Sample Size and Power Calculations for Additive Interactions.

The literature on power and sample size calculations for interaction has focused on the multiplicative scale (Lubin and Gail, 1990; Hwang et al., 1994; Foppa and Spiegelman, 1997; Yang et al., 1999; Garcia-Closas and Lubin, 1999; Qiu et al., 2000; Luan et al. 2001; Gauderman, 2002a b; Sturmer, 2002; Wang et al., 2003; Wang and Zhao, 2003; Demidenko, 2008; VanderWeele, 2011). However, interaction on the additive scale is more relevant for public health purposes (Rothman et al.,1980; Rothman et al., 2008) and is also more closely related to notions of mechanistic synergism within the sufficient cause framework (Rothman, 1976; VanderWeele and Robins, 2007, 2008; Rothman et al., 2008). Arguably, the reason interaction is most frequently assessed on the multiplicative scale is that this is what is most easily computed from the output of standard logistic regression software. In addition, in the context of case-control studies, odds ratios can be estimated but risk differences cannot, unless additional information concerning e.g. the prevalence of the outcome or exposures in the underlying population is available (Rothman et al., 2008). This again leads to the multiplicative scale as being the default for assessing interaction. That power and sample size calculations are better developed for multiplicative interaction than for additive interaction perhaps further encourages the use of the multiplicative scale for interaction assessment. However, measures of additive interaction based on risk ratios or odds ratios using the relative excess risk due to interaction (“RERI”; Rothman, 1986) can easily be calculated from logistic regression with either cohort or case-control data (Hosmer and Lemeshow, 1992) and in this paper we will derive power and sample size formulae for interaction on the additive scale. Power and sample size calculations for additive interaction were discussed in Greenland (1983, 1985) but no closed form expressions were provided. In this paper, we will consider measures of additive interaction based on absolute risks and also on the relative excess risk due to interaction for both cohort and case-control data and we will provide closed form analytic expressions for power and sample size in each of these cases. Analytically, we will for the most part follow the development of Demidenko (2008) who considered multiplicative interaction but we will be taking a similar approach for the additive scale. We will see that when main effects of both exposures are positive, power to detect positive interaction on the additive scale will be greater than that on the multiplicative scale, providing yet another reason, beyond public health relevance and relation to mechanistic synergism, for using the additive scale to assess interaction. The reader who is interested in only the application of the power and sample size formulae derived in this paper is referred to Appendix 2 at the end of the paper on epidemiologic practice. This appendix gives instructions on using Excel spreadsheets (included as an online supplement to this paper) to automatically carry out power and sample size calculations for additive and multiplicative interaction for cohort, case-control and case-only data.