Re-expressing coefficients from regression models for inclusion in a meta-analysis

Background: Quantitative summaries of study results, or meta-analyses, have proven useful in several fields, but pose special challenges when applied to non-randomized studies of exposure in relation to outcome. One such issue arises when summarizing regression results when some studies have log-transformed exposure while others have not. Methods of re-expression have recently been developed, with the goal of generating comparable results across studies regardless of data transformation. We examined the validity of three re-expression methods using simulations and real data examples. Methods: Simulations were developed to examine the validity of the re-expression methods when the exposure had a skewed distribution. We also identified 15 published studies for which we were able to either access the data and replicate the result, or which had provided regression coefficients (β) from analyses with both log-transformed and untransformed exposure data. For the simulated and real data we re-expressed β as though it had been based on untransformed data, and vice versa. We compared the re-expressed results to fitted regression coefficients. The proportional difference from the fitted β was used to quantitate bias. We also examined the effect of influential observations on the success of the re-expression using the real data examples. Results: In the simulated and real data examples, all three re-expression methods usually gave biased results. The method of Rodriguez-Barranco et al. (2017) tended to give less biased results but with a wider range of bias. For example, in the real data, when re-expressing results as if they had been fit to untransformed exposure, the median estimate was 22% too high, with a range of -40% to 1,904%. In some analyses, the presence of influential observations had a large effect on the validity of the re-expression. Conclusions: When the distribution of exposure is skewed, the re-expression methods examined are likely to give biased results. The bias varied by method, the direction of the re-expression, skewness, influential observations, and in some cases, the median exposure. Meta-analysts using any of these re-expression methods may want to consider the uncertainty, the likely direction and degree of bias, and conduct sensitivity analyses on the re-expressed results.