Additive Interaction in the Presence of a Mismeasured Outcome

Misclassification and measurement error are very common in studies of effects of exposures and are often ignored. The problem of exposure misclassification for interaction measures in the context of both multiplicative interaction (1–4) and additive interaction (5) has been discussed previously. Other investigators have discussed problems of outcome misclassification for a single exposure (6, 7). However, little is known about how interaction measures perform when the outcome is misclassified. In the present letter, we consider additive interactions with a mismeasured outcome. All the proofs are given in Web Appendix 1, available at http://aje.oxfordjournals.org/.

Let X₁ and X₂ denote 2 binary exposures and let D denote a binary outcome. Let $p_{x_1{x}_2}=P(D=1|X_1=x_1,X_2=x_2)$. The measure of interaction for risks on the linear additive scale is as follows:

$$p_{11}-p_{10}-p_{01}+p_{00}.$$

If p₁₁ − p₁₀ − p₀₁ – p₀₀ > 0, the additive interaction is said to be positive. If p₁₁ − p₁₀ − p₀₁ – p₀₀ < 0, the additive interaction is said to be negative. Now suppose that D is subject to misclassification, and let D* denote the observed outcome. We assume that the misclassification is nondifferential, that is, $P(D\ast |D,X_1,X_2)=P(D\ast |D)$. The probability of misclassification can be characterized by sensitivity $(\mathrm{SN}=P(D\ast =1|D=1))$ and specificity $(\text{SP}=P(D\ast =0|D=0))$. We assume that SN + SP > 1, which is plausible because the observed outcome is more likely to be 1 (or 0) if the true outcome is 1 (or 0). If we let $p_{x_1{x}_2}^{\ast }=P(D\ast =1|X_1=x_1,X_2=x_2)$ for x1, x2 = 0,1, we get the following result:

$$p_{11}^{\ast }-p_{10}^{\ast }-p_{01}^{\ast }+p_{00}^{\ast }=(\text{SN}+\text{SP}-1)(p_{11}-p_{10}-p_{01}+p_{00}).$$ (1)

Because SN + SP – 1 ≤ 1, the naive estimator that is determined using the observed outcome will always be biased towards the null. We can furthermore get corrected estimates and confidence intervals by dividing the estimate and both limits of the confidence interval of the naive estimator by (SN + SP –1). For corrected estimates, sensitivity and specificity will often be unknown but can be varied in a sensitivity analysis. The precise values of sensitivity and specificity, however, do not need to be known to obtain conservative estimates.

Sometimes, instead of using risk differences to measure interaction, we might use risk ratios or odds ratios. In general, measures of interaction on the odds ratio and risk ratio scale will be very close to one another whenever the outcome is rare (6). Thus, we will focus only on the risk ratio. Let $\text{R}{\text{R}}_{x_1{x}_2}=p_{x_1{x}_2}/p_{00}$ be the risk ratio effect measures. A measure of interaction on the additive scale for risk ratios (6) is as follows:

$$\text{RER}{\text{I}}_{\mathrm{RR}}=\text{R}{\text{R}}_{11}-\text{R}{\text{R}}_{10}-\text{R}{\text{R}}_{01}+1=\frac{p_{11}}{p_{00}}-\frac{p_{10}}{p_{00}}-\frac{p_{01}}{p_{00}}+1.$$

If we suppose the outcome is measured with error, we could use the naive estimator using the observed outcome:

$${\text{RERI}}_{\mathrm{RR}}^{\ast }=\frac{p_{11}^{\ast }}{p_{00}^{\ast }}-\frac{p_{10}^{\ast }}{p_{00}^{\ast }}-\frac{p_{01}^{\ast }}{p_{00}^{\ast }}+1.$$

Unfortunately, we cannot obtain a simple relation between ${\text{RERI}}_{\mathrm{RR}}^{\ast }$ and RERI_RR. Instead, we have the following qualitative result:

$$|{\text{RERI}}_{\mathrm{RR}}^{\ast }|\le |\text{RER}{\text{I}}_{\mathrm{RR}}|.$$ (2)

Equation 2 shows that the naive estimator for the additive interaction on the risk ratio scale will always estimate the true interaction towards the null. The interaction measure using the observed misclassified data will thus always be conservative, and tests using ${\text{RERI}}_{\mathrm{RR}}^{\ast }$ will thus be valid in the sense of having a type I error less than 5%. This conclusion enables us to draw qualitative conclusions in practice even when we cannot observe the true outcome. For example, if we estimate the interaction to be positive using the observed misclassified data, we can conclude that true additive interaction is positive. Further details on the mathematical relations between RERI_RR and ${\text{RERI}}_{\mathrm{RR}}^{\ast }$ are given in Web Appendix 1.

