Regression analysis of clustered interval-censored failure time data with the additive hazards model

Abstract

This paper discusses regression analysis of clustered failure time data, which means that the failure times of interest are clustered into small groups instead of being independent. Clustering occurs in many fields such as medical studies. For the problem, a number of methods have been proposed, but most of them apply only to clustered right-censored data. In reality, the failure time data is often interval-censored. That is, the failure times of interest are known only to lie in certain intervals. We propose an estimating equation-based approach for regression analysis of clustered interval-censored failure time data generated from the additive hazards model. A major advantage of the proposed method is that it does not involve the estimation of any baseline hazard function. Both asymptotic and finite sample properties of the proposed estimates of regression parameters are established and the method is illustrated by the data arising from a lymphatic filariasis study.

1. Introduction

This paper discusses regression analysis of clustered failure time data, which occur when the failure times of interest are clustered into small groups or some study subjects are related such as siblings, families, or communities. The subjects from the same cluster or group usually share certain unobserved characteristics and their failure times tend to be correlated as a result. Siblings, for example, share similar genetic and environmental influences. A key feature of the failure time data is censoring and one type of censoring that has been extensively discussed is right censoring. Another complex type is interval censoring that arises when the failure event of interest cannot be observed directly but is known only to have occurred over a time interval. This is common and natural in a clinical trial or longitudinal study in which there is a periodic follow-up. For instance, an individual who is monitored weekly for a response may miss visits for a few weeks and return in a changed response state, thus contributing an interval-censored observation. In the following, we will focus on the regression analysis of such data.

For regression analysis of clustered failure time data, several methods have been proposed when only right censoring is present (Cai and Prentice 1997; Cai, Wei and Wilcox 2000; Zeng, Lin and Lin 2008). For example, Cai et al. (2000) and Zeng et al. (2008). For example, Cai et al. (2000) and Zeng et al. (2008) investigated the fitting of semi-parametric linear transformation models to clustered right-censored data and Rossini and Moore (1999) considered the use of estimating equations and pseudolikelihood for the analysis. For the case of interval-censored data, there also exist some procedures when there is no clustering (Jewell 1994; Wang and Ding 2000; Jewell and van der Laan 2004). For example, Huang (1996) considered the maximum-likelihood approach for Cox model regression of current status data, a special case of interval-censored data, and Sun (2006) gave a comprehensive review of the existing literature on the topic.

The additive hazards model is one of the most commonly used models for regression analysis of failure time data (Lin, Oakes and Ying 1998; Ghosh 2001; Martinussen and Sheike 2002; Wang, Sun and Tong 2010; Cai and Zeng 2011). For instance, Lin et al. (1998) considered the fitting of the model to current status data and developed some estimating equation approaches for the estimation of regression parameters. Wang et al. (2010) discussed the fitting of the model to general interval-censored failure time data and Cai and Zeng (2011) investigated the additive mixed effect model for clustered right-censored failure time data. In this paper, an approach is developed for fitting the additive hazards model to clustered interval-censored failure time data.

The reminder of the paper is organised as follows. We will begin in Section 2 with describing notation and models that will be used throughout the paper. In particular, the failure time of interest is assumed to follow the additive hazards model marginally. In Section 3, an estimating equation-based approach is developed for estimating regression parameters of interest and the asymptotic properties of the proposed estimates are established. The approach can be easily implemented and does not require the estimation of the baseline hazard function. Section 4 presents some results obtained from a simulation study performed to evaluate the proposed estimation procedure and Section 5 illustrates the proposed approach by a set of clustered interval-censored data arising from a lymphatic filariasis (LF) study. Some concluding remarks and discussion are provided in Section 6.

2. Notation and models

Consider a survival study that involves n clusters of subjects with n_i denoting the size of cluster, where i = 1,…, n. Let T_ij denote the failure time of interest for subject j in cluster i, where j = 1,…, n_i, and Z_ij(t) is a possibly time-dependent p-dimensional vector of covariates that is associated with the subject and assumed to be completely observed. Throughout this paper, the T_ijs are considered to be independent for subjects in different clusters but could be dependent for subjects within the same cluster, and only interval-censored data on the T_ijs are observed. Specifically for each T_ij, there exist two monitoring times denoted by U_ij and V_ij with U_ij ≤ V_ij such that T_ij is known only to be smaller than U_ij, between U_ij and V_ij, or greater than V_ij. That is, we have clustered interval-censored data on the T_ijs. Furthermore, it will be assumed that T_ij is independent of the monitoring times U_ij and V_ij given covariate process Z_ij(t).

