Semiparametric Temporal Process Regression of Survival-Out-of-Hospital

Abstract

The recurrent/terminal event data structure has undergone considerable methodological development in the last 10–15 years. An example of the data structure that has arisen with increasing frequency involves the recurrent event being hospitalization and the terminal event being death. We consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-Out-of-Hospital is a useful alternative strategy for the analysis of hospitalization/survival in the chronic disease setting, with the response variate representing a refinement to survival time through the incorporation of an objective quality-of-life component. The semiparametric model we consider assumes multiplicative covariate effects and leaves unspecified the baseline probability of being alive-and-out-of-hospital. Using zero-mean estimating equations, the proposed regression parameter estimator can be computed without estimating the unspecified baseline probability process, although baseline probabilities can subsequently be estimated for any time point within the support of the censoring distribution. We demonstrate that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulation studies are performed to show that our estimating procedures have satisfactory finite sample performances. The proposed methods are applied to the Dialysis Outcomes and Practice Patterns Study (DOPPS), an international end-stage renal disease study.

1 Introduction

Often in clinical or epidemiological studies, both a recurrent event process and terminal event are of interest. This is particularly true given the recent proliferation of administrative databases available for secondary analysis. Correspondingly, a considerable number of methods have been developed for the recurrent/terminal event data structure. One option is to model the marginal mean/rate (Cook and Lawless, 1997; Ghosh and Lin, 2002; Schaubel et al, 2006; Cook et al, 2009), essentially averaging over the survival experience. Another class of approaches involves jointly modeling survival and the conditional recurrent event rate given survival (Huang and Wang, 2004; Liu et al, 2004; Ye et al, 2007; Zeng and Cai, 2010; Kalbeisch et al, 2013).

In this report, we study a useful alternative framework for the analysis of recurrent/terminal event data. In particular, a frequently arising example of this data structure involves hospitalization representing the recurrent event, with death serving as the terminal event. Although hospital admission can be regarded as a point process, the length of stay for a hospitalization may be several days and, therefore, should not be ignored in the analysis. This concept is recognized in the works of Hu et al (2011) and Zhu et al (2014), for example. Specifically, we consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-out-of-hospital may be viewed as a refinement of survival time in the study of chronic illness such as end-stage renal disease (ESRD), with the refinement being the incorporation of each patient’s hospital admission and length of stay information. An appealing characteristic of survival-out-of-hospital is that it incorporates morbidity data in an objective and easily understood manner. Data of this structure are increasingly available, given the relatively recent proliferation of publicly available health administrative data sets (Holland and Lam, 2000; Sands et al, 2006; Carson et al, 2009).

Since Survival-Out-of-Hospital is indexed by a continuous timeline, temporal process regression would appear to be a natural conceptualization. Fine et al (2004) developed functional generalized linear models, for which covariates effects are completely unspecified and are estimated nonparametrically over time. Such an approach has also been generalized to multivariate survival settings to model both mean and association structures (Yan and Fine, 2005). To increase precision, the partly functional temporal process model has been developed, with covariate effects being nonparametric for some covariate elements and parametric for others (Yan and Huang, 2009; Estes et al, 2015). These functional generalized linear models generally focus on time-varying covariate effects. In addition, martingale-based estimating equations have been proposed for directly modeling the survival function by solving a sequence of monotone equations (Peng and Huang, 2007). Approaches listed in this paragraph are part of the inspiration for the methods we propose in this report. However, as will become more clear later in our report, none of these approaches are applicable to our setting given the specifics of our analytic objectives, along with the assumed data structure and model of interest.

Various other existing methods are pertinent to the data structure of our interest, but not applicable to our research question. For instance, methods proposed in Andersen et al (2003) and Scheike and Zhang (2007) involved directly modeling a state transition probability or a state occupation probability in a multi-state model. In these approaches, each state is assumed to be visited not more than once. As such, none of these approaches are directly applicable to our particular research question and data structure, since patients can move in and out of hospital many times prior to death. In addition, pseudo-observation approaches (Andersen et al, 2003; Grand and Putter, 2016), despite their utility, generally only allow for censoring that does not depend on the covariate vector, an assumption that may be violated in observational studies. As described in Section 2, the methods we propose allow censoring to depend on the covariate vector.

In this report, we propose a semiparametric temporal process regression method, where covariates have multiplicative effects on a completely unspecified baseline probability function. This model can be thought of a process version of the generalized linear model with the process indexed by time. In terms of estimation, the regression parameter is estimated by the solution to an estimating equation which is free of the baseline probability function. We propose a nonparametric estimator for the baseline probability process, a closed form, which can be computed after estimating the regression parameter. The estimating functions do not require inverse weighting in the setting where censoring times are known (e.g., if all censoring was administrative, occurring at the same calendar date). To accommodate the more commonly occurring scenario where censoring is random, we employ multiple imputation (Little and Rubin, 2002) to recover censoring times unobserved due to death (Schaubel and Zhang, 2010).

Our method has several distinguishing features. First, covariate effects are on the relative risk (as opposed to odds ratio) scale. The baseline probability process is specified as a nonparametric function of follow-up time, to increase flexibility and robustness. Through a development that parallels the derivation of the Cox partial likelihood score function (Cox, 1972), we derive an estimating function for the regression parameter that is free of the baseline probability function. This reduces the complexity of computation and permits the use of standard statistical software. Moreover, the baseline probability function can be subsequently estimated at any specific time point before the maximum observation time. We also propose estimating the integral of the baseline probability, which can then be used to predict expected survival time out-of-hospital over a finite time interval.

In this article, we are interested in the joint event of being out-of-hospital and being alive. This response variable takes quality of life information into consideration, while leaving the dependence structure between the temporal indicator process and terminal event time completely unspecified. Our method is different from the current practice in clinical trials, which is to analyze the time to the first recurrent event or terminal event (Lewis, 1999; Pfeffer et al, 2003). The proposed approaches are also distinct from a weighted composite endpoint of all recurrent and terminal events (Neaton et al, 2005; Mao and Lin, 2016).

The remainder of the article is organized as follows. In Section 2, we introduce notation, formulate the model assumptions, and propose estimating procedures for the regression parameter and baseline probability function. In Section 3, we show that the regression parameter estimator converges to a Normal distribution, and estimator of the baseline probability function converges to a Gaussian process. Simulation studies are performed to evaluate our method under various scenarios in Section 4. We then illustrate our method through an analysis of survival-out-of-hospital using data from the Dialysis Outcomes and Practice Patterns Study (DOPPS), a long-running international prospective study of ESRD patients. Finally, concluding remarks are provided at Section 6.

