Category Archives: Most Published Research Findings are False

Ioannidis is Wrong Most of the Time

False Discovery Rate, Ioannidis, Most Published Research Findings are False, Science-Wise False Discovery Rate

John P. A. Ioannidis is a rock star in the world of science (wikipedia).

By traditional standards of science, he is one of the most prolific and influential scientists alive. He has published over 1,000 articles that have been cited over 100,000 times.

He is best known for the title of his article “Why most published research findings are false” that has been cited nearly 5,000 times. The irony of this title is that it may also apply to Ioannidis, especially because there is a trade-off between quality and quantity in publishing.

Fact Checking Ioannidis

The title of Ioannidis’s article implies a factual statement: “Most published results ARE false.” However, the actual article does not contain empirical data to support this claim. Rather, Ioannidis presents some hypothetical scenarios that show under what conditions published results MAY BE false.

To produce mostly false findings, a literature has to meet two conditions.

First, it has to test mostly false hypotheses.
Second, it has to test hypotheses in studies with low statistical power, that is a low probability of producing true positive results.

To give a simple example, imagine a field that tests only 10% true hypothesis with just 20% power. As power predicts the percentage of true discoveries, only 2 out of the 10 true hypothesis will be significant. Meanwhile, the alpha criterion of 5% implies that 5% of the false hypotheses will also produce a significant result. Thus, 5 of the 90 false hypotheses will also produce a significant result. As a result, there will be two times more false positives (4.5 over 100) than true positives (2 over 100).

These relatively simple calculations were well known by 2005 (Soric, 1989). Thus, why did Ioannidis article have such a big impact? The answer is that Ioannidis convinced many people that his hypothetical examples are realistic and describe most areas in science.

2020 has shown that Ioannidis’s claim does not apply to all areas of science. In amazing speed, bio-tech companies were able to make not just one but several successful vaccine’s with high effectiveness. Clearly some sciences are making real progress. On the other hand, other areas of science suggest that Ioannidis’s claims were accurate. For example, the whole literature on single-gene variations as predictors of human behavior has produced mostly false claims. Social psychology has a replication crisis where only 25% of published results could be replicated (OSC, 2015).

Aside from this sporadic and anecdotal evidence, it remains unclear how many false results are published in science as a whole. The reason is that it is impossible to quantify the number of false positive results in science. Fortunately, it is not necessary to know the actual rate of false positives to test Ioannidis’s prediction that most published results are false positives. All we need to know is the discovery rate of a field (Soric, 1989). The discovery rate makes it possible to quantify the maximum percentage of false positive discoveries. If the maximum false discovery rate is well below 50%, we can reject Ioannidis’s hypothesis that most published results are false.

The empirical problem is that the observed discovery rate in a field may be inflated by publication bias. It is therefore necessary to estimate the amount of publication bias and if necessary correct the discovery rate, if publication bias is present.

In 2005, Ioannidis and Trikalinos (2005) developed their own test for publication bias, but this test had a number of shortcomings. First, it could be biased in heterogeneous literatures. Second, it required effect sizes to compute power. Third, it only provided information about the presence of publication bias and did not quantify it. Fourth, it did not provide bias-corrected estimates of the true discovery rate.

When the replication crisis became apparent in psychology, I started to develop new bias tests that address these limitations (Bartos & Schimmack, 2020; Brunner & Schimmack, 2020; Schimmack, 2012). The newest tool, called z-curve.2.0 (and yes, there is a app for that), overcomes all of the limitations of Ioannidis’s approach. Most important, it makes it possible to compute a bias-corrected discovery rate that is called the expected discovery rate. The expected discovery rate can be used to examine and quantify publication bias by comparing it to the observed discovery rate. Moreover, the expected discovery rate can be used to compute the maximum false discovery rate.

The Data

The data were compiled by Simon Schwab from the Cochrane database (https://www.cochrane.org/) that covers results from thousands of clinical trials. The data are publicly available (https://osf.io/xjv9g/) under a CC-By Attribution 4.0 International license (“Re-estimating 400,000 treatment effects from intervention studies in the Cochrane Database of Systematic Reviews”; (see also van Zwet, Schwab, & Senn, 2020).

Studies often report results for several outcomes. I selected only results for the primary outcome. It is often suggested that researchers switch outcomes to produce significant results. Thus, primary outcomes are the most likely to show evidence of publication bias, while secondary outcomes might even be biased to show more negative results for the same reason. The choice of primary outcomes also ensures that the test statistics are statistically independent because they are based on independent samples.

