlap. This suggests systematic differences between the two fields. These results are consistent with other comparisons of the two fields, including actual replication outcomes (OSC, 2015). With an EDR of 44%, the false discovery risk for cognitive psychology is only 7% with an upper limit of the 95%CI at 12%. This suggests that the conventional criterion of .05 does keep the false positive risk at a reasonably low level or that an adjustment to alpha = .01 is sufficient. In sum, the results show that results published by cognitive researchers at NYU are more replicable than those published by social psychologists.

Position

Since 2015 research practices in some areas of psychology, especially social psychology, have changed to increase replicability. This would imply that research by younger researchers is more replicable than research by more senior researchers that have more publications before 2015. A generation effect would also imply that a department’s replicability increases when older faculty members retire. On the other hand, associate professors are relatively young and likely to influence the reputation of a department for a long time.

The figure above shows that most test statistics come from the (k = 13) professors. As a result, the z-curve looks similar to the z-curve for all test values in Figure 1. The results for the 6 associate professors (below) are more interesting. Although five of the six associate professors are in the social area, the z-curve results show a higher EDR and less selection bias than the plot for all social psychologists. This suggests that the department will improve when full professors in social psychology retire.

The table below shows the meta-statistics of all 19 faculty members. You can see the z-curve for each faculty member by clicking on their name.

Rank   Name ARP EDR ERR FDR
1 Jonathan B. Freeman 84 86 82 1
2 Karen E. Adolph 66 76 56 4
3 Bob Rehder 61 75 47 6
4 Marjorie Rhodes 58 68 48 6
5 David M. Amodio 53 65 40 8
6 Brian McElree 54 59 49 6
7 Todd M. Gureckis 49 75 23 17
8 Emily Balcetis 48 68 28 13
9 Eric D. Knowles 48 60 35 10
10 Jay J. van Bavel 55 66 44 7
11 Tessa V. West 46 55 37 9
12 Madeline E. Heilman 44 66 23 18
13 Catherine A. Hartley 45 70 19 23
14 John T. Jost 44 62 26 15
15 Andrei Cimpian 42 64 20 21
16 Peter M. Gollwitzer 36 54 18 25
17 Yaacov Trope 34 54 14 32
18 Gabriele Oettingen 30 46 14 32
19 Susan M. Andersen 30 47 13 35

Predicting Replication Outcomes: Prediction Markets vs. R-Index

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Conclusion

Gordon et al. (2021) conducted a meta-analysis of 103 studies that were included in prediction markets to forecast the outcome of replication studies. The results show that prediction markets can forecast replication outcomes above chance levels, but the value of this information is limited. Without actual replication studies, it remains unclear which published results can be trusted or not. Here I compare the performance of prediction markets to the R-Index and the closely related p < .005 rule. These statistical forecasts perform nearly as well as markets and are much easier to use to make sense of thousands of published articles. However, even these methods have a high failure rate. The best solution to this problem is to rely on meta-analyses of studies rather than to predict the outcome of a single study. In addition to meta-analyses, it will be necessary to conduct new studies that are conducted with high scientific integrity to provide solid empirical foundations for psychology. Claims that are not supported by bias-corrected meta-analyses or new preregistered studies are merely suggestive and currently lack empirical support.

Introduction

Since 2011, it became apparent that many published results in psychology, especially social psychology fail to replicate in direct replication studies (Open Science Collaboration, 2015). In social psychology the success rate of replication studies is so low (25%) that it makes sense to bet on replication failures. This would produce 75% successful outcomes, but it would also imply that an entire literature has to be discarded.

It is practically impossible to redo all of the published studies to assess their replicability. Thus, several projects have attempted to predict replication outcomes of individual studies. One strategy is to conduct prediction markets in which participants can earn real money by betting on replication outcomes. There have been four prediction markets with a total of 103 studies with known replication outcomes (Gordon et al., 2021). The key findings are summarized in Table 1.

Markets have a good overall success rate, (28+47)/103 = 73% that is above chance (flipping a coin). Prediction markets are better at predicting failures, 28/31 = 90%, than predicting successes, 47/72 = 65%. The modest success rate for success is a problem because it would be more valuable to be able to identify studies that will replicate and do not require a new study to verify the results.

Another strategy to predict replication outcomes relies on the fact that the p-values of original studies and the p-values of replication studies are influenced by the statistical power of a study (Brunner & Schimmack, 2020). Studies with higher power are more likely to produce lower p-values and more likely to produce significant p-values in replication studies. As a result, p-values also contain valuable information about replication outcomes. Gordon et al. (2021) used p < .005 as a rule to predict replication outcomes. Table 2 shows the performance of this simple rule.