2 Model and Methods

Suppose there are a total of n independent subjects. Let D_i denote the death (terminal event) time of subject i (i = 1, 2, …, n). We let C_i be the censoring time, and let Z_i(t) be a p-dimensional covariate vector which may contain time-varying elements (assumed to be external; Kalbeisch and Prentice (2002), p.g. 196). Here, we consider follow-up time t ∈ [0, τ], where τ is a pre-specified constant satisfying Pr(C_i ≥ τ) > 0 for i = 1, 2, …, n. In practice, τ could be chosen as the maximum of observation time, X_i = C_i ∧ D_i, where a ∧ b = min(a, b).

Let H_i(t) = 1 if subject i is in the hospital at time t, and 0 if out of hospital. The probability of interest is the probability that a subject i is alive and out-of-hospital at time t, i.e.,

$${\pi }_i(t)=P\{H_i(t)=0,D_i>t|{\mathit{Z}}_i(t)\}.$$ (1)

We assume that the covariates have multiplicative effects on an unspecified baseline probability function π₀(t), such that

$${\pi }_i(t)={\pi }_0(t)\text{exp}\{{\mathit{\beta }}_0^T{\mathit{Z}}_i(t)\},$$ (2)

where β₀ is a p-dimensional vector. This is similar to the Cox proportional hazard assumption (Cox, 1972), but π_i(t) is interpreted as the probability of being out of hospital and alive for subject i at time t. Note the distinction between (2) and an intensity (Andersen and Gill, 1982) or marginal rate function (Lin et al, 2000).

With respect to the motivation for (1), modeling Survival-Out-of-Hospital is essentially focusing on a composite event (surviving and not being hospitalized) rather than its components. For instance, in studying patients with a chronic disease, it is often of interest to determine which patient characteristics are associated with ‘good outcomes’, in general sense, such as surviving and thriving (e.g., not being in the hospital). This outcome is relevant with respect to addressing the question, “Who is healthy enough to survive, without the burden of being hospitalized?” Direct analysis of the composite event is an alternative to a joint analysis of the components, hospitalization and death (Liu et al, 2004; Ye et al, 2007). Various existing methods in the survival analysis literature have targeted an outcome that can be cast as an aggregate. For example, Ghosh and Lin (2000) and Ghosh and Lin (2002) proposed modeling the marginal mean number of recurrent events, implicitly averaging over the survival experience. In the competing risk setting, Gray (1988) and Fine and Gray (1999) analyzed the cumulative incidence (subdistribution) hazard, as opposed to the cause-specific hazards.

Regarding the specific formulation of model (2), the motivation for using semiparametric loglinear regression is two-fold. First, due to the high dimensionality of π₀(t) (since t is continuous), we will show that one can profile out π₀(t) when estimating β₀. Secondly, we will show that the estimating equation resembles the Cox regression score function, such that standard software can be leveraged for parameter estimation.

For brevity of notation, we define A_i(t) = I{D_i > t} as the survival indicator, and $A_i^0(t)=I\{H_i(t)=0,D_i>t\}$ as the survival-out-of-hospital indicator, of chief interest in this report. It is important to note that Z_i(t) and $A_i^0(t)$ are still available on subject i (i.e., before the censoring time) even after the terminal event has occurred. This is due to the fact that $A_i^0(t)$ is known to equal 0 for t > D_i, and since all time-dependent elements of Z_i(t) are assumed to be external (such that death does not preclude subsequent observation). In summary, observed data for subject i are given by {H_i(t), X_i, I(C_i ≤ D_i), Z_i(s)} for t ∈ [0, X_i] and s ∈ [0, τ].

2.1 Known Censoring

To begin, suppose that censoring time C_i is always known. This would be the case in a closely monitored prospective study from which patients could not be randomly lost to follow-up. Another example would be a retrospective study derived from a reportable-disease registry, which would typically feature staggered entry and the observation period ending on a fixed calendar date. In such cases, C_i would be known even for subjects with C_i > D_i. This set-up does not match most observational studies, but it is a useful starting point. We assume that censoring time is independent of our target event, $A_i^0(t)$, conditional on Z_i(t); more explicitly, we can express the assumption as follows,

$$E\{A_i^0(t)|{\mathit{Z}}_i(t),C_i\ge t\}=E\{A_i^0(t)|{\mathit{Z}}_i(t)\}.$$ (3)

Consider the following two estimating functions,

$$\sum _{i=1}^n{\int }_0^{\tau }{\mathit{Z}}_i(t)[A_i^0(t)-{\pi }_i(t)]I(C_i\ge t)\mathit{\text{dt}},$$ (4)

$$\sum _{i=1}^n[A_i^0(t)-{\pi }_i(t)]I(C_i\ge t).$$ (5)

These estimating functions have expectation zero under model (2) and assuming conditionally independent censoring (3). Note that equation (5) could be evaluated at any specific time point t ∈ [0, τ], and does not have to be an observed event time. Solving equation (5) for π₀(t) treating β₀ as known, then substituting the resulting π̂₀(t) back into equation (4) yields the following zero-mean estimating function for β₀,

$$\mathit{U}(\mathit{\beta })=\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t;\mathit{\beta })\}{A}_i^0(t)I(C_i\ge t)\mathit{\text{dt}},$$ (6)

where Z̄(t; β) = S⁽¹⁾(t; β)/S⁽⁰⁾(t; β), ${\mathit{S}}^{(k)}(t;\mathit{\beta })=n^{-1}{\sum }_{i=1}^n{\mathit{Z}}_i{(t)}^{\otimes k}I(C_i\ge t)\text{exp}\{{\mathit{\beta }}^T{\mathit{Z}}_i(t)\}$ for k = 0, 1, 2, where a^⊗0 = 1, a^⊗1 = a, and a^⊗2 = aa^T. We also define the following notations. Let s^(k)(t; β) = E[Z₁(t)^⊗kI(C₁ ≥ t)exp{β^TZ₁(t)}] and z̄(t; β) = s⁽¹⁾(t; β)/s⁽⁰⁾(t; β) be the limiting value of S^(k)(t; β) and Z̄(t; β), respectively.

