Regression analysis of doubly truncated data based on pseudo-observations

Abstract

Doubly truncated data arise when an individual is potentially observed only if its failure-time lies within a certain interval, unique to that individual. In this paper, we consider the pseudo-observations approach for estimating regression coefficients when data is subject to double truncation. The pseudo-observations generated from the nonparametric maximum likelihood estimates (NPMLE) of the survival function are used as response variables in a generalized estimating equation to estimate the parameters of the model. We look at two estimators for regression parameters of survival probabilities based on different ways of defining pseudo-observations, namely, the simple pseudo-observations (SPO) and stopped pseudo-observations (STPO). We establish asymptotic properties of the two estimators under some conditions. Simulations results show that the proportion of failed estimation based on STPO are smaller than that based on SPO. The estimator based on STPO performs adequately for finite samples while the estimator based on SPO can be very unstable when sample size is not large enough.

Appendix:

Proof of Theorem 1

Since ${{\mathcal {E}}}_n$ is a vector of empirical function, condition (3.5) of Overgaard et al. (2017) holds, i.e., $||{{\mathcal {E}}}_n-{{\mathcal {E}}}||_p=o_p(n^{-\lambda })$ for some $\lambda \in [{1\over 4},{1\over 2})$ and $p\in [1,2)$, where $||f||_p=\sup \sum _{i=2}^{k}|f(t_{i-1})-f(t_{i})|^p+||f||_{\infty }$, over $\zeta _1\le t_1<\dots ,<t_k\le \zeta _2$ in the interval $[\zeta _1,\zeta _2]$, where $||\cdot ||_{\infty }$ is the supremum norm. Let

$$\begin{aligned} {\hat{S}}_{n,i}^{*}(t)=\Psi ({{\mathcal {E}}})+{\dot{\Psi }}_j({X}_i) +{1\over {n-1}}\sum _{i\ne i^{'}}\ddot{\Psi }({X}_i,{X}_{i^{'}}) \end{aligned}$$

and

$$\begin{aligned} {\hat{U}}_{n}^{*}(\beta _{t,0})=\sum _{i=1}^{n} A(\beta _{t,0};Z_i)^T \left( {\hat{S}}_{n,i}^{*}(t)-E[{\hat{S}}_{n,i}^{*}(t)|Z_i]\right) . \end{aligned}$$

Under (C1) and model (3), we have

$$\begin{aligned}&E[{\hat{S}}_{n,i}^{*}(t)|Z_i]=\Psi ({{\mathcal {E}}})+E[{\dot{\Psi }}({X}_i)|Z_i] \\&\quad =\Psi ({{\mathcal {E}}})+E[{\hat{S}}_n(t)|Z_i]-S(t_j)=E[{\hat{S}}_n(t)|Z_i]=\phi ^{-1}(\beta _{t}^T Z_i)+o(n^{-1/2}). \end{aligned}$$

It follows that by (3.42) of Overgaard et al. (2017)

$$\begin{aligned}&n^{-1/2}|{\hat{U}}_{n}(\beta _{t,0})-{\hat{U}}_{n}^{*}(\beta _{t})| =\sum _{i=1}^{n}\left| A(\beta _{t,0};Z_i)^T \left( [{\hat{S}}_{n,i}(t)-{\hat{S}}_{n,i}^{*}(t)]+o(n^{-1/2})\right) \right| \\&\quad \le \max _{i}|{{\mathcal {R}}}_{ij}|n^{-1/2}\sum _{i=1}^{n}\left| A(\beta _{t,0};Z_i)^T+o(n^{-1/2})\right| = o_p(n^{1/2-2\lambda })+ o(1), \end{aligned}$$

