Improved Regression Calibration

Abstract

The likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form, which makes maximum likelihood estimation taxing. A popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation. We propose an improved regression calibration approach, a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood. It is both consistent and computationally efficient, and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood. Simulations suggest that improved regression calibration, which is easy to implement in standard software, works well in a range of situations.

Appendix: Obtaining \(\widehat{\boldsymbol{\mathcal{I}}}_{ \mathsf{ME},\mathsf{O}}\) in (18)

Here we describe the calculation of the estimate (18) of the matrix $\boldsymbol{\mathcal{I}}_{\mathsf{ME}, \mathsf{O}}$, which is used in the calculation of the variance matrix (17) of $\widehat{\boldsymbol {\vartheta }}_{\mathsf{O}}^{\mathsf{IRC}}$. Let us first introduce some convenient shorthand notation for the logarithm of the likelihood contribution (6):

Here g _xi and g _2i are multivariate normal density functions with parameters $\boldsymbol{\theta}_{1i}=(\boldsymbol{\xi}_{i}',\text {vec}(\boldsymbol{\Omega}_{i})')'$ and $\boldsymbol{\theta}_{2i}=(\boldsymbol{\mu}_{i}',\text {vec}(\boldsymbol{\Sigma}_{i})')'$ respectively, as defined by (11)–(12) and (9)–(10). These in turn are functions of the parameters χ=(ν′,vec(Λ)′,vec(Θ)′,vec(Γ)′,vec(Ψ)′)′, and ϑ _ME are the distinct, unknown elements of χ.

The required gradients for (18) are

(A.1)

(A.2)

where

(A.3)

(A.4)

(A.5)

Estimated values for these quantities, and thus for the estimated matrix $\widehat{\boldsymbol{\mathcal{I}}}_{\mathsf{ME},\mathsf{O}}$ given by (18), are obtained by substituting estimates $\widehat{\boldsymbol {\vartheta }}^{\mathsf{IRC}}$ of the parameters.

Starting with (A.2), we note that each element of χ is either a known constant or equal to a single element of ϑ _ME ; for illustration, consider Λ as shown in (2). Suppose that χ is of length t and ϑ _ME of length u. Then $\partial\boldsymbol{\chi}/\partial \boldsymbol {\vartheta }_{\mathsf {ME}}'$ is a t×u matrix whose (i,j)th element is 1 if the ith element of χ is equal to the jth element of ϑ _ME , and 0 otherwise.

Next, the elements of ∂ θ _2i /∂ χ′ in (A.2) are

and the elements of ∂ θ _1i /∂ χ′ are

where

and vec(⋅) denotes the column-by-column vectorization operator, ⊗ the Kronecker product, I _m an m×m identity matrix, and K _rm an rm×rm commutation matrix. The formulas are obtained through repeated application of rules of matrix differentiation (see, e.g., Lütkepohl 1996).

In the second term of (A.2), the elements of $\partial\log g_{2i}/\partial\boldsymbol{\theta}_{2i}'$ are $\partial\log g_{2i}/\partial\boldsymbol{\mu}_{i}'$ $= (\mathbf{w}_{i}-\boldsymbol{\mu}_{i})'\boldsymbol{\Sigma }_{i}^{-1}$ and ∂logg _2i /∂vec(Σ _i )′ $= \text{vec}[\boldsymbol{\Sigma}_{i}^{-1}(\mathbf{w}_{i}-\boldsymbol {\mu}_{i})(\mathbf{w}_{i}-\boldsymbol{\mu}_{i})' \boldsymbol{\Sigma}_{i}^{-1}-\boldsymbol{\Sigma}_{i}^{-1}]'/2$.

