Improved wrong-model inference for generalized linear models for binary responses in the presence of link misspecification

Abstract

In the framework of generalized linear models for binary responses, we develop parametric methods that yield estimators for regression coefficients less compromised by an inadequate posited link function. The improved inference are obtained without correcting a misspecified model, and thus are referred to as wrong-model inference. A byproduct of the proposed methods is a simple test for link misspecification in this class of models. Impressive bias reduction in estimators for the regression coefficients from the proposed methods and promising power of the proposed test to detect link misspecification are demonstrated in simulation studies. We also apply these methods to a classic data example frequently analyzed in the existing literature concerning this class of models.

Appendix: Proof of equation (2)

Because the assumed GLM is specified by $P(Y=1|X)=H(\eta )$, where $\eta =\beta _0+\beta _1 X$, and the reclassified response is generated according to $P(Y^*=Y|Y, X)=\pi$, one has

$$\begin{aligned} P(Y^*=1|X)&= P(Y^*=1, Y=0|X)+P(Y^*=1, Y=1|X) \\&= P(Y=0|X)P(Y^*\ne Y|Y, X)+P(Y=1|X)P(Y^*=Y|Y, X) \\&= \{1-H(\eta )\}(1-\pi )+H(\eta )\pi \\&= (2\pi -1)H(\eta )+1-\pi . \end{aligned}$$

(15)

It follows that the likelihood function based on the assumed primary model for $Y^*$ evaluated at one data point $(Y^*, X)$ is $L(\pi , \varvec{\beta })=P(Y^*=1|X)^{Y^*}\{1-P(Y^*=1|X)\}^{1-Y^*}$, and the log-likelihood function is $\ell (\pi , \varvec{\beta })=Y^*\log P(Y^*=1|X)+(1-Y^*)\log \{1-P(Y^*=1|X)\}$.

Differentiating (15) with respect to each element in $(\pi , \varvec{\beta })$ gives

$$\begin{aligned} \begin{aligned} \frac{\partial P(Y^*=1|X)}{\partial \pi }&=2H(\eta )-1, \\ \frac{\partial P(Y^*=1|X)}{\partial \beta _0}&=(2\pi -1)H'(\eta ), \\ \frac{\partial P(Y^*=1|X)}{\partial \beta _1}&=(2\pi -1)H'(\eta )X. \end{aligned} \end{aligned}$$

(16)

Using (16), one can show that the three normal score functions associated with $\ell (\pi , \varvec{\beta })$ are given by

$$\begin{aligned} \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \pi }&= Y^*\frac{2H(\eta )-1}{P(Y^*=1|X)}-(1-Y^*)\frac{2H(\eta )-1}{1-P(Y^*=1|X)},\\ \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \beta _0}&= Y^*\frac{(2\pi -1)H'(\eta )}{P(Y^*=1|X)}-(1-Y^*)\frac{(2\pi -1)H'(\eta )}{1-P(Y^*=1|X)}, \\ \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \beta _1}&= Y^*\frac{(2\pi -1)H'(\eta )X}{P(Y^*=1|X)}-(1-Y^*)\frac{(2\pi -1)H'(\eta )X}{1-P(Y^*=1|X)}. \end{aligned}$$

To further simplify notations, let $\mu =P(Y^*=1|X)$. The above three score functions can be re-expressed as

$$\begin{aligned} \begin{aligned} \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \pi }&= \frac{Y^*-\mu }{\mu (1-\mu )}\{2H(\eta )-1\}, \\ \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \beta _0}&= \frac{Y^*-\mu }{\mu (1-\mu )}(2\pi -1)H'(\eta ), \\ \frac{\partial \ell (\pi , \varvec{\beta })}{\partial \beta _1}&= \frac{Y^*-\mu }{\mu (1-\mu )}(2\pi -1)H'(\eta )X. \end{aligned} \end{aligned}$$

(17)

The expectation of the first score in (17) with respect to the true distribution of $(Y^*, X)$ is

$$\begin{aligned} E\left[ \frac{Y^*-\mu }{\mu (1-\mu )}\{2H(\eta )-1\}\right]&= E\left( E\left[ \left. \frac{Y^*-\mu }{\mu (1-\mu )}\{2H(\eta )-1\}\right| X\right] \right) \\&= E\left[ \frac{\mu _0-\mu }{\mu (1-\mu )}\{2H(\eta )-1\}\right] , \end{aligned}$$

where $\eta _0$ is equal to $\eta$ evaluated at the true value of $\varvec{\beta }$, and $\mu _0 =(2\pi -1)G(\eta _0)+1-\pi$, as defined in (4), is the mean of $Y^*$ given X under the correct model evaluated at the true parameter values. Setting this expectation equal to zero gives the first estimating equation in (2). Similarly, the expectations of the second and the third score functions in (17) with respect to the true distribution of $(Y^*, X)$ are given by

$$\begin{aligned} E\left[ \frac{Y^*-\mu }{\mu (1-\mu )}(2\pi -1)H'(\eta )\right]&= E\left[ \frac{\mu _0-\mu }{\mu (1-\mu )}H'(\eta )\right] (2\pi -1), \\ E\left[ \frac{Y^*-\mu }{\mu (1-\mu )}(2\pi -1)H'(\eta )X\right]&= E\left[ \frac{\mu _0-\mu }{\mu (1-\mu )}H'(\eta )X\right] (2\pi -1), \end{aligned}$$

respectively. Setting these two expectations equal to zero gives the second and the third equations in (2).