Another 2 measures of additive interaction that use risk ratios are called the attributable proportion and the synergy index. Rothman et al. (6) first defined the attributable proportion as

$$\text{A}{\text{P}}_1=\frac{\text{R}{\text{R}}_{11}-\text{R}{\text{R}}_{10}-\text{R}{\text{R}}_{01}+1}{\text{R}{\text{R}}_{11}}=\frac{p_{11}-p_{10}-p_{01}+p_{00}}{p_{11}},$$

which measures the proportion of the risk in the doubly exposed group that is due to the interaction. When the outcome is measured with error, the naive estimator using the misclassified data is

$${\text{AP}}_1^{\ast }=\frac{p_{11}^{\ast }-p_{10}^{\ast }-p_{01}^{\ast }+p_{00}^{\ast }}{p_{11}^{\ast }},$$

and we can obtain the following result:

$$|{\text{AP}}_1^{\ast }|\le |\text{A}{\text{P}}_1|.$$ (3)

Equation 3 has the same form as equation 2 in that the measure using the observed data is conservative for the true measure. An alternative definition of attributable proportion discussed by VanderWeele and Tchetgen Tchetgen (8) is:

$$\text{A}{\text{P}}_2=\frac{\text{R}{\text{R}}_{11}-\text{R}{\text{R}}_{10}-\text{R}{\text{R}}_{01}+1}{\text{R}{\text{R}}_{11}-1}=\frac{p_{11}-p_{10}-p_{01}+p_{00}}{p_{11}-p_{00}}.$$

This measure considers the proportion of the joint effects of both exposures together that is due to interaction. When the outcome is measured with error, the naive estimator using the observed outcome is

$${\text{AP}}_2^{\ast }=\frac{p_{11}^{\ast }-p_{10}^{\ast }-p_{01}^{\ast }+p_{00}^{\ast }}{p_{11}^{\ast }-p_{00}^{\ast }},$$

and we can obtain the following result:

$${\text{AP}}_2^{\ast }=\text{A}{\text{P}}_2.$$ (4)

Equation 4 shows that the attributable proportion discussed by VanderWeele and Tchetgen Tchetgen (8) remains the same when using a nondifferentially misclassified outcome. Therefore, using this measure, we can still obtain the magnitude of interaction without knowing the sensitivity and specificity parameters.

Rothman et al. (6) gave the definition of the synergy index as

$$S=\frac{\text{R}{\text{R}}_{11}-1}{(\text{R}{\text{R}}_{10}-1)+(R_{01}-1)}=\frac{p_{11}-p_{00}}{(p_{10}-p_{00})+(p_{01}-p_{00})},$$

which measures the extent to which the risk ratio for both exposures together exceeds 1 and whether this is greater than the sum of the extent to which each of the risk ratios considered separately exceeds 1. When the outcome is measured with error, we use the observed outcome to calculate the synergy index,

$$S^{\ast }=\frac{p_{11}^{\ast }-p_{00}^{\ast }}{(p_{10}^{\ast }-p_{00}^{\ast })+(p_{01}^{\ast }-p_{00}^{\ast })},$$

and we can obtain the following result:

$$S^{\ast }=S.$$ (5)

This shows that the synergy index does not change when estimated using the misclassified data. VanderWeele (9, 10) discussed how constraints of the form p₁₁ − p₁₀ − p₀₁ > 0 and p₁₁ − p₁₀ − p₀₁ – p₀₀ > 0 can be used to make inferences about causal sufficient cause interaction. We can use observed data with misclassified outcome to also make such inferences as stated in the following results:

$$If{p}_{11}^{\ast }-p_{10}^{\ast }-p_{01}^{\ast }>0,then{p}_{11}-p_{10}-p_{01}>0.$$ (6)

$$If{p}_{11}^{\ast }-p_{10}^{\ast }-p_{01}^{\ast }-p_{00}^{\ast }>0,then{p}_{11}-p_{10}-p_{01}-p_{00}>0.$$ (7)

Details are provided in Web Appendix 1. By similar arguments, analogous results also hold for n-way sufficient cause interactions (11) and for sufficient cause interactions with categorical and ordinal exposures.

Unfortunately, similar simple relations do not seem to hold for interaction on the multiplicative scale, and biases due to outcome misclassification can be in either direction. An exception occurs if there is perfect specificity (SP = 1) for the outcome, in which case measures of multiplicative interaction are unbiased even with imperfect sensitivity (Web Appendix 1). Further research could examine correction strategies and evaluate these through simulations.