As mentioned above, our focus will be on the estimation of the covariate effect on the T_ijs. For this, we assume that for each cluster, there exists a latent variable b_i and all T_ijs are independent given b_i. Also, the hazard function for T_ij is specified by the following additive hazard model:

$${\lambda }_{ij}(t\mid Z_{ij}(s),b_i,s\le t)={\lambda }_0\left(t\right)+{\beta }_0^{\mathrm{T}}{Z}_{ij}\left(t\right)+b_i$$ (1)

given b_i and the covariate process up to time t (Lin et al. 1998; Ghosh 2001). Here, λ₀(t) is an unknown baseline hazard function and β₀ denotes the p-dimensional vector of regression parameters.

In practice, the monitoring variables U_ij and V_ij could depend on covariates, too. To model this and due to the strict order restriction between them, it is natural to regard them as some failure models, too. Here, we consider employing the Cox-type models

$${\lambda }_{ij}^U(t\mid Z_{ij}(s),s\le t)={\lambda }_1\left(t\right)\exp \left\{{\gamma }_0^{\mathrm{T}}{Z}_{ij}\right(t\left)\right\}$$ (2)

and

$${\lambda }_{ij}^V(t\mid U_{ij},Z_{ij}(s),s\le t)=I(t>U_{ij}){\lambda }_2\left(t\right)\exp \left\{{\gamma }_0^{\mathrm{T}}{Z}_{ij}\right(t\left)\right\}$$ (3)

for their hazard functions. In the above, both λ₁(t) and λ₂(t) are unspecified baseline hazard functions and γ₀ is a p-dimensional vector of regression parameters. Note that model (3) essentially assumes that the gap time between the two monitoring times U_ij and V_ij follows a Cox-type model conditional on U_ij. There are several motivations for considering the models described. First, it is well known that the Cox model is one of the most widely used models partly due to its simplicity and it will be seen that under the models above, an easy estimation procedure can be developed for regression coefficients without the need to estimate the baseline hazard functions. Also, the model assumptions can be easily checked since we have complete data for the monitoring times. Furthermore, the same baseline hazard functions and covariate effects for all subjects were assumed in the models (1)–(3), respectively, for simplicity and the method developed below can be easily generalised to more general situations. Some comments on these will be given below.

For subject j in cluster i, define

$$\begin{array}{cc}\hfill {\delta }_{1ij}& =I(T_{ij}<U_{ij}),\hfill \\ \hfill {\delta }_{2ij}& =I(U_{ij}\le T_{ij}<V_{ij}),\hfill \\ \hfill {\delta }_{3ij}& =I(T_{ij}\ge V_{ij})=1-{\delta }_{1ij}-{\delta }_{2ij},\hfill \end{array}$$

where j = 1,… n_i; i = 1,…, n. Then, define the following counting processes:

$$N_{ij}^{\left(1\right)}\left(t\right)=(1-{\delta }_{1ij})I(U_{ij}\le t)={\delta }_{ij}^{\left(1\right)}I(U_{ij}\le t)$$

and conditional on U_ij,

$$N_{ij}^{\left(2\right)}\left(t\right)={\delta }_{3ij}I(V_{ij}\le t)={\delta }_{ij}^{\left(2\right)}I(V_{ij}\le t),$$

where ${\delta }_{ij}^{\left(1\right)}=1-{\delta }_{1ij}$ and ${\delta }_{ij}^{\left(2\right)}={\delta }_{3ij}$. Note that the definition of $N_{ij}^{\left(2\right)}$ is naturally based on the order restriction between U_ij and V_ij and indicates that V_ij is considered only after U_ij has been observed. Based on the models (1)–(3), following the same arguments as those in Wang et al. (2010), one can derive the intensity functions of $N_{ij}^{\left(1\right)}$ and $N_{ij}^{\left(2\right)}$ as

$${\lambda }_{ij}^{\left(1\right)}(t\mid Z_{ij})=I(U_{ij}\ge t)E_b\left({\mathrm{e}}^{-b_{i}t}\right){\mathrm{e}}^{{\Lambda }_0\left(t\right)}{\lambda }_1\left(t\right)\exp \{-{\beta }_0^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }_0^{\mathrm{T}}{Z}_{ij}(t\left)\right\}$$ (4)

and

$${\lambda }_{ij}^{\left(2\right)}(t\mid Z_{ij})=I(U_{ij}<t\le V_{ij})E_b\left({\mathrm{e}}^{-b_{i}t}\right){\mathrm{e}}^{{\Lambda }_0\left(t\right)}{\lambda }_2\left(t\right)\exp \{-{\beta }_0^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }_0^{\mathrm{T}}{Z}_{ij}(t\left)\right\},$$ (5)

respectively. It can be seen that models (4) and (5) are Cox-type ones similar to models (2) and (3) and model (5) is a conditional one since the starting time point is the observed monitoring time U_ij. In the following section, we will establish some estimating equations for the estimation of regression coefficients β₀ and γ₀.

