A Corrected Pseudo-score Approach for Additive Hazards Model with Longitudinal Covariates Measured with Error

Abstract

In medical studies, it is often of interest to characterize the relationship between a time-to-event and covariates, not only time-independent but also time-dependent. Time-dependent covariates are generally measured intermittently and with error. Recent interests focus on the proportional hazards framework, with longitudinal data jointly modeled through a mixed effects model. However, approaches under this framework depend on the normality assumption of the error, and might encounter intractable numerical difficulties in practice. This motivates us to consider an alternative framework, that is, the additive hazards model, about which little research has been done when time-dependent covariates are measured with error. We propose a simple corrected pseudo-score approach for the regression parameters with no assumptions on the distribution of the random effects and the error beyond those for the variance structure of the latter. The estimator has an explicit form and is shown to be consistent and asymptotically normal. We illustrate the method via simulations and by application to data from an HIV clinical trial.

Appendix: large sample properties of \(\hat{\varvec{\beta}}\)

For now, we assume the vector of error variances $\varvec{\xi}$ is known. Then $\hat{\varvec{\beta}}$ satisfies $\hat{\bf U}(\hat{\varvec{\beta}},\varvec{\xi}_0)=0.$ First, we show the consistency of $\hat{\varvec{\beta}}.$ Let B be a compact set including $\varvec{\beta}_0$ as an internal point. Since $\hat{\bf U}(\varvec{\beta}, \varvec{\xi})=\tilde{\bf U}(\varvec{\beta};\hat{\bf X})+n^{-1} \sum_{i=1}^n\int_0^\tau{\bf H}_i(u,\varvec{\xi})\varvec{\beta} du$, using the empirical mean operator $\hat{\mathcal E}$, at $\varvec{\xi}=\varvec{\xi}_0$, (7) can be rewritten as

$$\eqalign{\hat{\bf U}(\varvec{\beta},\varvec{\xi}_0)=& \int_{0}^{\tau}\left(d\hat{\mathcal E}\left\{\hat{\bf S}(u) N(u)\right\} -\hat{\mathcal E}\left\{{{d\hat{\bf S}(u)}\over{du}} N(u)\right\} du-\hat{\mathcal E}\left\{Y(u)\hat{\bf S}(u)\hat{\bf S}^T(u)\varvec{\beta} \right\} du\right. \cr&-{\hat{\mathcal E}\left\{ Y(u)\hat{\bf S}(u) \right\}\over\hat{\mathcal E} \left\{Y(u)\right\}}\left[ d\hat{\mathcal E}\left\{N(u)\right\}-\hat{\mathcal E}\left\{Y(u)\hat{\bf S}^{T}(u) \varvec{\beta}\right\} du\right]\cr & \left.\kern-70pt\phantom{\left\{{{d\hat{\bf S}(u)}\over{du}} N(u)\right\}}+\hat{\mathcal E}\{Y(u){\bf H}(u,\varvec{\xi}_0)\varvec{\beta}\}du\right).}$$

(11)

Under condition B, it is easy to show that

$$ \begin{array}{ll} \hbox{E}\left[\sup\limits_{u\in[0,\tau]}{\bf S}^{T}(u){\bf S}(u)\right]<\infty,& \hbox{E}\left[\sup\limits_{u\in[0,\tau]}{d{\bf X}^T(u)\over du} {d{\bf X}(u)\over du}\right] <\infty,\cr \hbox{E}\left[\sup\limits_{\mathop{u\in[0,\tau]}\limits_{\varvec{\beta}\in {\mathcal B}}}\{{\bf S}^{T}(u)\varvec{\beta} \}^{2}\right]<\infty,&\hbox{E}\left[\sup\limits_{\mathop{u\in[0,\tau]}\limits_{\varvec{\beta}\in {\mathcal B}}}\{{\bf S}^{T}(u)\varvec{\beta}\}^{2}{\bf S}^{T}(u){\bf S}(u)\right]<\infty,\cr \hbox{E}\left[\sup\limits_{\mathop{u\in[0,\tau]}\limits_{\varvec{\beta}\in {\mathcal B}}}\left\{\varvec{\beta}^{T}H^T(u,\varvec{\xi}_0){\bf H}(u,\varvec{\xi}_0)\varvec{\beta}\right\}\right]<\infty,& \end{array} $$

and condition A implies that E{Y(u)} is bounded away from zero for u [0,τ]. Hence, by using the extended law of large numbers in Appendix III of Anderson and Gill (1982), the empirical processes in (11) converge almost surely to their limits uniformly in u [0,τ] and $\varvec{\beta}\in{\mathcal B}$ and hence $\hat{\bf U}(\varvec{\beta},\varvec{\xi}_0)$ converges uniformly to ${\bf U}(\varvec{\beta};{\bf X})$ in $\varvec{\beta}\in{\mathcal B}.$ Since $\hat{\bf U}(\varvec{\beta},\varvec{\xi}_0)$ and ${\bf U}(\varvec{\beta};{\bf X})$ are linear functions of $\varvec{\beta}$ and ${\bf U}(\varvec{\beta}_0;{\bf X})=0,$ the consistency of $\hat{\varvec{\beta}}$ follows.

