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Lifetime Data Analysis

, Volume 12, Issue 1, pp 53–67 | Cite as

Nonparametric Estimation of Sojourn Time Distributions for Truncated Serial Event Data—a Weight-adjusted Approach

  • Shu-Hui Chang
  • Shinn-Jia Tzeng
Article

Abstract

In follow-up studies, survival data often include subjects who have had a certain event at recruitment and may potentially experience a series of subsequent events during the follow-up period. This kind of survival data collected under a cross-sectional sampling criterion is called truncated serial event data. The outcome variables of interest in this paper are serial sojourn times between successive events. To analyze the sojourn times in truncated serial event data, we need to confront two potential sampling biases arising simultaneously from a sampling criterion and induced informative censoring. In this study, nonparametric estimation of the joint probability function of serial sojourn times is developed by using inverse probabilities of the truncation and censoring times as weight functions to accommodate these two sampling biases under various situations of truncation and censoring. Relevant statistical properties of the proposed estimators are also discussed. Simulation studies and two real data are presented to illustrate the proposed methods.

Keywords

Bivariate distribution Informative censoring Multiple events Nonparametric estimation Recurrent events Truncation 

Introduction

In longitudinal studies, subjects may undergo a series of successive events. Data that involve serial events can be identified in various scientific areas, including biomedical sciences, reliability studies, demographical research, and possibly other fields. Examples in reliability studies include the repeated breakdowns of a machine. Medical examples of serial events include chronologically ordered clinical events such as HIV infection, AIDS, and death in studying a natural history of AIDS. To analyze serial event data, the sojourn times, which are defined as the times between two successive events, are often the time variables of interest. For the AIDS example, the first and second sojourn times of interest are the incubation time of HIV and the survival time with AIDS. In prospective studies, an incident cohort is a sample selected from subjects experiencing the initiating event within a recruiting period such as the HIV-incident cohort. Prevalent sampling is also a common sampling scheme in many follow-up studies in which subjects are recruited according to a certain sampling criterion. A prevalent cohort is thus defined as a sample of subjects who have experienced the initiating event but have not experienced the subsequent events at the calendar time of recruitment. For example, only subjects who have been infected with HIV and are still alive at the time of recruitment are included in the study of natural history of AIDS disease. In such HIV-prevalent cohort, either the first event time (time to the diagnosis of AIDS) or the second event time (time to death) is left-truncated and possibly right-censored in follow-up studies. Therefore, the main interest in this paper focuses on estimating the joint probability function of serial sojourn times under various situations of left truncation and right censoring.

Recently, several nonparametric methods for estimating the joint distribution function of serial sojourn times have been developed based on incident cohorts (Visser (1996), Wang and Wells (1998), and Lin, Sun and Ying (1999)). In particular, both nonparametric estimators considered by Wang and Wells (1998) and Lin et al. (1999) used the inverse probability of censoring as a weighted function to adjust the bias of induced informative censoring. On the other hand, when serial event data were collected from both incident and prevalent cohorts, Wang (1999) showed that the standard univariate survival methods could be used to analyze the second sojourn time provided that certain stationarity conditions were satisfied. Alternatively, for truncated bivariate survival data (i.e., ages of onset in affected parent-child pairs are retrospectively collected), nonparametric estimation of the bivariate distribution function has been extensively studied (Gürler, 1996, 1997; van der Laan, 1996; Quale and van der Laan, 2000; Huang, Vieland and Wang, 2001). Specifically, Gürler (1996, 1997) and van der Laan (1996) considered two different estimators of the weight function in estimation of the bivariate survival function to adjust the truncation effect.

While most existing methods for estimating the joint distribution function focused only on either serial event data from incident cohorts or bivariate survival data under truncation in the literature, less attention has been paid to studying the analysis of serial sojourn times in prevalent cohort studies. We note that two potential biases, truncation effect and induced informative censoring, must be resolved concurrently in the analysis of serial sojourn times from prevalent cohorts. In this paper, we develop nonparametrically weight-adjusted approaches to correct these two potential biases in various left truncation and right censoring models. In Section 2, we first introduce left truncation and right censoring models for serial events. The proposed estimators of the joint probability function of serial sojourn times are presented in Section 2. In addition, the asymptotic properties of the proposed estimators are discussed in Section 2. Simulation results and two empirical data examples are given in Section 3. Some possible generalization and remarks are elaborated in Section 4.

Estimation for left-truncated serial event data

Data structures and models

Suppose that a disease process consists of K successive events occurring in a chronological order. For simplicity, let K, say 3, represent three successive events including the initiating, first, and second events. Our proposed methods considered in the subsequent sections are also applicable to the setting of K>3. Let E0, E1, and E2, respectively, represent the calendar times of the initiating, first, and second events for a subject. We can subsequently define Y1=E1E0 and Y2=E2E1 as the first sojourn time and the second sojourn time, respectively. Let \({\tau}\) be the calendar time of recruitment and \({T=\tau -E_{0}}\) be the truncation time for a subject. First, we consider a type of left-truncated data including subjects who have experienced the initiating event and not experienced the first event before \({\tau}\). That is, subjects in this type of left-truncated data satisfy the recruiting criterion Y1T. For example, students over the age of 17 in a junior college in Taipei, Taiwan who never experienced any motorcycle crash belong to this type of left-truncated data (Lin et al., 2003). In the motorcycle crash example, two serial sojourn times of interest (Y1 and Y2) are age at the first occurrence of motorcycle crash and the duration between the first and second motorcycle crashes. Therefore, the truncation time (T) is age at entry into the study, in which birth is defined as the initiating event for controlling the potential birth cohort effect on the risk of motorcycle crash. Figure 1 provides a schematic description for the basic notations (Y1, Y2,T) and their relationship with \({(E_{0},E_{1},E_{2},\tau)}\) under the recruiting criterion Y1T.
Fig. 1

Notations for the recruiting criterion Y1T

In general, right censoring is present in left-truncated data from prospective studies. Let C be the censoring time from the initiating event to right censoring. Moreover, the condition, T<C, is always satisfied in the presence of left truncation and right censoring. In a classical left truncation and right censoring model, (T,C) is independent of (Y1, Y2) and has an arbitrary bivariate distribution function with the restriction \({\hbox{Pr}(T< C)=1}\), which is referred to as the general censoring setting. Two special types of the general censoring setting are first considered to develop nonparametric estimation of the joint probability function of (Y1, Y2) in the next section. First, when the exact censoring time C is always observed, this kind of censoring is referred to as type-A censoring. For instance, each student in the motorcycle crash study was followed to the end of study after recruitment. That is, the censoring time for each student is C=T+d0 in this study, where d0 is the length of study period. However, when right censoring is as a result of loss to follow-up, the exact censoring time may not be observed for uncensored cases. In this circumstance, we can define D=CT as the follow-up time from recruitment to right censoring and assume that D and T are independent (Wang, 1991). This special type of censoring (C=T+D) is called type-B censoring hereafter. More specifically, the general censoring setting includes type-A and type-B censoring because it allows incomplete observations of C in uncensored cases as well as arbitrary structures of dependence between T and C. Finally, we also derive a nonparametric estimate of the joint distribution of (Y1, Y2) under the general censoring setting.

