Regression analysis for bivariate gap time with missing first gap time data

Abstract

We consider ordered bivariate gap time while data on the first gap time are unobservable. This study is motivated by the HIV infection and AIDS study, where the initial HIV contracting time is unavailable, but the diagnosis times for HIV and AIDS are available. We are interested in studying the risk factors for the gap time between initial HIV contraction and HIV diagnosis, and gap time between HIV and AIDS diagnoses. Besides, the association between the two gap times is also of interest. Accordingly, in the data analysis we are faced with two-fold complexity, namely data on the first gap time is completely missing, and the second gap time is subject to induced informative censoring due to dependence between the two gap times. We propose a modeling framework for regression analysis of bivariate gap time under the complexity of the data. The estimating equations for the covariate effects on, as well as the association between, the two gap times are derived through maximum likelihood and suitable counting processes. Large sample properties of the resulting estimators are developed by martingale theory. Simulations are performed to examine the performance of the proposed analysis procedure. An application of data from the HIV and AIDS study mentioned above is reported for illustration.

Appendices

Appendix 1: Proof of Theorem 1

First we shall show that $\frac{1}{\sqrt{n}} \sum _{i=1}^n\varvec{U}_i(\varvec{\varOmega })$ at $\varvec{\varOmega }_0$ converges to a multivariate normal distribution, where

$$\begin{aligned} \varvec{U}_i (\varvec{\varOmega })= \varvec{S}_{1i}(\varvec{\varOmega }) + \int _0^\zeta \varvec{S}_{2i}(v,\varvec{\varOmega })\;dM_{2i}(v)+ \int _0^\zeta \varvec{S}_{Ci}(\varvec{\varOmega })\;dM_{Ci}(v). \end{aligned}$$

Note that $\varvec{S}_{ki}, k=1,2,3$, have been defined in (8). By the multivariate central limit theorem, we have $\frac{1}{\sqrt{n}} \sum _{i=1}^n\varvec{S}_{1i}(\varvec{\varOmega }_0) \rightarrow N\left( 0, E\left\{ -\frac{1}{n}\sum _{i=1}^n\frac{\partial \varvec{S}_{1i}(\varvec{\varOmega }_0)}{\partial \varvec{\varOmega }}\right\} \right)$. Since we assume that $dN_{2i}, dN_{Ci}$ cannot jump simultaneously and two counting processes are conditionally independent to $\delta _{1i}$ given $Z_i$,

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{i=1}^n\varvec{S}_{1i}(\varvec{\varOmega }_0), \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\zeta \varvec{S}_{2i}(v, \varvec{\varOmega }_0)\;dM_{2i}(v), \text{ and } \\&\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\zeta \varvec{S}_{Ci}(\varvec{\varOmega }_0)\;dM_{Ci}(v), \end{aligned}$$

are asymptotically independent. Based on martingale central limit theorem and regular assumptations $\text{(A1) }-\text{(A5) }$, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{i=1}^n\left\{ \varvec{S}_{1i}(\varvec{\varOmega }_0)+ \int _0^\zeta \varvec{S}_{2i}(v, \varvec{\varOmega }_0)\;dM_{2i}(v) + \varvec{S}_{Ci}(\varvec{\varOmega }_0)\;dM_{Ci}(v)\right\} \rightarrow N\left( 0, \;\mathscr {I}_0 \right) . \end{aligned}$$

Furthermore, the differential of $-\frac{1}{n}\sum _{i=1}^n\varvec{U}_{i}(\varvec{\varOmega })$ with respect to $\varvec{\varOmega }$ is

$$\begin{aligned}&-\frac{1}{n}\sum _{i=1}^n\frac{\partial \varvec{S}_{1i} (\varvec{\varOmega })}{\partial \varvec{\varOmega }}\nonumber \\&\quad +\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ Y_i(v)\tilde{\eta }_{2i}(v)\lambda _2\varvec{S}_{2i}(v, \varvec{\varOmega })^{\otimes 2} + Y_i(v)\tilde{\eta }_{Ci}(v)\lambda _C\varvec{S}_{Ci}(\varvec{\varOmega })^{\otimes 2} \right\} dv\nonumber \\&\quad - \frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{\partial \varvec{S}_{2i}(v, \varvec{\varOmega })}{\partial \varvec{\varOmega }} dM_{2i}(v) + \frac{\partial \varvec{S}_{Ci}(\varvec{\varOmega })}{\partial \varvec{\varOmega }} dM_{Ci}(v)\right\} . \end{aligned}$$

(9)