Results

I first fitted the default model to the data. The default model assumes that publication bias is present and only uses statistically significant results to fit the model. Z-curve.2.0 uses a finite mixture model to approximate the observed distribution of z-scores with a limited number of non-centrality parameters. After finding optimal weights for the components, power can be computed as the weighted average of the implied power of the components (Bartos & Schimmack, 2020). Bootstrapping is used to compute 95% confidence intervals that have shown to have good coverage in simulation studies (Bartos & Schimmack, 2020).

The main finding with the default model is that the model (grey curve) fits the observed distribution of z-scores very well in the range of significant results. However, z-curve has problems extrapolating from significant results to the distribution of non-significant results. In this case, the model (grey curve) underestimates the amount of non-significant results. Thus, there is no evidence of publication bias. This is seen in a comparison of the observed and expected discovery rates. The observed discovery rate of 26% is lower than the expected discovery rate of 38%.

When there is no evidence of publication bias, there is no reason to fit the model only to the significant results. Rather, the model can be fitted to the full distribution of all test statistics. The results are shown in Figure 2.

The key finding for this blog post is that the estimated discovery rate of 27% closely matches the observed discovery rate of 26%. Thus, there is no evidence of publication bias. In this case, simply counting the percentage of significant results provides a valid estimate of the discovery rate in clinical trials. Roughly one-quarter of trials end up with a positive result. The new question is how many of these results might be false positives.

To maximize the rate of false positives, we have to assume that true positives were obtained with maximum power (Soric, 1989). In this scenario, we could get as many as 14% (4 over 27) false positive results.

Even if we use the upper limit of the 95% confidence interval, we only get 19% false positives. Moreover, it is clear that Soric’s (1989) scenario overestimate the false discovery rate because it is unlikely that all tests of true hypotheses have 100% power.

In short, an empirical test of Ioannidis’s hypothesis that most published results in science are false shows that this claim is at best a wild overgeneralization. It is not true for clinical trials in medicine. In fact, the real problem is that many clinical trials may be underpowered to detect clinically relevant effects. This can be seen in the estimated replication rate of 61%, which is the mean power of studies with significant results. This estimate of power includes false positives with 5% power. If we assume that 14% of the significant results are false positives, the conditional power based on a true discovery is estimated to be 70% (14 * .05 + 86 * . 70 = .61).

With information about power, we can modify Soric’s worst case scenario and change power from 100% to 70%. This has only a small influence on the false positive discovery rate that decreases to 11% (3 over 27). However, the rate of false negatives increases from 0 to 14% (10 over 74). This also means that there are now three-times more false negatives than false positives (10 over 3).

Even this scenario overestimates power of studies that produced false negative results because power of studies with significant results is higher than power of studies that produced non-significant results when power is heterogenous (Brunner & Schimmack, 2020). In the worst case scenario, the null-hypothesis may rarely be true and power of studies with non-significant results could be as low as 14.5%. To explain, if we redo all of the studies, we expected that 61% of the significant studies produce a significant result again, producing 16.5% significant results. We also expect that the discovery rate will be 27% again. Thus, the remaining 73% of studies have to make up the difference between 27% and 16.5%, which is 10.5%. For 73 studies to produce 10.5 significant results, the studies have to have 14.5% power. 27 = 27 * .61 + 73 * .145.

In short, while Ioannidis predicted that most published results are false positives, it is much more likely that most published results are false negatives. This problem is of course not new. To make conclusions about effectiveness of treatments, medical researchers usually do not rely on a single clinical trial. Rather results of several studies are combined in a meta-analysis. As long as there is no publication bias, meta-analyses of original studies can boost power and reduce the risk of false negative results. It is therefore encouraging that the present results suggest that there is relatively little publication bias in these studies. Additional analyses for subgroups of studies can be conducted, but are beyond the main point of this blog post.

Conclusion

Ioannidis wrote an influential article that used hypothetical scenarios to make the prediction that most published results are false positives. Although this article is often cited as if it contained evidence to support this claim, the article contained no empirical evidence. Surprisingly, there also have been few attempts to test Ioannidis’s claim empirically. Probably the main reason is that nobody knew how to test it. Here I showed a way to test Ioannidis’s claim and I presented clear empirical evidence that contradicts this claim in Ioannidis’s own field of science, namely medicine.

The main feature that distinguishes science and fiction is not that science is always right. Rather, science is superior because proper use of the scientific method allows for science to correct itself, when better data become available. In 2005, Ioannidis had no data and no statistical method to prove his claim. Fifteen years later, we have good data and a scientific method to test his claim. It is time for science to correct itself and to stop making unfounded claims that science is more often wrong than right.