3. Estimation of regression parameters

Now we consider the estimation of regression parameters β₀ and γ₀ and for this, we will employ the estimating equation approach. For l = 0, 1 and 2, define

$$S_{1,\beta }^{\left(l\right)}(t;\beta ,\gamma )=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}I(t\le U_{ij})Z_{ij}^{\ast \left(l\right)}\left(t\right)\exp \{-{\beta }^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }^{\mathrm{T}}{Z}_{ij}(t\left)\right\}$$

and

$$S_{2,\beta }^{\left(l\right)}(t;\beta ,\gamma )=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}I(U_{ij}<t\le V_{ij})Z_{ij}^{\ast \left(l\right)}\left(t\right)\exp \{-{\beta }^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }^{\mathrm{T}}{Z}_{ij}(t\left)\right\},$$

where $Z_{ij}^{\ast \left(l\right)}\left(t\right)={\int }_0^t{Z}_{ij}^{\left(l\right)}\left(s\right)\mathrm{d}s$. Here for any vector a, a⁽⁰⁾ = 1, a⁽¹⁾ = a and a⁽²⁾ = aa^T. Based on the models and (5) and motivated by Zhu, Tong and Sun (2008) and Wang et al. (2010), we propose the following estimating function:

$$\begin{array}{cc}\hfill U_{\beta }(\beta ,\gamma )& =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}{\int }_0^{\infty }\left\{Z_{ij}^{\ast }\left(t\right)-\frac{S_{1,\beta }^{\left(1\right)}(t;\beta ,\gamma )}{S_{1,\beta }^{\left(0\right)}(t;\beta ,\gamma )}\right\}\mathrm{d}{N}_{ij}^{\left(1\right)}\left(t\right)\hfill \\ \hfill & +\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}{\int }_0^{\infty }\left\{Z_{ij}^{\ast }\left(t\right)-\frac{S_{2,\beta }^{\left(1\right)}(t;\beta ,\gamma )}{S_{2,\beta }^{\left(0\right)}(t;\beta ,\gamma )}\right\}\mathrm{d}{N}_{ij}^{\left(2\right)}\left(t\right)\hfill \\ \hfill & =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}\left\{Z_{ij}^{\ast }\left(W_{ij}^{\left(k\right)}\right)-\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}\right\}\hfill \end{array}$$

for the estimation of β₀ if γ₀ is known, where $W_{ij}^{\left(1\right)}=U_{ij}$ and $W_{ij}^{\left(2\right)}=V_{ij}$.

The key idea here is to reduce general interval-censored data to current status data (Zhu et al. 2008). In the expression of U_β(β, γ), the first term is the partial likelihood score function under model (4) if one observes only current status data, while the second term is the partial likelihood score function given under model (5) if one considers only current status data given by the observations on the V_ijs. Thus, U_β(β, γ) is an unbiased function. Similar estimating function can be developed for the estimation of γ₀. However, complete data are actually available for the monitoring times U_ij and V_ij and thus it is more efficient to estimate it based on models (2) and (3).

To this end, define ${\tilde{N}}_{ij}^{\left(1\right)}\left(t\right)=I(U_{ij}\le t)$ and ${\tilde{N}}_{ij}^{\left(2\right)}\left(t\right)=I(V_{ij}\le t)$ given U_ij. Also for l 0, 1 and 2, define

$$S_{1,\gamma }^{\left(l\right)}(t;\gamma )=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}I(t\le U_{ij})\exp \left\{{\gamma }^{\mathrm{T}}{Z}_{ij}\right(t\left)\right\}{Z}_{ij}^{\left(l\right)}\left(t\right)$$

and

$$S_{2,\gamma }^{\left(l\right)}(t;\gamma )=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}I(U_{ij}<t\le V_{ij})\exp \left\{{\gamma }^{\mathrm{T}}{Z}_{ij}\right(t\left)\right\}{Z}_{ij}^{\left(l\right)}\left(t\right).$$

Then, an estimating function for γ₀ can be constructed as

$$\begin{array}{cc}\hfill U_{\gamma }\left(\gamma \right)& =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}{\int }_0^{\infty }\left\{Z_{ij}\left(t\right)-\frac{S_{1,\gamma }^{\left(1\right)}(t;\gamma )}{S_{1,\gamma }^{\left(0\right)}(t;\gamma )}\right\}\mathrm{d}{\tilde{N}}_{ij}^{\left(1\right)}\left(t\right)\hfill \\ \hfill & +\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}{\int }_0^{\infty }\left\{Z_{ij}\left(t\right)-\frac{S_{2,\gamma }^{\left(1\right)}(t;\gamma )}{S_{2,\gamma }^{\left(0\right)}(t;\gamma )}\right\}\mathrm{d}{\tilde{N}}_{ij}^{\left(2\right)}\left(t\right)\hfill \\ \hfill & =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2\left\{Z_{ij}\left(W_{ij}^{\left(k\right)}\right)=\frac{S_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\gamma )}{S_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\gamma )}\right\}.\hfill \end{array}$$

