Regression analysis of current status data with latent variables

Abstract

Current status data occur in many fields including demographical, epidemiological, financial, medical, and sociological studies. We consider the regression analysis of current status data with latent variables. The proposed model consists of a factor analytic model for characterizing latent variables through their multiple surrogates and an additive hazard model for examining potential covariate effects on the hazards of interest in the presence of current status data. We develop a borrow-strength estimation procedure that incorporates the expectation–maximization algorithm and correlated estimating equations. The consistency and asymptotic normality of the proposed estimators are established. A simulation study is conducted to evaluate the finite sample performance of the proposed method. A real-life study on the chronic kidney disease of type 2 diabetic patients is presented.

Appendix

Let $\alpha _0$ and $\theta _0$ be the true values of $\alpha$ and $\theta$, respectively. Recall that $\varPi (\theta )=B \varPhi B^{T} + \varPsi$. The regularity conditions (C1)–(C4) that are required in the proof of Theorem 1 are as follows:

(C1): The matrix $\varPi (\theta _0)$ is positive definite; all partial derivatives of the first three orders of $\varPi (\theta _0)$ with respect to the elements of $\theta _0$ are continuous and bounded in a neighborhood of $\theta _0$; and the first derivative of $\varPi (\theta _0)$, ${\dot{\varPi }}(\theta _0)$, is of full rank in a neighborhood of $\theta _0$.

(C2): The function $\lambda _0(t)$ is continuous and integerable on $[0,\tau ]$, where $\tau$ is a prespecified positive constant, such that $P(C_{i}\ge \tau ) > 0$.

(C3): There exists a positive constant a such that $P(C_i\ge \tau , T_i \ge \tau \mid Z_i) > a$ with probability 1, and $Z_i$, $i=1,\cdots ,n$ are bounded almost surely.

(C4): The limit of matrix ${{\hat{A}}}$, A, is positive definite. and

$$\begin{aligned} {{\hat{A}}}=-\frac{1}{n}{\frac{{\partial U(\alpha ;{{\hat{\theta }}} )}}{{\partial \alpha }}} \bigg |_{\alpha = {{\hat{\alpha }}} }. \end{aligned}$$

We first define several notations as follows:

$$\begin{aligned}&{{S_Z}( {t,\alpha ,\theta }) = \sum \limits _{i=1}^n {Z_i^*\exp \left( { - {\beta ^T}Z_i^* - {\gamma ^T}\xi _i^*(\theta ) - \frac{1}{2}{\gamma ^T}D(\theta ){t^2}\gamma } \right) }{Y_i}\left( t \right) }, \end{aligned}$$

(A.1)

$$\begin{aligned}&\quad {{S_\xi }( {t,\alpha ,\theta }) = \sum \limits _{i=1}^n {\xi _i^*\exp \left( { - {\beta ^T}Z_i^* - {\gamma ^T}\xi _i^*(\theta ) - \frac{1}{2}{\gamma ^T}D(\theta ){t^2}\gamma } \right) } {Y_i}\left( t \right) }, \end{aligned}$$

(A.2)

$$\begin{aligned}&\quad {{S_0}( {t,\alpha ,\theta }) = \sum \limits _{i=1}^n {\exp \left( { - {\beta ^T}Z_i^* - {\gamma ^T}\xi _i^*(\theta ) - \frac{1}{2}{\gamma ^T}D(\theta ){t^2}\gamma } \right) } {Y_i}\left( t \right) }, \end{aligned}$$

(A.3)

$$\begin{aligned}&\quad S_\xi ^{(2)}\left( {t,\alpha ,\theta } \right) = \sum \limits _{i=1}^n {{Y_i}\left( t \right) \exp \left( { - {\beta ^T}Z_i^* - {\gamma ^T}\xi _i^*(\theta ) -\frac{1}{2}{\gamma ^T}D(\theta ){t^2}\gamma } \right) } \xi _j^{* \otimes 2}(\theta ), \end{aligned}$$

(A.4)

$$\begin{aligned}&\quad {Q}_1(\alpha ,\theta ) = \frac{1}{n}\sum \limits _{i = 1}^n {\int _0^\tau {\frac{{{t^2}}}{2}\left[ {1 + \frac{{{S_Z}\left( {t,\alpha , \theta } \right) }}{{{S_0}\left( {t,\alpha ,\theta } \right) }}} \right] d{H_0}(t)} }, \end{aligned}$$

