Simpson's paradox visualized: The example of the Rosiglitazone meta-analysis
Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Germany
BMC Medical Research Methodology 2008, 8:34 doi:10.1186/1471-2288-8-34Published: 30 May 2008
Simpson's paradox is sometimes referred to in the areas of epidemiology and clinical research. It can also be found in meta-analysis of randomized clinical trials. However, though readers are able to recalculate examples from hypothetical as well as real data, they may have problems to easily figure where it emerges from.
First, two kinds of plots are proposed to illustrate the phenomenon graphically, a scatter plot and a line graph. Subsequently, these can be overlaid, resulting in a overlay plot. The plots are applied to the recent large meta-analysis of adverse effects of rosiglitazone on myocardial infarction and to an example from the literature. A large set of meta-analyses is screened for further examples.
As noted earlier by others, occurrence of Simpson's paradox in the meta-analytic setting, if present, is associated with imbalance of treatment arm size. This is well illustrated by the proposed plots. The rosiglitazone meta-analysis shows an effect reversion if all trials are pooled. In a sample of 157 meta-analyses, nine showed an effect reversion after pooling, though non-significant in all cases.
The plots give insight on how the imbalance of trial arm size works as a confounder, thus producing Simpson's paradox. Readers can see why meta-analytic methods must be used and what is wrong with simple pooling.