The Fine–Gray Model Under Interval Censored Competing Risks Data

Abstract

We consider semiparametric analysis of competing risks data subject to mixed case interval censoring. The Fine–Gray model (Fine & Gray, 1999) is used to model the cumulative incidence function and is coupled with sieve semiparametric maximum likelihood estimation based on univariate or multivariate likelihood. The univariate likelihood of cause-specific data enables separate estimation of cumulative incidence function for each competing risk, in contrast with the multivariate likelihood of full data which estimates cumulative incidence functions for multiple competing risks jointly. Under both likelihoods and certain regularity conditions, we show that the regression parameter estimator is asymptotically normal and semiparametrically efficient, although the spline-based sieve estimator of the baseline cumulative subdistribution hazard converges at a rate slower than root-n. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated by a competing risk analysis of data from an dementia cohort study.

1. INTRODUCTION

Interval censored failure time data arise widely from longitudinal studies where the failure occurrence is only detectable at periodic assessments, yielding the time to failure being known up to an interval. An example of such data is from the PAQUID project (Letenneur et al., 1999), where the age of onset of dementia on incident cases was only known to be between the date of the visit last seen without dementia and the date of the visit first seen with dementia. Another example of such data is from the Breastfeeding, Antiretrovirals, and Nutrition (BAN) study (Chasela et al., 2010), where the time to HIV-1 infection of an infant bread-fed by its HIV-1 positive mother is only known up to being between two successive HIV tests. Besides interval censoring, competing risk is another common issue in the statistical analysis of failure time data. Competing risks data arise when subjects may fail from several dependent causes but the occurrence of failure from one cause precludes the occurrence of failure from the others or only time to the first failure occurence is of interest. For example, in the PAQUID study, a subject may die before the onset of dementia, so death is a competing risk that precludes the occurrence of dementia. Also, in the BAN study, an infant may be weaned before getting HIV infected, which greatly reduces the risk of infection; or an infant could die without being HIV infected, which totally precludes the infection. Therefore, only the time to the first event of weaning, death and HIV infection is of clinical interest with respect to mother-to-child HIV transmission.

Competing risks data have drawn a great deal of attention since 1970s. Most of the works in the literature summarize competing risks data by the cause-specific hazard function and/or the cumulative incidence function. The former quantifies the instantaneous risk of failure from a specific cause, while the latter quantifies the cumulative risk of failure from a competing risk. There have been a lot of works on the estimation of these two quantities with right censored competing risks data. The cause-specific hazard function may be estimated nonparametrically by the increment of Nelson–Aalen estimator censoring individuals who experience competing causes of failure, and the cumulative incidence function may be estimated nonparametrically by the Aalen– Johansen estimator (Aalen & Johansen, 1978). To incorporate covariates, one can estimate the cause-specific hazard function semiparametrically by the standard Cox proportional hazards model treating failures from competing causes as independent right censoring events (Prentice et al., 1978), or estimate the cumulative incidence function semiparametrically by the Fine–Gray model (Fine & Gray, 1999) or other semiparametric transformation models (Fine, 2001). However, there are not as many published works on analyzing interval censored competing risks data, which are seen quite often in longitudinal medical studies. For the estimation of the cumulative incidence function with interval censored competing risks data, Hudgens et al. (2001) derived an algorithm to compute the nonparametric maximum likelihood estimator, and Jewell et al. (2003) proposed a simpler naive nonparametric maximum likelihood estimator based on reduced current status data for the cause of interest, which could be naturally extended to mixed case interval censored data. Groeneboom et al. (2008a) and Groeneboom et al. (2008b) rigorously derived the asymptotic properties of these two nonparametric cumulative incidence estimators for current status competing risks data in terms of consistency, convergence rates and limiting distributions. For the estimation of the cause-specific hazard function with interval censored competing risks data, Li & Fine (2013) proposed two cause-specific hazard estimators for current status competing risks data based on smoothing the nonparametric cumulative incidence estimators and the plug-in principle, and derived their asymptotic distributions. The computation of the cause-specific hazard estimators in Li & Fine (2013) could be naturally extended to mixed case interval censored data. Despite these works, several important research areas on interval censored competing risks data are still open. For example, the large sample properties of the nonparametric maximum likelihood estimator of the cumulative incidence function (Hudgens et al., 2001) and the nonparametric estimators of the cause specific hazard (Li & Fine, 2013) have not been derived for mixed case interval censored data with competing risks; the semiparametric estimation of the cause-specific hazard and the cumulative incidence with interval censored competing risks data has not been rigorously studied yet.

Unlike right censored competing risks data, the likelihood of interval censored competing risks data can be formulated concisely by just cumulative incidence functions (Hudgens et al., 2014, Lemma 1). Furthermore, the likelihood of reduced mixed case interval censored data for the cause of interest involves only the cumulative incidence function of that cause (Hudgens et al., 2014, Lemma 2). Such properties make models that directly link the cumulative incidence to explanatory variables appealing in the analysis of interval censored competing risks data. The most popular model of this kind is the Fine–Gray model, which however has been only applied to competing risks data under right censoring so far.

In this article, we study the semiparametric inference for the Fine–Gray model with mixed case interval censored competing risks data where observation times are strictly separated. Inspired by Zhang et al. (2010), we employ a monotone B-spline (Schumaker, 1981) to approximate the log baseline cumulative subdistribution hazard, reducing the dimension of the parameter space to a finite number. With this approximation, maximum likelihood estimators of the regression parameters and the B-spline coefficients are obtained by an adaptive barrier algorithm (Lange, 1994). Under certain regularity conditions, we establish that the proposed estimators of the regression parameter and the baseline cumulative subdistribution hazard are consistent. Moreover, we show that the estimator of the baseline cumulative subdistribution hazard achieves the optimal convergence rate in the nonparametric regression setting, and the estimator of the regression parameter is asymptotically normal and semiparametrically efficient. When there are more than one competing risk under consideration, one could perform a univariate analysis modeling each competing risk separately or a multivariate analysis modeling multiple competing risks jointly. Through simulation studies, we show that the multivariate analysis is more efficient than the univariate analysis for estimating the regression parameters when sample size is large.

The rest of the article is organized as follows. Section 2 specifies the Fine–Gray model and its associated likelihood for interval censored competing risks data. Section 3 describes an spline-based maximum likelihood estimation for the model. Section 4 derives the information matrix for the regression parameter in the semiparametric inference setting. Section 5 presents the asymptotic properties of the maximum likelihood estimators of baseline cumulative subdistribution hazards and regression parameters. The finite sample performance of the proposed method is then evaluated by simulation studies in Section 6, followed by a real data example in Section 7 that illustrates the application of the method. Finally, we give some discussion on our method and future research directions in Section 8. The proofs of the asymptotic results are collected in the appendix.

2. MODEL SPECIFICATION

We assume that a Fine–Gray model holds for each of J competing risks under consideration. Specifically, the cumulative incidence function of cause k given a vector of covariates Z ∈ ℝ^d, F_k(t; Z), is modeled as:

$$F_k(t;\mathbf{Z})=1-\text{exp}\{-{\int }_0^t{\lambda }_{k0}(s)\text{exp}({\mathbf{Z}}^T{\mathit{\theta }}_k)ds\}k=1,\dots ,J,$$ (1)

i.e., the subdistribution hazard of F_k(t; Z) is λ_k(t; Z) = λ_k0(t)exp(Z^T θ_k). For the identifiability of the model parameters, we assume throughout:

• A0. ${\sum }_{k=1}^J{F}_k(\infty ;\mathbf{0})<1$

Remark 1

When J = 1, the upper bound one is allowed to be achieved, in which case there is actually no other failure cause and the Fine–Gray model becomes the proportional hazards model. The estimation and inferential procedures for that case are discussed in Zhang et al. (2010).

Remark 2

The total number of competing risks could be equal to or larger than J. When it’s equal to J, A0 means that there is a cured fraction of the population.

Define the following notations:

$$\mathit{\theta }={({\mathit{\theta }}_1^T,\dots ,{\mathit{\theta }}_J^T)}^T\in {ℝ}^{Jd},$$

$${\mathrm{\Lambda }}_{k0}(t)={\int }_0^t{\lambda }_{k0}(s)ds,{\varphi }_{k0}(t)=\text{log}{\mathrm{\Lambda }}_{k0}(t),k=1,\dots ,J,$$

and

$$\varphi =({\varphi }_{10}(·),\dots ,{\varphi }_{J0}(·)).$$

For subject i, i = 1, …, n, denote its underlying failure time by T_i, its failure cause by K_i and its covariate vector by Z_i. We consider the situation where the failure time is not observed exactly but subject to ‘mixed case’ interval censoring as defined in Schick & Yu (2000). Let M_i be a random positive integer denoting the number of examination times for failure on subject i and $W_i=\{W_{m,l}^{(i)}:l=1,\dots ,m,m=1,2,\dots \}$ be a triangular array of random examination times with $W_{m,l}^{(i)}<\cdots <W_{m,m}^{(i)}$ . The observed examination times are $W_{M_i,1}^{(i)},\dots ,W_{M_i,M_i}^{(i)}$ . From the observed examination times, we select a pair of examination times (U_i, V_i) bracketing T_i as follows. $(U_i,V_i)=(W_{M_i,l-1}^{(i)},W_{M_i,l}^{(i)})$ if $T_i\in (W_{M_i,l-1}^{(i)},W_{M_i,l}^{(i)}]$ for some l ∈ {2,…, M_i}; $(U_i,V_i)=(W_{M_i,1}^{(i)},W_{M_i,2}^{(i)})$ if $T_i\le W_{M_i,1}^{(i)};(U_i,V_i)=(W_{M_i,M_i-1}^{(i)},W_{M_i,M_i}^{(i)})$ if $T_i>W_{M_i,M_i}^{(i)}$ . Let ${\mathrm{\Delta }}_{k1}^{(i)}=\mathrm{I}\{T_i\le U_i,K_i=k\},{\mathrm{\Delta }}_{k2}^{(i)}=\mathrm{I}\{U_i<T_i\le V_i,K_i=k\}$ and ${\mathrm{\Delta }}_3^{(i)}=1-{\sum }_{k=1}^J({\mathrm{\Delta }}_{k1}^{(i)}+{\mathrm{\Delta }}_{k2}^{(i)})$ . The observable data related to the J competing risks from subject i consist of $(M_i,W_{M_i,1}^{(i)},\dots ,W_{M_i,M_i}^{(i)},{\mathbf{Z}}_i)$ , whether any of the J types of failure has occurred by the time of each examination, and K_i if a failure has occurred by the last examination time. In light of assumption A0, we define a convention that K_i = J + 1 if no failure from any of the J causes occurs to subject i, which happens with a positive probability.

