Nature Neuroscience: R-Index

R-Index of Nature Neuroscience

An article in nature review, neuroscience suggested that the median power in neuroscience studies is just 21% (Katherine S. Button, John P. A. Ioannidis, Claire Mokrysz, Brian A.Nosek, Jonathan Flint, Emma S.J. Robinson and Marcus R. Munafò, 2013).

The authors of this article examined meta-analyses of primary studies in neuroscience that were published in 2011. They analyzed 49 meta-analyses that were based on a total of 730 original studies (on average, 15 studies per meta-analysis, range 2 to 57).

For each primary study, the authors computed observed power based on the sample size and the estimated effect size in the meta-analysis.

Based on their analyses, the authors concluded that the median power in neuroscience is 21%.

Importantly, this is an estimate of the median observed power in neuroscience studies. There is a major problem with this estimate. It is incredibly low because a study with 21% observed power is not statistically significant, p = .25. If median power were 21%, it would mean that over 50% of the original studies in the meta-analysis reported a non-significant result (p > .05). This seems rather unlikely because journals tend to publish mostly significant results.

The estimate is even less plausible because it is based on meta-analytic averages without any correction for bias. These effect sizes are likely to be inflated, which means that median power estimate is inflated. Thus, true power is even less than 21%.

What could explain this implausible result?

1. A meta-analysis includes published and unpublished studies. It is possible that the published studies reported significant results with observed power greater than 50% (p < .05) and the unpublished studies reported non-significant results with power less than 50%. However, this would imply that meta-analysts were able to retrieve as many unpublished studies as published studies. The authors did not report whether power of published and unpublished studies differed.
2. A second possibility is that the power analyses produced false results. The authors relied on Ioannidis and Trikalinos’s (2007) approach to the estimation of power. This approach assumes that studies in a meta-analysis have the same true effect size and that the meta-analytic average (weighted mean) provides the best estimate of the true effect size. This estimate of the true effect size is then used to estimate power in individual studies based on the sample size of the study. As already noted by Ioannidis and Trikalinos (2007), this approach can produce biased results when effect sizes in a meta-analysis are heterogeneous.
3. Estimating power simply on the basis of effect size and sample size can be misleading when the design is not a simple comparison of two groups. Between-subject designs are common in animal studies in neuroscience. However, many fMRI studies use within-subject designs that achieve high statistical power with a few participants because participants serve as their own controls.

Schimmack (2012) proposed an alternative procedure that does not have this limitation. Power is estimated individually for each study based on the observed effect size in this study. This approach makes it possible to estimate median power for heterogeneous sets of studies with different effect sizes. Moreover, this approach makes it possible to compute power when power is not simply a function of sample size and effect size (e.g., within-subject designs).

R-Index of Nature Neuroscience: Analysis

To examine the replicability of research published in nature and neuroscience, I retrieved the most cited articles in this journal until I had a sample of 20 studies. I needed 14 articles to meet this goal. The number of studies in these articles ranged from 1 to 7.

The success rate for focal significance tests was 97%. This implies that the vast majority of significance tests reported a significant result. The median observed power was 84%. The inflation rate is 13% (97% – 84% = 13%). The R-Index is 71%. Based on these numbers, the R-Index predicts that the majority of studies in nature neuroscience would replicate in an exact replication study.

This conclusion differs dramatically from Button et al.’s (2013) conclusion. I therefore examined some of the articles that were used for Button et al.’s analyses.

A study by Davidson et al. (2003) examined treatment effects in 12 depressed patients and compared them to 5 healthy controls. The main findings in this article were three significant interactions between time of treatment and group with z-scores of 3.84, 4.60, and 4.08. Observed power for these values with p = .05 is over 95%. If a more conservative significance level of p = .001 is used, power is still over 70%. However, the meta-analysis focused on the correlation between brain activity at baseline and changes in depression over time. This correlation is shown with a scatterplot without reporting the actual correlation or testing it for significance. The text further states that a similar correlation was observed for an alternative depression measure with r = .46 and noting correctly that this correlation is not significant, t(10) = 1.64, p = .13, d = .95, obs. power = 32%. The meta-analysis found a mean effect size of .92. A power analysis with d = .92 and N = 12 yields a power estimate of 30%. Presumably, this is the value that Button et al. used to estimate power for the Davidson et al. (2003) article. However, the meta-analysis did not include the more powerful analyses that compared patients and controls over time.

