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Examining the Replicability of 66,212 Published Results in Social Psychology: A Post-Hoc-Power Analysis Informed by the Actual Success Rate in the OSF-Reproducibilty Project

The OSF-Reproducibility-Project examined the replicability of 99 statistical results published in three psychology journals. The journals covered mostly research in cognitive psychology and social psychology. An article in Science, reported that only 35% of the results were successfully replicated (i.e., produced a statistically significant result in the replication study).

I have conducted more detailed analyses of replication studies in social psychology and cognitive psychology. Cognitive psychology had a notably higher success rate (50%, 19 out of 38) than social psychology (8%, 3 out of 38). The main reason for this discrepancy is that social psychologists and cognitive psychologists use different designs. Whereas cognitive psychologists typically use within-subject designs with many repeated measurements of the same individual, social psychologists typically assign participants to different groups and compare behavior on a single measure. This so-called between-subject design makes it difficult to detect small experimental effects because it does not control the influence of other factors that influence participants’ behavior (e.g., personality dispositions, mood, etc.). To detect small effects in these noisy data, between-subject designs require large sample sizes.

It has been known for a long time that sample sizes in between-subject designs in psychology are too small to have a reasonable chance to detect an effect (less than 50% chance to find an effect that is actually there) (Cohen, 1962; Schimmack, 2012; Sedlmeier & Giegerenzer, 1989). As a result, many studies fail to find statistically significant results, but these studies are not submitted for publication. Thus, only studies that achieved statistical significance with the help of chance (the difference between two groups is inflated by uncontrolled factors such as personality) are reported in journals. The selective reporting of lucky results creates a bias in the published literature that gives a false impression of the replicability of published results. The OSF-results for social psychology make it possible to estimate the consequences of publication bias on the replicability of results published in social psychology journals.

A naïve estimate of the replicability of studies would rely on the actual success rate in journals. If journals would publish significant and non-significant results, this would be a reasonable approach. However, journals tend to publish exclusively significant results. As a result, the success rate in journals (over 90% significant results; Sterling, 1959; Sterling et al., 1995) gives a drastically inflated estimate of replicability.

A somewhat better estimate of replicability can be obtained by computing post-hoc power based on the observed effect sizes and sample sizes of published studies. Statistical power is the long-run probability that a series of exact replication studies with the same sample size would produce significant results. Cohen (1962) estimated that the typical power of psychological studies is about 60%. Thus, even for 100 studies that all reported significant results, only 60 are expected to produce a significant result again in the replication attempt.

The problem with Cohen’s (1962) estimate of replicability is that post-hoc-power analysis uses the reported effect sizes as an estimate of the effect size in the population. However, due to the selection bias in journals, the reported effect sizes and power estimates are inflated. In collaboration with Jerry Brunner, I have developed an improved method to estimate typical power of reported results that corrects for the inflation in reported effect sizes. I applied this method to results from 38 social psychology articles included in the OSF-reproducibility project and obtained a replicability estimate of 35%.

The OSF-reproducbility project provides another opportunity to estimate the replicability of results in social psychology. The OSF-project selected a representative set of studies from two journals and tried to reproduce the same experimental conditions as closely as possible. This should produce unbiased results and the success rate provides an estimate of replicability. The advantage of this method is that it does not rely on statistical assumptions. The disadvantage is that the success rate depends on the ability to exactly recreate the conditions of the original studies. Any differences between studies (e.g., recruiting participants from different populations) can change the success rate. The OSF replication studies also often changed the sample size of the replication study, which will also change the success rate. If sample sizes in a replication study are larger, power increases and the success rate no longer can be used as an estimate of the typical replicability of social psychology. To address this problem, it is possible to apply a statistical adjustment and use the success rate that would have occurred with the original sample sizes. I found that 5 out of 38 (13%) produced significant results and after correcting for the increase in sample size, replicability was only 8% (3 out of 38).

One important question is how how representative the 38 results from the OSF-project are for social psychology in general. Unfortunately, it is practically impossible and too expensive to conduct a large number of exact replication studies. In comparison, it is relatively easy to apply post-hoc power analysis to a large number of statistical results reported in social psychology. Thus, I examined the representativeness of the OSF-reproducibility results by comparing the results of my post-hoc power analysis based on the 38 results in the OSF to a post-hoc-power analysis of a much larger number of results reported in major social psychology journals .