Equation (6) is reminiscent of the Cox regression score function, a property which can be exploited computationally (i.e., to ease programming effort). For instance, if the time scale is days (like our real-data application in Section 5) or some other discrete measure, then standard proportional hazards software (e.g., coxph(·) in R, proc phreg in SAS) can be used after augmenting the data set. Specifically, the augmented data would contain one record for each time unit t a subject is uncensored, with the event indicator for time unit t would be $A_i^0(t)$. Such data would left truncated such that, within subject, the (j + 1)th record begins (i.e., its left subinterval boundary) where the jth records ended (i.e., its right interval boundary). Alternatively, if the time scale were truly continuous, then the augmented data would contain, for each subject, one record for every time at which Z_i(t), Z̄(t; β), or $A_i^0(t)$ changes. Additionally, for the jth record of subject i spanning, say, (t_j−1, t_j], a weight of (t_j − t_j−1) would be used, along with offset log(t_j − t_j−1). Note that the Breslow approximation (for tie-handling) would be used, as is the default in SAS’s phreg. Naturally, an alternative is to write explicit code to solve (6) using Newton-Raphson.

Denoting the solution to (6) by β̂, one can estimate π₀(t) through the closed form,

$${\hat{\pi }}_0(t)=\frac{{\sum }_{i=1}^n{A}_i^0(t)I(C_i\ge t)}{{\sum }_{i=1}^{n}I(C_i\ge t)\text{exp}[{\hat{\mathit{\beta }}}^T{\mathit{Z}}_i(t)]}.$$ (7)

Note that π̂₀(t) is closely related to the Breslow estimator of baseline cumulative hazard function (Breslow, 1972). With respect to leveraging standard software, π̂₀(t) can be computed directly through successive differencing in the previously described scenarios wherein the time scale is discrete (with one record per uncensored time unit per subject). If time is continuous, then π̂₀(t) would need to be computed explicitly, with the value changing each time any of $A_i^0(t)$, or I(C_i ≥ t) or Z_i(t) changes its value for any subject i in the data set.

Due to the use of the log link, π̂₀(t) > 1 is possible. This could occur, for example, in settings where both hospitalization and mortality rates are quite low. This does not induce bias for β̂ since, as described earlier, π₀(t) is profiled out from estimating equation (6). When subsequently estimating π₀(t) at specific time points from (7), one could cap π̂₀(t) by 1. Note that an analogous issue is encountered in the semiparametric additive hazards model (Lin and Ying, 1994), for which estimated increments in the cumulative baseline hazard are not bounded below by 0.

We define the integral of the baseline survival-out-of-hospital probability as

$${\Pi }_0(L)={\int }_0^L{\pi }_0(t)\mathit{\text{dt}},$$ (8)

which could be interpreted as the expected length of being alive and out-of-hospital up to time L for a subject with the baseline (i.e., 0) value of each covariate. We could estimate Π₀(L) by ${\hat{\Pi }}_0(L)={\int }_0^L{\hat{\pi }}_0(t)\mathit{\text{dt}}$. Note that π̂₀(t) in (7) would jump up when a subject is discharged from the hospital, and jump down when a subject either dies or is admitted to the hospital. This property would facilitate computation in real-data applications, since Π̂₀(L) would essentially be a sum of rectangles with height π̂₀(t) and (time) unit length.

2.2 Random Censoring

Now, consider the more typical scenario in which the censoring time C_i is not fixed at t = 0, with its randomness implying that C_i is not known in cases where D_i is observed for subject i. In this set-up, one cannot carry out estimation through (6) and (7), defined in the preceding subsection, due to C_i being missing for subjects observed to die. A simple solution is to set censoring time as the maximum follow-up time across all subjects, which may lead to substantial bias in estimating β₀ and π₀(t). Similarly, setting the censoring time to D_i for subjects with C_i > D_i would generally lead to inaccurate inference.

One could use weighting techniques to recover missing censoring time, essentially weights subjects with C_i > D_i by a conditional survival probability (Ghosh and Lin, 2002; Mao and Lin, 2016). However, a weighted version of estimating equation (6) is tedious to carry out, since the time line is continuous. In this report we consider an alternative imputation approach which is easy to implement using standard software.

Our solution is to impute C_i when D_i < C_i and, for this purpose, we assume the following proportional hazard model for the censoring time,

$${\lambda }_i^C(t)={\lambda }_0^C(t)\text{exp}\{{\mathit{\theta }}_0^T{\mathit{Z}}_i(t)\},$$ (9)

where ${\lambda }_0^C(t)$ is an unspecified baseline hazard function for C_i. In this manuscript, we only consider censoring which is independent of the process of interest, $A_i^0(t)$, per the assumption implied by (3). In light of the model for C_i, we now define the corresponding counting process $N_i^C(t)=I(C_i\le t\wedge D_i)$, and its increment ${\mathit{\text{dN}}}_i^C(t)=N_i^C(t^{-}+\mathit{\text{dt}})-N_i^C(t^{-})$. Similarly, the counting process for death time is defined by $N_i^D(t)=I(D_i\le t\wedge C_i)$. Let Y_i(t) = I(X_i ≥ t) be the at risk process, where (as indicated earlier) X_i = C_i ∧ D_i. Standard partial likelihood (Cox, 1975) techniques can be applied to the observed censoring time data ${\{X_i,I(C_i\le D_i),{\mathit{Z}}_i(t);t\in [0,\tau ]\}}_{i=1}^n$ to compute θ̂, which is known to be a strongly consistent estimator of θ₀ (Andersen and Gill, 1982). The baseline cumulative hazard function for ${\Lambda }_0^C(t)$ is estimated through the Breslow method (Breslow, 1972).

We will create M imputed datasets, where normally M = 10 would suffice. The proposed methods are also valid for M = 1, although this would be less efficient. Consider the mth imputed dataset, for a subject with C_i ≤ D_i, we set the imputed censoring time to the known censoring time. For a subject with C_i > D_i, we impute ${\hat{C}}_i^{\langle m\rangle }$ from the estimated conditional survival function,

$${\hat{G}}_i(t;\hat{\mathit{\theta }})=I(t\ge D_i)\text{exp}[-{\hat{\Lambda }}_i^C(t;\hat{\mathit{\theta }})+{\hat{\Lambda }}_i^C(D_i;\hat{\mathit{\theta }})].$$ (10)

More explicitly, we could express the imputed censoring time $C_i^{\langle m\rangle }$ as

$$C_i^{\langle m\rangle }=[1-N_i^D(X_i)]C_i+N_i^D(X_i){\hat{C}}_i^{\langle m\rangle }.$$

We provide further commentary on the proposed imputation procedure in Section 6.