The remaining elements of (A.1) and (A.2) depend also on the outcome model for y _i . For the logistic model, which is predominant in applications of generalized linear models with covariate measurement error, and which is also used in our simulations and example, $g_{yi}=\pi_{i}^{y_{i}}(1-\pi_{i})^{1-y_{i}}$ where π _i =exp(η _i )/[1+exp(η _i )] and $\eta_{i}=\mathbf{z}_{i}'\boldsymbol{\beta}_{z}+\mathbf {x}_{i}'\boldsymbol{\beta}_{x}$. For this model we employ the well-known closed-form approximation $g_{1i}\approx(\pi_{i}^{*})^{y_{i}}(1-\pi_{i}^{*})^{1-y_{i}}$, where $\pi_{i}^{*}=\exp(\eta^{*}_{i})/[1+\exp(\eta^{*}_{i})]$, $\eta^{*}_{i}=\eta_{1i}\eta_{2i}^{-1/2}$, $\eta_{1i}= \mathbf{z}_{i}'\boldsymbol{\beta}_{z}+ \boldsymbol{\xi}_{i}'\boldsymbol{\beta}_{x}$, $\eta_{2i}=1+d\boldsymbol{\beta}_{x}'\boldsymbol{\Omega }_{i}\boldsymbol{\beta}_{x}$, and d=1/1.7² (e.g., Liang & Liu 1991). For this approximation,

where $\boldsymbol{\xi}_{i}^{*}=\boldsymbol{\xi}_{i}-\eta_{1i}\eta _{2i}^{-1} \, d \boldsymbol{\Omega}_{i}\boldsymbol{\beta}_{x}$. These formulas complete explicit expressions for (A.1) and (A.2).

In our data analysis, we also apply a similar idea for the conventional regression calibration estimate of ϑ _O , which uses the first-order approximation $g_{1i}\approx (\pi_{i}^{\mathrm{RC}})^{y_{i}}(1-\pi_{i}^{ \mathrm{RC}})^{1-y_{i}}$ where $\pi_{i}^{\mathrm{RC}}=\exp(\eta_{1i})/[1+\exp(\eta_{1i})]$. We estimate its variance matrix analogously to (17)–(18), using in (A.1) and (A.2) $\partial g_{1i}/\partial \boldsymbol {\vartheta }_{\mathsf {O}}=(\partial g_{1i}/\partial\eta_{1i}) (\mathbf{z}_{i}',\boldsymbol{\xi}')'$ and $\partial g_{1i}/\partial\boldsymbol{\theta}_{1i}'=(\partial g_{1i}/\partial\eta_{1i}) [\boldsymbol{\beta}_{x}', \mathbf{0}']$, where $\partial g_{1i}/\partial\eta_{1i}=(-1)^{1-y_{i}} \pi_{i}^{\mathrm{RC}}\, (1-\pi_{i}^{\mathrm{RC}})$.

For other, less popular models, we must evaluate the integrals involved in (A.3)–(A.5). Note first that the partial derivatives $\partial g_{xi}/\partial \boldsymbol{\theta}_{1i}'$ are given by

Substituting these into (A.5), we see that each of the integrals there, and also in (A.3) and (A.4), are of the form ∫h _i (x _i )g _xi  dx _i for some function h _i (x _i ) of x _i , integrated over the multivariate normal density g _xi =g(x _i |w _i ,z _i ;ϑ _ME ). This suggests that the integrals can be evaluated through Monte Carlo integration, by first generating M independent draws x _ij , j=1,…,M, from $g(\mathbf{x}_{i}|\mathbf{w}_{i}, \mathbf{z}_{i}; \widehat{\boldsymbol {\vartheta }}_{\mathsf{ME}})$, and then approximating the integrals by the averages $M^{-1} \sum_{j=1}^{M} h_{i}(\mathbf{x}_{ij})$ for each of the h _i (⋅). Only one set of random draws is needed for all the observations i, if we first generate M uncorrelated m-vectors u _j of standard normal random variates and then calculate $\mathbf{x}_{ij}=\widetilde{\boldsymbol{\xi}}_{i}+\mathbf {B}_{i}\mathbf{u}_{j}$, where $\widehat{\boldsymbol{\Omega}}_{i}=\mathbf{B}_{i}\mathbf{B}_{i}'$.