I downloaded articles from 12 social psychology journals, which are the primary outlets for publishing experimental social psychology research: Basic and Applied Social Psychology, British Journal of Social Psychology, European Journal of Social Psychology, Journal of Experimental Social Psychology, Journal of Personality and Social Psychology: Attitudes and Social Cognition, Journal of Personality and Social Psychology: Interpersonal Relationships and Group Processes, Journal of Social and Personal Relationships, Personal Relationships, Personality and Social Psychology Bulletin, Social Cognition, Social Psychology and Personality Science, Social Psychology.

I converted pdf files into text files and searched for all reports of t-tests or F-tests and converted the reported test-statistic into exact two-tailed p-values. The two-tailed p-values were then converted into z-scores by finding the z-score corresponding to the probability of 1-p/2, with p equal the two-tailed p-value. The total number of z-scores included in the analysis is 134,929.

I limited my estimate of power to z-scores in the range between 2 and 4. Z-scores below 2 are not statistically significant (z = 1.96, p = .05). Sometimes these results are reported as marginal evidence for an effect, sometimes they are reported as evidence that an effect is not present, and sometimes they are reported without an inference about the population effect. It is more important to determine the replicability of results that are reported as statistically significant support for a prediction. Z-scores greater than 4 were excluded because z-scores greater than 4 imply that this test had high statistical power (> 99%). Many of these results replicated successfully in the OSF-project. Thus, a simple rule is to assign a success rate of 100% to these findings. The Figure below shows the distribution of z-scores in the range form z = 0 to6, but the power estimate is applied to z-scores in the range between 2 and 4 (n = 66,212).

PHP-Curve Social Journals

The power estimate based on the post-hoc-power curve for z-scores between 2 and 4 is 46%. It is important to realize that this estimate is based on 70% of all significant results that were reported. As z-scores greater than 4 essentially have a power of 100%, the overall power estimate for all statistical tests that were reported is .46*.70 + .30 = .62. It is also important to keep in mind that this analysis uses all statistical tests that were reported including manipulation checks (e.g., pleasant picture were rated as more pleasant than unpleasant pictures). For this reason, the range of z-scores is limited to values between 2 and 4, which is much more likely to reflect a test of a focal hypothesis.

46% power for z-scores between 2 and 4 of is a higher estimate than the estimate for the 38 studies in the OSF-reproducibility project (35%). This suggests that the estimated replicability based on the OSF-results is an underestimation of the true replicability. The discrepancy between predicted and observed replicability in social psychology (8 vs. 38) and cognitive psychology (50 vs. 75), suggests that the rate of actual successful replications is about 20 to 30% lower than the success rate based on statistical prediction. Thus, the present analysis suggests that actual replication attempts of results in social psychology would produce significant results in about a quarter of all attempts (46% – 20% = 26%).

The large sample of test results makes it possible to make more detailed predictions for results with different strength of evidence. To provide estimates of replicability for different levels of evidence, I conducted post-hoc power analysis for intervals of half a standard deviation (z = .5). The power estimates are:

Strength of Evidence      Power    

2.0 to 2.5                            33%

2.5 to 3.0                            46%

3.0 to 3.5                            58%

3.5 to 4.0                            72%

IMPLICATIONS FOR PLANNING OF REPLICATION STUDIES

These estimates are important for researchers who are aiming to replicate a published study in social psychology. The reported effect sizes are inflated and a replication study with the same sample size has a low chance to produce a significant result even if a smaller effect exists.   To conducted a properly powered replication study, researchers would have to increase sample sizes. To illustrate, imagine that a study demonstrate a significant difference between two groups with 40 participants (20 in each cell) with a z-score of 2.3 (p = .02, two-tailed). The observed power for this result is 65% and it would suggest that a slightly larger sample of N = 60 is sufficient to achieve 80% power (80% chance to get a significant result). However, after correcting for bias, the true power is more likely to be just 33% (see table above) and power for a study with N = 60 would still only be 50%. To achieve 80% power, the replication study would need a sample size of 130 participants. Sample sizes would need to be even larger taking into account that the actual probability of a successful replication is even lower than the probability based on post-hoc power analysis. In the OSF-project only 1 out of 30 studies with an original z-score between 2 and 3 was successfully replicated.

