Meta-analysis of skewed data: combining results reported on log-transformed or raw scales

Abstract

When literature-based meta-analyses involve outcomes with skewed distributions, the best available data can sometimes be a mixture of results presented on the raw scale and results presented on the logarithmic scale. We review and develop methods for transforming between these results for two-group studies, such as clinical trials and prospective or cross-sectional epidemiological studies. These allow meta-analyses to be conducted using all studies and on a common scale. The methods can also be used to produce a meta-analysis of ratios of geometric means when skewed data are reported on the raw scale for every study. We compare three methods, two of which have alternative standard error formulae, in an application and in a series of simulation studies. We conclude that an approach based on a log-normal assumption for the raw data is reasonably robust to different true distributions; and we provide new standard error approximations for this method. An assumption can be made that the standard deviations in the two groups are equal. This increases precision of the estimates, but if incorrect can lead to very misleading results.

Figure 1

Epidemiological studies of D9N polymorphism in the lipoprotein lipase gene and triglyceride levels: meta-analyses on the raw triglyceride scale, using various conversions from the logarithmic to the raw scale. Where conversions are made, methods are ordered as follows: Method 1 (separate variances, t-test SE); Method 1 (separate variances, Taylor SE); Method 2 (equal variances, t-test SE); Method 2 (equal variances, Taylor SE); Method 3.

Figure 2

Epidemiological studies of D9N polymorphism in the lipoprotein lipase gene and triglyceride levels: meta-analyses on the log triglyceride scale using various conversions from the raw to the logarithmic scale. Where conversions are made, methods are ordered as follows: Method 1 (separate variances, t-test SE); Method 1 (separate variances, Taylor SE); Method 2 (equal variances, t-test SE); Method 2 (equal variances, Taylor SE); Method 3.

Figure 3

Probability densities of all distributions used in the simulation study. Solid lines represent group 1 and dotted lines group 2.