The impact of bias due to exponentiation in the estimation of hazard, risk, and odds ratios: an empirical investigation from 1,495,059 effect sizes from MEDLINE/PubMed abstracts

Abstract

Background: Parameter estimation using regression methods plays a vital role in medical research. Often a non-linear transformation of a regression parameter is preferred for its more intuitive interpretation. Important examples in medical research are odds ratios, risk ratios, and hazard ratios, which are obtained by exponentiating the estimated regression coefficients of the logit link binomial generalized linear model, log link Poisson generalized linear model or Cox proportional hazards model, respectively. A lot of attention has been devoted to studying and removing the bias of the estimators on the scale of the regression, but the bias of the transformed parameters is rarely addressed.

Methods: Two approaches for reducing the bias due to the exponentiation are reviewed and applied to odds ratios, risk ratios, and hazard ratios reported in the abstracts published in the MEDLINE subset of English-language PubMed records.

Results: We show that correcting for the bias due to the exponentiation may yield substantially different estimates, potentially resulting in a large shrinkage of the reported effect size estimates.

Conclusion: Given the wide availability of methods to reduce the bias on the scale of regression, we encourage their routine use to improve estimation. In situations where the consequences of biased estimation are larger at the exponentiated scale than at the scale of regression, as for example in some policy and planning settings, we additionally encourage the removal of the bias due to the exponentiation.

Keywords: Bias reduction; Brglm2; Odds/risk/hazard ratios; Parameter estimation; Shrinkage; Text mining.

Fig. 1

Shrinkage percentage as a function of the estimated standard error

Fig. 2

Outline of the data cleaning process for the extracted effect sizes (ORs, RRs, and HRs), lower limits (LL) and upper limits (UL) of the confidence interval (CI), and the CI level

Fig. 3

Distribution of $\widehat{\theta }$ and $s_{\widehat{\theta }}$ in the first row and shrinkage percentage given ${\tilde{\psi }}^{\mathit{LA}}$ and ${\tilde{\psi }}^{\mathit{TA}}$ in the second and third row, respectively, for each effect type separately. The box plots in the first row represent the distribution of $\widehat{\theta }$ at the top and $s_{\widehat{\theta }}$ on the right side of the plot. To improve the visibility of the plot, the plots were cropped to the $\widehat{\theta }$ range [− 20, 20] and $s_{\widehat{\theta }}$ range [0, 7]. As a consequence, two observations, both odds ratios, were omitted. Only effect sizes with 95% confidence intervals are presented. The black line marks the decision boundary for the evidence of an effect before bias adjustment. The red and black colors denote values suggesting the evidence of an effect and values suggesting the evidence of no effect, respectively, before making the bias adjustment