Under mixed case interval censoring, (M, W) are independent of (T, K) conditional on Z and the joint distribution of (M, W, Z) does not depend on (θ, ϕ). Thus, by Lemma 1 and 2 in Hudgens et al. (2014), the likelihood of the data related to the J competing risks, omitting the multiplicative terms that don’t involve (θ, ϕ), can be written as

$$L_n(\mathit{\theta },\varphi )=\prod _{i=1}^n\left[\prod _{k=1}^J{F}_k{(U_i;{\mathbf{Z}}_i)}^{{\mathrm{\Delta }}_{k1}^{(i)}}{\{F_k(V_i;{\mathbf{Z}}_i)-F_k(U_i;{\mathbf{Z}}_i)\}}^{{\mathrm{\Delta }}_{k2}^{(i)}}\right]{\{1-\sum _{k=1}^J{F}_k(V_i;{\mathbf{Z}}_i)\}}^{{\mathrm{\Delta }}_3^{(i)}}.$$ (2)

Note that the data involved in the likelihood (2) are only composed of ${\mathbf{Y}}_i=({\mathrm{\Delta }}_{11}^{(i)},{\mathrm{\Delta }}_{12}^{(i)},{\mathrm{\Delta }}_{21}^{(i)},{\mathrm{\Delta }}_{22}^{(i)},\dots ,{\mathrm{\Delta }}_{J1}^{(i)},{\mathrm{\Delta }}_{J2}^{(i)},{\mathrm{\Delta }}_3^{(i)},U_i,V_i,{\mathbf{Z}}_i)$ , i = 1, …, n, not the full data, since Y_i’s contain sufficient information for the model parameters. In order to facilitate deriving the large sample properties of maximum likelihood estimators for (θ, ϕ), we make a working assumption that (U, V) are independent of (T, K) conditional on Z and the joint distribution of (U, V, Z) is uninformative about (θ, ϕ). This assumption was also assumed in Huang & Rossini (1997) and Zhang et al. (2010) for mixed case interval censored data except not involving failure cause K. It is satisfied under case 2 interval censoring but may not hold in general. However, in view of the equivalence between (2) and the likelihood written in terms of the full data, we can derive the same large sample properties under the likelihood (2) and the working assumption as derived under the likelihood in terms of the full data, and the former derivation would be less complicated.

The log likelihood of (2) is

$$l_n(\mathit{\theta },\varphi )=\sum _{i=1}^n\left[\sum _{k=1}^J\right[{\mathrm{\Delta }}_{k1}^{(i)}\text{log}{F}_k(U_i;{\mathbf{Z}}_i)+{\mathrm{\Delta }}_{k2}^{(i)}\text{log}\{F_k(V_i;{\mathbf{Z}}_i)-F_k(U_i;{\mathbf{Z}}_i)\}]+{\mathrm{\Delta }}_3^{(i)}\text{log}\{1-\sum _{k=1}^J{F}_k(V_i;{\mathbf{Z}}_i)\left\}\right],$$

which, under model (1), is

$$l_n(\mathit{\theta },\varphi )=\sum _{i=1}^n\left[\sum _{k=1}^J\right[{\mathrm{\Delta }}_{k1}^{(i)}\text{log}[1-\text{exp}\{-e^{{\mathbf{Z}}_i^T{\mathit{\theta }}_k+{\varphi }_{k0}(U_i)}\left\}\right]+{\mathrm{\Delta }}_{k2}^{(i)}\text{log}\left[\text{exp}\right\{-e^{{\mathbf{Z}}_i^T{\mathit{\theta }}_k+{\varphi }_{k0}(U_i)}\}-\text{exp}\{-e^{{\mathbf{Z}}_i^T{\mathit{\theta }}_k+{\varphi }_{k0}(V_i)}\left\}\right]]+{\mathrm{\Delta }}_3^{(i)}\text{log}[1-J+\sum _{k=1}^J\text{exp}\{-e^{{\mathbf{Z}}_i^T{\mathit{\theta }}_k+{\varphi }_{k0}(V_i)}\}\left]\right]\equiv \sum _{i=1}^{n}l({\mathbf{Y}}_i;\mathit{\theta },\varphi ).$$ (3)

3. SIEVE MAXIMUM LIKELIHOOD ESTIMATION

To estimate the parameters ζ = (θ, ϕ) in model (1), we consider a spline-based semiparametric maximum likelihood estimation. The estimation procedure extends that of Zhang et al. (2010) for univariate interval censored data to competing risks setting. Suppose τ₀ and τ₁ are respectively the lower and upper bounds of the bracketing examination times (U, V). Let τ₀ = d_k0 < d_k1 < ⋯ < d_k,Kkn < d_k,Kkn+1 = τ₁ (k = 1, …, J) be J partitions of [τ₀, τ₁], where K_kn = O(n^ν_k) for some ν_k > 0 is a positive integer such that max_{1≤j≤Kkn+1} |d_j − d_j−1| = O(n^−ν_k). Denote the kth set of partition points by D_kn = {d_k1, …, d_k,Kkn}. Let 𝒮_kn(D_kn, K_kn, m_k) be the space of polynomial splines of order m_k ≥ 1 comprising functions s satisfying: (i) the restriction of s to [d_kt, d_k,t+1), t = 0, …, K_kn, is a polynomial of order m_k, and (ii) for m_k ≥ 2 and $0\le m_k^{\prime }\le m_k-2$ , s is $m_k^{\prime }$ times continuously differentiable on [τ₀, τ₁]. According to Schumaker (1981, Corollary 4.10), there is a local basis ℬ_kn ≡ {b_kt(·), 1 ≤ t ≤ q_kn}, called B-spline, for 𝒮_kn(D_kn, K_kn, m_k), where q_kn ≡ K_kn + m_k. These basis functions are nonnegative and sum up to one at each point in [τ₀, τ₁], and every b_kt is zero outside the interval [d_k,t−mk, d_kt], where d_k,1−mk = d_k,2−mk = ⋯ = d_k,0 = τ₀ and d_k,Kkn+1 = d_k,Kkn+2 = ⋯ = d_k,Kkn+mk = τ₁.

Since ϕ_k0(t) is a non-decreasing function, it is desirable to restrict its estimate to be also non-decreasing. Let

$${ℳ}_{kn}(D_{kn},K_{kn},m_k)=\{{\varphi }_{kn}:{\varphi }_{kn}(t)=\sum _{j=1}^{q_{kn}}{\beta }_{kj}{b}_{kj}(t)\in {𝑆}_{kn}(D_{kn},K_{kn},m_k),{\mathit{\beta }}_k\in {\mathbf{B}}_{kn},t\in [{\tau }_0,{\tau }_1]\},$$

where B_kn = {β_k = (β_k1, …, β_k,qkn)^T : β_k1 ≤ β_k2 … ≤ β_k,qkn}. As a consequence of the variation diminishing properties of B-spline (see, for example, Schumaker, 1981, example 4.75 and Theorem 4.76), every element of ℳ_kn(D_kn, K_kn, m_k) is a nondecreasing function because of the monotonicity constraints on β_k1, …, β_k,qkn. Denote the feasible domain for the regression parameter θ by Θ ⊂ ℝ^Jd. We search for ζ̂_n = (θ̂_n, ϕ̂_n) that maximizes l_n(θ, ϕ) over $\mathbf{\Theta }\times {\prod }_{k=1}^J{ℳ}_{kn}$ , which is equivalent to maximizing $l_n(\mathit{\theta },{ℬ}_{1n}^T{\mathit{\beta }}_1,\dots ,{ℬ}_{Jn}^T{\mathit{\beta }}_J)$ over $\mathbf{\Theta }\times {\prod }_{k=1}^J{\mathbf{B}}_{kn}$ . In practice, we use an adaptive barrier algorithm (Lange, 1994) to maximize $l_n(\mathit{\theta },{ℬ}_{1n}^T{\mathit{\beta }}_1,\dots ,{ℬ}_{Jn}^T{\mathit{\beta }}_J)$ subject to the linear inequality constraints. It is implemented by the R function ‘constrOptim’. In the subsequent simulation study and real data example, we set the initial values of the regression parameters for the optimization to be all 0 and the initial value of β_kj to be μ + (j − 1)δ_k, where μ = log{−log(1 − 0.15J⁻¹)} and δ_k = [log{−log(1 − 0.85J⁻¹)} − μ]/(q_kn − 1) (j = 1, …, q_kn; k = 1, …, J).

4. INFORMATION FOR θ

We would like to show that the estimate of the regression parameter θ, θ̂_n, is semiparametrically efficient (Kosorok, 2008, Section 3.1). To prove that, we first derive the efficient score function and information matrix for θ (Kosorok, 2008, Section 3.2). Our derivation is similar to that in Huang & Wellner (1995).

By some algebra, the score function for θ is

$${\dot{l}}_{\mathit{\theta }}(\mathbf{Y})=\frac{\partial l(\mathbf{Y};\mathit{\theta },\varphi )}{\partial \mathit{\theta }}=\left(\begin{array}{c}\hfill [{\mathrm{\Delta }}_{11}{A}_{11}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{10}(U)-{\mathrm{\Delta }}_{12}\{A_{12}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{10}(U)-A_{13}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{10}(V)\}-{\mathrm{\Delta }}_3{A}_{14}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{10}(V)]e^{{\mathbf{Z}}^T{\mathit{\theta }}_1}\mathbf{Z}\hfill \\ \hfill ⋮\hfill \\ \hfill [{\mathrm{\Delta }}_{J1}{A}_{J1}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{J0}(U)-{\mathrm{\Delta }}_{J2}\{A_{J2}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{J0}(U)-A_{J3}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{J0}(V)\}-{\mathrm{\Delta }}_3{A}_{J4}(U,V,\mathbf{Z}){\mathrm{\Lambda }}_{J0}(V)]e^{{\mathbf{Z}}^T{\mathit{\theta }}_J}\mathbf{Z}\hfill \end{array}\right),$$ (4)

where

$$A_{k1}(U,V,\mathbf{Z})=\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(U)\}/[1-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(U)\left\}\right],$$

$$A_{k2}(U,V,\mathbf{Z})=\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(U)\}/\left[\text{exp}\right\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(U)\}-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(V)\left\}\right],$$

$$A_{k3}(U,V,\mathbf{Z})=\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(V)\}/\left[\text{exp}\right\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(U)\}-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(V)\left\}\right],$$

and

$$A_{k4}(U,V,\mathbf{Z})=\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}{\mathrm{\Lambda }}_{k0}(V)\}/[1-J+\sum _{j=1}^J\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_j}{\mathrm{\Lambda }}_{j0}(V)\left\}\right]$$

for k = 1, …, J.