Consider the uncensored indicator, $I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })$, where we include θ in the parenthesis to emphasize that $C_i^{\langle m\rangle }$ depends on the parameter θ. In finite samples, we impute ${\hat{C}}_i^{\langle m\rangle }$ from Ĝ_i(t; θ̂) for subjects with C_i > D_i to obtain $I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})$. Define $G_i(t;{\mathit{\theta }}_0)=I(t\ge D_i)\text{exp}[-{\Lambda }_i^C(t;{\mathit{\theta }}_0)+{\Lambda }_i^C(D_i;{\mathit{\theta }}_0)]$. Therefore, $I(C_i^{\langle m\rangle }\ge t;{\mathit{\theta }}_0)$ refers to the scenario where we impute ${\hat{C}}_i^{\langle m\rangle }$ from the true underlying G_i(t; θ₀). When (9) is correctly specified, I(C_i ≥ t) and $I(C_i^{\langle m\rangle }\ge t;{\mathit{\theta }}_0)$ follow the same distribution. Let ${\mathit{S}}^{(k)\langle m\rangle }(t;\mathit{\beta },\mathit{\theta })=n^{-1}{\sum }_{i=1}^n{\mathit{Z}}_i{(t)}^{\otimes k}I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })\text{exp}\{{\mathit{\beta }}^T{\mathit{Z}}_i(t)\}$ k = 0, 1, 2, and set Z̄^〈m〉(t; β, θ) = S^(1)〈m〉 (t; β, θ)/S^(0)〈m〉 (t; β, θ). For the mth imputed data set, the estimation of β₀ and π₀(t) is as defined in Section 2.1, but with C_i replaced by $C_i^{\langle m\rangle }$. More explicitly, β̂^〈m〉 is computed through the following estimating function,

$${\mathit{U}}^{\langle m\rangle }(\mathit{\beta },\hat{\mathit{\theta }})=\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{Z}}_i(t)-{\overline{\mathit{Z}}}^{\langle m\rangle }(t;\mathit{\beta },\hat{\mathit{\theta }})\}{A}_i^0(t)I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})\mathit{\text{dt}},$$ (11)

with ${\hat{\pi }}_0^{\langle m\rangle }(t)$ then given by

$${\hat{\pi }}_0^{\langle m\rangle }(t)=\frac{{\sum }_{i=1}^n{A}_i^0(t)I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})}{{\sum }_{i=1}^n\{I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})\text{exp}[{\mathit{Z}}_i^T(t){\hat{\mathit{\beta }}}^{\langle m\rangle }]\}}.$$ (12)

Having computed β̂^〈m〉 and ${\hat{\pi }}_0^{\langle m\rangle }(t)$ (m = 1, 2, …, M), β₀ and π₀(t) will be estimated by the following pooled estimators:

$${\hat{\mathit{\beta }}}^M=M^{-1}\sum _{m=1}^M{\hat{\mathit{\beta }}}^{\langle m\rangle },$$ (13)

$${\hat{\pi }}_0^M(t)=\frac{{\sum }_{i=1}^n{\sum }_{m=1}^M{A}_i^0(t)I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})}{{\sum }_{i=1}^n{\sum }_{m=1}^{M}I(C_i^{\langle m\rangle }\ge t;\hat{\mathit{\theta }})\text{exp}[{\mathit{Z}}_i^T(t){\hat{\mathit{\beta }}}^M]}.$$ (14)

Asymptotic properties for ${\hat{\pi }}_0^M(t)$ given in (14) are developed by expressing the numerator and denominator as empirical processes indexed by time t. Similar to π̂₀(t) defined in (7), ${\hat{\pi }}_0^M(t)$ would be larger then 1 at some time points; as such, we would cap this estimated probability by 1 after obtaining β̂^M. An imputation version of ${\hat{\Pi }}_0^M(L)$, denoted by ${\hat{\pi }}_0^M(t)$, is obtained by integrating (14).

3 Asymptotic Properties

We assume the following set of regularity conditions:

1. {H_i(t), X_i, I(C_i ≤ D_i), Z_i(s)} for t ∈ [0, X_i] and s ∈ [0, τ], i = 1, 2, …, n are independent and identically distributed.

2. Pr(C_i ≥ τ) > 0 for i = 1, 2, …, n, where τ is a pre-specified constant.

3. $|{\mathit{Z}}_{\mathit{\text{ij}}}(0)|+{\int }_0^{\tau }|d{\mathit{Z}}_{\mathit{\text{ij}}}(t)|<c_{\mathit{Z}}<\infty$ almost surely for i = 1, 2, …, n, j = 1, 2, …, p, i.e., Z_i(t) has bounded total variations.

4. $\mathit{\Omega }({\mathit{\beta }}_0)=E[{\int }_0^{\tau }\{{\mathit{s}}^{(2)}(t;{\mathit{\beta }}_0)/s^{(0)}(t;{\mathit{\beta }}_0)-\overline{\mathit{z}}{(t;{\mathit{\beta }}_0)}^{\otimes 2}\}{A}_1^0(t)I(C_1\ge t)\mathit{\text{dt}}]$ is positive definite.

5. For β ∈ ℬ_δ, where ℬ_δ is a small neighborhood around β₀, s⁽⁰⁾ (t; β), and s⁽¹⁾(t; β) are bounded away from zero.

6. For β ∈ ℬ_δ, k = 0, 1, 2, s^(k) (t; β) are continuous uniformly on t ∈ [0, τ], and are bounded on [0, τ] × ℬ_δ.

We summarize the essential asymptotic properties of β̂, π̂₀(t) and imputed versions β̂^M, ${\hat{\pi }}_0^M(t)$ in the following theorems. Proofs are sketched in the Supplemental Materials.

3.1 Known Censoring

We begin by describing the asymptotic properties of estimators applicable to the known-censoring set-up from Section 2.1.