Nonparametric estimation when the first event time is left-truncated

We first introduce notations and definitions to facilitate the development of nonparametric estimation of the joint probability function of (Y1, Y2) based on the type of left-truncated data under the recruiting criterion Y1 > T. Variables, Y1, Y2 T, and C, with superscript ‘o’ and subscript ‘i’ represent sampled variables for the ith subject satisfying the criterion Y1T. Variables, Y1, Y2, T, and C, without any additional superscript stand for population variables. Specifically, a random sample \((Y_1^o, Y_2^o, T^o, C^o)\) is selected from the random vector (Y1, Y2, T, C) given that Y1T. In the presence of right censoring, variables Y1 and Y2 with overscript ‘∼’ denote \(Y_1\wedge C\) and \(Y_{2}\wedge (C-\widetilde{Y_1})\), where \(A\wedge B\) is the minimum of A and B. Therefore, this type of left-truncated data includes \(\{(\widetilde Y_{1i}^o,\widetilde Y_{2i}^o,\delta _{1i}^o,\delta_{2i}^o, T_i^o): Y_{1i}^o \geq T_i^o,i=1,\ldots,n\} \), where \(\delta_{1i}^o = I(Y_{1i}^o \leq C_i^o )\), and \(\delta _{2i}^o = \delta_{1i}^o I(Y_{1i}^o + Y_{2i}^o \leq C_i^o)\).

We then define two bivariate probability functions, \(H_{Y_1,Y_2}(y_1,y_2)=\hbox{Pr}(Y_1 \leq y_1, Y_2 > y_2)\) and \(H_{T, C}(t,c)=\hbox{Pr}(T \le t, C > c)\). Let F W (·) and S W (·), respectively, denote the distribution and survival functions of a random variable (or vector) W. When both left truncation and right censoring occur, the range of the observed \((\widetilde Y_1^o, \widetilde Y_2^o)\) may not coincide with the complete support of \(H_{Y_{1}, Y_{2}}\). More specifically, only the conditional probability function \(H_{Y_1,Y_2}^\tau (y_1,y_2)=\hbox{Pr}(Y_1\leq y_1, Y_2 > y_2|Y_1 \geq\tau )\), which can be estimated over a restricted range \(\{(y_1,y_2): a \leq\tau\leq y_1\leq b_1, y_1+y_2\leq b_2\}\) for b2b1a, where \(a=\inf \{y: F_{T}(y)> 0\}> 0, S_{Y_1}(a)> 0\), and HT,C (a,b2)>0. Obviously, \(H^a_{Y_1,Y_2}=H_{Y_1,Y_2}\) if \(a \leq a_{Y_1}=\inf \{y: F_{Y_1}(y)> 0\}\). For notational convenience, nonparametric estimation of \(H_{Y_{1}, Y_{2}}(y_{1}, y_{2})\) is defined over \(\{ (y_1, y_2): a\leq y_1\leq b_1, y_1+y_2\leq b_2\}\) by assuming \(a\leq a_{Y_1}\). Furthermore, if \(a\leq a_{Y_1}, b_1=\hbox{sup}\{y: S_{Y_1}(y)> 0\}\) as well as \(b_2=\hbox{sup}\{y: S_{Y_1+Y_2}(y)> 0\}\), then \(H_{Y_{1}, Y_{2}}\) can be estimated completely. We use the convention 0/0=0 throughout the text.