Let ℘ be the statistical model consisting of probability measures on the sample space 𝒴 of Y, and it can be expressed in the form $\wp =\{P_{\mathit{\theta },\varphi }:(\mathit{\theta },\varphi )\in \mathbf{\Theta }\times {\prod }_{k=1}^J{\mathrm{\Phi }}_k\}$ , where Φ_k is a class of functions with bounded p_kth derivative in [τ₀, τ₁] for p_k ≥ 1 (k = 1, …, J). Consider its one-dimensional submodels of the form {P_θ,ϕ^(s) : s ∈ [0, ε)}, where the cumulative incidence function corresponding to ϕ^(s)’s kth component ${\varphi }_{k0}^{(s)}(k=1,\dots ,J)$ is $F_{k0}^{(s)}(t)={\int }_0^t(1+sa(w,k))d{F}_{k0}(w)$ for some function a(w, k) that is a function of w with bounded variation on ℝ₊ when holding k fixed, and ε > 0 depends on a(·, ·). Define a convention that

$$a(w,J+1)=\{\begin{array}{cc}{\{1-{\sum }_{k=1}^J{F}_{k0}(\infty )\}}^{-1}{\frac{\partial }{\partial s}|}_{s=0}\{1-{\sum }_{k=1}^J{F}_{k0}^{(s)}(\infty )\}\hfill & \text{if}w=\infty ,\hfill \\ 0\hfill & \text{if}w<\infty \hfill \end{array}$$

in order for the score operator for ϕ, derived from the score functions of the above one-dimensional submodels, to be a linear continuous map l̇_ϕ : $L_2^0(F_{(T,K)|\mathbf{Z}=0})\mapsto L_2^0(P_{\mathit{\theta },\varphi })$ , where F_(T,K)|Z denote the conditional distribution of (T, K) given Z, and $L_2^0(Q)$ denote the function space {f : ∫ f²dQ < ∞ and ∫ fdQ = 0} for any measure Q. Specifically, for any $a\in L_2^0(F_{(T,K)|\mathbf{Z}=0})$ ,

$${\dot{l}}_{\varphi }a(\mathbf{Y})={\frac{\partial }{\partial s}|}_{s=0}l(\mathbf{Y};\mathit{\theta },{\varphi }^{(s)})=\sum _{k=1}^J[{\mathrm{\Delta }}_{k1}{A}_{k1}(U,V,\mathbf{Z})h_k(U)-{\mathrm{\Delta }}_{k2}\{A_{k2}(U,V,\mathbf{Z})h_k(U)-A_{k3}(U,V,\mathbf{Z})h_k(V)\}-{\mathrm{\Delta }}_3{A}_{k4}(U,V,\mathbf{Z})h_k(V)]e^{{\mathbf{Z}}^T{\mathit{\theta }}_k}\equiv \sum _{k=1}^J{\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};\mathit{\theta },\varphi )({\eta }_k)\equiv {\dot{l}}_{\varphi }(\mathbf{Y};\mathit{\theta },\varphi )(\mathit{\eta }),$$ (5)

where $h_k(t)={\frac{\partial }{{\partial }_s}|}_{s=0}{\mathrm{\Lambda }}_{k0}^{(s)}(t)={\int }_0^{t}a(w,k)d{F}_{k0}(w)/\{1-F_{k0}(t)\},{\eta }_k(t)={\frac{\partial }{{\partial }_s}|}_{s=0}{\varphi }_{k0}^{(s)}(t)=h_k(t)/{\mathrm{\Lambda }}_{k0}(t)(k=1,\dots ,J)$ and η = (η₁(·), …, η_J (·)). Let 𝒩 be the class of functions η_k(t).

Since $L_2^0(F_{(T,K)|\mathbf{Z}=0})$ and $L_2^0(P_{\mathit{\theta },\varphi })$ are two Hilbert spaces with respective inner products < ·, · >_F(T,K)|Z=0 and < ·, · >_Pθ,ϕ, where < f, g >_Q= ∫ fgdQ, there is an adjoint operator ${\dot{l}}_{\varphi }^{*}:L_2^0(P_{\mathit{\theta },\varphi })\mapsto L_2^0(F_{(T,K)|\mathbf{Z}=0})$ satisfying ${<b,{\dot{l}}_{\varphi }a>}_{P_{\mathit{\theta },\varphi }}=<{\dot{l}}_{\varphi }^{*}b,a{>}_{F_{(T,K)|\mathbf{Z}=0}}$ for any $b\in L_2^0(P_{\mathit{\theta },\varphi })$ and $a\in L_2^0(F_{(T,K)|\mathbf{Z}=0})$ . To determine the efficient score function for θ, for each of the Jd components in l̇_θ, l̇_θlj (l = 1, …, J; j = 1, …, d), we need to find an $a_{lj}^{*}\in L_2^0(F_{(T,K)|\mathbf{Z}=0})$ such that ${<{\dot{l}}_{{\theta }_{lj}}-{\dot{l}}_{\varphi }{a}_{lj}^{*},{\dot{l}}_{\varphi }a>}_{P_{\mathit{\theta },\varphi }}=0$ for any $a\in L_2^0(F_{(T,K)|\mathbf{Z}=0})$ . This amounts to solving the following normal equations:

$${\dot{l}}_{\varphi }^{*}{\dot{l}}_{\varphi }{a}_{lj}^{*}={\dot{l}}_{\varphi }^{*}{\dot{l}}_{{\theta }_{lj}},l=1,\dots ,J;j=1,\dots ,d.$$ (6)

To solve for $a_{lj}^{*}$ , we need to evaluate both sides of (6). In the remainder of this section, we assume d = 1 to simplify the notations in the derivation of $a_{lj}^{*}$ . The derivation for the general case is similar. As a result, all the notations θ_lj (j = 1, …, d) can be simplified to a single scalar θ_l for l = 1, …, J, $a_{lj}^{*}(j=1,\dots ,d)$ can be simplified to $a_l^{*}$ , and we change the notation of one-dimensional Z to Z.

From Bickel et al. (1993, pp. 271–272), we know that

$${\dot{l}}_{\varphi }^{*}{\dot{l}}_{\varphi }a(t,k)=E[{\dot{l}}_{\varphi }a(Y)|T=t,K=k]=E_{Z}E[{\dot{l}}_{\varphi }a(Y)|T=t,K=k,Z].$$

By assumptions A3 and A7 in Section 5,

$$E[{\dot{l}}_{\varphi }a(Y)|T=t,K=k,Z=z]={\int }_{u=t}^{{\tau }_1-\eta }{\int }_{\upsilon =u+\eta }^{{\tau }_1}{e}^{z{\theta }_k}{h}_k(u)A_{k1}g(u,\upsilon |z)d\upsilon du-{\int }_{u={\tau }_0}^t{\int }_{\upsilon =t}^{{\tau }_1}{e}^{z{\theta }_k}\{h_k(u)A_{k2}-h_k(\upsilon )A_{k3}\}g(u,\upsilon |z)\mathbf{1}[\upsilon -u\ge \eta ]d\upsilon du-{\int }_{u={\tau }_0}^{t-\eta }{\int }_{\upsilon =u+\eta }^t\sum _{l=1}^J{e}^{z{\theta }_l}{A}_{l4}{h}_l(\upsilon )g(u,\upsilon |z)d\upsilon du,$$

where g(u, υ|z) is the conditional density of (U, V) given Z. Let B_kj(u, υ) = E_Z[e^Zθ_kA_kj(u, υ, Z)g(u, υ|Z)] (k = 1, …, J; j = 1, 2, 3, 4). We obtain

$$L(t,k)\equiv {\dot{l}}_{\varphi }^{*}{\dot{l}}_{\varphi }a(t,k)={\int }_{u=t}^{{\tau }_1-\eta }{\int }_{\upsilon =u+\eta }^{{\tau }_1}{h}_k(u)B_{k1}(u,\upsilon )d\upsilon du-{\int }_{u={\tau }_0}^t{\int }_{\upsilon =t}^{{\tau }_1}\{h_k(u)B_{k2}(u,\upsilon )-h_k(\upsilon )B_{k3}(u,\upsilon )\}\mathbf{1}[\upsilon -u\ge \eta ]d\upsilon du-{\int }_{u={\tau }_0}^{t-\eta }{\int }_{\upsilon =u+\eta }^t\sum _{l=1}^J{B}_{l4}(u,\upsilon )h_l(\upsilon )d\upsilon du.$$

Similar calculation yields

$$R_l(t,k)\equiv {\dot{l}}_{\varphi }^{*}{\dot{l}}_{{\theta }_l}(t,k)=\{\begin{array}{c}{\int }_{u=t}^{{\tau }_1-\eta }{\int }_{\upsilon =u+\eta }^{{\tau }_1}{\mathrm{\Lambda }}_{l0}(u)C_{l1}(u,\upsilon )d\upsilon du\hfill \\ -{\int }_{u={\tau }_0}^t{\int }_{\upsilon =t}^{{\tau }_1}\{{\mathrm{\Lambda }}_{l0}(u)C_{l2}(u,\upsilon )-{\mathrm{\Lambda }}_{l0}(\upsilon )C_{l3}(u,\upsilon )\}\mathbf{1}[\upsilon -u\ge \eta ]d\upsilon du\hfill & \text{if}k=l,\hfill \\ -{\int }_{u={\tau }_0}^{t-\eta }{\int }_{\upsilon =u+\eta }^t{\mathrm{\Lambda }}_{l0}(\upsilon )C_{l4}(u,\upsilon )d\upsilon du\hfill \\ -{\int }_{u={\tau }_0}^{t-\eta }{\int }_{\upsilon =u+\eta }^t{\mathrm{\Lambda }}_{l0}(\upsilon )C_{l4}(u,\upsilon )d\upsilon du\hfill & \text{if}k\ne l,\hfill \end{array}$$

where C_lj(u, υ) = E_Z[Ze^Zθ_lA_lj(u, υ, Z)g(u, υ|Z)] (l = 1, …, J; j = 1, 2, 3, 4). Equation (6) implies that

$$L(t,k)=R_l(t,k),t\in {ℝ}_{+},k=1,\dots ,J,\hspace{1em}l=1,\dots ,J.$$ (7)

Let $b_k(t)={\int }_{t+\eta }^{{\tau }_1}{B}_{k1}(t,x)dx+{\int }_{t+\eta }^{{\tau }_1}{B}_{k2}(t,x)dx+{\int }_{{\tau }_0}^{t-\eta }{B}_{k3}(x,t)dx+{\int }_{{\tau }_0}^{t-\eta }{B}_{k4}(x,t)dx$ . Then $\frac{\partial }{\partial t}L(t,k)=-b_k(t)h_k(t)+{\int }_{{\tau }_0}^{t-\eta }{h}_k(x)B_{k2}(x,t)dx+{\int }_{t+\eta }^{{\tau }_1}{h}_k(x)B_{k3}(t,x)dx-\sum _{m=1,m\ne k}^J{h}_m(t){\int }_{{\tau }_0}^{t-\eta }{B}_{m4}(x,t)dx$ Let $c_l(t)={\int }_{t+\eta }^{{\tau }_1}{C}_{l1}(t,x)dx+{\int }_{t+\eta }^{{\tau }_1}{C}_{l2}(t,x)dx+{\int }_{{\tau }_0}^{t-\eta }{C}_{l3}(x,t)dx+{\int }_{{\tau }_0}^{t-\eta }{C}_{l4}(x,t)dx$ Then

$$r_l(t,k)\equiv \frac{\partial }{\partial t}{R}_l(t,k)=\{\begin{array}{cc}-c_l(t){\mathrm{\Lambda }}_{l0}(t)+{\int }_{{\tau }_0}^{t-\eta }{\mathrm{\Lambda }}_{l0}(x)C_{l2}(x,t)dx+{\int }_{t+\eta }^{{\tau }_1}{\mathrm{\Lambda }}_{l0}(x)C_{l3}(t,x)dx\hfill & \text{if}k=l,\hfill \\ -{\mathrm{\Lambda }}_{l0}(t){\int }_{{\tau }_0}^{t-\eta }{C}_{l4}(x,t)dx\hfill & \text{if}k\ne l.\hfill \end{array}$$

Taking the derivative with respect to t on both sides of (7) and fixing l, we get

$$h_k(t)-\int K_k(t,x)h_k(x)dx+\frac{1}{b_k(t)}\sum _{m=1,m\ne k}^J{h}_m(t){\int }_{{\tau }_0}^{t-\eta }{B}_{m4}(x,t)dx=d_l(t,k),\hspace{1em}k=1,\dots ,J,$$ (8)

where K_k(t, x) = [B_k2(x, t)1[τ₀ ≤ x ≤ t − η] + B_k3(t, x)1[t + η ≤ x ≤ τ₁]]/b_k(t) and d_l(t, x) = −r_l(t, x)/b_k(t).