By $\text{(A3) }$–$\text{(A4) }$ we have the first two terms on the right-hand side of (9) converge to $\mathscr {I}_0$, and the third term is $o_p(1)$ because the sum is scaled by $n^{-1}$. So with the Taylor expansion of $\frac{1}{n}\sum _{i=1}^n\varvec{U}_{i}(\widehat{\varvec{\varOmega }})$ around $\varvec{\varOmega }_0$, we have

$$\begin{aligned} \sqrt{n}\Big (\varvec{\widehat{\alpha }}-\varvec{\alpha }_0, \varvec{\widehat{\beta }}-\varvec{\beta }_0, \hat{\theta }-\theta _0 \Big )= \Big \{-\frac{1}{n}\sum _{i=1}^n\frac{\partial \varvec{U}_{i}(\varvec{\varOmega }^*)}{\partial \varvec{\varOmega }}\Big \}^{-1} \frac{1}{\sqrt{n}} \sum _{i=1}^n\varvec{U}_i(\varvec{\varOmega }_0), \end{aligned}$$

where $\varvec{\varOmega }^{*}$ is on the line segment between $\varvec{\widehat{\varOmega }}$ and $\varvec{\varOmega }_0$. Therefore, by Slutsky’s theorem and $\varvec{\widehat{\varOmega }}\rightarrow \varvec{\varOmega }_0$ almost surely, Theorem 1 is established.

Appendix 2: Information matrix

For $s,l \in \{1,2,C\}$, the differential of $-\frac{1}{n}\sum _{i=1}^n\varvec{U}_{i}(\varvec{\varOmega })$ for each parameter are given

$$\begin{aligned} I_{\alpha _s\alpha _l}= & {} (-1)^{I(s\ne l)} I(s\ne 2, l\ne 2)\frac{1}{n}\sum _{i=1}^n\left[ \frac{\exp (\alpha _1+ \beta _1'Z_i)\exp (\alpha _C+ \beta _C'Z_i)}{\left\{ \exp (\alpha _1+ \beta _1'Z_i)+\exp (\alpha _C+ \beta _C'Z_i)\right\} ^2}\right] \\&+\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\left\{ \widetilde{X}_{si}(v)+I(s=2)\right\} \left\{ \widetilde{X}_{li}(v)+I(l=2)\right\} dv\\&+ \,\,I(s = l =C)\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{Ci}(v)\exp (\alpha _C) dv\\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial \alpha _s\partial \alpha _l}-\widetilde{X}_{si}(v)\widetilde{X}_{li}(v)\right\} dM_{2i}(v),\\ I_{\alpha _s\beta _l}= & {} (-1)^{I(s\ne l)} I(s\ne 2, l\ne 2)\frac{1}{n}\sum _{i=1}^n\left[ \frac{\exp (\alpha _1+ \beta _1'Z_i)\exp (\alpha _C+ \beta _C'Z_i)}{\left\{ \exp (\alpha _1+ \beta _1'Z_i)+\exp (\alpha _C+ \beta _C'Z_i)\right\} ^2}\right] Z_i\\&+\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\left\{ \widetilde{X}_{si}(v)+I(s=2)\right\} \widetilde{Z}_{li}(v)\;dv\\&+\,\, I(s = l =C)\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{Ci}(v)\exp (\alpha _C)Z_i\; dv\\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial \alpha _s\partial \beta _l}-\widetilde{X}_{si}(v)\widetilde{Z}_{li}(v)\right\} dM_{2i}(v),\\ I_{\alpha _s\theta }= & {} \frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\left\{ \widetilde{X}_{si}(v)+I(s=2)\right\} \widetilde{W}'_{i}(v)\;dv \\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial \alpha _s\partial \theta }-\widetilde{X}_{si}(v)\widetilde{W}'_{i}(v)\right\} dM_{2i}(v),\\ I_{\beta _s\beta _l}= & {} (-1)^{I(s\ne l)} I(s\ne 2, l\ne 2)\frac{1}{n}\sum _{i=1}^n\left[ \frac{\exp (\alpha _1+ \beta _1'Z_i)\exp (\alpha _C+ \beta _C'Z_i)}{\left\{ \exp (\alpha _1+ \beta _1'Z_i)+\exp (\alpha _C+ \beta _C'Z_i)\right\} ^2}\right] Z_i^{\otimes 2}\\&+\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\widetilde{Z}_{si}(v)\widetilde{Z}'_{li}(v)\;dv\\&+\,\, I(s = l =C)\frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{Ci}(v)\exp (\alpha _C)Z_i^{\otimes 2} dv\\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial \beta _s\partial \beta _l}-\widetilde{Z}_{si}(v)\widetilde{Z}'_{li}(v)\right\} dM_{2i}(v),\\ \end{aligned}$$

$$\begin{aligned} I_{\beta _s\theta }= & {} \frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\widetilde{Z}_{si}(v)\widetilde{W}'_{i}(v)\;dv\\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial \beta _s\partial \theta }-\widetilde{Z}_{si}(v)\widetilde{W}'_{i}(v)\right\} dM_{2i}(v),\\ I_{\theta \theta }= & {} \frac{1}{n}\sum _{i=1}^n\int _0^\zeta Y_{i}(v)\tilde{\eta }_{2i}(v)\exp (\alpha _2)\widetilde{W}^2_{i}(v)dv\\&-\frac{1}{n}\sum _{i=1}^n\int _0^\zeta \left\{ \frac{1}{\tilde{\eta }_{2i}(v)}\frac{\partial ^2\tilde{\eta }_{2i}(v)}{\partial ^2 \theta }-\widetilde{W}_{i}^2(v)\right\} dM_{2i}(v),\\ \end{aligned}$$