To develop the estimation of \(H_{Y_{1}, Y_{2}}\), we start with the left truncation model under the type-A censoring situation. Because \(\widetilde{Y}_{2}^o\) is observed only if Y 1 o <C o for serial event data in the presence of right censoring, we consider the representation of the joint sub-probability of \(\{\widetilde{Y}_1^o \le y_1,\delta_1^o=1,\widetilde{Y}_2^o> y_2\}\) denoted by \(H^*_{\widetilde{Y}_1^o,\widetilde{Y}_2^o}(y_1,y_2)\),
$$H^{*}_{\widetilde{Y}_1^o,\widetilde{Y}_2^o}(y_1,y_2) = \int I(u \leq y_1) \{ H_{T, C}(u, u+y_2)/\beta_1\} H_{Y_1, Y_2} (du, y_{2}),$$
(1)
where \(\beta_1=\hbox{Pr}(T\leq Y_1)\). Since \((T_i^o,C_i^o) (i=1,\ldots,n)\) are observed in the data, we can estimate HT, C(t,c)/β1 for \(t \leq c\) by
$$\widehat{H}_{T, C}(t,c)/\widehat{\beta}_1=\int {{I(u \le t)}\over {\widehat S_{Y_1}(u)}} \widehat{H}_{T^o,C^o}(du,c),$$
(2)
which is derived from the representation of the bivariate probability function \(H_{T^o, C^o}(t,c)=\int_0^t S_{Y_1}(u)H_{T, C}(du,c)/\beta_1\) (Wang, 1991). In (2), \(\widehat{H}_{T^o,C^o}(t,c)=\sum_{i=1}^n I( T_i^o \leq t, C_i^o> c)/n\) is the empirical estimate of \(H_{T^o,C^o}(t,c)\) and \(\widehat{S}_{Y_1}\) is the product-limit estimate of \(S_{Y_1}\) based on the univariate left-truncated data \(\{\widetilde{Y}_{1i}^o,\delta_{1i}^o,T_i^o \} (i=1,\ldots,n)\). Therefore, from (1) and (2), we obtain a nonparametric estimate of \(H_{Y_{1}, Y_{2}}(y_{1}, y_{2})\),
$$\widehat{H}^{L_1(A)}(y_1,y_2)= \int {{I(u \leq y_1)}\over {\widehat{H}_{T, C}(u,u+y_2)/\widehat{\beta}_1}} \widehat{H}^*_{\widetilde{Y}_{1}^o,\widetilde{Y}_{2}^o}(du,y_2),$$
(3)
where \(\widehat{H}^*_{\widetilde{Y}_{1}^o,\widetilde{Y}_{2}^o}(u,v)=\sum_{i=1}^n I(\widetilde{Y}_{1i}^o \leq u,\widetilde{Y}_{2i}^o > v, \delta_{1i}^o=1)/n\) is the empirical estimate of \(H_{\widetilde{Y}_{1}^o,\widetilde{Y}_{2}^o}^*(u,v)\). In (3), the inverse of \(\widehat{H}_{T, C}/\widehat{\beta}_1\) is used to adjust both left truncation and right censoring effects.
We now consider nonparametric estimation of \(H_{Y_{1}, Y_{2}}\) for left-truncated data in the presence of type-B censoring where D=CT is independent of T. From the relation that C=T+D, our interest then focuses on the expression of the joint sub-probability of \(\{T^o \leq t, \widetilde Y_1^o \leq y_1,\delta_1^o=1,\widetilde Y_2^o > y_2\}\) denoted by \(H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}(t,y_1,y_2)\)
$$H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}(t,y_1,y_2)= \int \int I(u \leq y_1, w \leq u \wedge t)\{F_T(dw)/\beta_1\} S_D(u+y_2-w)H_{Y_1,Y_2}(du,y_2), $$
(4)
where S D is the survival function of D. Therefore, the representation of \(H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}\) in (4) leads to the following estimate of \(H_{Y_{1}, Y_{2}}(y_{1}, y_{2})\)
$$\widehat H^{L_1(B)} (y_1, y_2) = \int \int {I(t \leq u \leq y_1)\over \{\widehat F_T(u)/\widehat \beta_1\}\widehat S_D(u+y_2-t)} \widehat H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}(dt,du,y_2), $$
(5)
where \(\widehat S_D\) is the product-limit estimate of S D based on the univariate survival data \(\{\widetilde Y_{1i}^o + \widetilde Y_{2i}^o - T_i^o,1 - \delta _{2i}^o \}\), and \(\widehat H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}(t,y_1,y_2)=\sum_{i=1}^n I(T_i^o \leq t,\widetilde Y_{1i}^o \leq y_1,\widetilde Y_{2i}^o > y_2, \delta_{1i}^o=1)/n\) is the empirical estimate of \(H^*_{T^o,\widetilde Y_1^o,\widetilde Y_2^o}(t,y_1,y_2)\). In (5), \(\widehat F_T(t)/\widehat \beta_1=\int \{I(u \leq t)/\widehat S_{Y_1}(u)\} \widehat F_{T^o}(du)\), an estimate of F T (t)/β1, is immediately obtained from expression (2) at c=0, where \(\widehat F_{T^o}(t)=\sum_{i=1}^n I(T_i^o \leq t)/n\). In the presence of type-B censoring, \((\widehat F_T(t)/\widehat \beta_1)^{-1}\) and \(1/\widehat S_D\) are regarded as two weight functions as shown in (5) to adjust the sampling biases from left truncation and right censoring, respectively.
In the general censoring setting, the exact censoring time may be unknown for uncensored data and the classical left truncation and right censoring model allows HT, C to be an arbitrary bivariate probability function with the restriction Pr(T<C)=1. Therefore, a nonparametric estimate of \(H_{Y_{1}, Y_{2}}\) will be derived in the general censoring setting. We note that, from the expression in (1), HT,C plays a key role in estimating \(H_{Y_{1}, Y_{2}}\). To derive an estimate of HT,C,HT, C(t,c)/β1, for tc, is developed as follows.
$$H_{T, C}(t,c)/\beta_1=\{F_T(t)/\beta_1\}\times S_{C|T \leq t} (c|t),$$
(6)
where \(S_{C|T \leq t}(c|t)=\hbox{Pr}(C> c|T \leq t)\) is the conditional survival function of C given \(T \leq t\) for \(t \leq c\). From the assumption that (T,C) is independent of (Y1, Y1+Y2), we can derive a product-limit estimate of \(S_{C|T \leq t}(c|t)\), which is expressed as
$$\widehat S_{C|T \leq t}(c)=\prod_{\{k : \widetilde Y_{1k}^o+\widetilde Y_{2k}^o \leq c\}}\{1-d \widehat\Lambda_{C|T \leq t}(\widetilde Y_{1k}^o+\widetilde Y_{2k}^o )\}. $$
(7)
In (7), \(\widehat \Lambda _{C|T \leq t(c)}\) is an estimate of \(\Lambda_{C|T \leq t}(c)\), the cumulative conditional hazard function of C given \(T \leq t\) at time c,
$$\widehat\Lambda_{C|T\leq t(c)} = \int_t^c {\sum\nolimits_{i = 1}^n {N_i^C} (du;t)\over \sum\nolimits_{i = 1}^n {R_i} (u;t)},$$
(8)
where \(N^{C}_{i}(y;t)= I(\widetilde Y_{1i}^o+\widetilde Y_{2i}^o \le y, \delta_{2i}^o=0, T_i^o \le t \le \widetilde Y_{1i}^o)\) and \(R_{i}(y;t)=I(\widetilde Y_{1i}^o+\widetilde Y_{2i}^o \ge y, T_i^o \le t \le \widetilde Y_{1i}^o)\). Thus, it follows from (1), (6) and (7) that a nonparametric estimate of \(H_{Y_{1}, Y_{2}}(y_{1}, y_{2})\) has the following expression,
$$\widehat{H}^{L_1 (G)} (y_1, y_2 ) = \int {I(u \leq y_1 )\over \{\widehat{F}_T (u)/\widehat{\beta}_1\}\widehat{S}_{C|T\leq u} (u + y_2)}\widehat{H}_{\tilde Y_1^o,\tilde Y_2^o}^* (du,y_2 ),$$
(9)
where \({\{\widehat F_T(u)/\widehat \beta_1\}\widehat S_{C|T\leq u} (u+y_2)}\) is an estimate of HT,C1 to correct the biases from left-truncation and right censoring. We note that \(\widehat{H}^{L_1(G)}\) is also an appropriate estimate of \(H_{Y_{1}, Y_{2}}\) under either type-A or type-B censoring situation because the general censoring setting subsumes these two special censoring situations. It is interesting to note that, in the absence of censoring (C=∞), all the estimates in (3), (5), and (9) are reduced to that considered by van der Laan (1996), and also note that, in the absence of truncation (T=0), \(H^{L_1(G)}\) in (5) is the same as that proposed by Lin et al.(1999).

To establish the asymptotic properties of the estimators, \(\widehat H^{L_1(A)}\), \(\widehat H^{L_1(B)}\), and \(\widehat H^{L_1(G)}\) on \(\{(y_1,y_2): a \leq y_1 \leq b_1, y_1+y_2 \leq b_2\}\), we first introduce the following two assumptions: (A1) \(\int \int F_{Y_1,Y_2}(du,dv)/H_{T, C}(u,u+v) < \infty\) and (A2) SY_1, Y_1+Y_2(b1, b2) is bounded away from zero. Note here that the product-limit estimates, \(\widehat S_{Y_1}\), \(\widehat S_D\), and \(\widehat S_{C|T \leq t}\), and \(\widehat F_T/\widehat \beta_1\) are strongly consistent and bounded away from zero given by standard results. The strong law of large numbers then implies the strong consistency of \(\widehat H^{L_1(A)}\), \(\widehat H^{L_1(B)}\), and \(\widehat H^{L_1(G)}\) for \(H_{Y_{1}, Y_{2}}\). We therefore have the weak convergence of \(\sqrt{n}(\widehat H^{L_1(A)}-H_{Y_1,Y_2})\), \(\sqrt{n}(\widehat H^{L_1(B)}-H_{Y_1,Y_2})\), and \(\sqrt{n}(\widehat H^{L_1(G)}-H_{Y_1,Y_2})\) to mean zero Gaussian processes, which follows from assumptions (A1) and (A2). The details of the proof of the weak convergence results are included in a technical report by Chang and Tzeng (2005).

Nonparametric estimation when the second event time is left-truncated

For a disease process with three successive events, another type of left-truncated data could be collected under a different recruiting criterion Y1+Y2T. An example of this kind of left-truncated data is a HIV-prevalent sample including individuals who are infected with HIV and alive at the recruitment of study, in which three serial events of interest are HIV infection, development of AIDS and then death. Left-truncated data for three serial events under the recruiting criterion Y1+Y2T is referred to below as the second type of left-truncated data. This section considers nonparametric estimation of the bivariate distribution function \(F_{Y_{1}, Y_{2}}(y_{1}, y_{2})\) based on the second type of left-truncated data. As discussed in Section 2.2, \(F_{Y_1,Y_2}(y_1,y_2)\) is estimable over {(y1,y2):ay1+y2b2 } for every b2a provided that \(a\le \inf\{y:F_{Y_1+Y_2}(y)> 0\}\) and HT,C(a,b2)>0.