Define the estimate of γ₀ and $\widehat{\gamma }$ to be the solution to U_γ (γ) = 0 and the estimate of β₀ and $\widehat{\beta }$ to be the solution to $U_{\beta }(\beta ,\widehat{\gamma })=0$. It can be easily shown that $\widehat{\gamma }$ is consistent and has an asymptotic normal distribution (Wei, Lin and Weissfeld 1989; Lin 1994). The consistency of $\widehat{\beta }$ can be similarly proved by noting that ${\widehat{A}}_{\beta }(\beta ,\widehat{\gamma })=-n^{-1}\partial U_{\beta }(\beta ,\widehat{\gamma })∕\partial \beta$ converges to a positive matrix at β₀. For the asymptotic distribution of $\widehat{\beta }$, one can first show that $n^{-1∕2}{U}_{\beta }({\beta }_0,\widehat{\gamma })$ converges in distribution to a vector of normal random variables with a zero mean vector and a covariance matrix that can be consistently estimated. It then follows by the Taylor series expansion of $U_{\beta }(\widehat{\beta },\widehat{\gamma })$ around β₀ that the distribution of $n^{1∕2}(\widehat{\beta }-{\beta }_0)$ can be asymptotically approximated by the normal distribution with mean zero and a covariance matrix that can be consistently estimated and provided in the appendix. The proof of the result is also sketched in the appendix.

4. A simulation study

We conducted an extensive simulation study to assess the finite sample performance of the estimation procedure proposed in the previous sections. To generate the failure times of interest in the study, it was assumed that T_ij followed the model

$${\lambda }_{ij}(t\mid Z_{ij},b_i)={\lambda }_0\left(t\right)+{\beta }_0^{\mathrm{T}}{Z}_{ij}+b_i$$

with λ₀(t = 2. Note that the covariate process was assumed to be time independent for simplicity and generated from the Bernoulli distribution with success probability p = 0.5. Also, the latent variables b_is were assumed to follow a normal distribution with zero mean and variance equal to $\frac{1}{4}$. For the monitoring variables U_ijs and V_ijs, they were generated based on the models (2) and (3) with λ₁(t) = 4 and λ₂(t) = 2, respectively. Finally, the cluster size n_i was assumed to follow the uniform distribution U{2, 3, 4}.

The following tables give the results based on 1000 replications, including the bias of the estimates given by the average of the estimates minus the true value (BIAS), the sample standard deviation (SSD) of the estimates, the average of the estimated standard errors (ESE) and the 95% empirical coverage probability (95%-CP). In particular, Table 1 presents the obtained results on the estimates $\widehat{\beta }$ and $\widehat{\gamma }$ based on the simulated data with the true values of β₀ being −0.25, 0, 0.25, 0.5 and 1, respectively with γ₀ = −0.25. Two choices of cluster sizes, n = 200 and 400, are considered in this simulation study. One can see that these results suggest that the proposed estimates seem to be unbiased and the SSD is close to the ESE, suggesting that the proposed variance estimate and the normality of the estimates are reasonable. It is interesting to find out that the parameter γ seems to be better estimated than the parameter β. This is reasonable as complete data are available for the former and one only observes incomplete data for the latter.

Table 1.

Estimation of γ and β with binary covariate and γ₀ = −0.25.

		n = 200				n = 400
TRUE	EST	BIAS	SSD	ESE	95%-CP	BIAS	SSD	ESE	95%-CP
γ0 = −0.25	$\widehat{\gamma }$	−0.0068	0.1107	0.1077	0.948	−0.0029	0.0772	0.0760	0.949
β0 = −0.25	$\widehat{\beta }$	−0.0638	0.8182	0.7605	0.937	−0.0299	0.5587	0.5331	0.940
γ0 = −0.25	$\widehat{\gamma }$	−0.0084	0.1087	0.1059	0.948	0.0058	0.0752	0.0748	0.965
β0 = 0.00	$\widehat{\beta }$	−0.0694	0.8162	0.7619	0.966	−0.0310	0.5616	0.5352	0.969
γ0 = −0.25	$\widehat{\gamma }$	0.0112	0.1074	0.1046	0.941	0.0082	0.0745	0.0739	0.952
β0 = 0.25	$\widehat{\beta }$	−0.0767	0.8226	0.7672	0.965	−0.0317	0.5658	0.5376	0.960
γ0 = −0.25	$\widehat{\gamma }$	0.0193	0.1063	0.1036	0.944	0.0120	0.0734	0.0732	0.945
β0 = 0.50	$\widehat{\beta }$	−0.0832	0.8256	0.7728	0.967	−0.0319	0.5640	0.5419	0.945
γ0 = −0.25	$\widehat{\gamma }$	0.0341	0.1044	0.1021	0.934	0.0259	0.0721	0.0720	0.937
β0 = 1.00	$\widehat{\beta }$	−0.0962	0.8518	0.7902	0.955	−0.0480	0.5797	0.5533	0.938