(8) is a system of Fredholm integral equations of the second kind. There exists a solution ${\mathbf{h}}_l^{*}(t)={(h_1^{(l),*}(t),\dots ,h_J^{(l),*}(t))}^T$ . The corresponding a is denoted by $a_l^{*}$ . Let ${\mathbf{a}}^{*}={(a_1^{*},\dots ,a_J^{*})}^T$ and ${\dot{l}}_{\varphi }{\mathbf{a}}^{*}(\mathbf{y})={({\dot{l}}_{\varphi }{a}_1^{*}(\mathbf{y}),\dots ,{\dot{l}}_{\varphi }{a}_J^{*}(\mathbf{y}))}^T$ . Then $l_{\mathit{\theta }}^{*}(\mathbf{y})\equiv {\dot{l}}_{\mathit{\theta }}(\mathbf{y})-{\dot{l}}_{\varphi }{\mathbf{a}}^{*}(\mathbf{y})$ is the efficient score function for θ, and its covariance matrix $I(\mathit{\theta })=E[l_{\mathit{\theta }}^{*}(\mathbf{Y})l_{\mathit{\theta }}^{*}{(\mathbf{Y})}^T]$ is the efficient information matrix for θ. The efficient influence function for θ is ${\tilde{l}}_{\mathit{\theta }}(\mathbf{y})\equiv I{(\mathit{\theta })}^{-1}{l}_{\mathit{\theta }}^{*}(\mathbf{y})$ .

5. LARGE SAMPLE PROPERTIES

To study the large sample properties of the semiparametric maximum likelihood estimator ζ̂_n = (θ̂_n, ϕ̂_n), we make the following assumptions similar to those in Zhang et al. (2010).

• A1.E(ZZ^T) is non-singular, and Z is bounded with probability 1.
• A2.Θ is a compact subset of ℝ^Jd.
• A3.There exists a positive number η such that Pr(V − U ≥ η) = 1, and the union of the supports of U and V is contained in an interval [τ₀, τ₁], where 0 < τ₀ < τ₁ < ∞, and 0 < Λ_k0(τ₀) < Λ_k0(τ₁) < ∞ (k = 1, …, J).
• A4.ϕ_k0 = logΛ_k0 belongs to Φ_k (k = 1, …, J), a class of functions with bounded p_kth derivative in [τ₀, τ₁] for p_k ≥ 1 and the first derivative of ϕ_k0 is uniformly positive and continuous on [τ₀, τ₁].
• A5.The conditional density g(u, υ|z) of (U, V) given Z has bounded partial derivatives with respect to (u, υ). The bounds of these partial derivatives do not depend on (u, υ, z).
• A6.For some κ ∈ (0, 1), x^T var(Z|U)x ≥ κx^T E(ZZ^T |U)x and x^T var(Z|V)x ≥ κx^T E(ZZ^T |V)x a.s. for all x ∈ ℝ^d.
• A7.(T, K) and (U, V) are conditionally independent given Z, and the distribution of (U, V, Z) does not involve (θ, ϕ).

These conditions, together with A0, guarantee the results in the forthcoming theorems.

Let $\mathbf{\Phi }={\prod }_{k=1}^J{\mathrm{\Phi }}_k$ . For any ${\varphi }_1=({\varphi }_{10}^{(1)},\dots ,{\varphi }_{J0}^{(1)}),{\varphi }_2=({\varphi }_{10}^{(2)},\dots ,{\varphi }_{J0}^{(2)})\in \mathbf{\Phi }$ , define

$${\Vert {\varphi }_1-{\varphi }_2\Vert }_{\mathbf{\Phi }}^2=\sum _{k=1}^J[E{\{{\varphi }_{k0}^{(1)}(U)-{\varphi }_{k0}^{(2)}(U)\}}^2+E{\{{\varphi }_{k0}^{(1)}(V)-{\varphi }_{k0}^{(2)}(V)\}}^2],$$

and for any ζ₁ = (θ⁽¹⁾, ϕ₁) and ζ₂ = (θ⁽²⁾, ϕ₂) in the space of 𝒯 = Θ × Φ, define an L₂-metric:

$$d({\mathit{\zeta }}_1,{\mathit{\zeta }}_2)=\Vert {\mathit{\zeta }}_1-{\mathit{\zeta }}_2\Vert \mathit{𝒯}={\{{\Vert {\mathit{\theta }}^{(1)}-{\mathit{\theta }}^{(2)}\Vert }^2+{\Vert {\varphi }_1-{\varphi }_2\Vert }_{\mathbf{\Phi }}^2\}}^{1/2}.$$

Theorem 1

For k = 1, …, J, let K_kn = O(n^ν_k), where ν_k satisfies the restriction 1/{2(1+p_k)} < ν_k < 1/(2p_k), and m_k ≥ p_k + 2. Furthermore, suppose that the assumptions A0–A7 hold. Denote the true parameters of model (1) by ζ₀ = (θ⁽⁰⁾, ϕ₀). Then d(ζ̂_n, ζ₀) = O_p(n^{− min{min_k(p_kν_k), (1 − max_k ν_k)/2}}).

This theorem implies that if ν_k = 1/(1 + 2p_k) (k = 1, …, J), d(ζ̂_n, ζ₀) = O_p(n^{− min_k{p_k/(1 + 2p_k)}}) which is the optimal convergence rate in the nonparametric regression setting.

Theorem 2

Under the conditions of Theorem 1, n^1/2(θ̂_n − θ⁽⁰⁾) ↦ N(0, I⁻¹(θ⁽⁰⁾)) in distribution, where I(θ⁽⁰⁾) is the efficient information matrix for θ⁽⁰⁾.

The theorem implies that, although the estimator of the baseline cumulative subdistribution hazard converges at a rate lower than n^1/2, the estimator of the regression parameter converges to the truth at the usual root-n rate and is semiparametrically efficient. Because I(θ⁽⁰⁾) is determined by a system of integral equations and does not have an explicit expression as shown in Section 4, direct plug-in estimator of I(θ⁽⁰⁾) is not available. We give an approach of estimating I(θ⁽⁰⁾) which is parallel to that of Zhang et al. (2010).

For a Jd-dimensional θ, l̇_θ(y) is the vector of partial derivatives of l(y; θ, ϕ) with respect to the components of θ. For each component of l̇_θ(y), we consider a corresponding score function for ϕ, l̇_ϕ(y; θ, ϕ)(η), as defined in (5). Thus the score vector for ϕ corresponding to l̇_θ(y) is

$${\dot{l}}_{\varphi }(\mathbf{y};\mathit{\theta },\varphi )(\overrightarrow{\mathit{\eta }})\equiv {\{{\dot{l}}_{\varphi }(\mathbf{y};\mathit{\theta },\varphi )({\mathit{\eta }}^{(11)}),{\dot{l}}_{\varphi }(\mathbf{y};\mathit{\theta },\varphi )({\mathit{\eta }}^{(12)}),\dots ,{\dot{l}}_{\varphi }(\mathbf{y};\mathit{\theta },\varphi )({\mathit{\eta }}^{(Jd)})\}}^T,$$ (9)

where η⃗ = {η⁽¹¹⁾, η⁽¹²⁾, …, η^(Jd)} and ${\mathit{\eta }}^{(lj)}=\{{\eta }_1^{(lj)}(·),\dots ,{\eta }_J^{(lj)}(·)\}$ with ${\eta }_k^{(lj)}\in 𝒩(k=1,\dots ,J;l=1,\dots ,J;j=1,\dots ,d)$ . According to Bickel et al. (1993, Theorem 1, pp. 70), the efficient score vector for θ is l̇_θ(y) − l̇_ϕ(y; θ, ϕ)(η⃗_*), where ${\overrightarrow{\mathit{\eta }}}_{*}=\{{\mathit{\eta }}_{*}^{(11)},{\mathit{\eta }}_{*}^{(12)},\dots ,{\mathit{\eta }}_{*}^{(Jd)}\}$ is an element of 𝒩^J2d that minimizes

$$\rho (\overrightarrow{\mathit{\eta }})\equiv E{\Vert {\dot{l}}_{\mathit{\theta }}(\mathbf{Y})-{\dot{l}}_{\varphi }(\mathbf{Y};\mathit{\theta },\varphi )(\overrightarrow{\mathit{\eta }})\Vert }^2$$ (10)

over 𝒩^J2d and is called the least favorable direction. Then the information for θ is

$$I(\mathit{\theta })=E[\{{\dot{l}}_{\mathit{\theta }}(\mathbf{Y})-{\dot{l}}_{\varphi }(\mathbf{Y};\mathit{\theta },\varphi )({\overrightarrow{\mathit{\eta }}}_{*})\}{\{{\dot{l}}_{\mathit{\theta }}(\mathbf{Y})-{\dot{l}}_{\varphi }(\mathbf{Y};\mathit{\theta },\varphi )({\overrightarrow{\mathit{\eta }}}_{*})\}}^T].$$

The definition of η⃗_* given by (10) leads to a least-squares estimator of η⃗_* based on the method of Huang et al. (2008). Specifically, with a random sample Y₁, …, Y_n and the consistent estimators θ̂_n and ϕ̂_n, we can estimate the least favorable direction η⃗_* corresponding to θ⁽⁰⁾ by the minimizer ${\widehat{\overrightarrow{\mathit{\eta }}}}_n$ of

$${\rho }_n(\overrightarrow{\mathit{\eta }})\equiv n^{-1}\sum _{i=1}^n{\Vert {\dot{l}}_{\mathit{\theta }}({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)-{\dot{l}}_{\varphi }({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)(\overrightarrow{\mathit{\eta }})\Vert }^2$$ (11)

over 𝒩^J2d. Then a natural estimator of I(θ⁽⁰⁾) is

$${Î}_n\equiv n^{-1}\sum _{i=1}^n\{{\dot{l}}_{\mathit{\theta }}({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)-{\dot{l}}_{\varphi }({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)({\widehat{\overrightarrow{\mathit{\eta }}}}_n)\}{\{{\dot{l}}_{\mathit{\theta }}({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)-{\dot{l}}_{\varphi }({\mathbf{Y}}_i;{\hat{\mathit{\theta }}}_n,{\hat{\varphi }}_n)({\widehat{\overrightarrow{\mathit{\eta }}}}_n)\}}^T.$$

In practice, one can compute the component of ${\widehat{\overrightarrow{\mathit{\eta }}}}_n$ corresponding to ${\mathit{\eta }}_{*}^{(lj)}(l=1,\dots ,J;j=1,\dots ,d)$ using the ordinary least-squares regression with its feasible region 𝒩^J approximated by the space ${\prod }_{k=1}^J{𝒩}_{kn}$ , where 𝒩_kn is the linear span of the B-spline basis functions ℬ_kn (k = 1, …, J).