Theorem 1

Under assumptions (2) and (3) and the afore-listed regularity conditions, β̂ is a consistent estimator of β₀, and n^1/2(β̂−β₀) converges in distribution to a mean-zero Normal random variable with a variance-covariance matrix

$$\mathit{\sum }({\mathit{\beta }}_0)=\mathit{\Omega }{({\mathit{\beta }}_0)}^{-1}E[{\mathit{u}}_1({\mathit{\beta }}_0){\mathit{u}}_1{({\mathit{\beta }}_0)}^T]\mathit{\Omega }{({\mathit{\beta }}_0)}^{-1},$$ (15)

where u_i(β) and Ω(β) are defined in Appendix 7. A consistent estimator of Σ(β₀) is obtained by replacing limiting values by its empirical counter parts.

We now describe the asymptotic behavior of π̂₀(t) from (7).

Theorem 2

Under assumptions (2) and (3), n^1/2(π̂₀−π₀) converges weakly to a mean-zero Gaussian process with a variance and covariance matrix between n^1/2[π̂₀(s) − π₀(s)] and n^1/2[π̂₀(t) − π₀(t)] given by σ(s, t) = E[ξ₁(s)ξ₁(t)], where

$${\xi }_i(t)=\frac{f_i^{{\pi }_1}(t;{\mathit{\beta }}_0)-f_i^{{\pi }_2}(t;{\mathit{\beta }}_0)}{s^{(0)}(t;{\mathit{\beta }}_0)}$$

and $f_i^{{\pi }_1}(t;\mathit{\beta })$ and $f_i^{{\pi }_2}(t;\mathit{\beta })$ are further defined in Appendix 7.

3.2 Random Censoring

Now we describe the limiting behavior of estimator for the random-censoring set-up from Section 2.2.

Theorem 3

If censoring mechanism (9), assumptions (2) and (3) are correctly specified, then under the previously listed regularity conditions, β̂^M is a consistent estimator of β₀, and n^1/2(β̂^M − β₀) converges in distribution to a mean-zero Normal random variable with a variance-covariance matrix

$${\mathit{\sum }}_M({\mathit{\beta }}_0,{\mathit{\theta }}_0)={[\mathit{\Omega }({\mathit{\beta }}_0)]}^{-1}E\{{[M^{-1}\sum _{m=1}^M{\mathit{u}}_1^{\langle m\rangle }({\mathit{\beta }}_0,{\mathit{\theta }}_0)]}^{\otimes 2}\}{[\mathit{\Omega }({\mathit{\beta }}_0)]}^{-1},$$

where ${\mathit{u}}_i^{\langle m\rangle }(\mathit{\beta },\mathit{\theta })$ is defined in Appendix 7.

The asymptotic behavior of imputed version ${\hat{\pi }}_0^M(t)$ from (14) is summarized in the following theorem.

Theorem 4

If censoring mechanism (9), assumptions (2) and (3) are correctly specified, then $n^{1/2}({\hat{\pi }}_0^M-{\pi }_0)$ converges weakly to a mean-zero Gaussian process with a variance and covariance matrix between $n^{1/2}[{\hat{\pi }}_0^M(s)-{\pi }_0(s)]$ and $n^{1/2}[{\hat{\pi }}_0^M(t)-{\pi }_0(t)]$ given by σ_M (s, t) = E[ξ_1M (s)ξ_1M (t)], where

$${\xi }_{\mathit{\text{iM}}}(t)=\frac{f_i^{{\pi }_1}(t;{\mathit{\beta }}_0,{\mathit{\theta }}_0,M)-f_i^{{\pi }_2}(t;{\mathit{\beta }}_0,{\mathit{\theta }}_0,M)}{s^{(0)\langle 1\rangle }(t;{\mathit{\beta }}_0,{\mathit{\theta }}_0)}$$

and $f_i^{{\pi }_1}(t;\mathit{\beta },\mathit{\theta },M)$ and $f_i^{{\pi }_2}(t;\mathit{\beta },\mathit{\theta },M)$ are further defined in Appendix 7.

4 Simulation Studies

We evaluate the finite-sample performance of our method through simulation studies. For each setting, n = 500 subjects are generated. The external covariate, Z(t) = Z₁ + Z₂(t), is generated from two components: Z₁ and Z₂(t), where Z₁ ~ Uniform(0.5, 1) and $Z_2(t)={\sum }_{j=1}^{10}I[10\times (j-1)<t\le 10\times j]Z_{3j}$ with Z_3j ~ Uniform(0, 1) for j = 1, 2, …, 10 and t = 1, 2, …, 100. The target model is survival-out-of-hospital probability, π(t) = π₀(t)exp{β₀Z(t)}, where π₀(t) = I(t ≤ 10)(1 − 0.07t) + I(t > 10)(0.3 − 0.0025t) or 0.3 − 0.0025t, for t = 1, 2, …, 100. This model of the joint outcome can be generated through the hazard function for death time D, λ^D(t), and P{H(t) = 0|D > t, Z(t)}. We set ${\lambda }^D(t)={\lambda }_0^D\text{exp}[{\alpha }_{0}Z(t)]$, where ${\lambda }_0^D=0.008$, α₀ = log(0.7) and log(1.2). In this case, the out-of hospital indicator given survival follows the conditional probability $P\{H(t)=0|D>t,Z(t)\}=\pi (t)\text{exp }[{\int }_0^t{\lambda }^D(s)\mathit{\text{ds}}]$. The censoring time, C, is generated from the hazard function ${\lambda }^C(t)={\lambda }_0^C\text{exp}[{\gamma }_{0}Z(t)]$, where ${\lambda }_0^C=0.008$ and γ₀ = log(1.5), such that C and the target event I{H(t) = 0, D > t} are conditionally independent given Z(t).

We consider 4 simulation settings, based on two versions of π₀(t) and two magnitudes of α₀ outlined in the previous paragraph. For π₀(t) = I(t ≤ 10)(1−0.07t) + I(t > 10)(0.3 − 0.0025t), we consider β₀ at −log(2), −log(1.5) and −log(1.3). In this scenario, the π̂₀(t) in (7) may be larger than 1, but we make no adjustments when estimating β₀. For π₀(t) = 0.3−0.0025t, we evaluate β₀ at log(1.5), −log(1.5) and log(1.3). Two magnitudes of α₀ at log(0.7) and log(1.2) result in 76% and 60% subjects being censored. In addition to the known censoring scenario, we evaluate the performance of the proposed imputation method with M = 1 and M = 10. Parameters used for each of the four simulation settings are listed in Table 1.