Some supplemental notation will be introduced for convenience of developing the estimation of \(F_{Y_1, Y_2}(y_1, y_2)\). Let variables, Y1, Y2, T, and C, with superscript ‘ \(\diamond\)’ be variables constrained by Y1+Y2T. That is, a random sample \((Y_1^\diamond, Y_2^\diamond,T^\diamond, C^\diamond)\) is selected from the random vector (Y1,Y2,T,C) satisfying the criterion Y1+Y2T. As mentioned in the previous section, variables Y1 and Y2 with overscript ‘∼’ represent \(Y_1 \wedge C\) and \(Y_2 \wedge (C-\widetilde Y_1)\), respectively. Thus, the second type of left-truncated data includes \(\{(\widetilde Y_{1i}^{\diamond},\widetilde Y_{2i}^{\diamond},\delta _{1i}^{\diamond},\delta _{2i}^{\diamond},T_i^{\diamond}):Y_{1i}^{\diamond}+ Y_{2i}^{\diamond}\ge T_i^{\diamond}, i=1,\cdots,n\}\), where \(\delta _{1i}^{\diamond} = I(Y_{1i}^{\diamond} \le C_i^\diamond)\), and \(\delta_{2i}^{\diamond}=\delta_{1i}^{\diamond} I(Y_{i1}^\diamond+Y_{i2}^\diamond \le C_i^\diamond)\).

In the second type of left truncation and right censoring model, the joint sub-distribution \(F^*_{\widetilde Y_1^\diamond, \widetilde Y_2^\diamond}(y_1,y_2)=\Pr(\widetilde Y_1^\diamond \le y_1, \widetilde Y_2^\diamond \le y_2, \delta_{2}^\diamond=1)\) has the following expression
$$F^*_{\widetilde Y_1^\diamond,\widetilde Y_2^\diamond}(y_1,y_2) =\int \int I(u \le y_1,v \le y_2)\{ H_{T, C}(u+v,(u+v)-)/\beta_2\} F_{Y_1,Y_2}(du,dv),$$
(10)
where \(\beta_2=\Pr (Y_1+Y_2 \ge T)\). When type-A censoring is present, we can obtain the following nonparametric estimator of \(F_{Y_1, Y_2}\),
$$\widehat {F}^{L_2(A)}(y_1,y_2)=\int \int {{I(u \le y_1, v \le y_2)}\over {\widetilde H_{T, C}(u+v,[u+v]-)/\widetilde \beta_2}} \widehat F^*_{\widetilde Y_{1}^\diamond,\widetilde Y_{2}^\diamond}(du,dv),$$
(11)
from (10). In (11), \(\widehat F^*_{\widetilde Y_{1}^\diamond,\widetilde Y_{2}^\diamond}(u,v)=\sum_{i=1}^n I(\widetilde Y_{1i}^\diamond \le u, \widetilde Y_{2i}^\diamond \le v, \delta_{2i}^\diamond=1)/n\) is the empirical estimate of \(F^*_{\widetilde Y_1^\diamond, \widetilde Y_2^\diamond}\), and \(\widetilde H_{T, C}(y,y-)/\widetilde \beta_2=\int \{ I(u \le y)/{\widehat S_{Y_1+Y_2}(u)}\}\widehat H_{T^\diamond,C^\diamond}(du,y-)\), where \(\widehat S_{Y_1+Y_2}\) is a product-limit estimate of \(S_{Y_1+Y_2}\) based on \(\{\widetilde Y_{1i}^{\diamond}+\widetilde Y_{2i}^{\diamond}, \delta_{2i}^{\diamond}, T_i^{\diamond}\}\) (i=1, ..., n) and \(\widehat H_{T^\diamond,C^\diamond}(u,v)=\sum_{i=1}^n I(T_{i}^\diamond \le u, C_{i}^\diamond > v)/n\).
In the presence of type-B censoring, it is analogous to the estimate \(\widehat H^{L_1(B)}\) in (5) that a nonparametric estimate of \(F_{Y_1,Y_2}(y_1,y_2)\) has the form
$$\widehat F^{L_2(B)} (y_1, y_2 ) =\int \int \int {{I(u \le y_1, v \le y_2, t \le u+v)}\over {\{ \widehat F_T (u+v)/\widehat \beta_2\}\widehat S_D(u+v-t)}}\widehat F^*_{T^\diamond,\widetilde Y_1^\diamond,\widetilde Y_2^\diamond}(dt,du,dv),$$
(12)
where \(\widehat F_T (t)/\widehat \beta_2 =\int_0^t \widehat F_{T^{\diamond}} (du) /\widehat S_{Y_1+Y_2} (u)\) and \(\widehat F^*_{T^\diamond,\widetilde Y_1^\diamond,\widetilde Y_2^\diamond}(t,u,v)=\sum_{i=1}^n I(T^\diamond_i \le t, \widetilde Y_{1i}^\diamond \le u, \widetilde Y_{2i}^\diamond \le v, \delta_{2i}^\diamond=1)/n\).
In the general censoring situation that P(T<C)=1, we can also derive a nonparametric estimate of F(y1,y2), which is parallel to \(\widehat H^{L_1(G)}\) in (9), given by
$$\widehat F^{L_2(G)}(y_1,y_2) = \int \int {{I(u \le y_1,v\le y_2)}\over {\widehat H_{T, C}(u+v,(u+v)-)/\widehat \beta_2}}\widehat F^*_{\widetilde Y_{1}^\diamond,\widetilde Y_{2}^\diamond}(du,dv).$$
(13)

In (13), we have \(\widehat H_{T, C}(y,y-)/\widehat \beta_2 =\int \{I(u \le y)/\widehat S_{Y_1+Y_2}(u-)\} \widehat K_{T^\diamond,\widetilde Y_1^\diamond+ \widetilde Y_2^\diamond}(du)\), which is derived from the equality \(H_{T^\diamond,\widetilde Y_1^\diamond+ \widetilde Y_2^\diamond}(y,y-)= H_{T, C}(y,y-) S_{Y_1+Y_2}(y-)/\beta_2\), where \(\widehat K_{T^\diamond,\widetilde Y_1^\diamond+ \widetilde Y_2^\diamond}(y)=\sum_{i=1}^n I(T^\diamond_i \le y \le \widetilde Y_{1i}^\diamond+ \widetilde Y_{2i}^\diamond )/n\) is the empirical estimate of \(H_{T^\diamond,\widetilde Y_1^\diamond+ \widetilde Y_2^\diamond}(y,y-)\). We note that in the absence of censoring the estimate \(\widehat F^{L_2(G)}\) is reduced to that proposed by Gürler (1996). We introduce two assumptions: (B1) \(\int \{F_{Y_1+Y_2}(du)/H_{T, C}(u,u-)\}< \infty\) and (B2) \(S_{Y_1+Y_2}(b_2)\) is bounded away from zero, which are similar to those given in the last paragraph of Section 2.2. Then, from these two assumptions, we also have that \(\widehat F^{L_2(A)}\), \(\widehat F^{L_2(B)}\), and \(\widehat F^{L_2(G)}\) are the strongly consistent estimators of \(F_{Y_1,Y_2}(y_1,y_2)\) and \(\sqrt{n}(\widehat F^{L_2(A)}-F_{Y_1,Y_2})\), \(\sqrt{n}(\widehat F^{L_2(B)}-F_{Y_1,Y_2})\) and \(\sqrt{n}(\widehat F^{L_2(G)}-F_{Y_1,Y_2})\) converge weakly to mean-zero Gaussian processes.