The danger of not trusting science has been on display this year, where millions of Americans ignored good scientific evidence, leading to the unnecessary death of many US Americans. So far, 330, 000 US Americans are estimated to have died of Covid-19. In a similar country like Canada, 14,000 Canadians have died so far. To adjust for population, we can compare the number of deaths per million, which is 1000 in the USA and 400 in Canada. The unscientific approach to the pandemic in the US may explain some of this discrepancy. Along with the development of vaccines, it is clear that science is not always wrong and can save lives. Iannaidis (2005) made unfounded claims that success stories are the exception rather than the norm. At least in medicine, intervention studies show real successes more often than false ones.

The Covid-19 pandemic also provides another example where Ioannidis used off-the-cuff calculations to make big claims without any evidence. In a popular article titled “A fiasco in the making” he speculated that the Covid-19 virus might be less deadly than the flu and suggested that policies to curb the spread of the virus were irrational.

As the evidence accumulated, it became clear that the Covid-19 virus is claiming many more lives than the flu, despite policies that Ioannidis considered to be irrational. Scientific estimates suggest that Covid-19 is 5 to 10 times more deadly than the flu (BNN), not less deadly as Ioannidis implied. Once more, Ioannidis quick, unempirical claims were contradicted by hard evidence. It is not clear how many of his other 1,000 plus articles are equally questionable.

To conclude, Ioannidis should be the last one to be surprised that several of his claims are wrong. Why should he be better than other scientists? The question is only how he deals with this information. However, for science it is not important whether scientists correct themselves. Science corrects itself by replacing old, false information with better information. One question is what science does with false and misleading information that is highly cited.

If YouTube can remove a video with Ioannidis’s false claims about Covid-19 (WP), maybe PLOS Medicine can retract an article with the false claim that “most published results in science are false”.

Washington Post

The attention-grabbing title is simply misleading because nothing in the article supports the claim. Moreover, actual empirical data contradict the claim at least in some domains. Most claims in science are not false and in a world with growing science skepticism spreading false claims about science may be just as deadly as spreading false claims about Covid-19.

If we learned anything from 2020, it is that science and democracy are not perfect, but a lot better than superstition and demagogy.

I wish you all a happier 2021.

Soric’s Maximum False Discovery Rate

Expected Discovery Rate, False Discovery Rate, Ioannidis, Most Published Research Findings are False, Observed Discovery Rate, Publication Bias, Science-Wise False Discovery Rate, Soric

Originally published January 31, 2020
Revised December 27, 2020

Psychologists, social scientists, and medical researchers often conduct empirical studies with the goal to demonstrate an effect (e.g., a drug is effective). They do so by rejecting the null-hypothesis that there is no effect, when a test statistic falls into a region of improbable test-statistics, p < .05. This is called null-hypothesis significance testing (NHST).

The utility of NHST has been a topic of debate. One of the oldest criticisms of NHST is that the null-hypothesis is likely to be false most of the time (Lykken, 1968). As a result, demonstrating a significant result adds little information, while failing to do so because studies have low power creates false information and confusion.

This changed in the 2000s, when the opinion emerged that most published significant results are false (Ioannidis, 2005; Simmons, Nelson, & Simonsohn, 2011). In response, there have been some attempts to estimate the actual number of false positive results (Jager & Leek, 2013). However, there has been surprisingly little progress towards this goal.

One problem for empirical tests of the false discovery rate is that the null-hypothesis is an abstraction. Just like it is impossible to say the number of points that make up the letter X, it is impossible to count null-hypotheses because the true population effect size is always unknown (Zhao, 2011, JASA).

An article by Soric (1989, JASA) provides a simple solution to this problem. Although this article was influential in stimulating methods for genome-wide association studies (Benjamin & Hochberg, 1995, over 40,000) citations, the article itself has garnered fewer than 100 citations. Yet, it provides a simple and attractive way to examine how often researchers may be obtaining significant results when the null-hypothesis is true. Rather than trying to estimate the actual false discovery rate, the method estimates the maximum false discovery rate. If a literature has a low maximum false discovery rate, readers can be assured that most significant results are true positives.

The method is simple because researchers do not have to determine whether a specific finding was a true or false positive result. Rather, the maximum false discovery rate can be computed from the actual discovery rate (i.e., the percentage of significant results for all tests).

The logic of Soric’s (1989) approach is illustrated in Tables 1.

