Multiplicative Interactions Under Differential Outcome Measurement Error with Perfect Specificity

To the Editor:

In this letter it is shown that multiplicative interactions on the risk ratio scale in which the outcome is subject to differential measurement error with respect to one of the exposures still yields consistent interaction estimates under perfect specificity, though differential and imperfect sensitivity, of the outcome measure.

Let Y denote the outcome and G and E two exposures. We will assume all analyses and probabilities are conditional on some set of measured covariates C. Let Y* be the measurement of the outcome which is subject to differential measurement error with respect to E so that P(y*|y,g,e₁,c)≠P(y*|y,g,e₀,c) for some values e₁ and e₀ of E, but is non-differential with respect to G so that P(y*|y,g,e,c)=P(y*|y,e,c) (since the measurement error is not differential with respect to G). Let s_e=P(Y*=1|Y=1,g,e,c)= P(Y*=1|Y=1,e,c) denote the sensitivity of the measurement Y* conditional on E=e,C=c; and let f_e=P(Y*=1|Y=0,g,e,c)= P(Y*=1|Y=0,e,c) denote the false positive probability of the measurement Y* conditional on E=e,C=c which is 1 minus the specificity. Let p_ge=P(Y=1|G=g,E=e,c) denote the outcome probability conditional on G=g,E=e,C=c and let p_ge*=P(Y*=1|G=g,E=e,c) denote the observed mismeasured outcome probability conditional on G=g,E=e,C=c. The multiplicative interaction on the risk ratio scale¹ is given by (p11/p00)/{(p10/p00)(p01/p00)}=(p₁₁p₀₀)/(p₁₀p₀₁) and the multiplicative interaction with the mismeasured outcome Y is given by (p₁₁*p₀₀*)/(p₁₀*p₀₁*). We will consider a setting in which the outcome has perfect specificity (f_e=0), which would hold if whenever the outcome is not present, it will be measured as not present, but when the outcome is present it is sometimes measured as present, but sometimes measured as absent. We then have the following result.

Theorem 1. If measurement error of Y is differential with respect to E but non-differential with respect to G, but with perfect specificity so that f_e=0 then (p₁₁*p₀₀*)/(p₁₀*p₀₁*)=(p₁₁p₀₀)/(p₁₀p₀₁).

To see that this result holds note that p_ge*= s_ep_ge+f_e(1-p_ge) which is equal to s_ep_ge if f_e=0 in which case we then have (p₁₁*p₀₀*)/(p₁₀*p₀₁*)=(s₁p₁₁s₀p₀₀)/(s₀p₁₀s₁p₀₁)=(p₁₁p₀₀)/(p₁₀p₀₁). With rare outcomes, odds ratios and hazard ratios will approximate risk ratios and so that the result is applicable in such settings as well. Note that although the interaction parameter remains unbiased with perfect specificity for E, the main effect for E in each stratum of G remains biased.

To illustrate the use of this result, VanderWeele et al.² examined associations between regular religious service attendance and completed suicide among 89,708 participants in the Nurses’ Health Study over 14 years from 1996 to 2010. Analyses also considered Protestant versus Catholic affiliation. Prior group-averaged³ and individual-level⁴ studies have noted lower suicide rates among Catholics as compared with Protestants. With the Nurses’ Health Study data², when analyses were stratified by Protestant vs. Catholic affiliation, the hazard ratio for suicide comparing religious service attendance at least once per week versus less than once per week was 0.34 (95% CI: 0.10, 1.10) for Protestants and 0.05 (95% CI: 0.01, 0.48) for Catholics. The measure of multiplicative interaction was 0.34/0.05=6.8 (95% CI: 1.0, 46; p=0.05). It was noted in the Discussion section that one potential concern in the analysis was differential under-reporting of suicide across affiliation due to Catholic beliefs about suicide constituting a grave mortal sin and possible implications for burial practices. This could introduce bias comparing Protestants and Catholic directly and would also be a concern in prior research and analyses^3,4. However, if under-reporting of suicide was differential with respect to Protestants versus Catholic affiliation, but non-differential with respect to service attendance, then we could apply the result above. That under-reporting of suicide, conditional on affiliation, may be non-differential with respect to service attendance is perhaps plausible if both attending and non-attending Catholic under-report similarly. Specificity would be perfect here if non-suicides were never reported as suicide. By the result above, we would then have that while a direct Catholic versus Protestant comparison would be biased, the interaction analysis would be unbiased. Thus, by the result above, one would retain the interpretation that the effect of service attendance is larger among Catholics than Protestants, even if the effect of Protestant versus Catholic affiliation itself is biased by the differential misclassification of suicide across affiliation. That the effect of service attendance itself on reducing suicide is larger among Catholics versus Protestants may be plausible due to weekly prayers during many Catholic liturgies promoting the value and sanctity of life “from conception to natural death”⁵ and would likewise be consistent with Durkheim’s hypotheses³ with regard to the role that social cohesion and social control may play in reducing suicide within Catholic communities, as these would presumably be principally operative for those attending services.

The result here is one of several others indicating that interaction measures will often be unbiased even when main effects are subject to bias.^6–9