(A.5)

$$\begin{aligned}&\quad {Q_2}(\alpha ,\theta ) = \frac{1}{n}\sum \limits _{i = 1}^n {\int _0^\tau {\left[ {\frac{{{{S_0}\left( {t,\alpha , \theta } \right) - {S_Z}\left( {t,\alpha ,\theta } \right) }}}{{{S_0}{{\left( {t,\alpha , \theta } \right) }^2}}}} {S^T_\xi }\left( {t,\alpha ,\theta } \right) \right] } d{H_0}(t)}, \end{aligned}$$

(A.6)

$$\begin{aligned}&\quad Q_3(\alpha ,\theta ) = \frac{1}{n}\sum \limits _{i = 1}^n {\int _0^\tau {\left[ {{\xi _i^*} - \frac{{{S_\xi }\left( {t,\alpha ,\theta } \right) }}{{{S_0}\left( {t,\alpha , \theta } \right) }}} \right] d{H_0}(t)} }, \end{aligned}$$

(A.7)

$$\begin{aligned}&\quad {Q_4}(\alpha ,\theta ) = \frac{1}{n}\sum \limits _{i = 1}^n {\int _0^\tau {\left[ S_\xi ^{(2)}( {t,\alpha ,\theta } )- {\frac{{{S^{\otimes 2}_\xi }( {t,\alpha ,\theta } )}}{{{S_0}{{( {t,\alpha ,\theta } )}^2}}}} \right] }d{H_0}(t)}, \end{aligned}$$

(A.8)

$$\begin{aligned}&\quad Q_5(\alpha ,\theta ) = \frac{1}{n}\sum \limits _{i = 1}^n {\int _0^\tau {\left[ {\frac{{{S_\xi } ({t,\alpha , \theta } )}}{{{S_0}( {t,\alpha ,\theta } )}}{\gamma ^T} + 2{I_{q}}} \right] D(\theta ){t^2}\gamma d{H_0}(t)}}, \end{aligned}$$

(A.9)

And $Q_d={Q}_d( \alpha _0, \theta _0)$ and ${{\hat{Q}}}_d={Q}_d({{\hat{\alpha }}},{{\hat{\theta }}})$, $d=1, \cdots , 5$.

$$\begin{aligned} K(\theta )= & {} {[{{\dot{\varPi }}} (\theta )\{ \varPi {(\theta )^{ - 1}} \otimes \varPi {(\theta )^{ - 1}}\} {{\dot{\varPi }}} {(\theta )^{ - 1}}]^{ - 1}}{{\dot{\varPi }}} (\theta )\{ \varPi {(\theta )^{ - 1}} \otimes \varPi {(\theta )^{ - 1}}\}, \end{aligned}$$

(A.10)

$$\begin{aligned} {R_i}(\theta )= & {} [{{{{\dot{\varGamma }}} }_1}(\theta )K(\theta )vec\{ V_i^{ \otimes 2} - \varPi (\theta )\} , \ldots ,{{{{\dot{\varGamma }}} }_p}(\theta )K(\theta )vec\{ V_i^{ \otimes 2} - \varPi (\theta )\}], \end{aligned}$$

(A.11)

$$\begin{aligned} {P_i}(\theta )= & {} [{{\dot{D}}_1}(\theta )K(\theta )vec\{ V_i^{ \otimes 2} - \varPi (\theta )\} , \ldots ,{{\dot{D}}_p}(\theta )K(\theta )vec\{ V_i^{ \otimes 2} - \varPi (\theta )\}],\nonumber \\ \end{aligned}$$

(A.12)

where $\otimes$ denotes the Kronecker product of two matrices, vec denotes the operation that converts a matrix into a column vector by stacking the rows sequentially, ${\dot{\varPi }} (\theta )= \partial {(vec\varPi (\theta ))^T}/\partial \theta ,$ ${\dot{\varGamma }}_j({\theta })\ (j=1,...,p)$ denote the derivatives of the jth column of $\varGamma (\theta )$ with respect to $\theta ^T$, and ${\dot{D}}_r({\theta }) \ (r=1,...,q)$ denote the derivatives of the rth column of ${D}({\theta })$ with respect to $\theta ^T$.

