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Confidence Intervals for the Interaction Contrast Ratio

Kuss, Oliver; Schmidt-Pokrzywniak, Andrea; Stang, Andreas

doi: 10.1097/EDE.0b013e3181cc9bfc

Institute of Medical Epidemiology, Biostatistics, and Informatics; Martin-Luther-University of Halle-Wittenberg; Halle (Saale), Germany (Kuss)

Institute of Clinical Epidemiology; Martin-Luther-University of Halle-Wittenberg; Halle (Saale), Germany (Schmidt-Pokrzywniak, Stang)

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To the Editor:

The interaction contrast ratio,1 formerly called the relative excess risk due to interaction, is a well-established quantitative measure for interaction in epidemiologic studies. Estimates for the interaction contrast ratio can be easily derived from an ordinary logistic regression model with an interaction term. However, calculation of confidence intervals is complicated by the fact that the interaction contrast ratio is a nonlinear combination of the parameters from this model. As such, bootstrapping or the multivariate delta method must be used to compute confidence intervals. Software code to accomplish this has been published in this journal2 and elsewhere.3–5 However, these codes are long and somewhat difficult to use. We show here that fitting the original model with SAS PROC NLMIXED or SAS PROC NLP provides a simple solution.

For illustration, we use an example on the association of sun protection, iris color, and uveal melanoma from a recent case-control study.6 Actually, this example initiated our research reported here, as the SAS code by Lundberg2 did not give confidence intervals for the interaction contrast ratio. SAS code 1 in the eAppendix ( provides the data set and PROC LOGISTIC code for fitting the ordinary logistic regression model. With this code we can use the published Lundberg code to compute the interaction contrast ratio and its corresponding confidence interval. We propose instead to fit the standard logistic regression with PROC NLMIXED, using code 2 in the eAppendix. Compared with PROC LOGISTIC this involves some additional effort, as we are forced to write down explicitly the parameters to be estimated and the model equation. However, this effort is more than compensated by the opportunity to calculate functions of model parameters via the ESTIMATE statement, where these functions do not have to be linear. As such, the additional ESTIMATE statement codes the standard formula7 for the interaction contrast ratio. PROC NLMIXED calculates confidence intervals for these additional estimates via the multivariate delta method. From the NLMIXED code we find an estimated interaction contrast ratio of 0.636 with a 95% confidence interval of −0.234 to 1.506.

As a limitation of this first proposal we note that PROC NLMIXED computes only Wald confidence intervals (or such that use the t distribution). Zou4 showed that these rely on the assumption of symmetric confidence intervals for risk ratios, and Richardson and Kaufman5 argue that likelihood-based confidence intervals should be used instead. To account for this, PROC NLP from the SAS/OR module can be used, as this procedure allows the computation of profile likelihood confidence intervals. Unfortunately, PROC NLP does not allow estimating nonlinear combinations of parameter estimates, and we are forced to reparametrize the model such that the interaction contrast ratio (for which we like to have profile likelihood confidence intervals) becomes an original model parameter. Straightforward calculation yields:

The reparametrized model can be fit via PROC NLP with code 3 in the eAppendix. Submitting this code, we again find an estimated interaction contrast ratio of 0.636, this time with a 95%-confidence interval of −0.365 to 1.529. As compared with the Wald interval, both profile likelihood interval bounds are larger in absolute value whereas this difference is larger for the lower bound.

In summary, we propose 2 methods for computing confidence intervals for the interaction contrast ratio in SAS that we feel are simpler and more robust (at least, in our case) than previously proposed methods. The ideas reported here could also be used to compute Wald and profile likelihood confidence intervals for the 2 other commonly used measures of interaction, the attributable proportion due to interaction and the synergy index.

Oliver Kuss

Institute of Medical Epidemiology, Biostatistics, and Informatics

Martin-Luther-University of Halle-Wittenberg

Halle (Saale), Germany

Andrea Schmidt-Pokrzywniak

Andreas Stang

Institute of Clinical Epidemiology

Martin-Luther-University of Halle-Wittenberg

Halle (Saale), Germany

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1. Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 3rd ed. Philadelphia: Wolters Kluwer, Lippincott Williams & Wilkins; 2008.
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6. Schmidt-Pokrzywniak A, Jockel KH, Bornfeld N, Sauerwein W, Stang A. Positive interaction between light iris color and ultraviolet radiation in relation to the risk of uveal melanoma: a case-control study. Ophthalmology. 2009;116:340–348.
7. Hosmer DW, Lemeshow S. Confidence interval estimation of interaction. Epidemiology. 1992;3:452–456.

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