Conclusion

In the current replication crisis, there is a lot of confusion about the replicability of published findings. Button et al. (2013) aimed to provide some objective information about the replicability of neuroscience research. They concluded that replicability is very low with a median estimate of 21%. In this post, I point out some problems with their statistical approach and the focus on meta-analyses as a way to make inferences about replicability of published studies. My own analysis shows a relatively high R-Index of 71%. To make sense of this index it is instructive to compare it to the following R-Indices.

In a replication project of psychological studies, I found an R-Index of 43% and 28% of studies were successfully replicated.

In the many-labs replication project, 10 out of 12 studies were successfully replicated, a replication rate of 83% and the R-Index was 72%.

Caveat

Neuroscience studies may have high observed power and still not replicate very well in exact replications. The reason is that measuring brain activity is difficult and requires many steps to convert and reduce observed data into measures of brain activity in specific regions. Actual replication studies are needed to examine the replicability of published results.

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Dr. Schnall’s R-Index

In several blog posts, Dr. Schnall made some critical comments about attempts to replicate her work and these blogs created a heated debate about replication studies. Heated debates are typically a reflection of insufficient information. Is the Earth flat? This question created heated debates hundreds of years ago. In the age of space travel it is no longer debated. In this blog, I presented some statistical information that sheds light on the debate about the replicability of Dr. Schnall’s research.

The Original Study

Dr. Schnall and colleagues conducted a study with 40 participants. A comparison of two groups on a dependent variable showed a significant difference, F(1,38) = 3.63. In these days, Psychological Science asked researchers to report P-Rep instead of p-values. P-rep was 90%. The interpretation of P-rep was that there is a 90% chance to find an effect with the SAME SIGN in an exact replication study with the same sample size. The conventional p-value for F(1,38) = 3.63 is p = .06, a finding that commonly is interpreted as marginally significant. The standardized effect size is d = .60, which is considered a moderate effect size. The 95% confidence interval is -.01 to 1.47.

The wide confidence interval makes it difficult to know the true effect size. A post-hoc power analysis, assuming the true effect size is d = .60 suggests that an exact replication study has a 46% chance to produce a significant results (p < .05, two-tailed). However, if the true effect size is lower, actual power is lower. For example, if the true effect size is small (d = .2), a study with N = 40 has only 9% power (that is a 9% chance) to produce a significant result.

The First Replication Study

Drs. Johnson, Cheung, and Donnellan conducted a replication study with 209 participants. Assuming the effect size in the original study is the true effect size, this replication study has 99% power. However, assuming the true effect size is only d = .2, the study has only 31% power to produce a significant result. The study produce a non-significant result, F(1, 206) = .004, p = .95. The effect size was d = .01 (in the same direction). Due to the larger sample, the confidence interval is smaller and ranges from -.26 to .28. The confidence interval includes d = 2. Thus, both studies are consistent with the hypothesis that the effect exists and that the effect size is small, d = .2.

The Second Replication Study

Dr. Huang conducted another replication study with N = 214 participants (Huang, 2004, Study 1). Based on the previous two studies, the true effect might be expected to be somewhere between -.01 and .28, which includes a small effect size of d = .20. A study with N = 214 participants has 31% power to produce a significant result. Not surprisingly, the study produce a non-significant result, t(212) = 1.22, p = .23. At the same time, the effect size fell within the confidence interval set by the previous two studies, d = .17.

A Third Replication Study

Dr. Hung conducted a replication study with N = 440 participants (Study 2). Maintaining the plausible effect size of d = .2 as the best estimate of the true effect size, the study has 55% power to produce a significant result, which means it is nearly as likely to produce a non-significant result as it is to produce a significant result, if the effect size is small (d = .2). The study failed to produce a significant result, t(438) = .042, p = 68. The effect size was d = .04 with a confidence interval ranging from -.14 to .23. Again, this confidence interval includes a small effect size of d = .2.