6. NUMERICAL EXPERIMENT

In this section, we present the results of a numerical investigation on the finite sample performance of our estimates of regression parameters and baseline subdistribution hazards in Model (1). In the numerical experiment, the underlying overall survival time and the observed failure cause were generated from Model (1) with J = 2, d = 3, Z₁ ~ Bin(1, 0.5), Z₂ ~ Unif(0, 5) and Z₃ ~ min{Γ(3, 1), 5}, the true regression parameters were (θ₁₁, θ₁₂, θ₁₃, θ₂₁, θ₂₂, θ₂₃) = (0.15, −0.10, −0.20, 0.39, −0.08, −0.13), and the true baseline cumulative incidence functions were F₀₁(t) = 1−exp[−0.15{1−exp(−0.25t)}/0.25] and F₀₂(t) = 1 − exp[−0.175{1 − exp(−0.5t)}/0.5], which are the Gompertz models considered in Jeong & Fine (2006). There were five successive inspections for each subject, and the inspection times were uniformly distributed from 0.8 to 1.2, 1.8 to 2.2, ⋯, and 4.8 to 5.2, respectively.

We computed the sieve maximum likelihood estimates using the cubic B-splines and estimated the variances of the regression parameter estimates using the least-squares method based on the same cubic B-splines as in the parameter estimation. For the B-splines for estimating ϕ₁₀(t) and ϕ₂₀(t), the knots were equally spaced, and the numbers of knots over the observation window (0.8, 5.2) were respectively chosen to be K_1n = ⌊n^1/3⌋ and K_2n = ⌊n^1/5⌋, where ⌊x⌋ denotes the largest integer not exceeding x and n is the sample size. For each simulated data set, we performed a multivariate analysis, which models the two competing risks jointly, and two univariate analyses, each of which models one competing risk based on the reduced interval censored data corresponding to it. We conducted the Monte Carlo simulation study with 1000 repetitions for sample size n = 150, 450 and 900 representing small, moderate and large sample size respectively.

The simulation results of the estimates of regression parameters are in Table 1, where Bias(θ̂) and Var(θ̂) are the empirical bias and variance of the parameter estimator respectively, E(vâr) is the average of the 1000 estimated asymptotic variances of the parameter estimator, MSE(θ̂) is the empirical mean squared error of the parameter estimator, and Pr(θ ∈ CI) is the proportion of the 1000 95% asymptotic confidence intervals that cover the true parameter. The estimation biases are negligible compared to the empirical standard errors, indicating that the asymptotic bias of θ̂ is on the order of $o(1/\sqrt{n})$ , which is consistent with the large sample theory. When the sample size is small, the estimated asympotic variances are larger than the empirical variances in both the univariate analysis and the multivariate analysis, leading to higher empirical coverage probabilities of the asymptotic confidence intervals. This variance overestimation lessens as sample size increases, as indicated by the coverage probabilities approaching their nominal level. In the univariate analysis, the overestimation is not worrisome when the sample size is 450 and does not exist for sample size 900. In the multivariate analysis, it is not worrisome when sample size is 900. Comparing the variance and MSE of the two types of analyses, the multivariate analysis is more efficient when the sample size is large. This is expected because the additional data in the multivariate analysis contain information about the cumulative incidence function of the competing risk estimated in the univariate analysis. However, since the multivariate analysis maximizes a likelihood in much higher dimension compared to the univariate analysis, its estimation variances could be bigger when the sample size is not large, as shown in the portions of Table 1 for n = 150 and 450. All the above findings suggest that compared to the univariate analysis, the multivariate one needs a larger sample size to perform well because of the higher dimension.

Table 1.

Simulation results of the estimates of regression parameters

		Univariate Analysis					Multivariate Analysis
n	θ	Bias(θ̂)	Var(θ̂) (×102)	E(vâr) (×102)	MSE(θ̂) (×102)	Pr(θ ∈ CI)	Bias(θ̂)	Var(θ̂) (×102)	E(vâr) (×102)	MSE(θ̂) (×102)	Pr(θ ∈ CI)
150	0.15	−0.001	17.105	20.538	17.088	0.973	−0.026	17.878	22.484	17.930	0.976
	−0.10	−0.006	1.761	2.507	1.763	0.977	−0.004	1.803	2.717	1.803	0.977
	−0.20	−0.019	2.171	3.162	2.204	0.977	−0.009	2.170	3.373	2.176	0.983
	0.39	0.005	15.831	17.702	15.817	0.967	−0.001	13.820	17.828	13.806	0.977
	−0.08	−0.002	1.707	2.072	1.706	0.975	−0.005	1.745	2.124	1.746	0.974
	−0.13	−0.014	2.037	2.494	2.054	0.961	−0.011	2.017	2.529	2.027	0.969
450	0.15	−0.017	4.799	5.323	4.823	0.960	0.005	4.497	5.350	4.495	0.968
	−0.10	−0.010	0.624	0.645	0.634	0.945	−0.006	0.506	0.650	0.509	0.978
	−0.20	−0.023	0.744	0.793	0.798	0.962	−0.012	0.611	0.798	0.624	0.971
	0.39	0.011	5.001	4.904	5.008	0.954	−0.017	3.982	4.952	4.006	0.968
	−0.08	−0.002	0.534	0.576	0.534	0.965	−0.009	0.560	0.581	0.568	0.958
	−0.13	−0.002	0.577	0.682	0.577	0.968	−0.010	0.636	0.685	0.646	0.961
900	0.15	−0.008	2.382	2.531	2.385	0.966	−0.029	1.885	2.467	1.965	0.969
	−0.10	−0.008	0.315	0.306	0.321	0.950	0.002	0.244	0.299	0.244	0.965
	−0.20	−0.012	0.410	0.373	0.423	0.947	0.019	0.285	0.364	0.319	0.957
	0.39	0.0005	2.217	2.335	2.214	0.961	−0.061	2.114	2.298	2.489	0.947
	−0.08	0.001	0.282	0.274	0.281	0.942	0.009	0.241	0.269	0.248	0.954
	−0.13	−0.003	0.322	0.323	0.323	0.957	0.012	0.260	0.320	0.273	0.959

The simulation results of the estimates of baseline cumulative subdistribution hazards are shown in Figures 1 and 2. Overall, the bias of the spline-based estimator is small. The estimator for the cumulative hazard of Cause 2 has a little bit smaller bias than the estimator for Cause 1. This is expected because the estimation assumed that the cumulative hazard of Cause 2 is one order more differentiable than that of Cause 1, yielding estimators with different convergence rates.

Figure 1.

Averaged estimates of the baseline cumulative subdistribution hazard functions for Cause 1 over the 1000 Monte Carlo samples.

Figure 2.

Averaged estimates of the baseline cumulative subdistribution hazard functions for Cause 2 over the 1000 Monte Carlo samples.

7. A TIME-TO-DEMENTIA DATA ANALYSIS

7.1 Backgrounds and data

Section 7 presents an application of the foregoing methods to the data from an ongoing cohort study of dementia: the Memory and Aging Project (MAP) (Bennett et al., 2012). The MAP study recruits older individuals without dementia who agree to receive clinical and psychological evaluation each year and to donate their brain for postmortem examination. The study began in 1997 and currently includes more than 1,400 participants from about 40 retirement communities and senior housing facilities in the Chicago metropolitan area. In the study, annual visits with diagnoses of mild cognitive impairment (MCI), Alzheimer’s disease (AD), and other types of dementia are scheduled for every participant. We used the MAP data to investigate the effects of years of education (categorized into two levels: ≤ 12 years and > 12 years), gender, and the presence of the apolipoprotein E ε4 allele (ApoE4) on time from baseline study visit to incident dementia (AD or other dementia). Two other covariates, baseline age and the presence of baseline MCI, were also included in the analysis to further explain inter-individual differences in time to dementia. In the MAP study, the time to incident dementia is known up to between two consecutive study visits, and death may occur before dementia in older people. This yields interval censored time-to-event data with dementia and death as two competing risks.

The data set we used from the MAP study was frozen in 2010. We excluded from that data set the participants who had only baseline visit and were not known to have died as well as the ones who have missing covariates. The resulting sample for the analysis has 956 subjects. A summary of characteristics of the sample is presented in Table 2.

Table 2.

Characteristics of the sample for the analysis

Characteristic	Analysis
Gender—no. (%)
Male	255 (26.7%)
Female	701 (73.3%)
Years of education—no. (%)
>12 (attended college)	651 (68.1%)
≤12 (not attended college)	305 (31.9%)
ApoE4 status—no. (%)
Carrier	214 (22.4%)
Non-carrier	742 (77.6%)
Age at baseline (yrs)—mean (sd)	80.28 (7.16)
MCI at baseline—no. (%)
MCI	267 (27.9%)
No MCI	689 (72.1%)
No. of study visits—mean (sd)	5.7 (2.4)
Years of follow-up—mean (sd)	4.8 (2.5)
Outcome event—no. (%)
Dementia	183 (19.1%)
Death before dementia	181 (18.9%)

7.2 Univariate analysis for cumulative incidence of dementia

In this subsection, we focus on estimating the cumulative incidence function for dementia and thus model the observed data related only to dementia, essentially treating deaths prior to dementia as right censored observations. The fact that death terminates the follow-up makes mixed case interval censoring model not directly applicable to the inspection process of the MAP study. However, according to the study design, those deceased subjects in our data set would have dementia examinations approximately one year apart until 2010 had they not died ahead of that year. Therefore, it is reasonable to treat the subjects who deceased before developing dementia as being right censored at 2010, and the resulting data are the time-to-dementia data under mixed case interval censoring, for which the likelihood is of form (2) with J equal to one. We used the Fine–Gray model to model the covariates’ effects on the cumulative incidence function of dementia. The sieve maximum likelihood estimation was performed using the cubic B-spline. The estimates of the regression parameters and their asymptotic standard errors are given in Table 3 along with the p-values for significance. All the covariates except gender have significant effects on the cumulative incidence for dementia at 0.05 level. Specifically, MCI at baseline, being older at baseline and ApoE4 allele all increase the cumulative risk for dementia, while having college education decreases the cumulative risk. Figure 3, which shows the estimated cumulative incidence functions for women with different values of the significant categorical covariates, reflects the magnitude of those covariates’ effects on the cumulative risk of dementia.

Table 3.

Inference for regression parameters in the univariate analysis of cumulative incidence of dementia

Covariate	Parameter Estimate	Standard Error	p-value
MCI at baseline (yes)	1.406	0.168	< 0.001
Baseline age (yrs)	0.039	0.013	0.003
College education (yes)	−0.318	0.158	0.044
Gender (Male)	0.015	0.158	0.926
ApoE4 (carrier)	0.521	0.163	0.001

Figure 3.

Cumulative incidence functions of dementia estimated from the univariate analysis for women with baseline age 80.28 years, the mean baseline age of the sample. Plots (a) and (b) show the cumulative incidence functions for no MCI at baseline and having MCI at baseline respectively. Solid lines are for no college education and no ApoE4 allele; Dashed lines are for no college education and having ApoE4 allele; Dotted lines are for having college education and no ApoE4 allele; Dotdash lines are for having college education and ApoE4 allele.