The overall success rate of this rule is nearly as good as the prediction markets, (39+35)/103 = 72%; a difference by k = 1 studies. The rule does not predict failures as well as the markets, 39/54 = 72% (vs. 90%), but it predicts successes slightly better than the markets, 35/49 = 71% (vs. 65%).

A logistic regression analysis showed that both predictors independently contribute to the prediction of replication outcomes, market b = 2.50, se = .68, p = .0002; p < .005 rule: b = 1.44, se = .48, p = .003.

In short, p-values provide valuable information about the outcome of replication studies.

The R-Index

Although a correlation between p-values and replication outcomes follows logically from the influence of power on p-values in original and replication studies, the cut-off value of .005 appears to be arbitrary. Gordon et al. (2017) justify its choice with an article by Benjamin et al. (2017) that recommended a lower significance level (alpha) to ensure a lower false positive risk. Moreover, they advocated for this rule for new studies that preregister hypotheses and do not suffer from selection bias. In contrast, the replication crisis was caused by selection for significance which produced success rates of 90% or more in psychology journals (Motyl et al., 2017; Sterling, 1959; Sterling et al., 1995). One main reason for replication failures is that selection for significance inflates effect sizes and due to regression to the mean, effect sizes in replication studies are bound to be weaker, resulting in non-significant results, especially if the original p-value was close to the threshold value of alpha = .05. The Open Science Collaboration (2015) replicability project showed that effect sizes are on average inflated by over 100%.

The R-Index provides a theoretical rational for the choice of a cut-off value for p-values. The theoretical cutoff value happens to be p = .0084. The fact that it is close to Benjamin et al.’s (2017) value of .005 is merely a coincidence.

P-values can be transformed into estimates of the statistical power of a study. These estimates rely on the observed effect size of a study and are sometimes called observed power or post-hoc power because power is computed after the results of a study are known. Figure 1 illustrates observed power with an example of a z-test that produced a z-statistic of 2.8 which corresponds to a two-sided p-value of .005.

A p-value of .005 corresponds to z-value of 2.8 for the standard normal distribution centered over zero (the nil-hypothesis). The standard level of statistical significance, alpha = .05 (two-sided) corresponds to z-value of 1.96. Figure 1 shows the sampling distribution of studies with a non-central z-score of 2.8. The green line cuts this distribution into a smaller area of 20% below the significance level and a larger area of 80% above the significance level. Thus, the observed power is 80%.

Selection for significance implies truncating the normal distribution at the level of significance. This means the 20% of non-significant results are discarded. As a result, the median of the truncated distribution is higher than the median of the full normal distribution. The new median can be found using the truncnorm package in R.

qtruncnorm(.5,a = qnorm(1-.05/2),mean=2.8) = 3.05

This value corresponds to an observed power of

qnorm(3.05,qnorm(1-.05/2) = .86

Thus, selection for significance inflates observed power of 80% to 86%. The amount of inflation is larger when power is lower. With 20% power, the inflated power after selection for significance is 67%.

Figure 3 shows the relationship between inflated power on the x-axis and adjusted power on the y-axis. The blue curve uses the truncnorm package. The green line shows the simplified R-Index that simply substracts the amount of inflation from the inflated power. For example, if inflated power is 86%, the inflation is 1-.86 = 14% and subtracting the inflation gives an R-Index of 86-14 = 82%. This is close to the actual value of 80% that produced the inflated value of 86%.

Figure 4 shows that the R-Index is conservative (underestimates power) when power is over 50%, but is liberal (overestimates power) when power is below 50%. The two methods are identical when power is 50% and inflated power is 75%. This is a fortunate co-incidence because studies with more than 50% power are expected to replicate and studies with less than 50% power are expected to fail in a replication attempt. Thus, the simple R-Index makes the same dichotomous predictions about replication outcomes as the more sophisticated approach to find the median of the truncated normal distribution.

The inflated power for actual power of 50% is 75% and 75% power corresponds to a z-score of 2.63, which in turn corresponds to a p-value of p = .0084.

Performance of the R-Index is slightly worse than the p < .005 rule because the R-Index predicts 5 more successes, but 4 of these predictions are failures. Given the small sample size, it is not clear whether this difference is reliable.

In sum, the R-Index is based on a transformation of p-values into estimates of statistical power, while taking into account that observed power is inflated when studies are selected for significance. It provides a theoretical rational for the atheoretical p < .005 rule, because this rule roughly cuts p-values into p-values with more or less than 50% power.