for $\lambda \in [1/4,1/2)$, where

$$\begin{aligned} {{\mathcal {R}}}_{ij}= & {} {1\over 2}\Psi _{{{\mathcal {E}}}}^{''}({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})+ {1\over {2(n-1)}}\Psi _{{{\mathcal {E}}}}^{''}(\delta _{{X}_i}-{{\mathcal {E}}}_n,\delta _{{X}_i}-{{\mathcal {E}}}_n) \\&+\Psi _{{{\mathcal {E}}}}^{''}({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_{n}^{-i}-{{\mathcal {E}}})+ \int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}}^{''}- \Psi _{{{\mathcal {E}}}}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})ds \\&+\int _{0}^{1}(1-s)(n-1)(\Psi _{{{\mathcal {E}}}_{n,s}}^{''}- \Psi _{{{\mathcal {E}}}_{n,s}^{-i},j}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})ds \\&+\int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}^{-i}}^{''} -\Psi _{{{\mathcal {E}}}}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},\delta _{\mathbf{X}_i}-{{\mathcal {E}}}_n)ds \\&+\int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}^{-i}}^{''} -\Psi _{{{\mathcal {E}}}}^{''})(\delta _{{X}_i}-{{\mathcal {E}}}_n,{{\mathcal {E}}}_n^{-i}-{{\mathcal {E}}})ds, \end{aligned}$$

where ${{\mathcal {E}}}_{n,s}={{\mathcal {E}}}+s({{\mathcal {E}}}_n-{{\mathcal {E}}})$ and ${{\mathcal {E}}}_{n,s}^{-i}={{\mathcal {E}}}+s({{\mathcal {E}}}_n^{-i}-{{\mathcal {E}}})$. Thus, $n^{-1/2}{\hat{U}}_{n}(\beta _{t,0})$ and $n^{-1/2}{\hat{U}}_{n}^{*}(\beta _{t})$ are asymptotically equivalent. Furthermore, ${\hat{U}}^{*}(\beta _{t,0})$ can be expressed as

$$\begin{aligned} {\hat{U}}^{*}(\beta _{t,0})=n{1\over {n\atopwithdelims ()2}}\sum _{i=1}^{n}\sum _{i^{'}<i} {1\over 2}h({X}_i,Z_i,{X}_{i^{'}},Z_{i^{'}}). \end{aligned}$$

The factor n aside, this is a U-statistic of order 2. It follows by Theorem 3.3 of Overgaard et al. (2017) that $n^{-1/2}{\hat{U}}_{n}(\beta _{t})$ converges in distribution to $N(0,\Sigma (\beta _{{t},0}))$, where $\Sigma (\beta _{t,0})) =E[h({X}_1,Z_1,{X}_2,Z_2)h({X}_1,Z_1,{X}_3,Z_3)^T]$. Under assumptions (A1)-(A5), it follows that $\sqrt{n}({\hat{\beta }}_{t}-\beta _{ t,0})$ converges in distribution to $N(0,M(\beta _{t,0})^{-1}\Sigma (\beta _{t,0})M(\beta _{t,0}))$ as $n\rightarrow \infty$. The proof is complete.

Proof of Theorem 2

Let

$$\begin{aligned} {\tilde{S}}_{n,i}^{*}(t)=\Psi ({{\mathcal {E}}})+{\dot{\Psi }}({X}_i) +{1\over {n_t-1}}\sum _{i\ne i^{'},i,i^{'}\in {{\mathcal {C}}}_d(t)}\ddot{\Psi }({X}_i,{ X}_{i^{'}}) \end{aligned}$$

and

$$\begin{aligned} {\tilde{U}}^{*}(\beta _{t,0})=\sum _{i\in {{\mathcal {C}}}_d(t)}A(\beta _{t,0};Z_i)^T \left( {\tilde{S}}_{n,i}^{*}(t)-E[{\tilde{S}}_{n,i}^{*}(t)|Z_i]\right) . \end{aligned}$$

