Selecting controls for assessing interaction in nested case-control studies

Abstract

Background: Two methods for selecting controls in nested case-control studies--matching on X and counter matching on X--are compared when interest is in interaction between a risk factor X measured in the full cohort and another risk factor Z measured only in the case-control sample. This is important because matching provides efficiency gains relative to random sampling when X is uncommon and the interaction is positive (greater than multiplicative), whereas counter matching is generally efficient compared to random sampling.

Methods: Matching and counter matching were compared to each other and to random sampling of controls for dichotomous X and Z. Comparison was by simulation, using as an example a published study of radiation and other risk factors for breast cancer in the Japanese atomic-bomb survivors, and by asymptotic relative efficiency calculations for a wide range of parameters specifying the prevalence of X and Z as well as the levels of correlation and interaction between them. Focus was on analyses utilizing general models for the joint risk of X and Z.

Results: Counter-matching performed better than matching or random sampling in terms of efficiency for inference about interaction in the case of a rare risk factor X and uncorrelated risk factor Z. Further, more general, efficiency calculations demonstrated that counter-matching is generally efficient relative to matched case-control designs for studying interaction.

Conclusions: Because counter-matched designs may be analyzed using standard statistical methods and allow investigation of confounding of the effect of X, whereas matched designs require a non-standard approach when fitting general risk models and do not allow investigating the adjusted risk of X, it is concluded that counter-matching on X can be a superior alternative to matching on X in nested case-control studies of interaction when X is known at the time of case-control sampling.

Figure 1.. Boxplots exhibiting 100 simulated estimates of the multiplicative interaction parameter γ_M (equation 2) and 95% likelihood-based confidence bounds for the three sampling methods: matched (M), counter-matched (C), and unmatched (U).
Estimates of the parameter and its confidence bounds were obtained by exponentiation of those for γ_M. Results are for case-control ratios of 1:1, 1:2, 1:3, 1:5, and 1:9. Results for counter matching with two controls are from the design with the extra control unexposed. Narrower boxplots reflect smaller sampling variation in the interaction estimates or confidence bounds. More precise estimation of interaction is evidenced by median bounds closer to the interaction estimate. For comparison, the cohort interaction parameter (1.14) and confidence limits (0.74, 1.73) are shown as dotted lines.

Figure 2.. Results of asymptotic relative efficiency (ARE) calculations versus three levels of prevalence of exposure [Pr(X=1)].
Points are the ratio of ARE for counter matching to that for matching (▲ ● ■) or random sampling (△ ○ □). Three levels of prevalence of the other factor, Z, are compared: Pr(Z=1) = 0.1 (▲ △), Pr(Z=1) = 0.25 (● ○), and Pr(Z=1) = 0.5 (■ □). Relative efficiencies were calculated for three levels of correlation between X and Z: odds ratio (OR) = 5.0, 1.0, or 0.2. Three values of the interaction between exposure X and other factor Z were examined: exp{γ}=5.0 (upper panel), exp{γ}=0.2 (lower panel), and exp{γ}=1.0 (not shown; results were intermediate to the other two). Results are for case-control ratios of 1:1, 1:2, 1:3, 1:5, and 1:9. With two controls, the AREs of counter matching for the two allocation strategies were averaged.