Predicting Success Rates

The overall success rate across the 103 replication studies was 50/103 = 49%. This percentage cannot be generalized to a specific population of studies because the 103 are not a representative sample of studies. Only the Open Science Collaboration project used somewhat representative sampling. However, the 49% success rate can be compared to the success rates of different prediction methods. For example, prediction markets predict a success rate of 72/103 = 70%, a significant difference (Gordon et al., 2021). In contrast, the R-Index predicts a success rate of 54/103 = 52%, which is closer to the actual success rate. The p < .005 rule does even better with a predicted success rate of 49/103 = 48%.

Another method that has been developed to estimate the expected replication rate is z-curve (Bartos & Schimmack, 2021; Brunner & Schimmack, 2020). Z-curve transforms p-values into z-scores and then fits a finite mixture model to the distribution of significant p-values. Figure 5 illustrates z-curve with the p-values from the 103 replicated studies.

The z-curve estimate of the expected replication rate is 60%. This is better than the prediction market, but worse than the R-Index or the p < .005 rule. However, the 95%CI around the ERR includes the true value of 49%. Thus, sampling error alone might explain this discrepancy. However, Bartos and Schimmack (2021) discussed several other reasons why the ERR may overestimate the success rate of actual replication studies. One reason is that actual replication studies are not perfect replicas of the original studies. So called, hidden moderators may create differences between original and replication studies. In this case, selection for significance produces even more inflation that the model assumes. In the worst case scenario, a better estimate of actual replication outcomes might be the expected discovery rate (EDR), which is the power of all studies that were conducted, including non-significant studies. The EDR for the 103 studies is 28%, but the 95%CI is wide and includes the actual rate of 49%. Thus, the dataset is too small to decide between the ERR or the EDR as best estimates of actual replication outcomes. At present it is best to consider the EDR the worst possible and the ERR the best possible scenario and to expect the actual replication rate to fall within this interval.

Social Psychology

The 103 studies cover studies from experimental economics, cognitive psychology, and social psychology. Social psychology has the largest set of studies (k = 54) and the lowest success rate, 33%. The prediction markets overpredict successes, 50%. The R-Index also overpredicted successes, 46%. The p < .005 rule had the least amount of bias, 41%.

Z-curve predicted an ERR of 55% s and the actual success rate fell outside the 95% confidence interval, 34% to 74%. The EDR of 22% underestimates the success rate, but the 95%CI is wide and includes the true value, 95%CI = 5% to 70%. Once more the actual success rate is between the EDR and the ERR estimates, 22% < 34% < 55%.

In short, prediction models appear to overpredict replication outcomes in social psychology. One reason for this might be that hidden moderators make it difficult to replicate studies in social psychology which adds additional uncertainty to the outcome of replication studies.

Regarding predictions of individual studies, prediction markets achieved an overall success rate of 76%. Prediction markets were good at predicting failures, 25/27 = 93%, but not so good in predicting successes, 16/27 = 59%.

The R-Index performed as well as the prediction markets with one more prediction of a replication failure.

The p < .005 rule was the best predictor because it predicted more replication failures.

Performance could be increased by combining prediction markets and the R-Index and only bet on successes when both predictors predicted a success. In particular, the prediction of success improved to 14/19 = 74%. However, due to the small sample size it is not clear whether this is a reliable finding.

Non-Social Studies

The remaining k = 56 studies had a higher success rate, 65%. The prediction markets overpredicted success, 92%. The R-Index underpredicted successes, 59%. The p < .005 rule underpredicted successes even more.

This time z-curve made the best prediction with an ERR of 67%, 95%CI = 45% to 86%. The EDR underestimates the replication rate, although the 95%CI is very wide and includes the actual success rate, 5% to 81%. The fact that z-curve overestimated replicability for social psychology, but not for other areas, suggests that hidden moderators may contribute to the replication problems in social psychology.

For predictions of individual outcomes, prediction markets had a success rate of (3 + 31)/49 = 76%. The good performance is due to the high success rate. Simply betting on success would have produced 32/49 = 65% successes. Predictions of failures had a s success rate of 3/4 = 75% and predictions of successes had a success rate of 31/45 = 69%.

The R-Index had a lower success rate of (9 +21)/49 = 61%. The R-Index was particularly poor at predicting failures, 9/20 = 45%, but was slightly better at predicting successes than the prediction markets, 21/29 = 72%.

