Analysis of Correlated Data with Error-Prone Response Under Generalized Linear Mixed Models

Abstract

Measurements of variables are often subject to error due to various reasons. Measurement error in covariates has been discussed extensively in the literature, while error in response has received much less attention. In this paper, we consider generalized linear mixed models for clustered data where measurement error is present in response variables. We investigate asymptotic bias induced by nonlinear error in response variables if such error is ignored, and evaluate the performance of an intuitively appealing approach for correction of response error effects. We develop likelihood methods to correct for effects induced from response error. Simulation studies are conducted to evaluate the performance of the proposed methods, and a real data set is analyzed with the proposed methods.

Appendix

Let $\varPsi _{i}(\boldsymbol{\theta },\boldsymbol{\eta }) = \left (\begin{matrix}\scriptstyle \mathbf{Q}_{i}(\boldsymbol{\eta })\\\scriptstyle \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })\end{matrix}\right )$. Because $(\hat{\boldsymbol{\theta }}_{p},\hat{\boldsymbol{\eta }})$ is a solution to $\varPsi _{i}(\boldsymbol{\theta },\boldsymbol{\eta }) = 0$, by first-order Taylor series approximation, we have

$$\displaystyle\begin{array}{rcl}n^{1/2}\left (\begin{array}{c} \hat{\boldsymbol{\eta }} -\boldsymbol{\eta }\\\hat{\boldsymbol{\theta }}_{p} -\boldsymbol{\theta } \end{array} \right )& =& -\left (\begin{array}{cc} \hspace{7.22743pt} E\left \{\partial \mathbf{Q}_{i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\right \} & \hspace{7.22743pt} 0\\E\left \{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\right \} & \hspace{7.22743pt} E\left \{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\right \} \end{array} \right )^{-1}{}\\ & & \times \;n^{-1/2}\sum _{i=1}^{n}\varPsi _{i}(\boldsymbol{\theta },\boldsymbol{\eta }) + o_{p}(1). {}\\\end{array}$$

It follows that $n^{1/2}(\hat{\boldsymbol{\theta }}_{p}-\boldsymbol{\theta })$ equals

$$\displaystyle\begin{array}{rcl}& & -n^{-1/2}\left [E\left \{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\right \}\right ]^{-1}\left \{\sum _{i=1}^{n}\mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta }) - E\left \{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\right \}\right.{}\\& & \times \left.\left [E\left \{\partial \mathbf{Q}_{i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\right \}\right ]^{-1}\sum _{i=1}^{n}\mathbf{Q}_{i}(\boldsymbol{\eta })\right \} + o_{p}(1) = -n^{-1/2}\varGamma ^{-1}(\boldsymbol{\theta },\boldsymbol{\eta }){}\\& & \quad \sum _{i=1}^{n}\varOmega _{i}(\boldsymbol{\theta },\boldsymbol{\eta }) + o_{p}(1), {}\\\end{array}$$

where $\varOmega _{i}(\boldsymbol{\theta },\boldsymbol{\eta }) = \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta }) - E\{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}[E\{\partial \mathbf{Q}_{i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}]^{-1}\mathbf{Q}_{i}(\boldsymbol{\eta })$, and $\varGamma (\boldsymbol{\theta },\boldsymbol{\eta }) = E\{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\}$.

Applying the Central Limit Theorem, we can show that $n^{1/2}(\hat{\boldsymbol{\theta }}_{p}-\boldsymbol{\theta })$ is asymptotically normally distributed with mean 0 and asymptotic covariance matrix given by Γ ⁻¹ Σ(Γ ⁻¹)^T, where $\varSigma = E\{\varOmega _{i}(\boldsymbol{\theta },\boldsymbol{\eta })\varOmega _{i}^{\mathrm{T}}(\boldsymbol{\theta },\boldsymbol{\eta })\}$. But under suitable regularity conditions and correct model specification, $E\{\mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })\mathbf{U}_{i}^{{\ast}\mathrm{T}}(\boldsymbol{\theta },\boldsymbol{\eta })\} = E\{ - \partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\}$, $E\{\mathbf{Q}_{i}(\boldsymbol{\eta })\mathbf{Q}_{i}^{\mathrm{T}}(\boldsymbol{\eta })\} = E\{ - \partial \mathbf{Q}_{i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}$, and $E\{\mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })\mathbf{Q}_{i}^{\mathrm{T}}(\boldsymbol{\eta })\} = E\{ - \partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}$. Thus,

$$\displaystyle\begin{array}{rcl} \varSigma & =& E\{ - \partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\} + E\{\partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}\left [E\{\partial \mathbf{Q}_{ i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}\right ]^{-1} {}\\ & & \times \;\left [E\{\partial \mathbf{U}_{i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}\right ]^{\mathrm{T}}. {}\\ \end{array}$$

Therefore, the asymptotic covariance matrix for $n^{1/2}(\hat{\boldsymbol{\theta }}_{p}-\boldsymbol{\theta })$ is

$$\displaystyle\begin{array}{rcl} \varSigma ^{{\ast}}& =& \left [E\{ - \partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\}\right ]^{-1} + \left [E\{ - \partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\}\right ]^{-1}E\{\partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\} {}\\ & & \times \;\left [E\{\partial \mathbf{Q}_{i}(\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}\right ]^{-1}\left [E\{\partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\eta }^{\mathrm{T}}\}\right ]^{\mathrm{T}}\left [E\{ - \partial \mathbf{U}_{ i}^{{\ast}}(\boldsymbol{\theta },\boldsymbol{\eta })/\partial \boldsymbol{\theta }^{\mathrm{T}}\}\right ]^{-1}. {}\\ \end{array}$$