Our proposed estimation methods can also be generalized to estimate the joint sub-distribution of serial sojourn times from left-truncated data in the presence of competing risks, which is an identifiable function as the cumulative incidence function in typical competing risks settings. For example, consider a set of colon cancer data including residents over age 60 in a community of Taiwan. For those residents with colon cancer, their ages at the first diagnosis of colon cancer were retrospectively collected from the cancer registry. At the end of follow-up, some residents had died free of colon cancer, but some other residents died from colon cancer. In this example, the truncation time T is a resident’s age at entry into the study and two potential types of the first event for a resident are the first diagnosis of colon cancer and death without experiencing colon cancer. For those residents having colon cancer, their cause of death (referred to as the second event) could be attributed to either colon cancer or other causes. Thus, we now define \(Y_1=Y_{11}\wedge Y_{12}\) and \(Y_2=\{Y_{21} \wedge Y_{22}\}I(Y_{11}< Y_{12})\), where Y jk represents the jth sojourn time for the kth type of the jth event for k and j=1,2, and Y2=0 if Y11>Y12. In the colon cancer example, Y11 and Y12, respectively, correspond to age at the first diagnosis of colon cancer and age at death without experiencing colon cancer. For a resident with colon cancer, the survival time from his/her first colon cancer diagnosis to death from this disease (or other causes) is denoted by Y21 (or Y22). Thus, the recruiting criterion is Y1+Y2T in this example. It is often of interest to estimate the joint sub-distribution of two sojourn times for the incidence and mortality of colon cancer, denoted by \(F^{**}_{Y_1,Y_2}(y_1,y_2)=\hbox{Pr}(Y_1 \le y_1,Y_2 \le y_2, Y_{11} \le Y_{12}, Y_{21} \le Y_{22})\). It is noted that \(F^{**}_{Y_{1}, Y_{2}}\) is estimable because it can be expressed in terms of the joint distribution of the sampled triplet \((\widetilde Y_{1i}^\diamond,\widetilde Y_{2i}^\diamond,\delta_{21i}^\diamond)\), where \(\delta_{11i}^\diamond=\delta_{1i}^\diamond I( Y_{11i} ^\diamond \le Y_{12i}^\diamond)\) and \(\delta_{21i}^\diamond=\delta_{11i}^\diamond \delta_{2i}^\diamond I( Y_{21i} ^\diamond \le Y_{22i}^\diamond)\). More specifically, from the expression of the joint distribution of \((\widetilde Y_{1i}^\diamond,\widetilde Y_{2i}^\diamond,\delta_{21i}^\diamond=1)\) as \(\int \int I(u \le y_1, v \le y_2)\{H_{T, C}(u+v,(u+v)-)/\beta_2\}F_{Y_1,Y_2}^{**}(du,dv)\), we can obtain nonparametric estimators of \(F^{**}_{Y_1,Y_2}(y_1,y_2)\) simply by substituting \(I\{\widetilde Y_{1i}^\diamond \le u,\widetilde Y_{2i}^\diamond \le v, \delta_{21i}^\diamond =1\}\) for \(I\{\widetilde Y_{1i}^\diamond \le u,\widetilde Y_{2i}^\diamond \le v, \delta_{2i}^\diamond =1\}\) into (11) and (13). Similarly, when type-B censoring is present, \(I\{T^\diamond_1 \le t,\widetilde Y_{1i}^\diamond \le u,\widetilde Y_{2i}^\diamond \le v, \delta_{2i}^\diamond =1\}\) can be placed with \(I\{T^\diamond_1 \le t,\widetilde Y_{1i}^\diamond \le u,\widetilde Y_{2i}^\diamond \le v, \delta_{21i}^\diamond =1\}\) in (12) to yield an estimate of \(F^{**}_{Y_1,Y_2}(y_1,y_2)\).

Examples

Simulation

In this section we conduct simulations to assess the performance of the proposed estimators for left-truncated data under the recruiting criterion Y1T in the previous section. In each simulation sample, the bivariate sojourn times (y1i, y2i) are generated from Clayton’s (1987) bivariate exponential survival function with an association parameter \(\theta,S(y_1, y_2 )= \left\{ {\exp (y_1/ \theta) + \exp (y_2/ \theta) - 1} \right\}^{-\theta}\), where \(\theta=1.5\) is chosen to represent a moderate (0.51) correlation of (Y1,Y2). Two left-truncation and right-censoring models are considered for (Y1,Y2) and (T,C). The first truncation and censoring model satisfies the sampling criterion Y1T and type-B censoring assumption that CT is independent of T. In this model, the truncation time t i is exponentially distributed with mean μ T =10 and the censoring time c i =t i +d i , where d i is uniformly distributed on (0,4) and is independent of (t i , y1i, y2i). Next, we consider various levels of correlation between T and CT in the second truncation and censoring model. In the second model, \(t_i= \mu_{w_1} w_{1i}\) and \(c_i=t_i+\mu_{w_2} w_{2i}\) are generated, where (w1i,w2i) are obtained from Clayton’s bivariate exponential survival function with an association parameter \(\theta_w\). We choose three sets of \((\mu_{w_1}, \theta_w,\mu_{w_2})=(4,6,5.5), (1,3,3),\) and (0.5,2,5.5), respectively, to give (β1T,C-T,cp2)=(0.2, 0.16, 0.3), (0.5, 0.3, 0.5), and (0.67, 0.42, 0.34), where ρT,C-T and cp2 denote the correlation between T and CT and the censoring rate \(\Pr(Y_1^o+Y_2^o> C^o)\), respectively. In each simulation sample, 100 quadruples (t i ,c i , y1i, y2i) are retained as y1it i . Tables 1 and 2 summarize the simulation results including the empirical biases and empirical Monte-Carlo standard deviations from 1000 samples given size n equal to 100.
Table 1

Simulation results for the left-truncated data with Y1T

(y1,y2)

H(y1,y2)

\(\triangle\widehat H^{L_1(A)}(y_1,y_2)\) (sd)

\(\triangle\widehat H^{L_1(B)}(y_1,y_2)\) (sd)

\(\triangle\widehat H^{L_1(G)}(y_1,y_2)\) (sd)

β1=9%, ρT,C-T=0.,cp1=0.24, cp2=0.47

(0.223, 0.223)

0.141

−0.002 (0.075)

−0.002 (0.076)

−0.004 (0.075)

(0.223, 0.511)

0.090

−0.003 (0.060)

−0.003 (0.061)

