To the Editor:
In a recent letter to the editor, Juárez García et al.1 suggest that: “Testing alternative ways of defining effort–reward imbalance does not result in a bias if all tests are reported.” This statement is problematic because it implies that there are many different ways to test the effort–reward imbalance theory.
Before going into details, we must establish a basic fact: a ratio is a multiplication where the denominator is inverted:
Thus, effort–reward imbalance is a special case of the standard interaction model, where reward has been inverted to form the denominator. Using this notation, and assuming additive/linear data, a popular operationalization of effort–reward imbalance is presented in equation 2 below. The standard interaction model,2 which also adjusts for the main effects, is presented in equation 3:
Juárez García et al.1 refer to a study applying a dichotomized version of equation 2, together with other covariates and a link function to model exponential data and estimate a hazard ratio, indicating an association with coronary heart disease. We have argued elsewhere3 that this association is biased because it is not adjusted for confounding by the main effects; it is likely that the reported association is explained by a single univariate association with reward, which was demonstrated on data in the same study. Only equation 3 produces an estimate of the interaction that is not confounded by the main effects2; thus, only equation 3 provides an unbiased estimate of effort–reward imbalance.
We agree with Juárez García et al.1 that researchers should commit to report all models fitted in a study regardless of the result to minimize selective reporting bias. However, none of the alternatives they suggest produce unbiased estimates in observational studies because they are all susceptible to confounding by the main effects; and we cannot assume a priori that the (omitted) main effects are not associated with the outcome variable. One of their proposals is ambiguous: “An interaction term of effort and reward.” If it refers to the standard interaction model,2 similar to equation 3 above, it would indeed produce an unbiased estimate of the interaction, but if the interaction term is fitted alone, without including the constituent main effects, it is susceptible to bias just like equation 2.4
Department of Clinical Neuroscience
Institute for Globally Distributed Open Research and Education Stockholm
Sweden, [email protected]
Johan Hviid Andersen
Department of Occupational Medicine
Danish Ramazzini Centre
Regional Hospital West Jutland University Research Clinic
Department of Occupational and Environmental Medicine
Bispebjerg University Hospital
1. Juárez García A, Choi B, Landsbergis P. Re: effort-reward imbalance at work and incident coronary heart disease. Epidemiology. 2018;29:e12e13.
2. Kronmal RA. Spurious correlation and the fallacy of the ratio standard revisited. J R Stat Soc Ser A Stat Soc. 1993;156:379392.
3. Mikkelsen S, Andersen JH, Ingre M. Artefacts Re. Effort–Reward Imbalance at Work and Incident Coronary Heart Disease. Epidemiology. 2018; 29:e35
4. Ingre M. P-hacking in academic research: a critical review of the job strain model and of the association between night work and breast cancer in women. 2017. Available at: http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-141136
. Accessed 10 April 2018