Tables 2 and 3 give the results obtained for the estimation of the regression parameters β and γ with γ₀ = 0 and 0.25, respectively, and all other set-ups being the same as those in Table 1. They are similar to those given in Table 1 and again suggest that the proposed estimation approach seems to work well for the situations considered.

Table 2.

Estimation of γ and β with binary covariate and γ₀ = 0.00.

		n = 200				n = 400
TRUE	EST	BIAS	SSD	ESE	95%-CP	BIAS	SSD	ESE	95%-CP
γ0 = 0.00	$\widehat{\gamma }$	−0.0051	0.1126	0.1100	0.952	−0.0034	0.0799	0.0777	0.945
β0 = −0.25	$\widehat{\beta }$	0.0172	0.8776	0.8255	0.938	0.0155	0.6065	0.5795	0.942
γ0 = 0.00	$\widehat{\gamma }$	0.0019	0.1106	0.1081	0.948	0.0039	0.0785	0.0764	0.945
β0 = 0.00	$\widehat{\beta }$	0.0315	0.8804	0.8273	0.975	0.0065	0.6158	0.5810	0.968
γ0 = 0.00	$\widehat{\gamma }$	0.0073	0.1102	0.1065	0.946	0.0122	0.0768	0.0753	0.943
β00 = 0.25	$\widehat{\beta }$	−0.0225	0.8835	0.8303	0.973	−0.0061	0.6106	0.5838	0.963
γ0 = 0.00	$\widehat{\gamma }$	0.0122	0.1087	0.1053	0.941	0.0143	0.0755	0.0744	0.946
β00 = 0.50	$\widehat{\beta }$	−0.0422	0.8903	0.8389	0.931	−0.0169	0.6110	0.5892	0.941
γ0 = 0.00	$\widehat{\gamma }$	0.0263	0.1063	0.1035	0.932	0.0212	0.0746	0.0732	0.940
β0 = 1.00	$\widehat{\beta }$	−0.0748	0.9116	0.8583	0.933	−0.0498	0.6539	0.6036	0.960

Table 3.

Estimation of γ and β with binary covariate and γ₀ = 0.25.

		n = 200				n = 400
TRUE	EST	BIAS	SSD	ESE	95%-CP	BIAS	SSD	ESE	95%-CP
γ0 = 0.25	$\widehat{\gamma }$	−0.0036	0.1177	0.1151	0.941	−0.0009	0.0829	0.0831	0.956
β0 = −0.25	$\widehat{\beta }$	0.0992	0.9643	0.9112	0.963	0.0328	0.7017	0.6768	0.956
γ0 = 0.25	$\widehat{\gamma }$	0.0035	0.1154	0.1126	0.941	0.0054	0.0812	0.0762	0.948
β0 = 0.00	$\widehat{\beta }$	0.0850	0.9610	0.9150	0.940	0.0353	0.6957	0.6747	0.956
γ0 = 0.25	$\widehat{\gamma }$	0.0122	0.1138	0.1107	0.948	0.0118	0.0802	0.0783	0.945
β0 = 0.25	$\widehat{\beta }$	0.0731	0.9634	0.9133	0.971	0.0307	0.7038	0.6763	0.959
γ0 = 0.25	$\widehat{\gamma }$	0.0193	0.1124	0.1093	0.941	0.0186	0.0792	0.0772	0.938
β0 = 0.50	$\widehat{\beta }$	0.0627	0.9690	0.9216	0.936	0.0262	0.7102	0.6822	0.960
γ0 = 0.25	$\widehat{\gamma }$	0.0286	0.1112	0.1070	0.926	0.0208	0.0769	0.0757	0.935
β0 = 1.00	$\widehat{\beta }$	0.0432	0.9721	0.9289	0.972	0.0299	0.7129	0.6964	0.969

5. An application

In this section, we apply the estimation procedure proposed in the previous sections to a set of clustered interval-censored failure time data arising from an LF study (Williamson, Kim, Manatunga and Addiss 2008). The study followed 47 men with LF, a debilitating parasitic disease in which several worms live together in several nests. An effective treatment is expected to kill the worms in all of the nests. The goal of the study was to compare the effect of the co-administration of diethylcarbamazine (DEC) and albendazole (ALB) (new treatment) versus DEC alone (standard treatment) for the treatment of LF. The patients in the study were followed for a year since their treatment and periodically examined by ultrasound to see if the worms were still alive. Thus, for the times to the clearance of the worms in each nest, the variables of interest, only clustered interval-censored data were observed with each patient serving as a cluster and the cluster size being the number of nests of adult filial worms in the body of each patient.