−0.005 (0.060)

(0.223, 0.916)

0.047

−0.002 (0.039)

−0.002 (0.040)

−0.003 (0.039)

(0.223, 1.609)

0.015

0.000 (0.025)

0.000 (0.024)

0.000 (0.026)

(0.511, 0.223)

0.290

−0.002 (0.074)

−0.001 (0.075)

−0.012 (0.073)

(0.511, 0.511)

0.190

−0.002 (0.068)

−0.002 (0.068)

−0.009 (0.067)

(0.511, 0.916)

0.103

−0.001 (0.051)

−0.001 (0.051)

−0.005 (0.050)

(0.511, 1.609)

0.035

0.000 (0.030)

0.000 (0.029)

−0.002 (0.030)

(0.916, 0.223)

0.447

−0.002 (0.073)

−0.002 (0.073)

−0.032 (0.071)

(0.916, 0.511)

0.303

−0.002 (0.070)

−0.002 (0.071)

−0.023 (0.069)

(0.916, 0.916)

0.173

−0.002 (0.057)

−0.001 (0.057)

−0.014 (0.055)

(0.916, 1.609)

0.063

−0.001 (0.039)

−0.001 (0.038)

−0.006 (0.037)

(1.609, 0.223)

0.615

−0.001 (0.068)

−0.001 (0.068)

−0.072 (0.066)

(1.609, 0.511)

0.435

−0.001 (0.073)

−0.001 (0.072)

−0.055 (0.069)

(1.609, 0.916)

0.263

−0.001 (0.064)

−0.001 (0.064)

−0.036 (0.059)

(1.609, 1.609)

0.106

−0.001 (0.051)

−0.001 (0.050)

−0.017 (0.046)

β1=20%, ρT,C-T=0.162, cp1=0.168, cp2=0.344

(0.223, 0.223)

0.141

−0.007 (0.111)

−0.007 (0.108)

−0.009 (0.107)

(0.223, 0.511)

0.090

−0.002 (0.088)

−0.002 (0.089)

−0.003 (0.089)

(0.223, 0.916)

0.047

−0.002 (0.070)

−0.003 (0.068)

−0.003 (0.071)

(0.223, 1.609)

0.015

0.000 (0.049)

−0.001 (0.048)

−0.001 (0.048)

(0.511, 0.223)

0.290

−0.006 (0.121)

−0.007 (0.119)

−0.014 (0.118)

(0.511, 0.511)

0.190

−0.003 (0.105)

−0.003 (0.106)

−0.007 (0.105)

(0.511, 0.916)

0.103

−0.003 (0.081)

−0.004 (0.080)

−0.005 (0.082)

(0.511, 1.609)

0.035

0.000 (0.052)

0.000 (0.050)

−0.001 (0.050)

(0.916, 0.223)

0.447

−0.004 (0.108)

−0.004 (0.107)

−0.022 (0.106)

(0.916, 0.511)

0.303

−0.001 (0.103)

−0.002 (0.104)

−0.014 (0.103)

(0.916, 0.916)

0.173

−0.002 (0.090)

−0.004 (0.088)

−0.010 (0.090)

(0.916, 1.609)

0.063

0.000 (0.059)

−0.001 (0.057)

−0.003 (0.057)

(1.609, 0.223)

0.615

−0.001 (0.095)

−0.002 (0.093)

−0.040 (0.093)

(1.609, 0.511)

0.435

0.001 (0.101)

0.000 (0.101)

−0.029 (0.100)

(1.609, 0.916)

0.263

−0.001 (0.092)

−0.003 (0.090)

−0.020 (0.091)

(1.609, 1.609)

0.106

0.001 (0.067)

0.000 (0.065)

−0.008 (0.064)

\(\triangle \widehat H(y_1,y_2)= \widehat H(y_1,y_2)-H(y_1,y_2)\)

sd=empirical Monte-Carlo standard deviation of 1000 estimates

cp1=Pr(Y 1 o >C o ), cp2=Pr(Y 1 o +Y 2 o >C o )

Table 2

Simulation results for the first left-truncated data with Y1T

(y1,y2)

H(y1,y2)

\(\triangle\widehat H^{L_1(A)}(y_1,y_2)\) (sd)

\(\triangle\widehat H^{L_1(B)}(y_1,y_2)\) (sd)

\(\triangle\widehat H^{L_1(G)}(y_1,y_2)\) (sd)

β1=50%, ρT,C-T=0.30, cp1=0.28, cp2=0.50

(0.223, 0.223)

0.141

−0.005 (0.094)

−0.007 (0.091)

−0.008 (0.093)

(0.223, 0.511)

0.090

0.000 (0.082)

−0.003 (0.075)

−0.002 (0.083)

(0.223, 0.916)

0.047

0.000 (0.070)

−0.002 (0.062)

−0.001 (0.066)

(0.223, 1.609)

0.015

0.000 (0.045)

−0.001 (0.044)

−0.001 (0.045)

(0.511, 0.223)

0.290

−0.007 (0.104)

−0.012 (0.102)

−0.022 (0.103)

(0.511, 0.511)

0.190

−0.001 (0.098)

−0.006 (0.092)

−0.011 (0.098)

(0.511, 0.916)

0.103

0.001 (0.082)

−0.004 (0.074)

−0.005 (0.077)

(0.511, 1.609)

0.035

0.000 (0.048)

−0.002 (0.046)

−0.002 (0.047)

(0.916, 0.223)

0.447

−0.006 (0.097)

−0.014 (0.096)

−0.044 (0.097)

(0.916, 0.511)

0.303

−0.001 (0.099)

−0.009 (0.093)

−0.027 (0.098)

(0.916, 0.916)

0.173

0.000 (0.089)

−0.007 (0.082)

−0.016 (0.083)

(0.916, 1.609)

0.063

0.000 (0.054)

−0.004 (0.052)

−0.007 (0.052)

(1.609, 0.223)

0.615

−0.003 (0.089)

−0.013 (0.087)

−0.083 (0.089)

(1.609, 0.511)

0.435

0.002 (0.097)

−0.008 (0.092)

−0.057 (0.096)

(1.609, 0.916)

0.263

0.003 (0.091)

−0.006 (0.084)

−0.037 (0.085)

(1.609, 1.609)

0.106

0.001 (0.063)

−0.004 (0.059)

−0.017 (0.058)

β1=80%, ρT,C-T=0.420, cp1=0.163, cp2=0.299

(0.223, 0.223)

0.141

−0.002 (0.074)

−0.004 (0.074)

−0.004 (0.074)

(0.223, 0.511)

0.090

−0.001 (0.063)

−0.004 (0.063)

−0.002 (0.063)

(0.223, 0.916)

0.047

−0.002 (0.042)

−0.004 (0.041)

−0.002 (0.042)

(0.223, 1.609)

0.015

−0.001 (0.028)

−0.002 (0.030)

−0.001 (0.029)

(0.511, 0.223)

0.290

−0.002 (0.075)

−0.006 (0.075)

−0.012 (0.075)

(0.511, 0.511)

0.190

−0.001 (0.068)

−0.005 (0.068)

−0.007 (0.068)

(0.511, 0.916)

0.103

−0.001 (0.052)

−0.004 (0.050)