Next, we show that $n^{1/2}\{\hat{\bf U}(\varvec{\beta}_0,\varvec{\xi}_0)-{\bf U}(\varvec{\beta}_0;{\bf X})\}$ converges to a normal distribution. Let $\varvec{\Lambda}(u,\varvec{\beta},\varvec{\xi}_0)$ denote the vector composed of the empirical processes in (11). Following the proof of Lemma 5.1 in Tsiatis (1981), under condition B, $n^{1/2}[\varvec{\Lambda}(u,\varvec{\beta}_0,\varvec{\xi}_0)-\hbox{E} \{\varvec{\Lambda}(u,\varvec{\beta}_0,\varvec{\xi}_0)\}]$ converges to a Gaussian process. Coupled with $\hat{\bf U}(\varvec{\beta}_0,\varvec{\xi}_0)$ being Hadamard differentiable as a functional of $\varvec{\Lambda}$, the asymptotic normality of $n^{1/2}\hat{\bf U}(\varvec{\beta}_0,\varvec{\xi}_0)$ follows by the functional delta method. Using the functional Taylor expansion, with some algebra, we can show that

$$ n^{1/2}\hat{\bf U}(\varvec{\beta}_0,\varvec{\xi}_0)=n^{-1/2} \sum_{i=1}^{n}\varvec{\omega}_{i}(\varvec{\beta}_0,\varvec{\xi}_0) +o_{p}(1), $$

(12)

where

$$ \eqalign{\varvec{\omega}_{i}(\varvec{\beta},\varvec{\xi}) =&\int_{0}^{\tau}\left(\left[\hat{\bf S}_{i}(u)-{\hbox{E}\left\{Y(u)\hat{\bf S}(u)\right\}\over \hbox{E}\left\{Y(u)\right\}} \right]\left\{dN_{i}(u)-Y_{i}(u)\hat{\bf S}_{i}^{T}(u) \varvec{\beta} du\right\}\right. \cr &-\left[{Y_{i}(u)\hat{\bf S}_{i}(u)\over \hbox{E}\left\{Y(u)\right\}}-{Y_{i}(u)\hbox{E}\left\{Y(u)\hat{\bf S}(u)\right\}\over \hbox{E}^{2}\left\{Y(u)\right\}}\right]\hbox{E} \left[dN(u)-Y(u)\hat{\bf S}^{T}(u)\varvec{\beta} du\right] \cr &\left.+Y_i(u){\bf H}_i(u,\varvec{\xi})\varvec{\beta} du\right).} $$

By the central limit theorem, $n^{1/2}\left\{\hat{\bf U}(\varvec{\beta}_0,\varvec{\xi}_0)-{\bf U}(\varvec{\beta}_0;{\bf X})\right\}$ converges to a normal distribution with mean 0 and variance $\Omega(\varvec{\beta}_0,\varvec{\xi}_0)=\hbox{var}\{\varvec{\omega}_{i} (\varvec{\beta}_0,\varvec{\xi}_0)\}$. By empirical processes theory, we can show that $\hat{\Omega}(\varvec{\beta},\varvec{\xi}_0)=n^{-1}\sum_{i=1}^n \left\{\hat{\varvec{\omega}_i}(\varvec{\beta},\varvec{\xi}_0) -\overline{\hat{\varvec{\omega}}}(\varvec{\beta}, \varvec{\xi}_0)\right\}^{\otimes 2}$ converges almost surely to $\Omega(\varvec{\beta},\varvec{\xi}_0)$ uniformly in $\varvec{\beta}\in{\mathcal B}$, where $\overline{\hat{\varvec{\omega}}}(\hat{\varvec{\beta}}, \varvec{\xi})=\hat{\mathcal E}\{\hat{\varvec{\omega}} (\hat{\varvec{\beta}},\varvec{\xi})\}$. With simple algebra, we can show $\hat{\Omega}(\hat{\varvec{\beta}},\varvec{\xi}_0)=n^{-1}\sum_{i=1}^n \hat{\varvec{\omega}}_i(\hat{\varvec{\beta}},\varvec{\xi}_0) \hat{\varvec{\omega}}_i^T(\hat{\varvec{\beta}},\varvec{\xi}_0)$.

Since $\hat{\bf U}(\varvec{\beta},\varvec{\xi})$ is a linear function of $\varvec{\beta}$,

$$ 0=n^{1/2}\hat{\bf U}(\hat{\varvec{\beta}},\varvec{\xi}_0)=n^{1/2} \hat{\bf U}(\varvec{\beta}_{0},\varvec{\xi}_0)-\hat{\varvec{\Gamma}}(\varvec{\xi}_0) n^{1/2}(\hat{\varvec{\beta}}-\varvec{\beta}_0 ), $$