It follows that by (3.42) of Overgaard et al. (2017)

$$\begin{aligned}&n_t^{-1/2}|{\tilde{U}}(\beta _{t,0})-{\tilde{U}}^{*}(\beta _{t,0})| =\sum _{i\in {{\mathcal {C}}}_d(t)}^{n}\left| A(\beta _{t,0};Z_i)^T \left( [{\tilde{S}}_{n,i}(t)-{\tilde{S}}_{n,i}^{*}(t)]+o(n_t^{-1/2})\right) \right| \\&\quad \le \max _{i\in {{\mathcal {C}}}_d(t)}|{{\mathcal {D}}}_{ij}|n^{-1/2}\sum _{i\in {{\mathcal {C}}}_d(t)}^{n}\left| A(\beta _{t,0};Z_i)^T+o(n_t^{-1/2})\right| = o_p(n_t^{1/2-2\lambda })+ o(1), \end{aligned}$$

for $\lambda \in [1/4,1/2)$, where

$$\begin{aligned} {{\mathcal {D}}}_{ij}= & {} {1\over 2}\Psi _{{{\mathcal {E}}}}^{''}({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})+ {{n_t-1}\over {2(n-1)^2}}\Psi _{{{\mathcal {E}}}}^{''}(\delta _{\mathbf{X}_i}-{{\mathcal {E}}}_n,\delta _{\mathbf{X}_i}-{{\mathcal {E}}}_n) \\&+{{n_t-1}\over {n-1}}\Psi _{{{\mathcal {E}}}}^{''}({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_{n}^{-i}-{{\mathcal {E}}})+{{n_t-1}\over {n-1}} \int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}}^{''}- \Psi _{{{\mathcal {E}}}}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})ds \\&+\int _{0}^{1}(1-s)(n_t-1)(\Psi _{{{\mathcal {E}}}_{n,s}}^{''}- \Psi _{{{\mathcal {E}}}_{n,s}^{-i}}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},{{\mathcal {E}}}_n-{{\mathcal {E}}})ds \\&+{{n_t-1}\over {n-1}}\int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}^{-i}}^{''} -\Psi _{{{\mathcal {E}}}}^{''})({{\mathcal {E}}}_n-{{\mathcal {E}}},\delta _{\mathbf{X}_i}-{{\mathcal {E}}}_n)ds \\&+{{n_t-1}\over {n-1}}\int _{0}^{1}(1-s)(\Psi _{{{\mathcal {E}}}_{n,s}^{-i}}^{''} -\Psi _{{{\mathcal {E}}}}^{''})(\delta _{\mathbf{X}_i}-{{\mathcal {E}}}_n,{{\mathcal {E}}}_n^{-i}-{{\mathcal {E}}})ds. \end{aligned}$$

Thus, $n_t^{-1/2}{\tilde{U}}(\beta _{t,0})$ and $n_t^{-1/2}{\tilde{U}}^{*}(\beta _{t,0})$ are asymptotically equivalent. Furthermore, ${\tilde{U}}^{*}(\beta _{t,0})$ can be expressed as

$$\begin{aligned} {\tilde{U}}^{*}(\beta _{t,0})=n_t{1\over {n_t\atopwithdelims ()2}}\sum _{i\in {{\mathcal {C}}}_d(t)}\sum _{i^{'}<i, i^{'}\in {{\mathcal {C}}}_d(t)}{1\over 2}h({X}_i,Z_i,{X}_{i^{'}},Z_{i^{'}}). \end{aligned}$$

It follows that $n_t^{-1/2}{\tilde{U}}(\beta _{t})$ converges in distribution to $N(0,\Sigma _d(\beta _{t,0}))$. Under assumptions (A1)–(A5), $\sqrt{n_t}({\tilde{\beta }}_{t}-\beta _{{t},0})$ converges in distribution to $N(0,M(\beta _{{t},0})^{-1}\Sigma _d(\beta _{{t},0})M(\beta _{{t_j},0}))$ as $n_t\rightarrow \infty$. The proof is complete.