Let ${{{\hat{U}}}_i} = { ({{\hat{U}}}_{i1}^T, {{\hat{U}}}_{i2}^T)^T}$, ${{\hat{\varSigma }}}=\frac{1}{n}\sum \limits _{i = 1}^n {{\hat{U}}}_i^{ \otimes 2}$ and

In the above, $d{{{\hat{M}}}^c_i}(t) = dN_i^c(t) - {Y_i}(t)\exp \big \{- {{\hat{\beta }}} ^TZ_i^*(t) - {{\hat{\gamma }}}^T \varGamma ({{\hat{\theta }}}){V_i}t \big \}d{{{\hat{H}}}_0}(t)$ and ${{{\hat{H}}}_0}(t) = \sum \limits _{i = 1}^n \int _0^t \frac{{dN_i^c(s)}}{{\sum \nolimits _{j = 1}^n {{Y_j}(s)\exp ( - {{\hat{\beta }}}^TZ_j^*(t) - {{\hat{\gamma }}} ^T \varGamma ({{\hat{\theta }}}){V_j}t- \frac{1}{2}{{{\hat{\gamma }}} ^T}D({{\hat{\theta }}} ){t^2} {{\hat{\gamma }}})} }}$.

Now, we are ready to prove the consistency and asymptotic normality of ${{\hat{\alpha }}}$. For the consistency, it is easy to obtain from the following two facts. One is that parameter ${{\hat{\theta }}}$ and its functions $\varGamma ({{\hat{\theta }}})=({{\hat{B}}}^{T}{{\hat{\varPsi }}}^{-1}{{\hat{B}}})^{-1}{{\hat{B}}}^{T}{{\hat{\varPsi }}}^{-1}$ and $D({{\hat{\theta }}})=({{\hat{B}}}^T {{\hat{\varPsi }}}^{-1}{{\hat{B}}})^{-1}$ involved in the CFA model are consistent. The consistency of ${{\hat{\theta }}}$ has been well established in the literature (e.g., Amemiya, Fuller and Pantula, 1987; Anderson and Amemiya 1988; Lee 2007). The consistency of $\varGamma ({{\hat{\theta }}})$ and $D({{\hat{\theta }}})$ can be obtained in a similar manner as in Pan et al. (2015). The second fact is that the working corrected estimating equations $U_1(\alpha ; {\hat{\theta }}) = 0$ and $U_2(\alpha ; {\hat{\theta }}) = 0$ can be written as the summation of n independently and identically distributed mean zero random variables plus some negligible errors.

Hence, based on the lemma of Pan et al. (2015), we have

$$\begin{aligned}&{{\hat{\theta }}} - {\theta _0} = K({\theta _0})vec\{ {S^*} - \varPi ({\theta _0})\} + o_p({1 /{\sqrt{n} }}) , \end{aligned}$$

(A.13)

$$\begin{aligned}&\quad \varGamma ({{\hat{\theta }}} ) - \varGamma ({\theta _0}) = \frac{1}{n}\sum \limits _{i = 1}^n {{R_i}({\theta _0})} + o_p({1 / {\sqrt{n} }}) , \end{aligned}$$

(A.14)

$$\begin{aligned}&\quad D({{\hat{\theta }}} ) - D({\theta _0}) = \frac{1}{n}\sum \limits _{i = 1}^n {{P_i}({\theta _0})} + o_p({1 / {\sqrt{n} }}) , \end{aligned}$$

(A.15)

where $K(\theta )$, $R_i(\theta )$, and $P_i(\theta )$ are defined by (A.10), (A.11), and (A.12), respectively.