where

$$ \eqalign{\hat{\varvec{\Gamma}}(\varvec{\xi})&=-{\partial\hat{\bf U}(\varvec{\beta},\varvec{\xi})\over \partial \varvec{\beta}}=n^{-1}\sum_{i=1}^n\int_0^\tau\left(Y_{i}(u) \left[\left\{\hat{\bf S}_{i}(u)-\overline{\hat{\bf S}}(u) \right\}^{\otimes 2}-{\bf H}_{i}(u,\varvec{\xi})\right]\right)du \cr &=\int_0^\tau \left[\hat{\mathcal E}\left\{Y(u)\hat{\bf S}(u) \hat{\bf S}^T(u)\right\}-{\hat{\mathcal{E}}\left\{Y(u)\hat{\bf S}(u) \right\}\hat{\mathcal{E}}\left\{Y(u)\hat{\bf S}^T(u)\right\}\over \hat{\mathcal{E}}\left\{Y(u)\right\}}- \hat{\mathcal{E}}\left\{Y(u){\bf H}(u,\varvec{\xi})\right\}\right]du.} $$

By the extended strong law of large numbers, $\hat{\varvec{\Gamma}}(\varvec{\xi})$ converges almost surely to

$$ \eqalign{\varvec{\Gamma}(\varvec{\xi})=&\int_0^\tau \left[\hbox{E}\left\{Y(u)\hat{\bf S}(u)\hat{\bf S}^T(u)\right\}-{\hbox{E}\left\{Y(u)\hat{\bf S}(u)\right\} \hbox{E}\left\{Y(u)\hat{\bf S}^T(u)\right\}\over \hbox{E}\left\{Y(u)\right\}}\right. \cr &\left.-\hbox{E}\left\{Y(u){\bf H}(u,\varvec{\xi})\right\}\right]du} $$

uniformly in a neighborhood of $\varvec{\xi}_0$. It is easy to show that $\varvec{\Gamma}(\varvec{\xi}_0)=\varvec{\Gamma}_0$. Thus, under condition C, the asymptotic normality of $n^{1/2}(\hat{\varvec{\beta}}-\varvec{\beta}_0)$ follows from that of $n^{1/2}\hat{\bf U}(\varvec{\beta}_{0},\varvec{\xi}_0)$ with asymptotic variance $\Sigma (\varvec{\beta}_{0})=\varvec{\Gamma}_0^{-1}\Omega (\varvec{\beta}_{0},\varvec{\xi}_0)\left\{\varvec{\Gamma}_0^{-1} \right\}^{T}$. An estimator for $\Sigma(\varvec{\beta}_0)$ is $\hat{\Sigma}(\hat{\varvec{\beta}})=\hat{\varvec{\Gamma}} (\varvec{\xi}_0)^{-1}\hat{\Omega}(\hat{\varvec{\beta}},\varvec{\xi}_0) \left\{\hat{\varvec{\Gamma}}(\varvec{\xi}_0)^{-1}\right\}^{T}$. The consistency of $\Sigma(\varvec{\beta}_0)$ then follows from the almost sure convergence of $\hat{\Omega}(\varvec{\beta},\varvec{\xi}_0)$ uniformly in $\varvec{\beta}\in \mathcal{B}$.

When $\varvec{\xi}$ is unknown, following Song et al. (2002a), $\hat{\varvec{\xi}}$ is a consistent estimator of $\varvec{\xi}_0$ under condition D. Now $\hat{\varvec{\beta}}$ is obtained by replacing $\varvec{\xi}$ with $\hat{\varvec{\xi}}$ in (8), the denominator of which is equal to $n\hat{\varvec{\Gamma}}(\hat{\varvec{\xi}})$. It has been shown that $\hat{\varvec{\Gamma}}(\hat{\varvec{\xi}})$ converges almost surely to $\varvec{\Gamma}(\varvec{\xi})$ uniformly in a neighborhood of $\varvec{\xi}_0$. This, together with the continuity of $\varvec{\Gamma}(\varvec{\xi})$ and consistency of $\hat{\varvec{\xi}}$, implies that $\hat{\varvec{\Gamma}}(\hat{\varvec{\xi}})$ converges to $\varvec{\gamma}_0$. The consistency of $\hat{\varvec{\beta}}$ then follows. Using arguments similar to those when $\varvec{\xi}$ is known, we can show that $n^{1/2}\left\{(\hat{\varvec{\beta}}^T,\hat{\varvec{\xi}}^T)^T -(\varvec{\beta}_0^T,\varvec{\xi}_0^T)^T\right\}$ converges to a normal distribution with variance V=A⁻¹B{A⁻¹}^T, where ${\bf A}=\hbox{E}\left\{\partial{\varvec{\Phi}(\varvec{\beta}_0,\varvec{\xi}_0)}/ \partial{(\varvec{\beta}^T,\varvec{\xi}^T)}\right\}$ and ${\bf B}=\hbox{var}\{{\varvec{\varphi}}^\ast(\varvec{\beta}_0,\varvec{\xi}_0)\}$, $\varvec{\varphi}_i^{\ast}(\varvec{\beta}_0,\varvec{\xi}_0) =\{\varvec{\omega}_i^T(\varvec{\beta}_0,\varvec{\xi}_0),{\bf h}_i^T(\varvec{\beta}_0,\varvec{\xi}_0)\}^T$, and V is consistently estimated by $\hat{\bf V}$.