The p < .500 rule had a success rate equal to the R-Index, (10 + 20)/49 = 61%, with one more correctly predicted failure and one less correctly predicted success.

Discussion

The present results reproduce the key findings of Gordon et al. (2021). First, prediction markets overestimate the success of actual replication studies. Second, prediction markets have some predictive validity in forecasting the outcome of individual replication studies. Third, a simple rule based on p-values also can forecast replication outcomes.

The present results also extend Gordon et al.’s (2021) findings based on additional analyses. First, I compared the performance of prediction markets to z-curve as a method for the prediction of the success rates of replication outcomes (Bartos & Schimmack, 2021; Brunner & Schimmack, 2021). Z-curve overpredicted success rates for all studies and for social psychology, but was very accurate for the remaining studies (economics, cognition). In all three comparisons, z-curve performed better than prediction markets. Z-curve also has several additional advantages over prediction markets. First, it is much easier to code a large set of test statistics than to run prediction markets. As a result, z-curve has already been used to estimate the replication rates for social psychology based on thousands of test statistics, whereas estimates of prediction markets are based on just over 50 studies. Second, z-curve is based on sound statistical principles that link the outcomes of original studies to the outcomes of replication studies (Brunner & Schimmack, 2020). In contrast, prediction markets rest on unknown knowledge of market participants that can vary across markets. Third, z-curve estimates are provided with validated information about the uncertainty in the estimates, whereas prediction markets provide no information about uncertainty and uncertainty is large because markets tend to be small. In conclusion, z-curve is more efficient and provides better estimates of replication rates than prediction markets.

The main goal of prediction markets is to assess the credibility of individual studies. Ideally, prediction markets would help consumers of published research to distinguish between studies that produced real findings (true positives) and studies that produced false findings (false positives) without the need to run additional studies. The encouraging finding is that prediction markets have some predictive validity and can distinguish between studies that replicate and studies that do not replicate. However, to be practically useful it is necessary to assess the practical usefulness of the information that is provided by prediction markets. Here we need to distinguish the practical consequences of replication failures and successes. Within the statistical framework of nil-hypothesis significance testing, successes and failures have different consequences.

A replication failure increases uncertainty about the original finding. Thus, more research is needed to understand why the results diverged. This is also true for market predictions. Predictions that a study would fail to replicate cast doubt about the original study, but do not provide conclusive evidence that the original study reported a false positive result. Thus, further studies are needed, even if a market predicts a failure. In contrast, successes are more informative. Replicating a previous finding successfully strengthens the original findings and provides fairly strong evidence that a finding was not a false positive result. Unfortunately, the mere prediction that a finding will replicate does not provide the same reassurance because markets only have an accuracy of about 70% when they predict a successful replication. The p < .500 rule is much easier to implement, but its ability to forecast successes is also around 70%. Thus, neither markets nor a simple statistical rule are accurate enough to avoid actual replication studies.

Meta-Analysis

The main problem of prediction markets and other forecasting projects is that single studies are rarely enough to provide evidence that is strong enough to evaluate theoretical claims. It is therefore not particularly important whether one study can be replicated successfully or not, especially when direct replications are difficult or impossible. For this reason, psychologists have relied for a long time on meta-analyses of similar studies to evaluate theoretical claims.

It is surprising that prediction markets have forecasted the outcome of studies that have been replicated many times before the outcome of a new replication study was predicted. Take the replication of Schwarz, Strack, and Mai (1991) in Many Labs 2 as an example. This study manipulated the item-order of questions about marital satisfaction and life-satisfaction and suggested that a question about marital satisfaction can prime information that is used in life-satisfaction judgments. Schimmack and Oishi (2005) conducted a meta-analysis of the literature and showed that the results by Schwarz et al. (1991) were unusual and that the actual effect size is much smaller. Apparently, the market participants were unaware of this meta-analysis and predicted that the original result would replicate successfully (probability of success = 72%). Contrary to the market, the study failed to replicate. This example suggests that meta-analyses might be more valuable than prediction markets or the p-value of a single study.