Among 47 patients, 22 received the co-administration of DEC and ALB, while the others were given DEC alone. In total, 78 adult worm nests were detected by ultrasound with the cluster size n_i ranging from 1 to 5. In addition to the treatment indicator, the age of each subject in years was also observed, ranging from 16 to 66. In the analysis below, we define X_1i to be 0 if subject i was given the co-administration of DEC and ALB and 1 otherwise and let X_2i be the age of the corresponding patient. Note that here we only have cluster-specific covariates.

To apply the proposed method, we assume that the times to the clearance of the worms and the monitoring times can be described by models (1)–(3), respectively. Since the observed data were given in the form T_ij ∈ [L_ij, R_ij), that is, we have a mixture of left-, interval-, and right-censored observations, we need to transfer them to the form expressed by U_ij and V_ij to implement the proposed estimation procedure. For this and an observed interval [L_ij, R_ij), we set U_ij = R_ij and V_ij to be the largest observation time in the study if L_ij = 0; if 0 < L_ij < R_ij < +∞, we let U_ij = L_ij and V_ij = R_ij; if R_ij = +∞, we take V_ij = L_ij and U_ij to be the smallest observation time in the study. Correspondingly, the estimating function U_γ (γ) needs to be adjusted to

$$U_{\gamma }\left(\gamma \right)=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2{\tilde{\delta }}_{ij}^{\left(k\right)}\left\{Z_{ij}\left(W_{ij}^k\right)-\frac{S_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\gamma )}{S_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\gamma )}\right\},$$

where ${\tilde{\delta }}_{ij}^{\left(1\right)}=1-{\delta }_{3ij}$ and ${\tilde{\delta }}_{ij}^{\left(2\right)}=1-{\delta }_{1ij}$. In contrast, the estimating function U_β(β, γ) remains the same. This essentially treats U_ij as missing in the right-censored case, and V_ij as missing in the left-censored case.

The results obtained by the application of the proposed estimation procedure to the data are presented in Table 4 and it includes the estimated treatment and age effects on the time to the clearance of the worms, the estimated standard deviation (SD), and the p-values for testing the covariate effects equal to zero. They suggest that the two treatments seem to have no significant difference in killing or cleaning the worms and also the clearance of the worms did not seem to be significantly related to the age of the patient. However, one may be careful about the conclusions due to the small number of subjects.

Table 4.

Estimations from the LF study.

Covariate	Estimate	SD	p-Value
Treatment (γ1)	−0.375	0.383	0.327
Age (γ1)	0.0061	0.020	0.755
Treatment (β1)	−0.522	0.442	0.237
Age (β2)	0.0108	0.415	0.794

6. Concluding remarks and discussion

As mentioned before, clustered failure time data occurs in a study if study subjects are related through being clustered into small groups. In this case, one has to take into account the correlation among them to perform a valid analysis. In the previous sections, an estimation procedure was developed for the problem in the presence of interval censoring for the data arising from the additive hazards model and the asymptotic properties of the proposed estimates were established. One major advantage of the presented method is that it does not involve the estimation of the baseline hazard functions.

For the problem considered here, an alternative approach is to employ the full likelihood approach and a main advantage of this method is that one can expect that it could be more efficient than the estimating equation method proposed here. However, the full likelihood approach would be time-consuming and may be infeasible if a nonparametric or semi-parametric approach was adopted since it involves the estimation of the infinite-dimensional functions. Also, it could be difficult to derive the asymptotic properties of the resulting estimates. In contrast, the proposed method can be easily implemented.

The proposed methodology involves modelling gap times between the adjacent monitoring times using the Cox model. An alternative choice is to model the monitoring times using the Cox model marginally. However, such modelling requires stricter conditions due to the order relationship. In contrast, the gap time modelling approach is more flexible. In the presented approach, the latent variables b_is are assumed to follow the normal distribution for simplicity and the methodology still applies for other distributions.

