Tag Archives: Meta-Regression

Using the R-index to detect questionable research practices in SSRI studies

Amna Shakil and Ulrich Schimmack

In 2008, Turner and colleagues (2008) examined the presence of publication bias in clinical trials of antidepressants. They found that out of 74 FDA-registered studies, 51% showed positive results. However, positive results were much more likely to be published, as 94% of the published results were positive. There were two reasons for the inflated percentage of positive results. First, negative results were not published. Second, negative results were published as positive results. Turner and colleagues’ (2008) results received a lot of attention and cast doubt on the effectiveness of anti-depressants.

A year after Turner and colleagues (2008) published their study, Moreno, Sutton, Turner, Abrams, Cooper and Palmer (2009) examined the influence of publication bias on the effect-size estimate in clinical trials of antidepressants. They found no evidence of publication bias in the FDA-registered trials, leading the researchers to conclude that the FDA data provide an unbiased gold standard to examine biases in the published literature.

The effect size for treatment with anti-depressants in the FDA data was g = 0.31, 95% confidence interval 0.27 to 0.35. In contrast, the uncorrected average effect size in the published studies was g = 0.41, 95% confidence interval 0.37 to 0.45. This finding shows that publication bias inflates effect size estimates by 32% ((0.41 – 0.31)/0.31).

Moreno et al. (2009) also used regression analysis to obtain a corrected effect size estimate based on the biased effect sizes in the published literature. In this method, effect sizes are regressed on sampling error under the assumption that studies with smaller samples (and larger sampling error) have more bias. The intercept is used as an estimate of the population effect size when sampling error is zero. This correction method yielded an effect size estimate of g = 0.29, 95% confidence interval 0.23 to 0.35, which is similar to the gold standard estimate (.31).

The main limitation of the regression method is that other factors can produce a correlation between sample size and effect size (e.g., higher quality studies are more costly and use smaller samples). To avoid this problem, we used an alternative correction method that does not make this assumption.

The method uses the R-Index to examine bias in a published data set. The R-Index increases as statistical power increases and it decreases when publication bias is present. To obtain an unbiased effect size estimate, studies are selected to maximize the R-Index.

Since the actual data files were not available, graphs A and B from Moreno et al.’s (2009) study were used to obtain information about effect size and sample error of all the FDA-registered and the published journal articles.

The FDA-registered studies had the success rate of 53% and the observed power of 56%, resulting in an inflation of close to 0. The close match between the success rate and observed confirms FDA studies are not biased. Given the lack of bias (inflation), the most accurate estimate of the effect size is obtained by using all studies.

The published journal articles had a success rate of 86% and the observed power of 73%, resulting in the inflation rate of 12%. The inflation rate of 12% confirms that the published data set is biased. The R-Index subtracts the inflation rate from observed power to correct for inflation. Thus, the R-Index for the published studies is 73-12 = 61. The weighted effect size estimate was d = .40.

The next step was to select sets of studies to maximize the R-Index. As most studies were significant, the success rate could not change much. As a result, most of the increase would be achieved by selecting studies with higher sample sizes in order to increase power. The maximum R-Index was obtained for a cut-off point of N = 225. This left 14 studies with a total sample size of 4,170 participants. The success rate was 100% with median observed power of 85%. The Inflation was still 15%, but the R-Index was higher than it was for the full set of studies (70 vs. 61). The weighted average effect size in the selected set of powerful studies was d = .34. This result is very similar to the gold standard in the FDA data. The small discrepancy can be attributed to the fact that even studies with 85% power still have a small bias in the estimation of the true effect size.

In conclusion, our alternative effect size estimation procedure confirms Moreno et al.’s (2009) results using an alternative bias-correction method and shows that the R-Index can be a valuable tool to detect and correct for publication bias in other meta-analyses.

These results have important practical implications. The R-Index confirms that published clinical trials are biased and can provide false information about the effectiveness of drugs. It is therefore important to ensure that clinical trials are preregistered and that all results of clinical trials are published. The R-Index can be used to detect violations of these practices that lead to biased evidence. Another important finding is that clinical trials of antidepressants do show effectiveness and that antidepressants can be used as effective treatments of depression. The presence of publication bias should not be used to claim that antidepressants lack effectiveness.

References

Moreno, S. G., Sutton, A. J., Turner, E. H., Abrams, K. R., Cooper, N. J., Palmer, T. M., & Ades, A. E. (2009). Novel methods to deal with publication biases: secondary analysis of antidepressant trials in the FDA trial registry database and related journal publications. Bmj, 339, b2981.

Turner, E. H., Matthews, A. M., Linardatos, E., Tell, R. A., & Rosenthal, R. (2008). Selective publication of antidepressant trials and its influence on apparent efficacy. New England Journal of Medicine, 358(3), 252-260.

The R-Index of Nicotine-Replacement-Therapy Studies: An Alternative Approach to Meta-Regression

Stanley and Doucouliagos (2013) demonstrated how meta-regression can be used to obtain unbiased estimates of effect sizes from a biased set of original studies. The regression approach relies on the fact that small samples often need luck or questionable practices to produce significant results, whereas large samples can show true effects without the help of luck and questionable practices. If questionable practices or publication bias are present, effect sizes in small samples are inflated and this bias is evident in a regression of effect sizes on sampling error. When bias is present, the intercept of the regression equation can provide a better estimate of the average effect size in a set of studies.

