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Is the Risk Difference Really a More Heterogeneous Measure?

Schmidt, Amand F.; Dudbridge, Frank; Groenwold, Rolf H. H.

doi: 10.1097/EDE.0000000000000444

Faculty of Population Health, Institute of Cardiovascular Science, University College London, London, United Kingdom,

Department of Non-Communicable Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom

Julius Center for Health Sciences and Primary Care, University Medical Center Utrecht, Utrecht, The Netherlands

Division of Pharmacoepidemiology and Clinical Pharmacology, Utrecht Institute for Pharmaceutical Sciences, Utrecht, The Netherlands

The authors report no conflicts of interest.

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To the Editor:

With great interest did we read the much-needed commentary by Poole et al.1 on the empirical evidence behind claims that the risk difference (RD) is more heterogeneous (i.e., the difference in effects between studies or groups of patients) than the odds ratio (OR). In their contribution, the authors show that the previously reported higher rejection rates of RD homogeneity may be explained by differences in power between measurement scales, an issue we previously addressed as well.2 Without detracting from their contribution or conclusions, with which we agree, we were surprised that the authors omitted the theoretical grounds why the OR is thought to be a less heterogeneous measure.

Essentially, our comment is that the OR can be homogeneous given any conceivable distribution of risk in patient subgroups, while the risk ratio (RR) and RD cannot. Take, for example, the hypothetical study mentioned by Poole et al., where the average risk in the control group was 0.27, and the average risk in the treated group 0.46 (scenario 1), resulting in an OR of 2.30, a RR of 1.70, and a RD of 0.19. If we now think it acceptable that the risk for any subgroup of control subjects is only bounded by 0 and 1, given the RD of 0.19, there cannot be any subgroup of control subjects with a risk higher than 0.81 unless the RD is heterogeneous. In general, the RD can be homogenous if the control group risk is bounded between max(0, 0-RD) and min(1-RD, 1). Similarly, for the RR to be homogeneous, the risk in any subgroup of control subjects should be bounded between 0/RR and 1/RR, for the current example 0 and 0.59. The OR, however, can be homogeneous for any control group risk between 0 and 1. Given that the OR never “forces” heterogeneity, one might expect the OR to be the least heterogeneous in empirical settings as well. However, one may question whether these bounds are actually violated often in empirical settings; therefore, we agree with Poole et al. that there is insufficient empirical evidence to claim any effect measure induced heterogeneity, and furthermore that comparisons between scales are difficult.

In the end, sound biological reasoning on potential pathways may provide the most suitable grounds for choosing an effect measure, not mathematical or statistical properties. However, to facilitate such a choice, knowledge of these properties is, we feel, essential.

Amand F. Schmidt

Faculty of Population Health

Institute of Cardiovascular Science

University College London

London, United Kingdom

Frank Dudbridge

Department of Non-Communicable Disease Epidemiology

London School of Hygiene and Tropical Medicine

London, United Kingdom

Rolf H. H. Groenwold

Julius Center for Health Sciences and Primary Care

University Medical Center Utrecht

Utrecht, The Netherlands

Division of Pharmacoepidemiology and Clinical Pharmacology

Utrecht Institute for Pharmaceutical Sciences

Utrecht, The Netherlands

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1. Poole C, Shrier I, VanderWeele TJ.. Is the risk difference really a more heterogeneous measure? Epidemiology. 2015;26:714–718
2. Schmidt AF, Groenwold RH, Knol MJ, et al. Exploring interaction effects in small samples increases rates of false-positive and false-negative findings: results from a systematic review and simulation study. J Clin Epidemiol. 2014;67:821–829
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