7.3 Multivariate analysis for cumulative incidences of dementia and deaths

In this subsection, we estimate the cumulative incidence functions of dementia and death jointly. Keep in mind that death considered here is death prior to dementia. As mentioned above, the deceased subjects in our data set would have dementia examinations approximately one year apart until 2010 had they not died ahead of that year. Thus, it is reasonable to treat the subjects who deceased before developing dementia as being interval censored between the last visit before death and the first visit after death, which did not happen but was scheduled. This scheduled visit is m years apart from the last visit before death where m is a positive integer and most often equal to one. The resulting data are interval censored time-to-event data of the competing risks, dementia and death, for which the likelihood is of the form (2) with J = 2. It loses statistical efficiency to treat death as being interval censored rather than use its exact time. This interval censoring treatment was performed in order to use the proposed method to estimate the cumulative incidences of dementia and death jointly. Since the time intervals bracketing death are mostly one year in length in the analytic sample (138 out of 181 death bracketing intervals), the efficiency loss is expected to be small. Annually scheduled dementia examinations also decrease the chance that dementia incidence was not caught in subjects who died after developing dementia. The analysis results of regression parameters are given in Table 4. We can see that the results of the covariate effects on cumulative incidence of dementia are similar to the univariate analysis except that the effect of college education became non-significant. For the cumulative incidence of death, baseline age has a significant increasing effect, men have significantly higher cumulative risk than women, and the other covariates are not significant. Figure 4 is the counterpart of Figure 3 obtained from the multivariate analysis. The two figures look similar. Figure 5 shows the estimated cumulative incidence functions of death for people who attended college and whose baseline age is 80.28 years.

Table 4.

Inference for regression parameters in the multivariate analysis for cumulative incidences of dementia and death

Cause	Covariate	Parameter Estimate	Standard Error	p-value
Dementia	MCI at baseline (yes)	1.341	0.160	< 0.001
	Baseline age (yrs)	0.063	0.014	< 0.001
	College education (yes)	−0.207	0.156	0.182
	Gender (Male)	0.048	0.162	0.786
	ApoE4 (carrier)	0.496	0.162	0.002
Death	MCI at baseline (yes)	−0.187	0.172	0.278
	Baseline age (yrs)	0.062	0.013	< 0.001
	College education (yes)	0.058	0.161	0.716
	Gender (Male)	0.418	0.162	0.010
	ApoE4 (carrier)	−0.280	0.188	0.137

Figure 4.

Cumulative incidence functions of dementia estimated from the multivariate analysis for women with baseline age 80.28 years, the mean baseline age of the sample. Plots (a) and (b) show the cumulative incidence functions for no MCI at baseline and having MCI at baseline respectively. Solid lines are for no college education and no ApoE4 allele; Dashed lines are for no college education and having ApoE4 allele; Dotted lines are for having college education and no ApoE4 allele; Dotdash lines are for having college education and ApoE4 allele.

Figure 5.

Cumulative incidence functions of death estimated from the multivariate analysis for people with college education and baseline age 80.28 years, the mean baseline age of the sample. Plots (a) and (b) show the cumulative incidence functions for female and male respectively. Solid lines are for no MCI at baseline and no ApoE4 allele; Dashed lines are for no MCI at baseline and having ApoE4 allele; Dotted lines are for having MCI at baseline and no ApoE4 allele; Dotdash lines are for having MCI at baseline and ApoE4 allele.

8. DISCUSSION

In this article, we proposed a spline-based sieve maximum likelihood estimation for the Fine–Gray model to analyze interval-censored competing risks data. We showed that, under certain regularity conditions, the sieve estimator of the baseline cumulative subdistribution hazard is consistent and converges with the optimal rate in the nonparametric regression setting, and the regression parameter estimator is asymptotically normal and semiparametrically efficient. The efficient information matrix for the regression parameter can be consistently estimated by a least-squares method.

One could also use the nonparametric maximum likelihood estimator (NPMLE) for the baseline cumulative subdistribution hazard in the Fine–Gray model under interval censored competing risks data. The NPMLE approach assumes that the baseline cumulative subdistribution hazard jumps only at the effective observation times of (U_i, V_i) (i = 1, …, n) as defined in Definition 1.1, page 45, of Groeneboom & Wellner (1992). In the univariate analysis, this way is equivalent to the method of Huang & Wellner (1995). But we recommend the spline-based estimator over the NPMLE for the following reasons. First, the dimension of spline-based estimator is actually smaller than that of the NPMLE, making the computation simpler. The former is of order ${\sum }_{k=1}^J{n}^{{\nu }_k}({\nu }_k<1/(2p_k))$ , while the latter is on the order of n. Second, if the number of knots is selected properly, e.g. ν_k = 1/(1 + 2p_k), the spline-based estimator will converge at a rate, which is n^p_k/(1+2p_k) when ν_k = 1/(1 + 2p_k), no less than that of the NPMLE, which is always n^1/3. Third, a smooth cumulative hazard estimator is more realistic than an estimator that is a step function.

Though the joint analysis of multiple competing risks improves statistical efficiency when sample size is large, it is more vulnerable to model mis-specification because it requires that the Fine–Gray model holds for every competing risk under consideration. If a competing risk is of much more interest compared to the others, the univariate analysis of that specific competing risk is preferred to the joint analysis since it would be more robust. Take the MAP study for example, the investigators are more interested in dementia incidence in alive people compared to death prior to dementia. Thus the univariate analysis in Section 7.2 should be used to report study findings. As a result, we would conclude that college education has a decreasing effect on the cumulative risk of dementia, which was not reflected by the joint analysis in Section 7.3.

It should not take much effort to generalize the developed inferential approach to the class of semiparametric transformation models (Fine, 2001) for the analysis of interval censored competing risks data, since the Fine–Gray model is a member of that class. Motivated by the late entry of the MAP participants to the study for the analysis of age to dementia, extending our method to accommodate left truncation is warranted. Also, the fact that time to some competing risk such as death was exactly known in the MAP study as well as many other longitudinal studies makes it worthwhile to extend our method to partly interval censored competing risks data. Computationally, these two extensions should be straightforward, since one just needs to respectively divide the likelihood (2) by the overall survival function at the entry time and replace F_k(V_i; Z_i) − F_k(U_i; Z_i) in (2) by the corresponding subdensity at the exactly observed occurrence time, and the resulting likelihoods can be maximized using the same model for F_k(t; Z) as (1) and the same sieve method as in Section 3. The derivation of the relevant asymptotic theory is not that straightforward and needs further investigation.

APPENDIX

PROOFS

The proofs of Theorems 1 and 2 followed similar arguments in Zhang et al. (2010). Throughout the following proofs, P_ζ f = ∫ f(y)dP_{θ, ϕ}(y), ${\mathbb{P}}_{n}f=n^{-1}{\sum }_{i=1}^{n}f({\mathbf{Y}}_i)$ , ⌊x⌋ denotes the largest integer below x, and we let C represent a generic constant that may vary from place to place.

Proof of Theorem 1

We first prove the consistency of ζ̂_n in the metric d. This can be accomplished by verifying the conditions of Theorem 5.7 in van der Vaart (1998).

Let 𝕄(ζ) = P_ζ0 l(Y; ζ) = P_ζ0 l(Y; θ, ϕ) and 𝕄_n(ζ) = ℙ_nl(Y; ζ) = ℙ_nl(Y; θ, ϕ). Therefore, for any $\mathit{\zeta }\in {\mathit{𝒯}}_n=\mathbf{\Theta }\times {\prod }_{k=1}^J{ℳ}_{kn}$ , 𝕄_n(ζ) − 𝕄 (ζ) = (ℙ_n − P_ζ0)l(Y; ζ).

Let ℒ₁ = {l(Y; ζ) : ζ ∈ 𝒯_n}. By the calculation of Shen & Wong (1994, pp. 597), for all ε > 0, there exist J sets of brackets $\{[{\varphi }_{ki}^L,{\varphi }_{ki}^U]:i=1,2,\dots ,\lfloor {(1/\epsilon )}^{C{q}_{kn}}\rfloor \}$ (k = 1, …, J) such that for any $\varphi =({\varphi }_{10}(·),\dots ,{\varphi }_{J0}(·))\in {\prod }_{k=1}^J{ℳ}_{kn}$ , one has ${\varphi }_{k{i}_k}^L(t)\le {\varphi }_{k0}(t)\le {\varphi }_{k{i}_k}^U(t)$ (k = 1, …, J) for some 1 ≤ i_k ≤ ⌊(1/ε)^Cq_kn⌋ and all t ∈ [τ₀, τ₁], and ${\mathbb{P}}_n|{\varphi }_{ki}^U(Y)-{\varphi }_{ki}^L(Y)|\le \epsilon$ with Y = U and V for all 1 ≤ i ≤ ⌊(1/ε)^Cq_kn⌋ and 1 ≤ k ≤ J. As Θ ⊂ ℝ^Jd is compact, Θ can be covered by ⌊C(1/ε)^Jd⌋ balls with radius ε and centers being denoted by θ^(s) (s = 1, …, ⌊C(1/ε)^Jd⌋) respectively; that is for any θ ∈ Θ, there exists a 1 ≤ s ≤ ⌊C(1/ε)^Jd⌋ such that ‖θ − θ^(s)‖ ≤ ε and hence $|{\mathbf{Z}}^T{\mathit{\theta }}_k-{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}|\le C\epsilon$ for all Z and 1 ≤ k ≤ J because of A1. This implies that ${\mathbf{Z}}^T{\mathit{\theta }}_k\in [{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}-C\epsilon ,{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}+C\epsilon ]$ for all Z and 1 ≤ k ≤ J. Hence we can easily construct such a set of brackets $\{[l_{s,i_1,\dots ,i_J}^L(\mathbf{Y}),l_{s,i_1,\dots ,i_J}^U(\mathbf{Y})]:s=1,2,\dots ,\lfloor C{(1/\epsilon )}^{Jd}\rfloor ;i_k=1,\dots ,\lfloor {(1/\epsilon )}^{C{q}_{kn}}\rfloor ,k=1,\dots ,J\}$ that for any l(Y; ζ) ∈ ℒ₁, there exist a 1 ≤ s ≤ ⌊C(1/ε)^Jd⌋ and a 1 ≤ i_k ≤ ⌊(1/ε)^Cq_kn⌋ (k = 1, …, J) such that $l(\mathbf{Y};\mathit{\zeta })\in [l_{s,i_1,\dots ,i_J}^L(\mathbf{Y}),l_{s,i_1,\dots ,i_J}^U(\mathbf{Y})]$ for any sample point Y, where

$$l_{s,i_1,\dots ,i_J}^L\left(\mathbf{Y}\right)=\sum _{k=1}^J\left[{\mathrm{\Delta }}_{k1}\text{log}\right[1-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{\left(s\right)}-C\epsilon +{\varphi }_{k{i}_k}^L(U)}\}]+{\mathrm{\Delta }}_{k2}\text{log}[\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{\left(s\right)}+C\epsilon +{\varphi }_{k{i}_k}^U(U)}\}-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{\left(s\right)}-C\epsilon +{\varphi }_{k{i}_k}^L(V)}\}\left]\right]+{\mathrm{\Delta }}_3\text{log}[1-J+\sum _{k=1}^J\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{\left(s\right)}+C\epsilon +{\varphi }_{k{i}_k}^U(V)}\left\}\right]$$

and

$$l_{s,i_1,\dots ,i_J}^U(\mathbf{Y})=\sum _{k=1}^J\left[{\mathrm{\Delta }}_{k1}\text{log}\right[1-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}+C\epsilon +{\varphi }_{k{i}_k}^U(U)}\}]+{\mathrm{\Delta }}_{k2}\text{log}[\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}-C\epsilon +{\varphi }_{k{i}_k}^L(U)}\}-\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}+C\epsilon +{\varphi }_{k{i}_k}^U(V)}\}\left]\right]+{\mathrm{\Delta }}_3\text{log}[1-J+\sum _{k=1}^J\text{exp}\{-e^{{\mathbf{Z}}^T{\mathit{\theta }}_k^{(s)}-C\epsilon +{\varphi }_{k{i}_k}^L(V)}\left\}\right]$$