The main obstacle for the use of meta-analyses is that many published meta-analyses fail to take selection for significance into account and overestimate replicability. However, new statistical methods that correct for selection bias may address this problem. The R-Index is a rather simple tool that allows to correct for selection bias in small sets of studies. I use the article by Nairne et al. (2008) that was used for the OSC project as an example. The replication project focused on Study 2 that produced a p-value of .026. Based on this weak evidence alone, the R-Index would predict a replication failure (observed power = .61, inflation = .39, R-Index = .61 – .39 = .22). However, Study 1 produced much more convincing evidence for the effect, p = .0007. If this study had been picked for the replication attempt, the R-Index would have predicted a successful outcome (observed power = .92, inflation = .08, R-Index = .84). A meta-analysis would average across the two power estimates and also predict a successful replication outcome (mean observed power = .77, inflation = .23, R-Index = .53). The actual replication study was significant with p = .007 (observed power = .77, inflation = .23, R-Index = .53). A meta-analysis across all three studies also suggests that the next study will be a successful replication (R-Index = .53), but the R-Index also shows that replication failures are likely because the studies have relatively low power. In short, prediction markets may be useful when only a single study is available, but meta-analysis are likely to be superior predictors of replication outcomes when prior replication studies are available.

Conclusion

Gordon et al. (2021) conducted a meta-analysis of 103 studies that were included in prediction markets to forecast the outcome of replication studies. The results show that prediction markets can forecast replication outcomes above chance levels, but the value of this information is limited. Without actual replication studies, it remains unclear which published results can be trusted or not. Statistical methods that simply focus on the strength of evidence in original studies perform nearly as well and are much easier to use to make sense of thousands of published articles. However, even these methods have a high failure rate. The best solution to this problem is to rely on meta-analyses of studies rather than to predict the outcome of a single study. In addition to meta-analyses, it will be necessary to conduct new studies that are conducted with high scientific integrity to provide solid empirical foundations for psychology.

Prediction Markets of Replicability

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Abstract

I reinvestigate the performance of prediction markets for the Open Science Collaboration replicability project. I show that performance of prediction markets varied considerably across the two markets, with the second market failing to replicate the excellent performance of the first market. I also show that the markets did not perform significantly better than a “burn everything to the ground” rule that bets on failure every time. Finally, I suggest a simple rule that can be easily applied to published studies that only treats results with p-values below .005 as significant. Finally, I discuss betting on future studies as a way to optimize resource allocation for future studies.

Introduction

For decades, psychologists failed to properly test their hypotheses. Statistically significant results in journals are meaningless because published results are selected for significance. A replication project with 100 studies from three journals that reported significant results found that only 37% (36/97) of published significant results could be replicated (Open Science Collaboration, 2015).

Unfortunately, it is impossible to rely on actual replication studies to examine the credibility of thousands of findings that have been reported over the years. Dreber, Pfeiffer, Almenberg, Isakssona, Wilson, Chen, Nosek, and Johannesson (2015) proposed prediction markets as a solution to this problem. Prediction markets rely on a small number of traders to bet on the outcome of replication studies. They can earn a small amount of real money for betting on studies that actually replicate.

To examine the forecasting abilities of prediction markets, Dreber et al. (2015) conducted two studies. The first study with 23 studies started in November 2012 and lasted two month (N = 47 participants). The second study with 21 studies started in October 2014 (N = 45 participants). The studies are very similar to each other. Thus, we can consider Study 2 a close replication of Study 1.

Studies were selected from the set of 100 studies based on time of completion. To pay participants, studies were chosen that were scheduled to be completed within two month after the completion of the prediction market. It is unclear how completion time may influence the type of study that was included or the outcome of the included studies.

The published article reports the aggregated results across the two studies. A market price above 50% was considered to be a prediction of a successful replication and a market price below 50% was considered to be a prediction of a replication failure. The key finding was that “the prediction markets correctly predict the outcome of 71% of the replications (29 of 41 studies” (p. 15344). The authors compare this finding to a proverbial coin flip which implies a replication rate of 50% and find that 71% is [statistically] significantly higher than than 50%.

Below I am conducting some additional statistical analyses of the open data. First, we can compare the performance of the prediction market with a different prediction rule. Given the higher prevalence of replication failures than successes, a simple rule is to use the higher base rate of failures to predict that all studies will fail to replicate. As the failure rate for the total set of 97 studies was 37%, this prediction rule has a success rate of 1-.37 = 63%. For the 43 studies with significant results, the success rate of replication studies was also 37% (15/41). Somewhat surprisingly, the success rates were also close to 37% for Prediction Market 1, 32% (7/22) and Prediction Market 2, 42% (8 / 19).

In comparison to a simple prediction rule that everything in psychology journals does not replicate, prediction markets are no longer statistically significantly better, chi2(1) = 1.82, p = .177.