Table 1.

Simulation Study: Data Configurations

Parameter	Setting 1	Setting 2	Setting 3	Setting 4
${\lambda }_0^D$	0.008	0.008	0.008	0.008
α0	log(0.7)	log(1.2)	log(0.7)	log(1.2)
π0(t)	I(t ≤ 10)(1 − 0.07t)+I(t > 10)(0.3 − 0.0025t)	I(t ≤ 10)(1 − 0.07t)+I(t > 10)(0.3 − 0.0025t)	0.3 − 0.0025t	0.3 − 0.0025t
β	−log(2), −log(1.5) or −log(1.3)	−log(2), −log(1.5) or −log(1.3)	log(1.5), −log(1.5) or log(1.3)	log(1.5), −log(1.5) or log(1.3)
${\lambda }_0^C$	0.008	0.008	0.008	0.008
γ0	log(1.5)	log(1.5)	log(1.5)	log(1.5)

Table 2 and Table 3 summarize the simulation results for β̂ and Π̂₀(50), respectively. In each setting, the bias of β̂ and Π̂₀(50) are very small, indicating that our estimators are consistent even based on M = 1. Moreover, empirical standard deviations (ESDs) are generally close to the average asymptotic standard errors (ASEs), showing that our proposed variance estimator appear to be applicable to finite samples. The empirical coverage probabilities (ECPs) are also around 0.95, implying the accuracy of large-sample confidence intervals. For the random-censoring settings, we also calculate asymptotic relative efficiencies (AREs), defined as the ratio of mean square error (i.e., squared bias plus variance) between M = 1 and 10. The larger than 1 AREs indicate that multiple imputation is generally more efficient than single imputation.

Table 2.

Simulation results for β̂

Censoring	Setting	β0	BIAS	ASE	ESD	ECP	ARE
C known	Setting 1	−0.693	−0.005	0.149	0.149	0.955	-
		−0.405	0.001	0.133	0.136	0.946	-
		−0.262	0.001	0.126	0.127	0.952	-
	Setting 2	−0.693	0.003	0.180	0.185	0.950	-
		−0.405	0.005	0.167	0.173	0.944	-
		−0.262	−0.002	0.161	0.159	0.950	-
	Setting 3	0.405	0.000	0.131	0.131	0.955	-
		−0.405	0.001	0.162	0.161	0.947	-
		0.262	0.000	0.134	0.135	0.949	-
	Setting 4	0.405	0.004	0.182	0.188	0.934	-
		−0.405	−0.010	0.206	0.205	0.955	-
		0.262	−0.001	0.184	0.186	0.951	-
C random	Setting 1	−0.693	0.004	0.149	0.153	0.946	1.021
M = 1		−0.405	0.007	0.132	0.134	0.951	1.03
		−0.262	−0.002	0.125	0.124	0.948	1.003
	Setting 2	−0.693	0.001	0.180	0.179	0.955	0.931
		−0.405	0.000	0.167	0.168	0.953	0.972
		−0.262	0.008	0.161	0.162	0.947	0.964
	Setting 3	0.405	−0.002	0.130	0.132	0.947	0.963
		−0.405	0.006	0.162	0.165	0.947	1.01
		0.262	0.000	0.135	0.128	0.963	0.886
	Setting 4	0.405	0.001	0.182	0.187	0.942	1.039
		−0.405	−0.001	0.206	0.209	0.943	1.028
		0.262	0.005	0.185	0.180	0.956	0.913

Table 3.

Simulation results for Π̂₀(50)

Censoring	Setting	Π0(50)	BIAS	ASE	ESD	ECP	ARE
C known	Setting 1	15.100	0.371	2.882	2.972	0.948	-
		15.100	0.222	2.582	2.697	0.945	-
		15.100	0.186	2.424	2.489	0.946	-
	Setting 2	15.100	0.430	3.520	3.788	0.941	-
		15.100	0.273	3.221	3.399	0.930	-
		15.100	0.332	3.100	3.066	0.943	-
	Setting 3	11.812	0.151	1.964	2.030	0.943	-
		11.812	0.246	2.455	2.481	0.952	-
		11.812	0.188	2.027	2.096	0.942	-
	Setting 4	11.812	0.270	2.755	2.919	0.926	-
		11.812	0.564	3.177	3.274	0.952	-
		11.812	0.347	2.813	2.872	0.937	-
C random	Setting 1	15.100	0.196	2.869	3.022	0.942	1.009
M = 1		15.100	0.100	2.529	2.622	0.944	1.028
		15.100	0.241	2.433	2.438	0.945	1.017
	Setting 2	15.100	0.390	3.468	3.546	0.943	0.905
		15.100	0.350	3.243	3.295	0.949	0.954
		15.100	0.158	3.073	3.135	0.929	0.91
	Setting 3	11.812	0.218	1.980	2.028	0.938	1.004
		11.812	0.156	2.426	2.473	0.935	0.949
		11.812	0.158	2.039	1.980	0.963	0.906
	Setting 4	11.812	0.339	2.769	3.010	0.943	1.092
		11.812	0.429	3.144	3.319	0.941	1.104
		11.812	0.219	2.781	2.733	0.957	0.896

5 Real Data Analysis

We applied the proposed methods to data from the Dialysis Outcomes and Practice Patterns Study (DOPPS) Phase 5. The DOPPS is a prospective, observational study designed to elucidate aspects of hemodialysis practice that are associated with the best outcomes for hemodialysis patients (Young et al, 2000). In particular, Phase 5 data were collected between 2012 and 2015. Our research interests include identifying demographic and clinical variables that are associated with survival-out-of-hospital probability, and characterizing the underlying survival-out-of-hospital process.

The study population for DOPPS is of prevalent patients. We restricted our study sample to include the n = 6, 032 patients who entered DOPPS within 3 months of initiating dialysis since the interest is in time-since-dialysis-initiation instead of time-since-DOPPS-entry. Patients included in our analysis were from 470 hemodialysis units across 11 different countries, with the counties including: Belgium, Canada, China, Gulf Coast Consortium, Germany, Italy, Japan, Spain, Sweden, the United Kingdom and the U.S.. Covariates include age, race, gender, height, time on dialysis at study entry, as well as the following list of comorbid conditions: coronary artery disease (CAD), cancer, cardiovascular disease (CVD), stroke, congestive heart failure (CHF), diabetes, hypertension, chronic obstructive pulmonary disease (COPD), psychiatric disorder and peripheral vascular disease (PVD).