Appendix

In the following, we will sketch the proofs of the asymptotic normality of the estimates $\widehat{\gamma }$ and $\widehat{\beta }$. First, we will discuss the asymptotic normality of $\widehat{\gamma }$. For this and l = 0, 1, 2, let $s_{0,\gamma }^{\left(l\right)}$ and $s_{1,\gamma }^{\left(l\right)}$ denote the limits of $S_{0,\gamma }^{\left(l\right)}$ and $S_{1,\gamma }^{\left(l\right)}$, respectively. By following Wang et al. (2010), one can obtain that

$$\widehat{\gamma }-{\gamma }_0=U_{\gamma \gamma }^{-1}\frac{1}{n}\sum _{i=1}^n{U}_i\left({\gamma }_0\right)+o_p\left(n^{-1∕2}\right),$$ (1)

where

$$U_i\left(\gamma \right)=\sum _{j=1}^{n_i}\sum _{k=1}^2\left\{Z_{ij}(W_{ij}^{\left(k\right)}-\frac{s_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\gamma )}{s_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\gamma )}\right\}$$

and

$$U_{\gamma \gamma }=E\left\{\sum _{j=1}^{n_i}\sum _{k=1}^2\frac{s_{k,\gamma }^{\left(2\right)}(W_{ij}^{\left(k\right)};{\gamma }_0)}{s_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};{\gamma }_0)}-\frac{s_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};{\gamma }_0)s_{k,\gamma }^{\left(1\right)}{(W_{ij}^{\left(k\right)};{\gamma }_0)}^{\mathrm{T}}}{s_{k,\gamma }^{\left(0\right)}{(W_{ij}^{\left(k\right)};{\gamma }_0)}^2}\right\}.$$

with $W_{ij}^{\left(1\right)}=U_{ij}$ and $W_{ij}^{\left(2\right)}=V_{ij}$. Thus, the distribution of $n^{1∕2}(\widehat{\gamma }-{\gamma }_0)$ can be approximated by the normal distribution with mean zero and covariance matrix $U_{\gamma \gamma }^{-1}{\Sigma }_{\gamma }{U}_{\gamma \gamma }^{-1}$ which can be consistently estimated by ${\widehat{U}}_{\gamma \gamma }^{-1}{\widehat{\Sigma }}_{\gamma }{\widehat{U}}_{\gamma \gamma }^{-1}$, In the above,

$${\widehat{\Sigma }}_{\gamma }=\frac{1}{n}\sum _{i=1}^n{U}_i\left(\widehat{\gamma }\right)U_i{\left(\widehat{\gamma }\right)}^{\mathrm{T}},$$

$$U_i\left(\widehat{\gamma }\right)=\sum _{j=1}^{n_i}\sum _{k=1}^2\left\{Z_{ij}\left(W_{ij}^{\left(k\right)}\right)-\frac{s_{k,\widehat{\gamma }}^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}{s_{k,\widehat{\gamma }}^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}\right\}$$

and

$${\widehat{U}}_{\gamma \gamma }=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2\left\{\frac{S_{k,\gamma }^{\left(2\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}{S_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}-\frac{S_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })S_{k,\gamma }^{\left(1\right)}{(W_{ij}^{\left(k\right)};\widehat{\gamma })}^{\mathrm{T}}}{S_{k,\gamma }^{\left(0\right)}{(W_{ij}^{\left(k\right)};\widehat{\gamma })}^2}\right\}.$$

For the asymptotic normality of $\widehat{\beta }$, let $s_{k,\beta }^{\left(l\right)}$ denote the limit of $S_{k,\beta }^{\left(l\right)},l=0,1,2$ and k = 1, 2. By some calculation and Equation (1), we have

$$\begin{array}{cc}\hfill U(\beta ,{\gamma }_0)-U(\beta ,\widehat{\gamma })& =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}\left\{\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma }-S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma })}-\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)\left[S_{k,\beta }^{\left(0\right)}\right(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma })-S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0\left)\right]}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma })S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0}\right\}\hfill \\ \hfill & =\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}\left\{\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma }-S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}-\frac{s_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)\left[S_{k,\beta }^{\left(0\right)}\right(W_{ij}^{\left(k\right)};\beta ,\widehat{\gamma })-S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0\left)\right]}{s_{k,\beta }^{\left(0\right)}{(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}^2}\right\}+o_p\left(n^{-1∕2}\right).\hfill \end{array}$$