−0.004 (0.051)

(0.511, 1.609)

0.035

−0.001 (0.032)

−0.002 (0.033)

−0.002 (0.032)

(0.916, 0.223)

0.447

−0.004 (0.074)

−0.008 (0.073)

−0.028 (0.073)

(0.916, 0.511)

0.303

−0.002 (0.068)

−0.007 (0.068)

−0.019 (0.067)

(0.916, 0.916)

0.173

−0.001 (0.056)

−0.005 (0.055)

−0.011 (0.055)

(0.916, 1.609)

0.063

−0.001 (0.038)

−0.003 (0.039)

−0.005 (0.037)

(1.609, 0.223)

0.615

−0.002 (0.069)

−0.006 (0.068)

−0.055 (0.068)

(1.609, 0.511)

0.435

0.000 (0.070)

−0.005 (0.070)

−0.041 (0.067)

(1.609, 0.916)

0.263

0.000 (0.060)

−0.003 (0.059)

−0.026 (0.057)

(1.609, 1.609)

0.106

0.001 (0.044)

−0.002 (0.045)

−0.013 (0.042)

\(\triangle \widehat H(y_1,y_2)= \widehat H(y_1,y_2)-H(y_1,y_2)\)

sd=empirical Monte-Carlo standard deviation of 1000 estimates

cp1=Pr(Y 1 o >C o ), cp2=Pr(Y 1 o +Y 2 o >C o )

From the results in Tables 1 and 2, \(\widehat H^{L_1(A)}\) which requires knowledge for all the potential censoring times has smaller bias and variance than \(\widehat H^{L_1(B)}\) and \(\widehat H^{L_1(G)}\) at almost every point (y1,y2). It is also notable that the differences between the variances of \(\widehat H^{L_1(A)}\), \(\widehat H^{L_1(B)}\) and \(\widehat H^{L_1(G)}\) are very small. The results for the first model (ρT,C-T=0) in Table 1 indicate that the performance of \(\widehat H^{L_1(B)}\) in terms of bias and variance is comparable to that of \(\widehat H^{L_1(A)}\). Tables 1 and 2 also show that \(\widehat H^{L_1(B)}\) is virtually unbiased even when there exists a mild correlation between T and CT. In addition, the bias of \(\widehat H^{L_1(B)}\) becomes larger as ρT,C-T increases. Regardless of the magnitude of ρT,C-T, \(\widehat H^{L_1(G)}(y_1,y_2)\) appears to be unbiased at small values of y1 and y2. However, the bias of \(\widehat H^{L_1}(G)\) increases substantially at large values of y1 and y2 as censoring becomes more severe. Additional simulation results indicate that the effect of the correlation between Y1 and Y2 is not significant on the performance of each estimate; and a high truncation probability produces an increase in the bias and variance for each estimator. We note that the negative mass problem occurs in most bivariate distribution estimation for censored data (Pruitt, 1991). In addition, the weight function with the censoring effect adjusted in the estimators may make major contribution to the negative mass. Simulations are conducted to compute the number and magnitude of negative mass points in each estimator. The total amount of negative mass assigned by each estimate ranges from 0 to 0.6 as 50% of Y 2 o is censored and tends to be zero as the censoring proportion decreases. The average percentage of points with negative mass is from 15% to 25% while 25–50% of Y 2 o is censored. The simulation results for \(\widehat F^{L_2(A)}\), \(\widehat F^{L_2(B)}\), and \(\widehat F^{L_2(G)}\) derived from the second type of left-truncated data are very close to the above results, so they are omitted here.

Analysis of motorcycle crash data

Motorcycle crash data was analyzed to illustrate the proposed estimates in Section 2.2. This motorcycle crash data consisted of 524 male students and 693 female students. Among 524 male students, 250 and 87 had experienced their first and second motorcycle crashes, respectively, in the follow-up period from February 1, 1995 to June 30, 1996. During the follow-up period, 239 of 693 female students had their first motorcycle crash and 84 of them had the second crash. Let d0 (=1.42 years) be the length of study period. In this study, each student had a censoring time C=T+d0 because each student was completely followed up to the end of study. The minimum age at the occurrence of the first crash (Y1) is 17.83 for male students and 17.59 for female students. It is interesting to note that the minimum age of getting a license to ride a motorcycle in Taiwan is 18-years old. To compare the risk of having motorcycle crashes for male students with the risk for female students, we first present the product-limit estimates of the conditional survival functions of Y1 given Y1 ≥ 18 ( \(S_{Y_1}^{18}(y)\)) for male and female students in Figure 2. Figure 2 shows that male students always have a higher risk of having their initial crash than female students among those who did not have any motorcycle crash before age 18. The median ages at the first motorcycle incidence for male and female students are 19.3 and 20.2 years old, respectively. This result may be explained by the fact that males in general start to ride motorcycles at younger ages than females.
Fig. 2

Estimated conditional survival function of age at the first motorcycle crash

Table 3 shows the estimates of the conditional joint distribution functions given Y1 ≥ 18 ( \(\widehat F^{18}(y_1,y_2)\)) for male and female students and their estimated standard error, where the conditional joint distribution functions given Y1 ≥ 18 is \(F^{18}_{Y_1,Y_2}(y_1,y_2)=\{H_{Y_1,Y_2}(y_1,y_2)-H_{Y_1,Y_2}(18,y_2)\}/S_{Y_1}(18)\). Table 3 shows that the conditional joint distribution function estimates of male students are considerably higher than those of female students. We notice that \(\widehat F^{18}(y_1,y_2)\) for female students is not a proper distribution, for example, \(\widehat F^{18}(20,y_2) > \widehat F^{18}(21,y_2)\) at y2=1 and 1.25 in Table 3. It is hard to tackle the negative mass problem in \(\widehat F^{18}(y_1,y_2)\) (Pruitt, 1991). We also consider the conditional survival function of Y2 given 18 ≤ Y1 ≤ 20, \(S_{Y_2|18 \le Y_1 \le 20}(y)=\{H_{Y_1,Y_2}(20,y_2)-H_{Y_1,Y_2}(18,y_2)\}/\{S_{Y_1}(18)-S_{Y_1}(20)\}\). As suggested by Lin and Ying (1994), the negative mass issue in the estimate of SY_2|18 ≤ Y_1 ≤ 20(y), can be resolved by converting this estimate into a proper estimate \(\widehat S_{Y_2|18 \le Y_1 \le 20}^{*}(y)\). Specifically, \(\widehat S_{Y_2|18 \le Y_1 \le 20}^{*}(y)=\inf_{v \le y} \widehat S_{Y_2|18 \le Y_1 \le 20}(v)\) and \(\widehat S_{Y_2|18 \le Y_1 \le 20}^{*}(0)=1\). Figure 3 gives the modified estimate \(\widehat S_{Y_2|18 \le Y_1 \le 20}^{*}(y)\) and the corresponding 95% confidence interval. It shows that female and male students, who had their first motorcycle crashes between 18 and 20 years of age, had a similar risk of having the second motorcycle crash. For both male and female students, the median sojourn times between the first and second crashes are about 0.80 years.
Table 3

Estimated joint distributions for motorcycle crash data

(y1,y2)