IMPLICATIONS FOR THE EVALUATION OF PUBLISHED RESULTS

The results also have implications for the way social psychologists should conduct and evaluate new research. The main reason why z-scores between 2 and 3 provide untrustworthy evidence for an effect is that they are obtained with underpowered studies and publication bias. As a result, it is likely that the strength of evidence is inflated. If, however, the same z-scores were obtained in studies with high power, a z-score of 2.5 would provide more credible evidence for an effect. The strength of evidence in a single study would still be subject to random sampling error, but it would no longer be subject to systematic bias. Therefore, the evidence would be more likely to reveal a true effect and it would be less like to be a false positive.   This implies that z-scores should be interpreted in the context of other information about the likelihood of selection bias. For example, a z-score of 2.5 in a pre-registered study provides stronger evidence for an effect than the same z-score in a study where researchers may have had a chance to conduct multiple studies and to select the most favorable results for publication.

The same logic can also be applied to journals and labs. A z-score of 2.5 in a journal with an average z-score of 2.3 is less trustworthy than a z-score of 2.5 in a journal with an average z-score of 3.5. In the former journal, a z-score of 2.5 is likely to be inflated, whereas in the latter journal a z-score of 2.5 is more likely to be negatively biased by sampling error. For example, currently a z-score of 2.5 is more likely to reveal a true effect if it is published in a cognitive journal than a social journal (see ranking of psychology journals).

The same logic applies even more strongly to labs because labs have a distinct research culture (MO). Some labs conduct many underpowered studies and publish only the studies that worked. Other labs may conduct fewer studies with high power. A z-score of 2.5 is more trustworthy if it comes from a lab with high average power than from a lab with low average power. Thus, providing information about the post-hoc-power of individual researchers can help readers to evaluate the strength of evidence of individual studies in the context of the typical strength of evidence that is obtained in a specific lab. This will create an incentive to publish results with strong evidence rather than fishing for significant results because a low replicability index increases the criterion at which results from a lab provide evidence for an effect.

REPLICABILITY RANKING OF 26 PSYCHOLOGY JOURNALS

THEORETICAL BACKGROUND

Neyman & Pearson (1933) developed the theory of type-I and type-II errors in statistical hypothesis testing.

A type-I error is defined as the probability of rejecting the null-hypothesis (i.e., the effect size is zero) when the null-hypothesis is true.

A type-II error is defined as the probability of failing to reject the null-hypothesis when the null-hypothesis is false (i.e., there is an effect).

A common application of statistics is to provide empirical evidence for a theoretically predicted relationship between two variables (cause-effect or covariation). The results of an empirical study can produce two outcomes. Either the result is statistically significant or it is not statistically significant. Statistically significant results are interpreted as support for a theoretically predicted effect.

Statistically non-significant results are difficult to interpret because the prediction may be false (the null-hypothesis is true) or a type-II error occurred (the theoretical prediction is correct, but the results fail to provide sufficient evidence for it).

To avoid type-II errors, researchers can design studies that reduce the type-II error probability. The probability of avoiding a type-II error when a predicted effect exists is called power. It could also be called the probability of success because a significant result can be used to provide empirical support for a hypothesis.

Ideally researchers would want to maximize power to avoid type-II errors. However, powerful studies require more resources. Thus, researchers face a trade-off between the allocation of resources and their probability to obtain a statistically significant result.

Jacob Cohen dedicated a large portion of his career to help researchers with the task of planning studies that can produce a successful result, if the theoretical prediction is true. He suggested that researchers should plan studies to have 80% power. With 80% power, the type-II error rate is still 20%, which means that 1 out of 5 studies in which a theoretical prediction is true would fail to produce a statistically significant result.

Cohen (1962) examined the typical effect sizes in psychology and found that the typical effect size for the mean difference between two groups (e.g., men and women or experimental vs. control group) is about half-of a standard deviation. The standardized effect size measure is called Cohen’s d in his honor. Based on his review of the literature, Cohen suggested that an effect size of d = .2 is small, d = .5 moderate, and d = .8. Importantly, a statistically small effect size can have huge practical importance. Thus, these labels should not be used to make claims about the practical importance of effects. The main purpose of these labels is that researchers can better plan their studies. If researchers expect a large effect (d = .8), they need a relatively small sample to have high power. If researchers expect a small effect (d = .2), they need a large sample to have high power.   Cohen (1992) provided information about effect sizes and sample sizes for different statistical tests (chi-square, correlation, ANOVA, etc.).

