# Multiplicative Interactions Under Differential Outcome Measurement Error with Perfect Specificity

VanderWeele, Tyler J.

doi: 10.1097/EDE.0000000000000979
Letters
Free

Departments of Epidemiology and Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA.

Supported by National Institutes of Health grant CA222147.

The author reports no conflicts of interest.

tvanderw@hsph.harvard.edu

## 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 nondifferential 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 se = 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 fe = 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 pge = P(Y = 1|G = g,E = e,c) denote the outcome probability conditional on G = g,E = e,C = c and let pge* = 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/p0 0)(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 (fe = 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 nondifferential with respect to G, but with perfect specificity so that fe = 0, then

.

To see that this result holds note that

, which is equal to sepge if fe = 0 in which case we then have

. 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 al2 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 data,2 when analyses were stratified by Protestant versus 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 underreporting 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 underreporting of suicide was differential with respect to Protestants versus Catholic affiliation, but nondifferential with respect to service attendance, then we could apply the result above. That underreporting of suicide, conditional on affiliation, may be nondifferential with respect to service attendance is perhaps plausible if both attending and nonattending Catholic underreport similarly. Specificity would be perfect here if nonsuicides 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 owing 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

Tyler J. VanderWeele

Departments of Epidemiology and Biostatistics

Harvard T.H. Chan School of Public Health

Boston, MA.

## REFERENCES

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2. VanderWeele TJ, Li S, Tsai AC, Kawachi I. Association between religious service attendance and lower suicide rates among US women. JAMA Psychiatry. 2016;73:845–851.
3. Durkheim E. Suicide: A Study in Sociology. 1966.1st Free Press Paperback ed. New York, NY: Free Press.
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9. García-Closas M, Thompson WD, Robins JM. Differential misclassification and the assessment of gene-environment interactions in case-control studies. Am J Epidemiol. 1998;147:426–433.
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