A Fourth Replication Study

Dr. Hung published a replication study in the supplementary materials to the article. The study again failed to demonstrate a main effect, t(434) = 0.42, p = .38. The effect size is d = .08 with a confidence interval of -.11 to .27. Again, the confidence interval is consistent with a small true effect size of d = .2. However, the study with 436 participants had only a 55% chance to produce a significant result.

If Dr. Huang had combined the two samples to conduct a more powerful study, a study with 878 participants would have 80% power to detect a small effect size of d = .2. However, the combined effect size of d = .06 for the combined samples is still not significant, t(876) = .89. The confidence interval ranges from -.07 to .19. It no longer includes d = .20, but the results are still consistent with a positive, yet small effect in the range between 0 and .20.

Conclusion

In sum, nobody has been able to replicate Schnall’s finding that a simple priming manipulation with cleanliness related words has a moderate to strong effect (d = .6) on moral judgments of hypothetical scenarios. However, all replication studies show a trend in the same direction. This suggests that the effect exists, but that the effect size is much smaller than in the original study; somewhere between 0 and .2 rather than .6.

Now there are three possible explanations for the much larger effect size in Schnall’s original study.

1. The replication studies were not exact replications and the true effect size in Schnall’s version of the experiment is stronger than in the other studies.

2. The true effect size is the same in all studies, but Dr. Schnall was lucky to observe an effect size that was three times as large as the true effect size and large enough to produce a marginally significant result.

3. It is possible that Dr. Schnall did not disclose all of the information about her original study. For example, she may have conducted additional studies that produced smaller and non-significant results and did not report these results. Importantly, this practice is common and legal and in an anonymous survey many researchers admitted using practices that produce inflated effect sizes in published studies. However, it is extremely rare for researchers to admit that these common practices explain one of their own findings and Dr. Schnall has attributed the discrepancy in effect sizes to problems with replication studies.

Dr. Schnall’s Replicability Index

Based on Dr. Schnall’s original study it is impossible to say which of these explanations accounts for her results. However, additional evidence makes it possible to test the third hypothesis that Dr. Schnall knows more than she was reporting in her article. The reason is that luck does not repeat itself. If Dr. Schnall was just lucky, other studies by her should have failed because Lady Luck is only on your side half the time. If, however, disconfirming evidence is systematically excluded from a manuscript, the rate of successful studies is higher than the observed statistical power in published studies (Schimmack, 2012).

To test this hypothesis, I downloaded Dr. Schnall’s 10 most cited articles (in Web of Science, July, 2014). These 10 articles contained 23 independent studies. For each study, I computed the median observed power of statistical tests that tested a theoretically important hypothesis. I also calculated the success rate for each study. The average success rate was 91% (ranging from 45% to 100%, median = 100%). The median observed power was 61%. The inflation rate is 30% (91%-61%). Importantly, observed power is an inflated estimate of replicability when the success rate is inflated. I created the replicability index (R-index) to take this inflation into account. The R-Index subtracts the inflation rate from observed median power.

Dr. Schnall’s R-Index is 31% (61% – 30%).

What does an R-Index of 31% mean? Here are some comparisons that can help to interpret the Index.

Imagine the null-hypothesis is always true, and a researcher publishes only type-I errors. In this case, observed power is 61% and the success rate is 100%. The R-Index is 22%.

Dr. Baumeister admitted that his publications select studies that report the most favorable results. His R-Index is 49%.

The Open Science Framework conducted replication studies of psychological studies published in 2008. A set of 25 completed studies in November 2014 had an R-Index of 43%. The actual rate of successful replications was 28%.

Given this comparison standards, it is hardly surprising that one of Dr. Schnall’s study did not replicate even when the sample size and power of replication studies were considerably higher.

Conclusion

Dr. Schnall’s R-Index suggests that the omission of failed studies provides the most parsimonious explanation for the discrepancy between Dr. Schnall’s original effect size and effect sizes in the replication studies.