Cohen (1962) conducted a meta-analysis of studies published in a prominent psychology journal. Based on the typical effect size and sample size in these studies, Cohen estimated that the average power in studies is about 60%. Importantly, this also means that the typical power to detect small effects is less than 60%. Thus, many studies in psychology have low power and a high type-II error probability. As a result, one would expect that journals often report that studies failed to support theoretical predictions. However, the success rate in psychological journals is over 90% (Sterling, 1959; Sterling, Rosenbaum, & Weinkam, 1995). There are two explanations for discrepancies between the reported success rate and the success probability (power) in psychology. One explanation is that researchers conduct multiple studies and only report successful studies. The other studies remain unreported in a proverbial file-drawer (Rosenthal, 1979). The other explanation is that researchers use questionable research practices to produce significant results in a study (John, Loewenstein, & Prelec, 2012). Both practices have undesirable consequences for the credibility and replicability of published results in psychological journals.

A simple solution to the problem would be to increase the statistical power of studies. If the power of psychological studies in psychology were over 90%, a success rate of 90% would be justified by the actual probability of obtaining significant results. However, meta-analysis and method articles have repeatedly pointed out that psychologists do not consider statistical power in the planning of their studies and that studies continue to be underpowered (Maxwell, 2004; Schimmack, 2012; Sedlmeier & Giegerenzer, 1989).

One reason for the persistent neglect of power could be that researchers have no awareness of the typical power of their studies. This could happen because observed power in a single study is an imperfect indicator of true power (Yuan & Maxwell, 2005). If a study produced a significant result, the observed power is at least 50%, even if the true power is only 30%. Even if the null-hypothesis is true, and researchers publish only type-I errors, observed power is dramatically inflated to 62%, when the true power is only 5% (the type-I error rate). Thus, Cohen’s estimate of 60% power is not very reassuring.

Over the past years, Schimmack and Brunner have developed a method to estimate power for sets of studies with heterogeneous designs, sample sizes, and effect sizes. A technical report is in preparation. The basic logic of this approach is to convert results of all statistical tests into z-scores using the one-tailed p-value of a statistical test.  The z-scores provide a common metric for observed statistical results. The standard normal distribution predicts the distribution of observed z-scores for a fixed value of true power.   However, for heterogeneous sets of studies the distribution of z-scores is a mixture of standard normal distributions with different weights attached to various power values. To illustrate this method, the histograms of z-scores below show simulated data with 10,000 observations with varying levels of true power: 20% null-hypotheses being true (5% power), 20% of studies with 33% power, 20% of studies with 50% power, 20% of studies with 66% power, and 20% of studies with 80% power.

RepRankSimulation

The plot shows the distribution of absolute z-scores (there are no negative effect sizes). The plot is limited to z-scores below 6 (N = 99,985 out of 10,000). Z-scores above 6 standard deviations from zero are extremely unlikely to occur by chance. Even with a conservative estimate of effect size (lower bound of 95% confidence interval), observed power is well above 99%. Moreover, quantum physics uses Z = 5 as a criterion to claim success (e.g., discovery of Higgs-Boson Particle). Thus, Z-scores above 6 can be expected to be highly replicable effects.

Z-scores below 1.96 (the vertical dotted red line) are not significant for the standard criterion of (p < .05, two-tailed). These values are excluded from the calculation of power because these results are either not reported or not interpreted as evidence for an effect. It is still important to realize that true power of all experiments would be lower if these studies were included because many of the non-significant results are produced by studies with 33% power. These non-significant results create two problems. Researchers wasted resources on studies with inconclusive results and readers may be tempted to misinterpret these results as evidence that an effect does not exist (e.g., a drug does not have side effects) when an effect is actually present. In practice, it is difficult to estimate power for non-significant results because the size of the file-drawer is difficult to estimate.

It is possible to estimate power for any range of z-scores, but I prefer the range of z-scores from 2 (just significant) to 4. A z-score of 4 has a 95% confidence interval that ranges from 2 to 6. Thus, even if the observed effect size is inflated, there is still a high chance that a replication study would produce a significant result (Z > 2). Thus, all z-scores greater than 4 can be treated as cases with 100% power. The plot also shows that conclusions are unlikely to change by using a wider range of z-scores because most of the significant results correspond to z-scores between 2 and 4 (89%).

The typical power of studies is estimated based on the distribution of z-scores between 2 and 4. A steep decrease from left to right suggests low power. A steep increase suggests high power. If the peak (mode) of the distribution were centered over Z = 2.8, the data would conform to Cohen’s recommendation to have 80% power.

Using the known distribution of power to estimate power in the critical range gives a power estimate of 61%. A simpler model that assumes a fixed power value for all studies produces a slightly inflated estimate of 63%. Although the heterogeneous model is correct, the plot shows that the homogeneous model provides a reasonable approximation when estimates are limited to a narrow range of Z-scores. Thus, I used the homogeneous model to estimate the typical power of significant results reported in psychological journals.