Since hospitalization and death times are recorded in days, t represents day (i.e, day post DOPPS entry) in our analysis. The mean number of hospital admissions was 0.525 per patient, while the median length of stay per visit was 5 days. Observed follow-up time had a median of 409 days. Around 16% of subjects were observed to die (i.e., for whom C_i > D_i). Censoring time cannot realistically be considered to be fixed, since patients are frequently lost to follow-up for reasons other than death or closure of the DOPPS database. To recover missing censoring time, we created M = 10 imputed data sets as described in Section 2.2.

As shown in Table 4, coronary artery disease (p = 0.037), cancer (p = 0.010), cardiovascular disease (p = 0.031), congestive heart failure (p < 0.001) and peripheral vascular disease (p = 0.012) have significant negative effects on survival-out-of-hospital, while hypertension (p = 0.006) has significant positive effect. Patients from Canada (p = 0.009), Japan (p < 0.001) and Spain (p < 0.001) have significantly higher survival-out-of-hospital probability than patients from the U.S. (reference). For continuous variables, age had a significant negative effect (p < 0.001), while height had significant positive effect (p < 0.001). The estimated baseline survival-out-of-hospital probability ${\hat{\pi }}_0^{10}(t)$ is shown in Figure 1. The curve generally decreases as follow-up time increases, although the decrease is not monotone.

Table 4.

Analysis of DOPPS data: Covariate effects on survival-out-of-hospital probability (based on M = 10 imputations)

Covariate	β̂	$\widehat{\mathit{\text{SE}}}(\stackrel{⌢}{\beta })$	p	exp(β̂)
(Age in years)/5	−0.019	0.002	< 0.001*	0.982
Time on dialysis (years)	−0.044	0.068	0.444	0.957
(Height in cm)/10	0.039	0.006	< 0.001*	1.040
Female	0.008	0.014	0.550	1.008
CADa	−0.040	0.018	0.037*	0.961
Cancer	−0.062	0.023	0.010*	0.940
CVDb	−0.055	0.022	0.031*	0.946
Stroke	0.005	0.024	0.711	1.005
CHFc	−0.073	0.017	< 0.001*	0.930
Diabetes	0.012	0.011	0.329	1.012
Hypertension	0.037	0.013	0.006*	1.038
COPDd	−0.028	0.027	0.325	0.972
Psychiatric Disorder	−0.027	0.022	0.249	0.973
PVDe	−0.050	0.019	0.012*	0.951
Belgium	0.002	0.034	0.715	1.002
Canada	0.074	0.025	0.009*	1.076
China	−0.010	0.035	0.654	0.990
Gulf	−0.012	0.024	0.620	0.988
Germany	0.013	0.026	0.610	1.013
Italy	0.035	0.027	0.255	1.035
Japan	0.106	0.019	< 0.001*	1.112
Spain	0.093	0.024	< 0.001*	1.097
Sweden	0.033	0.029	0.301	1.033
UK	0.039	0.030	0.266	1.040

^a: Coronary artery disease

^b: Cardiovascular disease

^c: Congestive heart failure

^d: Chronic obstructive pulmonary disease

^e: Peripheral vascular disease

^*: p < 0.05

Fig. 1.

Fitted baseline survival-out-of-hospital probability

6 Concluding Remarks

In this report, we propose semiparametric methods for analyzing the probability of survival-out-of-hospital, a novel end-point pertinent to a frequently arising instance of the recurrent/terminal event data structure. Estimation proceeds through estimating equations which are analogous to those employed in Cox regression. Multiple imputation is implemented to accommodate missing censoring times that are unobserved due to the subject dying. Asymptotic properties of estimators are derived, and simulation studies show the proposed methods have satisfactory finite sample performance.

We applied the methods to data from Phase 5 of the Dialysis Outcomes and Practice Patterns Study in order to identify significant predictors of survival-out-of-hospital. Coronary artery disease, cancer, cardiovascular disease, congestive heart failure and peripheral vascular disease are found to be comorbidity factors with significant negative effects on survival-out-of-hospital events, while hypertension has significant positive effect. Patients from Canada, Japan and Spain have significantly higher survival-out-of-hospital probability than patients in the U.S.. Moreover, increasing age had a significantly negative effect, while height had significantly positive effect on survival-out-of-hospital.

A key advantage of the proposed methods is that the baseline probability process does not need to be specified. The ability to profile out π₀(t) in the estimation of β₀ is facilitated by the use of a log link function in model (2). This was the reason for using the log link is lieu of its more frequently used alternatives. This is important, since the baseline probability function will often represent a nuisance parameter. Modeling the baseline parametrically could be tedious to carry out accurately, and could lead to bias in the regression parameter (of chief interest) if an incorrect parametric form is assumed. Note that, although the proposed baseline estimator shares a structure similar to that of the Breslow (1972) estimator, there are some important differences. First, our baseline estimator is not monotone, consistent with the temporal process it is targeting. Second, the estimator can be computed for any value of t, since the baseline probability reflects prevalence, as opposed to an intensity or occurrence rate.

Under the proposed methods, if C > D then C is considered to be missing data. Of the two most frequently employed techniques for handling missing data (namely inverse weighting and imputation), we chose to impute unobserved censoring times. Note that a related approach was developed by Schaubel and Zhang (2010). The multiple imputation method we use for unobserved censoring times represents so-called improper imputation, in the sense that the imputation model parameters are fixed at their estimated values. For an alternative strategy, proper imputation could also be considered, where the imputation model parameters are drawn from their estimated predictive posterior distribution (Little and Rubin, 2002). Both imputation methods would lead to consistent estimators of covariate effects but, potentially, with slightly different efficiency (Schaubel and Cai, 2006). The proposed methods are valid under single imputation, M = 1, which would not be the case under proper imputation. More specifically, Rubin’s renown variance formula for multiple imputation requires M > 1 since it explicitly utilizes the variability across the M estimates. As with its proper counterpart, it is well-known that improper imputation leads to consistent estimation. However, due to the inapplicability of Rubin’s formula under improper imputation, one must derive the variance explicitly (per Theorems 3–4 and their corresponding proofs in the Supplementary Materials). Examples of other methods that have handled imputation in a similar fashion include Lu and Tsiatis (2001), who imputed cause of death when missing. Note that, although M = 1 is valid, we recommend M = 5 or M = 10; using the mean of M imputed estimators is generally more efficient than using a single impute, although returns do diminish.