Importantly, the selective reporting of favorable results was and still is an accepted practice in psychology. It is a statistical fact that these practices reduce the replicability of published results. So why do failed replication studies that are entirely predictable create so much heated debate? Why does Dr. Schnall fear that her reputation is tarnished when a replication study reveals that her effect sizes were inflated? The reason is that psychologists are collectively motivated to exaggerate the importance and robustness of empirical results. Replication studies break with the code to maintain an image that psychology is a successful science that produces stunning novel insights. Nobody was supposed to test whether published findings are actually true.

However, Bem (2011) let the cat out of the bag and there is no turning back. Many researchers have recognized that the public is losing trust in science. To regain trust, science has to be transparent and empirical findings have to be replicable. The R-Index can be used to show that researchers reported all the evidence and that significant results are based on true effect sizes rather than gambling with sampling error.

In this new world of transparency, researchers still need to publish significant results. Fortunately, there is a simple and honest way to do so that was proposed by Jacob Cohen over 50 years ago. Conduct a power analysis and invest resources only in studies that have high statistical power. If your expertise led you to make a correct prediction, the force of the true effect size will be with you and you do not have to rely on Lady Luck or witchcraft to get a significant result.

P.S. I nearly forgot to comment on Dr. Huang’s moderator effects. Dr. Huang claims that the effect of the cleanliness manipulation depends on how much effort participants exert on the priming task.

First, as noted above, no moderator hypothesis is needed because all studies are consistent with a true effect size in the range between 0 and .2.

Second, Dr. Huang found significant interaction effects in two studies. In Study 2, the effect was F(1,438) = 6.05, p = .014, observed power = 69%. In Study 2a, the effect was F(1,434) = 7.53, p = .006, observed power = 78%. The R-Index for these two studies is 74% – 26% = 48%.   I am waiting for an open science replication with 95% power before I believe in the moderator effect.

Third, even if the moderator effect exists, it doesn’t explain Dr. Schnall’s main effect of d = .6.

Roy Baumeister’s R-Index

“We did run multiple studies, some of which did not work, and some of which worked better than others. You may think that not reporting the less successful studies is wrong, but that is how the field works.” (Roy Baumeister, personal email communication)

The R-Index can be used to evaluate the replicability of a set of statistical results. It can be used to evaluate the statistical research integrity of journals, articles on a specific topic (meta-analysis), and researchers. Just like the H-Index has become a popular metric of research excellence, the R-Index of individual researchers can be used to evaluate the replicability of their findings.

I chose Roy Baumeister as an example for several reasons. First, the R-Index is based on my earlier work on the incredibility-index (Schimmack, 2012). In this article, I demonstrated how power analysis can be used to reveal that researchers used questionable research practices to produce statistically significant results. I illustrated this approach with two articles. One article published 10 experiments that appeared to demonstrate time-reversed causality. Independent replication studies failed to replicate this incredible finding. The Incredibility-Index predicted this failure. The second article was a study on glucose consumption and will-power with Roy Baumeister as the senior author. The Incredibility-Index showed that the statistical results reported in this article were even less credible than the time-travel studies in Bem’s (2011) article.

Not surprisingly, Roy Baumeister was a reviewer of the incredibility article. During the review process, Roy Baumeister explained why his article reported more significant results than one would expect on the basis of the statistical power of these studies.

“My paper with Gailliot et al. (2007) is used as an illustration here. Of course, I am quite familiar with the process and history of that one. We initially submitted it with more studies, some of which had weaker results. The editor said to delete those. He wanted the paper shorter so as not to use up a lot of journal space with mediocre results. It worked: the resulting paper is shorter and stronger. Does that count as magic? The studies deleted at the editor’s request are not the only story. I am pretty sure there were other studies that did not work. Let us suppose that our hypotheses were correct and that our research was impeccable. Then several of our studies would have failed, simply given the realities of low power and random fluctuations. Is anyone surprised that those studies were not included in the draft we submitted for publication? If we had included them, certainly the editor and reviewers would have criticized them and formed a more negative impression of the paper. Let us suppose that they still thought the work deserved publication (after all, as I said, we are assuming here that the research was impeccable and the hypotheses correct). Do you think the editor would have wanted to include those studies in the published version?”