Under the AH model (2), we redefine a zero-mean stochastic process as follows: for $i = 1, \ldots ,n$,

$$\begin{aligned} d{M^c_i}(t) = dN_i^c(t) - {Y_i}(t)\exp \big \{- \beta _0^TZ_i^*(t) - \gamma _0^T\varGamma (\theta _0 ){V_i}t \big \}d{H_0}(t). \end{aligned}$$

Denote ${{\bar{Z}}}(t,\alpha _0,\theta _0)={{\bar{Z}}}(t)$. We can obtain the following:

$$\begin{aligned} {U_1}({\alpha _0};{\theta _0})&= \sum \limits _{i = 1}^n {\int _0^\tau {\{ Z_i^*(t) - {{\bar{Z}}}(t)\} } d{M^c_i}(t)} \\&= \sum \limits _{i = 1}^n {\int _0^\tau {\{ Z_i^*(t) - {e_Z}(t)\} } d{M^c_i}(t)} + o_p({n^{{1 / 2}}}),\\ {U_2}({\alpha _0};{\theta _0})&= \sum \limits _{i = 1}^n {\int _0^\tau {\{ {{{{\hat{\xi }}} }_i}({\theta _0})t - {{{{\bar{\xi }}} }_i}(t;{\theta _0})\} d{M^c_i}(t)} } + \sum \limits _{i = 1}^n {\int _0^\tau {D({\theta _0}){\gamma _0}{t^2}dN_{i}^{c}(t)}} \\&= \sum \limits _{i = 1}^n {\int _0^\tau {\{ \varGamma ({\theta _0}){V_i}t - {e_\xi }(t)\} d{M^c_i}(t)}} + \sum \limits _{i = 1}^n {\int _0^\tau {D({\theta _0}){\gamma _0}{t^2}dN_{i}^{c}(t)}} + o_p({n^{{1 /2}}}), \end{aligned}$$

where

$$\begin{aligned} {e_Z}(t)={\frac{ES_Z( {t,\alpha _0,\theta _0})}{ES_0( {t,\alpha _0,\theta _0})}}, ~~\text{ and }~~ {e_\xi }(t) = \frac{{E{S_\xi }(t,\alpha _0,\theta _0)}}{{E{S_0}(t,\alpha _0 ,\theta _0)}}. \end{aligned}$$

By the Taylor expansion, we have

$$\begin{aligned}&{U_1}({\alpha _0},{{\hat{\theta }}} ) - {U_1}({\alpha _0},{\theta _0}) \nonumber \\&\quad = - n{{{{\hat{Q}}}}_1}{\gamma _0 ^T}\left[ {D({{\hat{\theta }}} ) - D({\theta _0})} \right] \gamma _0 + n{{{{\hat{Q}}}}_2}{\left[ {\varGamma ({{\hat{\theta }}} ) - \varGamma ({\theta _0})} \right] ^T}\gamma _0 + o_p({n^{{1 / 2}}})\nonumber \\ \end{aligned}$$

(A.16)

and

$$\begin{aligned}&{U_2}({\alpha _0},{{\hat{\theta }}} ) - {U_2}({\alpha _0},{\theta _0})\nonumber \\&\quad = - n{{{{\hat{Q}}}}_4}{\left[ {\varGamma ({{\hat{\theta }}} ) - \varGamma ({\theta _0})} \right] ^T}\gamma _0 + n\left[ {\varGamma ({{\hat{\theta }}} ) - \varGamma ({\theta _0})} \right] {{{{\hat{Q}}}}_3}+ n{{{{\hat{Q}}}}_5}\left[ {D({{\hat{\theta }}} ) - D({\theta _0})} \right] \gamma _0 + o_p({n^{{1 / 2}}}).\nonumber \\ \end{aligned}$$

(A.17)

Based on (A.14) and (A.15), equation (A.16) can be rewritten as

$$\begin{aligned}&{U_1}({\alpha _0},{{\hat{\theta }}} ) - {U_1}({\alpha _0},{\theta _0}) \nonumber \\&\quad = - {Q_1}\sum \limits _{i = 1}^n {{\gamma _0 ^T}\left[ {{P_i}( \theta _0 )} \right] \gamma _0 } + {Q_2}\sum \limits _{i = 1}^n {{{\left[ {{R_i}( \theta _0 )} \right] }^T}\gamma _0 } + o_p({n^{{1 / 2}}}). \end{aligned}$$

(A.18)

Similarly, equation (A.17) can be rewritten as

$$\begin{aligned}&{U_2}({\alpha _0},{{\hat{\theta }}} ) - {U_2}({\alpha _0},{\theta _0})= - {Q_4}\sum \limits _{i = 1}^n {{{\left[ {{R_i}(\theta _0 )} \right] }^T}\gamma _0 } + \sum \limits _{i = 1}^n {\left[ {{R_i}(\theta _0 )} \right] {Q_3}} \nonumber \\&\quad {} + {Q_5}\sum \limits _{i = 1}^n {\left[ {{P_i}( \theta _0 )} \right] \gamma _0 } + o_p({n^{{1 / 2}}}). \end{aligned}$$