Note that here ${\delta }_{ij}^{\left(1\right)}=1-{\delta }_{1ij}$ and ${\delta }_{ij}^{\left(2\right)}={\delta }_{3ij}$. Furthermore, define the following for l = 0, 1,

$$A_1^{\left(l\right)}(t;\beta ,\gamma )=E\left[\sum _{j=1}^{n_i}I(t\le U_{ij})Z_{ij}^{\ast \left(l\right)}\left(t\right)Z_{ij}{\left(t\right)}^{\mathrm{T}}\exp \{-{\beta }^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }^{\mathrm{T}}{Z}_{ij}(t\left)\right\}\right],$$

$$A_2^{\left(l\right)}(t;\beta ,\gamma )=E\left[\sum _{j=1}^{n_i}I(U_{ij}<t\le V_{ij})Z_{ij}^{\ast \left(l\right)}\left(t\right)Z_{ij}{\left(t\right)}^{\mathrm{T}}\exp \{-{\beta }^{\mathrm{T}}{Z}_{ij}^{\ast }(t)+{\gamma }^{\mathrm{T}}{Z}_{ij}(t\left)\right\}\right]$$

and

$$A_3(\beta ,\gamma )=-E\left[\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}\left\{\frac{A_k^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}-\frac{s_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )A_k^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}\right\}\right].$$

By the Taylor series expansion and Equation (1), one obtains that

$$\begin{array}{cc}\hfill U(\beta ,\widehat{\gamma })-U(\beta ,{\gamma }_0)& =\frac{1}{n}\sum _{i=1}\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}(\widehat{\gamma }-{\gamma }_0)\left\{\frac{A_k^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}-\frac{s_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)A_k^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}{s_{k,\beta }^{\left(0\right)}{(W_{ij}^k;\beta ,{\gamma }_0)}^2}\right\}+o_p\left(n^{-1∕2}\right)\hfill \\ \hfill & =-E\left[\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}(\widehat{\gamma }-{\gamma }_0)\left\{\frac{A_k^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}-\frac{s_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)A_k^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,{\gamma }_0)}{s_{k,\beta }^{\left(0\right)}{(W_{ij}^k;\beta ,{\gamma }_0)}^2}\right\}\right]+o_p\left(n^{-1∕2}\right).\hfill \end{array}$$

This yields that

$$U(\beta ,\widehat{\gamma })=\frac{1}{n}\sum _{i=1}^n{a}_i(\beta ,{\gamma }_0)+o_p\left(n^{-1∕2}\right),$$

where

$$a_i(\beta ,\gamma )=\sum _{j=1}^{n_i}\sum _{k=1}^2\left[{\delta }_{ij}^{\left(k\right)}\left\{Z_{ij}^{\ast }\left(W_{ij}^{\left(k\right)}\right)-\frac{s_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}{s_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\beta ,\gamma )}\right\}+A_3(\beta ,\gamma )U_{\gamma \gamma }^{-1}\left\{Z_{ij}(W_{ij}^{\left(k\right)}-\frac{s_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\gamma )}{s_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\gamma )}\right\}\right].$$

This shows that the distribution of $n^{1∕2}(\widehat{\beta }-{\beta }_0)$ can be approximated by the normal distribution with zero mean and the covariance matrix that can be consistently estimated by ${\widehat{\Gamma }}^{-1}\widehat{\Sigma }{\widehat{\Gamma }}^{-1}$, where

$$\widehat{\Gamma }=-\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{n_i}\sum _{k=1}^2\left\{\frac{S_{k,\beta }^{\left(2\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}-\frac{S_{k,\beta }^{\left(1\right)}{(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}^{\mathrm{T}}}{S_{k,\beta }^{\left(0\right)}{(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}^2}\right\},$$

$$\widehat{\Sigma }=\frac{1}{n}\sum _{i=1}^n{a}_i(\widehat{\beta },\widehat{\gamma })a_i{(\widehat{\beta },\widehat{\gamma })}^{\mathrm{T}},$$

$$a_i(\widehat{\beta },\widehat{\gamma })=\sum _{j=1}^{n_i}\sum _{k=1}^2\left[{\delta }_{ij}^{\left(k\right)}\left\{Z_{ij}^{\ast }\left(W_{ij}^{\left(k\right)}\right)-\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}\right\}+A_3(\widehat{\beta },\widehat{\gamma }){\widehat{U}}_{\gamma \gamma }^{-1}\left\{Z_{ij}\left(W_{ij}^{\left(k\right)}\right)-\frac{S_{k,\gamma }^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}{S_{k,\gamma }^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\gamma })}\right\}\right]$$

and

$$A_3(\widehat{\beta },\widehat{\gamma })=-\frac{1}{n}\sum _{i=1}^n\left[\sum _{j=1}^{n_i}\sum _{k=1}^2{\delta }_{ij}^{\left(k\right)}\left\{\frac{A_k^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}{S_{k,\beta }^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}-\frac{S_{k,\beta }^{\left(1\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })A_k^{\left(0\right)}(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}{S_{k,\beta }^{\left(0\right)}{(W_{ij}^{\left(k\right)};\widehat{\beta },\widehat{\gamma })}^2}\right\}\right].$$