To my knowledge this is one of the few frank acknowledgements that questionable research practices (i.e., excluding evidence that does not support an author’s theory) contributed to the picture-perfect results in a published article. It is therefore instructive to examine the R-Index of a researcher who openly acknowledged that the reported results are a biased selection of the empirical evidence.

A tricky issue in any statistical analysis is the sampling of studies. In this case it would be possible to conduct the analysis on the full set of articles published by Roy Baumeister. However, for my analysis I selected a sample. To ensure that the sample is unbiased, I chose a sampling strategy that makes a priori sense and does not involve random sampling because I have control over the random generator. My sampling strategy was to focus on the Top 10 most cited original research articles.

To evaluate the R-Index, it is instructive to keep the following scenarios in mind.

1. The null-hypothesis is true and a researcher uses questionable research practices to obtain just significant results (p = .049999). The observed power for this set of studies is 50%, but all statistical results are significant, 100% success rate. The success rate is inflated by 50%. The R-Index is observed power minus inflation rate, which is 0%.
2. The null-hypothesis is true and a researcher drops non-significant results and/or uses questionable research methods that capitalize on chance. In this case, p-values above .05 are not reported and p-values below .05 have a uniform distribution with a median of .025. A p-value of .025 corresponds to 61% observed power. With 100% significant results, the inflation rate is 39%, and the R-Index is 22% (61%-39%).
3. The null-hypothesis is false and researcher conducts studies with 30% power. The non-significant studies are not published. In this case, observed power is 70%. With 100% success rate, the inflation rate is 30%. The R-Index is 40%.
4. The null-hypothesis is false and researcher conducts studies with 50% power. The non-significant studies are not published. In this case, observed power is 75%. With 100% success rate, the inflation rate is 25%. The R-Index is 50%.
5. The null-hypothesis is false and researchers conduct studies with 80% power, as recommended by Cohen. The non-significant results are not published (20% missing). In this case, observed power is 90% with 100% significant results. With 10% inflation rate, the R-Index is 80% (90% – 10%).
6. A sample of psychological studies published in 2008 produced an R-Index of 43% (Observed Power = 72%, Success Rate = 100%, Inflation Rate = 28%). Exact replications of these studies produced a success rate of 28%.

Roy Baumeister’s Top-10 articles contained 40 studies. Each study reported multiple statistical tests. I computed the median observed power of statistical tests that tested a theoretically relevant hypothesis. I also recorded whether the test was considered supportive of the theoretical hypothesis (typically, p < .05). The median observed power in this set of 40 studies was 69%. The success rate was 89%. The inflation rate is 20% and the R-Index is 49% (69% – 20%).

Roy Baumeister’s R-Index of 49% is consistent with his statement that his articles do not contain all of the studies that tested a theoretical prediction. Studies that tested theoretical predictions and failed to support them are missing. An R-Index of 49% is also consistent with Roy Baumeister’s claim that his practices reflect the common practices in the field. Other sets of studies in social psychology produce similar indices (e.g., replicability project of psychological studies, R-Index = 43%; success rate in empirical replication studies 28%).

In conclusion, Roy Baumeister’s acknowledged the use of questionable research practices (i.e., excluding evidence that does not support a theoretical hypothesis) and his R-Index is 49%. The R-Index of a representative set of studies in psychology in 2008 produced an R-Index of 42%. This suggests that the use of questionable research practices in psychology is widespread and the R-Index is able to detect the use of these practices. A set of studies that were subjected to empirical replication attempts produced a R-Index of 38%, and 28% of replication attempts were successful (72% failed).

The R-Index makes it possible to quantify and compare the use of questionable research practices and I hope it will encourage researchers to conduct fewer and more powerful studies. I also hope that a quantitative index makes it possible to make replicability an evaluation criterion for scientists.

So what could Roy Baumeister have done? He published 9 studies that supported his hypothesis and excluded several more studies because they were underpowered.  I suggest running fewer studies with higher power so that all studies can produce significant results, assuming the null-hypothesis is really false.