Closer inspection of the original data also revealed a notable difference in the performance of the two prediction markets. Table 1 shows the results separately for prediction markets 1 and 2. Whereas the performance of the first prediction market is nearly perfect, 91% (20/22), the replication market performed only at chance levels (flip a coin), 53% (10/19). Despite the small sample size, the success rates in the two studies are statistically significantly different, chi2(1) = 5.78, p = .016.

There is no explanation for these discrepancies and the average result reported in the article can still be considered the best estimate of prediction markets’ performance, but trust in their ability is diminished by the fact that a close replication of excellent performance failed to replicate. Not reporting the different outcomes for two separate studies could be considered a questionable decision.

The main appeal of prediction markets over the nihilistic trash-everything rule is that decades of research would have produced some successes. However, the disadvantage of prediction markets is that they take a long time, cost money, and the success rates are currently uncertain. A better solution might be to find rules that can be applied easily to large sets of studies (Yang, Wu, & Uzzi, 2020). One simple rule is suggested by the simple relationship between strength of evidence and replicability. The stronger the evidence against the null-hypothesis is (i.e., lower p-values), the more likely it is that the original study had high power and that the results will replicate in a replication study. There is no clear criterion for a cut-off point to optimize prediction, but the results of the replication project can be used to validate cut-off points empirically.

One suggests has been to consider only p-values below .005 as statistically significant (Benjamin et al., 2017). This rule is especially useful when selection bias is present. Selection bias can produce many results with p-values between .05 and .005 that have low power. However, p-values below .005 are more difficult to produce with statistical tricks.

The overall success rate for the 41 studies included in the Prediction Markets was 63% (26/41), a difference of 4 studies. The rule also did better for the first market, 81% (18.22) than for the second market, 42% (8/19).

Table 2 shows that the main difference between the two markets was that the first market contained more studies with questionable p-values between .05 and .005 (15 vs. 6). For the second market, the rule overpredicts successes and there are more false (8) than correct (5) predictions. This finding is consistent with examinations of the total set of replication studies in the replicability project (Schimmack, 2015). Based on this observation, I recommended a 4-sigma rule, p < .00006. The overall success rate increases to 68% (28/41) and improvement by 2 studies. However, an inspection of correct predictions of successes shows that the rule only correctly predicts 5 of the 15 successes (33%), whereas the p < .005 rule correctly predicted 10 of the 15 successes (67%). Thus, the p < .005 rule has the advantage that it salvages more studies.

Conclusion

Meta-scientists are still scientists and scientists are motivated to present their results in the best possible light. This is also true for Derber et al.’s (2015) article. The authors enthusiastically recommend prediction markets as a tool “to quickly identify findings
that are unlikely to replicate” Based on their meta-analysis of two prediction markets with a total of just 41 studies, the authors conclude that “prediction markets are well suited” to assess the replicability of published results in psychology. I am not the only one to point out that this conclusion is exaggerated (Yang, Wu, & Uzzi, 2020). First, prediction markets are not quick at identifying replicable results, especially when we compare the method to a simple computation of the exact p-values to decide whether the p-value is below .005 or not. It is telling that nobody has used prediction markets to forecast the outcome of new replication studies. One problem is that a market requires a set of studies, which makes it difficult to use them to predict outcomes of single studies. It is also unclear how well prediction markets really work. The original article omitted the fact that it worked extremely well in the first market and not at all in the second market, a statistically significant difference. The outcome seems to depend a lot on the selection of studies in the market. Finally, I showed that a simple statistical rule alone can predict replication outcomes nearly as well as prediction markets.

There is no reason to use markets for multiple studies. One could also set up betting for individual studies, just like individuals can bet on the outcome of a single match in sports or a single election outcome. Betting might be more usefully employed for the prediction of original studies than to vet the outcome of replication studies. For example, if everybody bets that a study will produce a significant result, there appears to be little uncertainty about the outcome, and the actual study may be a waste of resources. One concerns in psychology is that many studies merely produce significant p-values for obvious predictions. Betting on effect sizes would help to make effect sizes more important. If everybody bets on a very small effect size a study might not be useful to run because the expected effect size is trivial, even if the effect is greater than zero. Betting on effect sizes could also be used for informal power analyses to determine the sample size of the actual study.

References

Dreber, A., Pfeiffer, T., Almenberg, J., Isaksson, S., Wilson, B., Chen, Y., Nosek, B. A., & Johannesson, M. (2015). Using prediction markets to estimate the reproducibility of scientific research. PNAS Proceedings of the National Academy of Sciences of the United States of America, 112(50), 15343–15347. https://doi.org/10.1073/pnas.1516179112 1 commen