We proposed the Cox (1972) model for imputing censoring times, due to its flexibility and dominance in the analysis of censored epidemiologic data. Parameters estimated through the proposed methods are consistent only provided the censoring model is correctly specified. Assumptions on the censoring model could be assessed through standard techniques, such as Schoenfeld and Martingale residuals (Kalbfleisch and Prentice, 2002). Since dM_i (t; β) defined at Section 3.1 is the difference between observed prevalence and predicted values, one could use the cumulative sum of dM_i (t; β), analogous to the diagnostic techniques developed in Lin et al (2000) for recurrent event data.

7 Appendix

We first list notation that appeared in Section 3.1:

$$\hat{\mathit{\Omega }}(\mathit{\beta })=n^{-1}{\sum }_{i=1}^n{\int }_0^{\tau }\{{\mathit{S}}^{(2)}(t;\mathit{\beta })/S^{(0)}(t;\mathit{\beta })-\overline{\mathit{Z}}{(t;\mathit{\beta })}^{\otimes 2}\}{A}_i^0(t)I(C_i\ge t)\mathit{\text{dt}}.$$

$$\mathit{\Omega }(\mathit{\beta })=E[{\int }_0^{\tau }\{{\mathit{s}}^{(2)}(t;\mathit{\beta })/s^{(0)}(t;\mathit{\beta })-\overline{\mathit{z}}{(t;\mathit{\beta })}^{\otimes 2}\}{A}_1^0(t)I(C_1\ge t)\mathit{\text{dt}}].$$

$${\mathit{u}}_i(\mathit{\beta })={\int }_0^{\tau }\{{\mathit{Z}}_i(t)-\overline{\mathit{z}}(t;\mathit{\beta })\}{\mathit{\text{dM}}}_i(t;\mathit{\beta }).$$

$${\mathit{\text{dM}}}_i(t;\mathit{\beta })=A_i^0(t)I(C_i\ge t)\mathit{\text{dt}}-\text{exp}[{\mathit{\beta }}^T{\mathit{Z}}_i(t)]I(C_i\ge t)\mathit{\text{dt}}.$$

$$f_i^{{\pi }_1}(t;\mathit{\beta })=A_i^0(t)I(C_i\ge t)-\text{exp}[{\mathit{\beta }}^T{\mathit{Z}}_i(t)]I(C_i\ge t){\pi }_0(t).$$

$$f_i^{{\pi }_2}(t;\mathit{\beta })={\mathit{s}}^{(1)}{(t;\mathit{\beta })}^T{\pi }_0(t){\mathit{f}}_i^{\mathit{\beta }}(\mathit{\beta }).$$

$${\mathit{f}}_i^{\mathit{\beta }}(\mathit{\beta })=\mathit{\Omega }{(\mathit{\beta })}^{-1}{\mathit{u}}_i(\mathit{\beta }).$$

Formulas pertaining to Section 3.2 are as follows:

$${\hat{\mathit{\Omega }}}^{\langle m\rangle }(\mathit{\beta },\mathit{\theta })=n^{-1}{\sum }_{i=1}^n{\int }_0^{\tau }\{{\mathit{S}}^{(2)\langle m\rangle }(t;\mathit{\beta },\mathit{\theta })/S^{(0)\langle m\rangle }(t;\mathit{\beta },\mathit{\theta })-{\overline{\mathit{Z}}}^{\langle m\rangle }{(t;\mathit{\beta };\mathit{\theta })}^{\otimes 2}\}{A}_i^0(t)I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })\mathit{\text{dt}}.$$

$${\mathit{\Omega }}^{\langle 1\rangle }(\mathit{\beta },\mathit{\theta })=E[{\int }_0^{\tau }\{{\mathit{s}}^{(2)\langle 1\rangle }(t;\mathit{\beta },\mathit{\theta })/s^{(0)\langle 1\rangle }(t;\mathit{\beta },\mathit{\theta })-{\overline{\mathit{z}}}^{\langle 1\rangle }{(t;\mathit{\beta },\mathit{\theta })}^{\otimes 2}\}{A}_1^0(t)I(C_1^{\langle 1\rangle }\ge t;\mathit{\theta })\mathit{\text{dt}}].$$

$${\mathit{u}}_i^{\langle m\rangle }(\mathit{\beta },\mathit{\theta })={\int }_0^{\tau }\{{\mathit{Z}}_i(t)-{\overline{\mathit{z}}}^{\langle 1\rangle }(t;\mathit{\beta },\mathit{\theta })\}{\mathit{\text{dM}}}_i^{\langle m\rangle }(t;\mathit{\beta },\mathit{\theta }).$$

$${\mathit{\text{dM}}}_i^{\langle m\rangle }(t;\mathit{\beta },\mathit{\theta })=A_i^0(t)I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })\mathit{\text{dt}}-\text{exp}[{\mathit{\beta }}^T{\mathit{Z}}_i(t)]I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })\mathit{\text{dt}}.$$

$$f_i^{{\pi }_1}(t;\mathit{\beta },\mathit{\theta },M)=M^{-1}{\sum }_{m=1}^{M}I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })A_i^0(t)-M^{-1}{\sum }_{m=1}^{M}I(C_i^{\langle m\rangle }\ge t;\mathit{\theta })\text{exp}[{\mathit{\beta }}^T{\mathit{Z}}_i(t)]{\pi }_0(t).$$

$$f_i^{{\pi }_2}(t;\mathit{\beta },\mathit{\theta },M)={\mathit{s}}^{(1)\langle 1\rangle }{(t;\mathit{\beta },\mathit{\theta })}^T{\pi }_0(t){\mathit{f}}_i^{\mathit{\beta }}(\mathit{\beta },\mathit{\theta },M).$$

$${\mathit{f}}_i^{\mathit{\beta }}(\mathit{\beta },\mathit{\theta },M)={[\mathit{\Omega }(\mathit{\beta })]}^{-1}{M}^{-1}{\sum }_{m=1}^M{\mathit{u}}_i^{\langle m\rangle }(\mathit{\beta },\mathit{\theta }).$$