Using Taylor expansion along with assumptions A1–A3, we can easily demonstrate that ${\mathbb{P}}_n|l_{s,i_1,\dots ,i_J}^U(\mathbf{Y})-l_{s,i_1,\dots ,i_J}^L(\mathbf{Y})|\le C\epsilon$ for all 1 ≤ s ≤ ⌊C(1/ε)^Jd⌋ and 1 ≤ i_k ≤ ⌊(1/ε)^Cq_kn⌋ (k = 1, …, J), which leads to the conclusion that the ε-bracketing number for ℒ₁ with L₁(ℙ_n)-norm is bounded by $C{(1/\epsilon )}^{C{\sum }_{k=1}^J{q}_{kn}+Jd}$ . As N(ε, ℒ₁, L₁(ℙ_n)) ≤ N_[](ε, ℒ₁, L₁(ℙ_n)), ℒ₁ is Glivenko-Cantelli by van der Vaart & Wellner (1996, Theorem 2.4.3). Therefore, sup_ζ∈𝒯n |𝕄_n(ζ) − 𝕄(ζ)| →a.s. 0. Let g_k(Z, t) = exp{Z^T θ_k + ϕ_k0(t)} and $g_{k0}(\mathbf{Z},t)=\text{exp}({\mathbf{Z}}^T{\mathit{\theta }}_k^{(0)}+{\varphi }_{k0}^{(0)}(t))$ (k = 1, …, J). Some algebra yields that

$$𝕄({\mathit{\zeta }}_0)-𝕄(\mathit{\zeta })=E\left[\sum _{k=1}^J\right[[1-\text{exp}\{-g_{k0}(\mathit{Z},U)\}]\text{log}\frac{1-\text{exp}\{-g_{k0}(\mathit{Z},U)\}}{1-\text{exp}\{-g_k(\mathit{Z},U)\}}+[\text{exp}\{-g_{k0}(\mathit{Z},U)\}-\text{exp}\{-g_{k0}(\mathbf{Z},V)\}]\text{log}\frac{\text{exp}\{-g_{k0}(\mathbf{Z},U)\}-\text{exp}\{-g_{k0}(\mathbf{Z},V)\}}{\text{exp}\{-g_k(\mathbf{Z},U)\}-\text{exp}\{-g_k(\mathbf{Z},V)\}}]+[1-J+\sum _{k=1}^J\text{exp}\{-g_{k0}(\mathbf{Z},V)\}\left]\text{log}\frac{1-J+{\sum }_{k=1}^J\text{exp}\{-g_{k0}(\mathbf{Z},V)\}}{1-J+{\sum }_{k=1}^J\text{exp}\{-g_k(\mathbf{Z},V)\}}\right]=E\left[\sum _{k=1}^J\right[[1-\text{exp}\{-g_k(\mathbf{Z},U)\}]w\left[\frac{1-\text{exp}\{-g_{k0}(\mathbf{Z},U)\}}{1-\text{exp}\{-g_k(\mathbf{Z},U)\}}\right]+[\text{exp}\{-g_k(\mathbf{Z},U)\}-\text{exp}\{-g_k(\mathbf{Z},V)\}]w\left[\frac{\text{exp}\{-g_{k0}(\mathbf{Z},U)\}-\text{exp}\{-g_{k0}(\mathbf{Z},V)\}}{\text{exp}\{-g_k(\mathbf{Z},U)\}-\text{exp}\{-g_k(\mathbf{Z},V)\}}\right]]+[1-J+\sum _{k=1}^J\text{exp}\{-g_k(\mathbf{Z},V)\}]w[\frac{1-J+{\sum }_{k=1}^J\text{exp}\{-g_{k0}(\mathbf{Z},V)\}}{1-J+{\sum }_{k=1}^J\text{exp}\{-g_k(\mathbf{Z},V)\}}]],$$

where w(x) = xlogx − x + 1 ≥ (x − 1)²/4 for 0 ≤ x ≤ 5. Further analysis by using Taylor expansion leads to

$$𝕄({\mathit{\zeta }}_0)-𝕄(\mathit{\zeta })\ge \frac{1}{4}E[\sum _{k=1}^J\frac{1}{1-\text{exp}\{-g_k(\mathbf{Z},U)\}}{[\text{exp}\{-g_k(Z,U)\}-\text{exp}\{-g_{k0}(\mathbf{Z},U)\}]}^2+\frac{1}{1-J+{\sum }_{k=1}^J\text{exp}\{-g_k(\mathbf{Z},V)\}}{[\sum _{k=1}^J[\text{exp}\{-g_{k0}(\mathbf{Z},V)\}-\text{exp}\{-g_k(\mathbf{Z},V)\}]]}^2]=\frac{1}{4}E[\sum _{k=1}^J\frac{1}{1-\text{exp}\{-g_k(\mathbf{Z},U)\}}{\gamma }^2[\xi \{{\mathit{\theta }}_k^{(0)},{\mathit{\theta }}_k,\mathbf{Z},{\varphi }_{k0}^{(0)}(U),{\varphi }_{k0}(U)\}]{\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(U)\}}^2+\frac{{[{\sum }_{k=1}^J\gamma [\xi \{{\mathit{\theta }}_k^{(0)},{\mathit{\theta }}_k,\mathbf{Z},{\varphi }_{k0}^{(0)}(V),{\varphi }_{k0}(V)\}]\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(V)\}]}^2}{1-J+{\sum }_{k=1}^J\text{exp}\{-g_k(\mathbf{Z},V)\}}],$$

where γ(x) = d{exp(−e^x)}/dx = −exp(−e^x + x), $\xi \{{\mathit{\theta }}_k^{(0)},{\mathit{\theta }}_k,\mathbf{Z},{\varphi }_{k0}^{(0)}(U),{\varphi }_{k0}(U)\}$ is some number between ${\mathbf{Z}}^T{\mathit{\theta }}_k^{(0)}+{\varphi }_{k0}^{(0)}(U)$ and Z^T θ_k + ϕ_k0(U), and $\xi \{{\mathit{\theta }}_k^{(0)},{\mathit{\theta }}_k,\mathbf{Z},{\varphi }_{k0}^{(0)}(V),{\varphi }_{k0}(V)\}$ is some number between ${\mathbf{Z}}^T{\mathit{\theta }}_k^{(0)}+{\varphi }_{k0}^{(0)}(V)$ and Z^T θ_k + ϕ_k0(V). Let

$$\mathbb{Z}={\left(\begin{array}{ccc}\hfill {\mathbf{Z}}^T\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\mathbf{Z}}^T\hfill \end{array}\right)}_{J\times Jd},\mathrm{\Gamma }=\left(\begin{array}{c}\hfill \gamma [\xi \{{\mathit{\theta }}_1^{(0)},{\mathit{\theta }}_1,\mathbf{Z},{\varphi }_{10}^{(0)}(V),{\varphi }_{10}(V)\}]\hfill \\ \hfill ⋮\hfill \\ \hfill \gamma [\xi \{{\mathit{\theta }}_J^{(0)},{\mathit{\theta }}_J,\mathbf{Z},{\varphi }_{J0}^{(0)}(V),{\varphi }_{J0}(V)\}]\hfill \end{array}\right)$$

and λ_min = the minimum eigenvalue of ΓΓ^T. Further analysis using assumptions A1–A3 leads to

$$𝕄({\mathit{\zeta }}_0)-𝕄(\mathit{\zeta })\ge CE[\sum _{k=1}^J{\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(U)\}}^2+{\{\mathbb{Z}({\mathit{\theta }}^{(0)}-\mathit{\theta })+({\varphi }_0-\varphi )(V)\}}^T\mathrm{\Gamma }{\mathrm{\Gamma }}^T\{\mathbb{Z}({\mathit{\theta }}^{(0)}-\mathit{\theta })+({\varphi }_0-\varphi )(V)\}]\ge CE[\sum _{k=1}^J{\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(U)\}}^2+{\lambda }_{\text{min}}{\{\mathbb{Z}({\mathit{\theta }}^{(0)}-\mathit{\theta })+({\varphi }_0-\varphi )(V)\}}^T\{\mathbb{Z}({\mathit{\theta }}^{(0)}-\mathit{\theta })+({\varphi }_0-\varphi )(V)\}]\ge CE[\sum _{k=1}^J{\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(U)\}}^2+\sum _{k=1}^J{\{{\mathbf{Z}}^T({\mathit{\theta }}_k^{(0)}-{\mathit{\theta }}_k)+({\varphi }_{k0}^{(0)}-{\varphi }_{k0})(V)\}}^2].$$

With assumptions A1–A4 and A6, using the same arguments as those in Wellner & Zhang (2007, pp. 2126–2127) leads to

$$𝕄({\mathit{\zeta }}_0)-𝕄(\mathit{\zeta })\ge C({\Vert {\mathit{\theta }}^{(0)-}\mathit{\theta }\Vert }^2+{\Vert {\varphi }_0-\varphi \Vert }_{\mathbf{\Phi }}^2)=C{d}^2({\mathit{\zeta }}_0,\mathit{\zeta }).$$

Then it implies that sup_{ζ:d(ζ,ζ0)≥ε} 𝕄(ζ) ≤ 𝕄(ζ₀) − Cε² < 𝕄(ζ₀).

For ϕ₀ ∈ Φ, Lu (2007) has shown that there exists a ϕ_k0,n ∈ ℳ_kn of order m_k ≥ p_k + 2 such that ${\Vert {\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}\Vert }_{\infty }\le C{q}_{kn}^{-p_k}=O(n^{-p_k{\nu }_k})$ (k = 1, …, J). This also implies that ${\Vert {\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}\Vert }_{{\mathrm{\Phi }}_k}\le C{q}_{kn}^{-p_k}=O(n^{-p_k{\nu }_k})$ (k = 1, …, J). Now let ϕ_0,n = (ϕ_10,n(·), …, ϕ_J0,n(·)) and ζ_0,n = (θ⁽⁰⁾, ϕ_0,n), we have

$${𝕄}_n({\hat{\mathit{\zeta }}}_n)-{𝕄}_n({\mathit{\zeta }}_0)={𝕄}_n({\hat{\mathit{\zeta }}}_n)-{𝕄}_n({\mathit{\zeta }}_{0,n})+{𝕄}_n({\mathit{\zeta }}_{0,n})-{𝕄}_n({\mathit{\zeta }}_0)\ge {\mathbb{P}}_{n}l(\mathbf{Y};{\mathit{\zeta }}_{0,n})-{\mathbb{P}}_{n}l(\mathbf{Y};{\mathit{\zeta }}_0)=({\mathbb{P}}_n-P)\{l(\mathbf{Y};{\mathit{\zeta }}_{0,n})-l(\mathbf{Y};{\mathit{\zeta }}_0)\}+𝕄({\mathit{\zeta }}_{0,n})-𝕄({\zeta }_0).$$