Male

Female

\(\widehat F^{18}(y_1,y_2)\) (se)

\(\widehat F^{18}(y_1,y_2)\) (se)

(19, 0.25)

0.073 (0.022)

0.045 (0.015)

(19, 0.50)

0.150 (0.032)

0.089 (0.019)

(19, 0.75)

0.204 (0.038)

0.141 (0.027)

(19, 1.00)

0.286 (0.045)

0.205 (0.032)

(19, 1.25)

0.329 (0.054)

0.203 (0.043)

(20, 0.25)

0.108 (0.024)

0.080 (0.017)

(20, 0.50)

0.207 (0.034)

0.159 (0.023)

(20, 0.75)

0.287 (0.042)

0.231 (0.030)

(20, 1.00)

0.407 (0.049)

0.293 (0.038)

(20, 1.25)

0.423 (0.073)

0.303 (0.058)

(21, 0.25)

0.117 (0.027)

0.077 (0.023)

(21, 0.50)

0.232 (0.037)

0.167 (0.033)

(21, 0.75)

0.313 (0.047)

0.279 (0.043)

(21, 1.00)

0.489 (0.050)

0.278 (0.099)

(21, 1.25)

0.520 (0.072)

0.270 (0.137)

y1 = age at the first motorcycle crash

y2 = time between the first and second crashes

Fig. 3

Estimated conditional survival function of the second sojourn time between the first and second motorcycle crashes

Analysis of colon cancer data

We now analyze the colon cancer data to illustrate our proposed estimate from the second type of left-truncated data in the presence of competing risks. In the colon cancer data, we identified 47,881 residents over 60 years old in a community of Taiwan at the beginning of 2001 (Chen et al., 2004). Of these 47,881 residents, 440 were retrospectively identified as colon cancer patients from Taiwan cancer registry before December 31, 2002. A total of 3,286 deaths were ascertained from Taiwan national mortality registry between January 1, 2001 and December 31, 2002. In this 2-year period, 114 of 440 colon cancer patients died of this disease and 21 died from other causes. In this data, we have the censoring time C=T+d0, where d0=2 years. We also found that age at death ( \(Y_1^\diamond+Y_2^\diamond\)) is between the ages of 60 and 105 and age at the first diagnosis of colon cancer ( \(Y_1^\diamond\)) is between the ages of 46 and 97.

For analyzing the colon cancer data in the presence of competing events, we estimate the joint subdistribution function for the incidence and mortality of colon cancer given that Y1+Y2 > 60 ( \(F^{**(60)}_{Y_1,Y_2}(y_1,y_2)=\Pr(Y_1 \le y_1, Y_2 \le y_2, Y_{11} \le Y_{12}, Y_{21} \le Y_{22}|Y_1+Y_2 > 60)\)). Table 4 gives the estimates of \(F^{**(60)}_{Y_{1}, Y_{2}}(y_{1}, y_{2})\), their estimated standard errors and 95% confidence intervals. For example, there is 0.62% (1.98%, 2.56%) chance that a resident aged 60 or older was diagnosed as a colon cancer patient before age 70 (before age 80, before age 90) and died from this illness within 5 years after diagnosis. The small values of the estimates of \(F^{**(60)}_{Y_1,Y_2}(y_1,y_2)\) in Table 4 are ascribed to a large proportion of residents who had never experienced colon cancer in their lifetime. More explicitly, for those residents over 60 years old, the probability of having colon cancer in their lifetime is approximately 3.68% and the chance of having colon cancer and eventually dying from this disease is about 3.12%.
Table 4

Estimated joint subdistributions for colon cancer data

(y1,y2)

\(\widehat F^{**(60)}(y_1,y_2)\)

se

95% CI

(70, 2)

0.0035

(0.0008)

(0.0022, 0.0056)

(70, 5)

0.0062

(0.0011)

(0.0044, 0.0088)

(70, 10)

0.0064

(0.0011)

(0.0046, 0.0090)

(70, 15)

0.0064

(0.0011)

(0.0046, 0.0090)

(80, 2)

0.0037

(0.0009)

(0.0023, 0.0058)

(80, 5)

0.0198

(0.0021)

(0.0161, 0.0244)

(80, 10)

0.0210

(0.0022)

(0.0171, 0.0258)

(80, 15)

0.0214

(0.0022)

(0.0174, 0.0263)

(90, 2)

0.0047

(0.0011)

(0.0030, 0.0075)

(90, 5)

0.0256

(0.0027)

(0.0209, 0.0314)

(90, 10)

0.0283

(0.0029)

(0.0231, 0.0347)

(90, 15)

0.0287

(0.0030)

(0.0234, 0.0351)

y1=age at initial diagnosis of colon cancer

y2= time between diagnosis and death from colon cancer

Remarks

This paper provides an inverse-probability weighted approach for estimating the bivariate probability functions for serial sojourn times under various right-censoring situations in prevalent cohort studies, where the inverse bivariate probability function of truncation and censoring, H T, C −1 , is exploited as a weight function to adjust both left truncation and right censoring effects concurrently. Our proposed approach can be extended to the general case of multiple serial events in the presence of competing risks under left truncation. In addition, when serial event data are collected from both incident and prevalent cohorts (Wang, 1999), our proposed method is a valid estimation approach by using inverse probability of censoring time to adjust induced informative censoring for the incident cases and the inverse of HT,C to adjust the biases for the prevalent cases, respectively.

This study presents several nonparametric estimators of bivariate probability function for sojourn times in various truncation and censoring models. The performance of each estimator is mainly determined by the behavior of the corresponding estimator of HT,C. Our simulation results indicate that the most preferable estimate is \(\widehat H^{L_1(A)}\), which requires additional censoring information in uncensored cases (type-A censoring) to estimate HT,C nonparametrically without specifying the dependence structure between T and C. Note that type-A censoring is not uncommon in short-term follow-up studies to observe the exact censoring times in uncensored cases. Although the legitimacy of \(\widehat H^{L_1(B)}\) relies on the additional assumption that T and CT are independent, \(\widehat H^{L_1(B)}\) still performs well under a mild correlation of (T,CT) as seen from simulation results. We develop \(\widehat H^{L_1(G)}\) without entailing any additional assumption about the bivariate distribution of (T,C). However, its moderate-sample bias becomes more significant as the censoring proportion increases. This phenomenon is attributed to the fact that the estimators for the tails of SC|Tt(·) may not be consistent due to very small risk sets at the tails of SC|Tt(·) in the presence of left truncation and right censoring.

Notes

Acknowledgments

We are very grateful to Associate Editor and two referees for their constructive comments that led to a significant improvement of this paper. We thank Dr. Mau-Roung Lin at Taipei Medical University Institute of Injury Prevention and Control for providing the anonymous motorcycle crash data. We also thank the Department of Health, Taiwan, R.O.C. for providing the Taiwan Cancer registry and Death registry data. Part of this work was supported by NSC-91-2118-M-002-003 from National Science Council in Taiwan.

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Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Division of Biostatistics, Graduate Institute of Epidemiology, College of Public HealthNational Taiwan UniversityTaipeiTaiwan
  2. 2.Department of Agronomy, College of AgricultureNational Chiayi UniversityChiayiTaiwan

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