One limitation of this approach is that other factors can also produce a correlation between effect size and sampling error. Another problem is that the regression equation can only approximate the effect of bias on effect size estimates.

The R-Index can complement meta-regression in several ways. First, it can be used to examine whether a correlation between effect size and sampling error reflects bias. If small samples have higher effect sizes due to bias, they should also yield more significant results than the power of these studies justifies. If this is not the case, the correlation may simply show that smaller samples examined stronger effects. Second, the R-Index can be used as an alternative way to estimate unbiased effect sizes that does not rely on the relationship between sample size and effect size.

The usefulness of the R-Index is illustrated with Stanley and Doucouliagos (2013) meta-analysis of the effectiveness of nicotine replacement therapy (the patch). Table A1 lists sampling errors and t-values of 42 studies. Stanley and Doucouliagos (2013) found that the 42 studies suggested a reduction in smoking by 93%, but that effectiveness decreased to 22% in a regression that controlled for biased reporting of results. This suggests that published studies inflate the true effect by more than 300%.

I entered the t-values and standard errors into the R-Index spreadsheet. I used sampling error to estimate sample sizes and degrees of freedom (2 / sqrt [N]). I used one-tailed t-tests to allow for negative t-values because the sign of effects is known in a meta-analysis of studies that try to show treatment effects. Significance was tested using p = .025, which is equivalent to using .050 in the test of significance for two-tailed tests (z > 1.96).

The R-Index for all 42 studies was 27%. The low R-Index was mostly explained by the low power of studies with small samples. Median observed power was just 34%. The number of significant results was only slightly higher 40%. The inflation rate was only 7%.

As studies with low power add mostly noise, Stanley (2010) showed that it can be preferable to exclude them from estimates of actual effect sizes. The problem is that it is difficult to find a principled way to determine which studies should be included or excluded. One solution is to retain only studies with large samples. The problem with this approach is that this often limits a meta-analysis to a small set of studies.

One solution is to compute the R-Index for different sets of studies and to base conclusions on the largest unbiased set of studies. For the 42 studies of nicotine replacement therapy, the following effect size estimates were obtained (effect sizes are d-values, d = t * se).

NicotinePatch

The results show the highest R-Index for studies with more than 80 participants. For these studies, observed power is 83% and the percentage of significant results is also 83%, suggesting that this set of studies is an unbiased sample of studies. The weighted average effect size for this set of studies is d = .44. The results also show that the weighted average effect size does not change much as a function of the selection of studies. When all studies are included, there is evidence of bias (8% inflation) and the weighted average effect size is inflated, but the amount of inflation is small (d = .56 vs. d = .44, difference d = .12).

The small amount of bias appears to be inconsistent with Stanley and Doucouliagos (2013) estimate that an uncorrected meta-analysis overestimates the true effect size by over 300% (93% vs. 22% RR). I therefore also examined the log(RR) values in Table 1a.

The average is .68 (compared to the simple mean reported as .66); the median is .53 and the weighted average is .49.   The regression-corrected estimate reported by Stanley and Doucouliagos (2013) is .31. The weighted mean for studies with more than 80 participants is .43. It is now clear why Stanley and Doucouliagos (2013) reported a large effect of the bias correction. First, they used the simple mean as a comparison standard (.68 vs. 31). The effect would be smaller if they had used the weighted mean as a comparison standard (.49 vs. .31). Another factor is that the regression procedure produces a lower estimate than the R-Index approach (.31 vs. 43). More research is needed to compare these results, but the R-Index has a simple logic. When there is no evidence of bias, the weighted average provides a reasonable estimate of the true effect size.

Conclusion

Stanley and Doucouliagos (2013) used regression of effect sizes on sampling error to reveal biases and to obtain an unbiased estimate of the typical effect size in a set of studies. This approach provides a useful tool in the fight against biased reporting of research results. One limitation of this approach is that other factors can produce a correlation between sampling error and effect size. The R-Index can be used to examine how much reporting biases contribute to this correlation. The R-Index can also be used to obtain an unbiased estimate of effect size by computing a weighted average for a select set of studies with a high R-Index.

A meta-analysis of 42 studies of nicotine replacement theory illustrates this approach. The R-Index for the full set of studies was low (24%). This reveals that many studies had low power to demonstrate an effect. These studies provide little information about effectiveness because non-significant results are just as likely to be type-II errors as demonstrations of low effectiveness.

The R-Index increased when studies with larger samples were selected. The maximum R-Index was obtained for studies with at least 80 participants. In this case, observed power was above 80% and there was no evidence of bias. The weighted average effect size for this set of studies was only slightly lower than the weighted average effect size for all studies (log(RR) = .43 vs. .49, RR = 54% vs. 63%, respectively). This finding suggests that smokers who use a nicotine patch are about 50% more likely to quit smoking than smokers without a nicotine patch.

The estimate of 50% risk reduction challenges Stanley and Doucouliagos’s (2013) preferred estimate that bias correction “reduces the efficacy of the patch to only 22%.” The R-Index suggests that this bias-corrected estimate is itself biased.

Another important conclusion is that studies with low power are wasteful and uninformative. They generate a lot of noise and are likely to be systematically biased and they contribute little to a meta-analysis that weights studies by sample size. The best estimate of effect size was based on only 6 out of 42 studies. Researchers should not conduct studies with low power and editors should not publish studies with low power.