(A.19)

Thus, we obtain

$$\begin{aligned}&{U_1}({\alpha _0},{{\hat{\theta }}} ) = {U_1}({\alpha _0},{\theta _0}) + \mathrm{{\{ }}{U_1}({\alpha _0},{{\hat{\theta }}} ) - {U_1}({\alpha _0},{\theta _0})\mathrm{{\} }}\nonumber \\&\quad {} = \sum \limits _{i = 1}^n {{U_{i1}}} + o_p({n^{{1 / 2}}}), \end{aligned}$$

(A.20)

and

$$\begin{aligned}&{U_2}({\alpha _0},{{\hat{\theta }}} ) = {U_2}({\alpha _0},{\theta _0}) + \mathrm{{\{ }}{U_2}({\alpha _0},{{\hat{\theta }}} ) - {U_2}({\alpha _0},{\theta _0})\mathrm{{\} }}\nonumber \\&\quad {} = \sum \limits _{i = 1}^n {{U_{i2}}} + o_p({n^{{1 / 2}}}), \end{aligned}$$

(A.21)

where

$$\begin{aligned} {U_{i1}} = \int _0^\tau {(Z_i^* - {e_Z})d{M^c_i}(t)} - {Q_1}{\gamma _0 ^T}\left[ {{P_i}(\theta _0 )} \right] \gamma _0 + {Q_2}{\left[ {{R_i}( \theta _0 )} \right] ^T}\gamma _0 , \end{aligned}$$

and

$$\begin{aligned} {U_{i2}}= & {} \int _0^\tau {(\varGamma ({\theta _0}){V_i}t - {e_\xi })d{M^c_i}(t)} + \int _0^\tau {D(\theta _0 ) \gamma _0 {t^2}dN_i^c(t)} - {} {Q_4}{\left[ {{R_i}(\theta _0 )} \right] ^T}\gamma _0 \\&+ \left[ {{R_i}(\theta _0 )} \right] {Q_3} + {Q_5}\left[ {{P_i}(\theta _0 )} \right] \gamma _0 . \end{aligned}$$

Let ${U_i} = {(U_{i1}^T,U_{i2}^T)^T}$. Then, it follows from (A.20) and (A.21) that

$$\begin{aligned} U({\alpha _0};{{\hat{\theta }}} ) = \sum \limits _{i = 1}^n {{U_i}} + o_p({n^{{1 / 2}}}) , \end{aligned}$$

(A.22)

which is a sum of independently and identically distributed zero-mean random vectors plus an asymptotically negligible term. The law of large numbers and the multivariate central limit theorem show that $\frac{1}{n}U({\alpha _0};{{\hat{\theta }}} ) \rightarrow 0$ in probability and $\frac{1}{{\sqrt{n} }}U({\alpha _0};{{\hat{\theta }}} )$ converges in distribution to a normal random vector with mean zero and covariance matrix $\varSigma = E(U_i^{ \otimes 2})$. Note that

$$\begin{aligned} {{\hat{\alpha }}} - {\alpha _0} = {n^{ - 1}}{{{{\hat{A}}}}^{ - 1}}U({\alpha _0};{{\hat{\theta }}} ) +o_p({n^{{-1 / 2}}}) , \end{aligned}$$

(A.23)

and ${{\hat{A}}} \rightarrow A$ in probability by the consistency of $\varGamma ({{\hat{\theta }}})$ and $D({{\hat{\theta }}})$. Then, based on (A.23), ${{\hat{\alpha }}}$ converges in probability to $\alpha _0$, and $\sqrt{n}({{\hat{\alpha }}}- \alpha _0)$ is asymptotically normal with mean zero and covariance matrix ${A^{ - 1}}\varSigma {A^{ - T}}$, and $A^{-1}\varSigma {A^{-T}}$ can be consistently estimated by ${{{{\hat{A}}}}^{ - 1}}{{\hat{\varSigma }}} {{{{\hat{A}}}}^{ - T}}.$