NSSIG
TRUE06060
FALSE76040800
760100860
Table 1

To maximize the false discovery rate, we make the simplifying assumption that all tests of true hypotheses (i.e., the null-hypothesis is false) are conducted with 100% power (i.e., all tests of true hypotheses produce a significant result). In Table 1, this leads to 60 significant results for 60 true hypotheses. The percentage of significant results for false hypotheses (i.e., the null-hypothesis is true) is given by the significance criterion, which is set at the typical level of 5%. This means that for every 20 tests, there are 19 non-significant results and one false positive result. In Table 1 this leads to 40 false positive results for 800 tests.

In this example, the discovery rate is (40 + 60)/860 = 11.6%. Out of these 100 discoveries, 60 are true discoveries and 40 are false discoveries. Thus, the false discovery rate is 40/100 = 40%.

Soric’s (1989) insight makes it easy to examine empirically whether a literature tests many false hypotheses, using a simple formula to compute the maximum false discovery rate from the observed discovery rate; that is, the percentage of significant results. All we need to do is count and use simple math to obtain valuable information about the false discovery rate.

However, a major problem with Soric’s approach is that the observed discovery rate in a literature may be misleading because journals are more likely to publish significant results than non-significant results. This is known as publication bias or the file-drawer problem (Rosenthal, 1979). In some sciences, publication bias is a big problem. Sterling (1959; also Sterling et al., 1995) found that the observed discovery rated in psychology is over 90%. Rather than suggesting that psychologists never test false hypotheses, it rather suggests that publication bias is particularly strong in psychology (Fanelli, 2010). Using these inflated discovery rates to estimate the maximum FDR would severely understimate the actual risk of false positive results.

Recently, Bartoš and Schimmack (2020) developed a statistical model that can correct for publication bias and produce a bias-corrected estimate of the discovery rate. This is called the expected discovery rate. A comparison of the observed discovery rate (ODR) and the expected discovery rate (EDR) can be used to assess the presence and extent of publication bias. In addition, the EDR can be used to compute Soric’s maximum false discovery rate when publication bias is present and inflates the ODR.

To demonstrate this approach, I I use test-statistics from the journal Psychonomic Bulletin and Review. The choice of this journal is motivated by prior meta-psychological investigations of results published in this journal. Gronau, Duizer, Bakker, and Wagenmakers (2017) used a Bayesian Mixture Model to estimate that about 40% of results published in this journal are false positive results. Using Soric’s formula in reverse shows that this estimate implies that cognitive psychologists test only 10% true hypotheses (Table 3; 72/172 = 42%). This is close to Dreber, Pfeiffer, Almenber, Isakssona, Wilsone, Chen, Nosek, and Johannesson’s (2015) estimate of only 9% true hypothesis in cognitive psychology.

NSSIG
TRUE0100100
FALSE136872900
13681721000
Table 3

These results are implausible because rather different results are obtained when Soric’s method is applied to the results from the Open Science Collaboration (2015) project that conducted actual replication studies and found that 50% of published significant results could be replicated; that is, produced a significant results again in the replication study. As there was no publication bias in the replication studies, the ODR of 50% can be used to compute the maximum false discovery rate, which is only 5%. This is much lower than the estimate obtained with Gronau et al.’s (2018) mixture model.

I used an R-script to automatically extract test-statistics from articles that were published in Psychonomic Bulletin and Review from 2000 to 2010. I limited the analysis to this period because concerns about replicability and false positives might have changed research practices after 2010. The program extracted 13,571 test statistics.

Figure 1 shows clear evidence of selection bias. The observed discovery rate of 70% is much higher than the estimated discovery rate of 35% and the 95%CI of the EDR, 25% to 53% does not include the ODR. As a result, the ODR produces an inflated estimate of the actual discover rate and cannot be used to compute the maximum false discovery rate.

However, even with a much lower estimated discovery rate of 36%, the maximum false discovery rate is only 10%. Even with the lower bound of the confidence interval for the EDR of 25%, the maximum FDR is only 16%.

Figure 2 shows the results for a replication with test statistics from 2011 to 2019. Although changes in research practices could have produced different results, the results are unchanged. The ODR is 69% vs. 70%; the EDR is 38% vs. 35% and the point estimate of the maximum FDR is 9% vs. 10%. This close replication also implies that research practices in cognitive psychology have not changed over the past decade.

The maximum FDR estimates of 10% confirms the results based on the replication rate in a small set of actual replication studies (OSC, 2015) with a much larger sample of test statistics. The results also show that Gronau et al.’s mixture model produces dramatically inflated estimates of the false discovery rate (see also Brunner & Schimmack, 2019, for a detailed discussion of their flawed model).