DATA

The results presented below are based on an ongoing project that examines power in psychological journals (see results section for the list of journals included so far). The set of journals does not include journals that primarily publish reviews and meta-analysis or clinical and applied journals. The data analysis is limited to the years from 2009 to 2015 to provide information about the typical power in contemporary research. Results regarding historic trends will be reported in a forthcoming article.

I downloaded pdf files of all articles published in the selected journals and converted the pdf files to text files. I then extracted all t-tests and F-tests that were reported in the text of the results section searching for t(df) or F(df1,df2). All t and F statistics were converted into one-tailed p-values and then converted into z-scores.

RepRankAll

The plot above shows the results based on 218,698 t and F tests reported between 2009 and 2015 in the selected psychology journals. Unlike the simulated data, the plot shows a steep drop for z-scores just below the threshold of significance (z = 1.96). This drop is due to the tendency not to publish or report non-significant results. The heterogeneous model uses the distribution of non-significant results to estimate the size of the file-drawer (unpublished non-significant results). However, for the present purpose the size of the file-drawer is irrelevant because power is estimated only for significant results for Z-scores between 2 and 4.

The green line shows the best fitting estimate for the homogeneous model. The red curve shows fit of the heterogeneous model. The heterogeneous model is doing a much better job at fitting the long tail of highly significant results, but for the critical interval of z-scores between 2 and 4, the two models provide similar estimates of power (55% homogeneous & 53% heterogeneous model).   If the range is extended to z-scores between 2 and 6, power estimates diverge (82% homogenous, 61% heterogeneous). The plot indicates that the heterogeneous model fits the data better and that the 61% estimate is a better estimate of true power for significant results in this range. Thus, the results are in line with Cohen (1962) estimate that psychological studies average 60% power.

REPLICABILITY RANKING

The distribution of z-scores between 2 and 4 was used to estimate the average power separately for each journal. As power is the probability to obtain a significant result, this measure estimates the replicability of results published in a particular journal if researchers would reproduce the studies under identical conditions with the same sample size (exact replication). Thus, even though the selection criterion ensured that all tests produced a significant result (100% success rate), the replication rate is expected to be only about 50%, even if the replication studies successfully reproduce the conditions of the published studies. The table below shows the replicability ranking of the journals, the replicability score, and a grade. Journals are graded based on a scheme that is similar to grading schemes for undergraduate students (below 50 = F, 50-59 = E, 60-69 = D, 70-79 = C, 80-89 = B, 90+ = A).

ReplicabilityRanking

The average value in 2000-2014 is 57 (D+). The average value in 2015 is 58 (D+). The correlation for the values in 2010-2014 and those in 2015 is r = .66.   These findings show that the replicability scores are reliable and that journals differ systematically in the power of published studies.

LIMITATIONS

The main limitation of the method is that focuses on t and F-tests. The results might change when other statistics are included in the analysis. The next goal is to incorporate correlations and regression coefficients.

The second limitation is that the analysis does not discriminate between primary hypothesis tests and secondary analyses. For example, an article may find a significant main effect for gender, but the critical test is whether gender interacts with an experimental manipulation. It is possible that some journals have lower scores because they report more secondary analyses with lower power. To address this issue, it will be necessary to code articles in terms of the importance of statistical test.

The ranking for 2015 is based on the currently available data and may change when more data become available. Readers should also avoid interpreting small differences in replicability scores as these scores are likely to fluctuate. However, the strong correlation over time suggests that there are meaningful differences in the replicability and credibility of published results across journals.

CONCLUSION

This article provides objective information about the replicability of published findings in psychology journals. None of the journals reaches Cohen’s recommended level of 80% replicability. Average replicability is just about 50%. This finding is largely consistent with Cohen’s analysis of power over 50 years ago. The publication of the first replicability analysis by journal should provide an incentive to editors to increase the reputation of their journal by paying more attention to the quality of the published data. In this regard, it is noteworthy that replicability scores diverge from traditional indicators of journal prestige such as impact factors. Ideally, the impact of an empirical article should be aligned with the replicability of the empirical results. Thus, the replicability index may also help researchers to base their own research on credible results that are published in journals with a high replicability score and to avoid incredible results that are published in journals with a low replicability score. Ultimately, I can only hope that journals will start competing with each other for a top spot in the replicability rankings and as a by-product increase the replicability of published findings and the credibility of psychological science.