In contrast to cognitive psychology, social psychology has seen more replication failures. The OSC project estimated a discovery rate of only 25%. Even this low rate would imply that a maximum of 16% of discoveries in social psychology are false positives. A z-curve analysis of a representative sample of 678 focal tests in social psychology produced an estimated discovery rate of 19% with a 95%CI ranging from 6% to 36% (Schimmack, 2020). The point estimate implies a maximum FDR of 22%, but the lower limit of the confidence interval allows for a maximum FDR of 82%. Thus, social psychology may be a literature where most published results are false. However, the replication crisis in social psychology should not be generalized to other disciplines.

Conclusion

Numerous articles have made claims that false discoveries are rampant (Dreber et al., 2015; Gronau et al., 2015; Ioannidis, 2005; Simmons et al., 2011). However, these articles did not provide empirical data to support their claim. In contrast, empirical studies of the false discovery risk usually show much lower rates of false discoveries (Jager & Leek, 2013), but this finding has been dismissed (Ioannidis, 2014) or ignored (Gronau et al., 2018). Here I used a simpler approach to estimate the maximum false discovery rate and showed that most significant results in cognitive psychology are true discoveries. I hope that this demonstration revives attempts to estimate the science-wise false discovery rate (Jager & Leek, 2013) rather than relying on hypothetical scenarios or models that reflect researchers’ prior beliefs that may not match actual data (Gronau et al., 2018; Ioannidis, 2005).

References

Bartoš, F., & Schimmack, U. (2020, January 10). Z-Curve.2.0: Estimating Replication Rates and Discovery Rates. https://doi.org/10.31234/osf.io/urgtn

Dreber A., Pfeiffer T., Almenberg, J., Isaksson S., Wilson B., Chen Y., Nosek B. A.,  Johannesson, M. (2015). Prediction markets in science. Proceedings of the National Academy of Sciences, 50, 15343-15347. DOI: 10.1073/pnas.1516179112

Fanelli D (2010) Positive” Results Increase Down the Hierarchy of the Sciences. PLOS ONE 5(4): e10068. https://doi.org/10.1371/journal.pone.0010068

Gronau, Q. F., Duizer, M., Bakker, M., & Wagenmakers, E.-J. (2017). Bayesian mixture modeling of significant p values: A meta-analytic method to estimate the degree of contamination from H₀. Journal of Experimental Psychology: General, 146(9), 1223–1233. https://doi.org/10.1037/xge0000324

Ioannidis JPA (2005) Why Most Published Research Findings Are False. PLOS Medicine 2(8): e124. https://doi.org/10.1371/journal.pmed.0020124

Ioannidis JP. (2014). Why “An estimate of the science-wise false discovery rate and application to the top medical literature” is false. Biostatistics, 15(1), 28-36.
DOI: 10.1093/biostatistics/kxt036.

Jager, L. R., & Leek, J. T. (2014). An estimate of the science-wise false discovery rate and application to the top medical literature. Biostatistics, 15(1), 1-12.
DOI: 10.1093/biostatistics/kxt007

Lykken, D. T. (1968). Statistical significance in psychological research. Psychological Bulletin, 70(3, Pt.1), 151–159. https://doi.org/10.1037/h0026141

Open Science Collaboration. (2015). Estimating the reproducibility of psychological science. Science, 349(6251), 1–8.

Schimmack, U. (2019). The Bayesian Mixture Model is fundamentally flawed. https://replicationindex.com/2019/04/01/the-bayesian-mixture-model-is-fundamentally-flawed/

Schimmack, U. (2020). A meta-psychological perspective on the decade of replication failures in social psychology. Canadian Psychology/Psychologie canadienne, 61(4), 364–376. https://doi.org/10.1037/cap0000246

Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2011). False-Positive Psychology: Undisclosed Flexibility in Data Collection and Analysis Allows Presenting Anything as Significant. Psychological Science, 22(11), 1359–1366. 
https://doi.org/10.1177/0956797611417632

Soric, B. (1989). Statistical “Discoveries” and Effect-Size Estimation. Journal of the American Statistical Association, 84(406), 608-610. doi:10.2307/2289950

Zhao, Y. (2011). Posterior Probability of Discovery and Expected Rate of Discovery for Multiple Hypothesis Testing and High Throughput Assays. Journal of the American Statistical Association, 106, 984-996, DOI: 10.1198/jasa.2011.tm09737