Using the brackets for ℳ_kn (k = 1, …, J) given before, we can similarily construct a set of brackets for the class ${ℒ}_2-\{l(\mathbf{Y};{\mathit{\theta }}^{(0)},\varphi )-l(\mathbf{Y};{\mathit{\theta }}^{(0)},{\varphi }_0):\varphi \in {\prod }_{k=1}^J{ℳ}_{kn}\text{and}{\Vert {\varphi }_{k0}-{\varphi }_{k0}^{(0)}\Vert }_{{\mathrm{\Phi }}_k}\le C{n}^{-p_k{\nu }_k}\text{for}k=1,\dots ,J\}$ with the ε-bracketing number associated with L₂(P)-norm bounded by ${(1/\epsilon )}^{C{\sum }_{k=1}^J{q}_{kn}}$ . This yields a finite-valued bracketing integral defined in van der Vaart (1998, p. 270). Hence the class ℒ₂ is P-Donsker. By the dominated convergence theorem, it is obvious that in this class P{l(Y; θ⁽⁰⁾, ϕ) − l(Y; θ⁽⁰⁾, ϕ₀)}² → 0 as n → ∞. Hence

$$({\mathbb{P}}_n-P)\{l(\mathbf{Y};{\mathit{\theta }}^{(0)},{\varphi }_{0,n})-l(\mathbf{Y};{\mathit{\theta }}^{(0)},{\varphi }_0)\}=o_p(n^{-1/2})$$

by the relationship between Donsker and asymptotic equicontinuity given by van der Vaart & Wellner (1996, Corollary 2.3.12). By the dominated convergence theorem again, it is easy to see that 𝕄(ζ_0,n) − 𝕄(ζ₀) > −o(1) as n → ∞. Therefore,

$${𝕄}_n({\hat{\mathit{\zeta }}}_n)-{𝕄}_n({\mathit{\zeta }}_0)\ge o_p(n^{-1/2})-o(1)=-o_p(1).$$

This completes the proof of d(ζ̂_n, ζ₀) → 0 in probability.

Next, we verify the conditions of van der Vaart & Wellner (1996, Theorem 3.2.5) to derive the convergence rate. First, we have already shown in the proof of consistency that 𝕄(ζ₀) − 𝕄(ζ) ≥ Cd²(ζ₀, ζ).

Second, we further explore 𝕄_n(ζ̂_n) − 𝕄_n(ζ₀). In the proof of consistency, we know that 𝕄_n(ζ̂_n) − 𝕄_n(ζ₀) ≥ I_1,n + I_2,n, where I_1,n = (ℙ_n − P){l(Y; θ⁽⁰⁾, ϕ_0,n) − l(Y; θ⁽⁰⁾, ϕ₀)} and I_2,n = P{l(Y; θ⁽⁰⁾, ϕ_0,n) − l(Y; θ⁽⁰⁾, ϕ₀)}. By Taylor expansion, we have, for any 0 < ε < 1/2 − min_1≤k≤J{p_kν_k},

$$I_{1,n}=({\mathbb{P}}_n-P)\{\sum _{k=1}^J{\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};{\mathit{\theta }}^{(0)},\tilde{\varphi })({\varphi }_{k0,n}-{\varphi }_{k0}^{(0)})\}=\sum _{k=1}^J{n}^{-p_k{\nu }_k+\epsilon }({\mathbb{P}}_n-P)\{{\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};{\mathit{\theta }}^{(0)},\tilde{\varphi })\frac{{\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}}{n^{-p_k{\nu }_k+\epsilon }}\},$$

where ϕ̃ = (ϕ̃₁₀, …, ϕ̃_J0) and ${\tilde{\varphi }}_{k0}={\varphi }_{k0}^{(0)}+s^{*}({\varphi }_{k0,n}-{\varphi }_{k0}^{(0)})$ for some s* ∈ (0, 1) and all 1 ≤ k ≤ J. Because ${\Vert {\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}\Vert }_{\infty }=O(n^{-p_k{\nu }_k})$ and ${\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};{\mathit{\theta }}^{(0)},\tilde{\varphi })$ is uniformly bounded as a result of assumptions A1–A4, we can easily obtain that $P{\{{\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};{\mathit{\theta }}^{(0)},\tilde{\varphi })\frac{{\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}}{n^{-p_k{\nu }_k+\epsilon }}\}}^2\to 0$ . As a result of ℒ₂ being Donsker, using van der Vaart & Wellner (1996, Corollary 2.3.12) again, we can conclude that $({\mathbb{P}}_n-P)\{{\dot{l}}_{\varphi }^{(k)}(\mathbf{Y};{\mathit{\theta }}^{(0)},\tilde{\varphi })\frac{{\varphi }_{k0,n}-{\varphi }_{k0}^{(0)}}{n^{-p_k{\nu }_k+\epsilon }}\}=o_p(n^{-1/2})$ (k = 1, …, J). Hence,

$$I_{1,n}=\sum _{k=1}^J{o}_p(n^{-p_k{\nu }_k+\epsilon }{n}^{-1/2})=o_p(n^{-2{\text{min}}_k\{p_k{\nu }_k\}}),$$

owing to the selection of ε. Using the fact that the function m(x) = xlog(x) − x + 1 ≤ (x − 1)² in the neighborhood of x = 1, it can be easily argued that $𝕄({\mathit{\zeta }}_0)-𝕄({\mathit{\zeta }}_{0,n})\le C{\sum }_{k=1}^J{\Vert {\varphi }_{k0,n}-{\varphi }_{k0}\Vert }_{{\mathrm{\Phi }}_k}^2=C{\sum }_{k=1}^{J}O(n^{-2p_k{\nu }_k})=CO(n^{-2{\text{min}}_k\{p_k{\nu }_k\}})$ , which implies that I_2,n = 𝕄(ζ_0,n) − 𝕄(ζ₀) ≥ −O(n^{−2 min_k{p_kν_k}}). Thus, we conclude that

$${𝕄}_n({\hat{\mathit{\zeta }}}_n)-{𝕄}_n({\mathit{\zeta }}_0)\ge -O_p(n^{-2{\text{min}}_k\{p_k{\nu }_k\}})=-O_p\left(n^{-2\text{min}\{{\text{min}}_k(p_k{\nu }_k),(1-{\text{max}}_k{\nu }_k)/2\}}\right).$$

.

Let ${ℒ}_3(\eta )=\{l(\mathbf{Y};\mathit{\zeta })-l(\mathbf{Y};{\mathit{\zeta }}_0):\varphi \in {\prod }_{k=1}^J{ℳ}_{kn}\text{and}d(\mathit{\zeta },{\mathit{\zeta }}_0)\le \eta \}$ . Using the same arguments as in the proof of consistency, we obtain that the logarithm of the ε-bracketing number of ℒ₃(η), logN_[]{ε, ℒ₃(η), L₂(P)}, is bounded by $C({\sum }_{k=1}^J{q}_{kn})\text{log}(\eta /\epsilon )$ . This leads to

$$J_{[]}\{\eta ,{ℒ}_3(\eta ),L_2(P)\}={\int }_0^{\eta }\sqrt{1+\text{log}{N}_{[]}\{\epsilon ,{ℒ}_3(\eta ),L_2(P)\}}d\epsilon \le C{(\sum _{k=1}^J{q}_{kn})}^{1/2}\eta .$$

Because assumptions A1–A3 guarantee the uniform boundedness of l(Y; ζ), using van der Vaart & Wellner (1996, Lemma 3.4.2), the key function ϕ_n(η) in van der Vaart & Wellner (1996, Theorem 3.2.5) is given by ${\varphi }_n(\eta )={({\sum }_{k=1}^J{q}_{kn})}^{1/2}\eta +{\sum }_{k=1}^J{q}_{kn}/\sqrt{n}$ . Note that

$$n^{2{\text{min}}_k\{p_k{\nu }_k\}}{\varphi }_n(1/n^{{\text{min}}_k\{p_k{\nu }_k\}})=n^{{\text{min}}_k\{p_k{\nu }_k\}}O(n^{{\text{max}}_k{\nu }_k/2})+n^{2{\text{min}}_k\{p_k{\nu }_k\}}{n}^{-1/2}O(n^{{\text{max}}_k{\nu }_k})\le C{n}^{1/2}\{n^{{\text{min}}_k(p_k{\nu }_k)-(1-{\text{max}}_k{\nu }_k)/2}+n^{2{\text{min}}_k(p_k{\nu }_k)-(1-{\text{max}}_k{\nu }_k)}\}.$$

Therefore, if min_k{p_kν_k} ≤ (1 − max_k ν_k)/2, n^{2 min_k{p_kν_k}} ϕ_n(1/n^{min_k{p_kν_k}}) ≤ n^1/2. This implies that if we choose r_n = n^{min{min_k(p_kν_k),(1−max_k ν_k)/2}}, it follows that $r_n^2{\varphi }_n(1/r_n)\le n^{1/2}$ and ${𝕄}_n({\hat{\mathit{\zeta }}}_n)-{𝕄}_n({\mathit{\zeta }}_0)\ge -O_p(r_n^{-2})$ . Hence, r_nd(ζ̂_n, ζ₀) = O_p(1).

Proof of Theorem 2

To derive the asymptotic normality for θ̂_n, we just need to verify the conditions (B1)–(B3) of the general theorem given in Appendix B of Zhang et al. (2010). For condition (B1), we only need to verify that ℙ_nl̇_ϕ(Y; θ̂_n, ϕ̂_n)(η*) = o_p(n^1/2) as ℙ_nl̇_θ(Y; θ̂_n, ϕ̂_n) ≡ 0. Because each component of η* has a bounded derivative, it is also a function with bounded variation. Then it can be easily shown using the argument in Billingsley (1986, pp. 435–436) that there exists a ${\mathit{\eta }}_n^{*}\in {({\prod }_{k=1}^J{𝒮}_{kn}(D_{kn},K_{kn},m_k))}^{\otimes Jd}$ such that ${\Vert {\eta }_{n,jk}^{*}-{\eta }_{jk}^{*}\Vert }_{{\mathrm{\Phi }}_k}=O(q_{kn}^{-1})=O(n^{-{\nu }_k})$ for j = 1, …, Jd and ${\mathbb{P}}_n{\dot{l}}_{\varphi }(\mathbf{Y};{\hat{\mathit{\zeta }}}_n)({\mathit{\eta }}_n^{*})=0$ . Therefore, we can write ℙ_nl̇_ϕ(Y; ζ̂_n)(η*) = I_3,n + I_4,n, where

$$I_{3,n}=({\mathbb{P}}_n-P){\dot{l}}_{\varphi }(\mathbf{Y};{\hat{\mathit{\zeta }}}_n)({\mathit{\eta }}^{*}-{\mathit{\eta }}_n^{*})$$

and

$$I_{4,n}=P\{{\dot{l}}_{\varphi }(\mathbf{Y};{\hat{\mathit{\zeta }}}_n)({\mathit{\eta }}^{*}-{\mathit{\eta }}_n^{*})-{\dot{l}}_{\varphi }(\mathbf{Y};{\mathit{\zeta }}_0)({\mathit{\eta }}^{*}-{\mathit{\eta }}_n^{*})\}.$$

. Let ${ℒ}_4=\{{\dot{l}}_{\varphi }(\mathbf{Y};\mathit{\zeta })({\mathit{\eta }}^{*}-\mathit{\eta }):\mathit{\zeta }\in {\mathit{𝒯}}_n,\mathit{\eta }\in {({\prod }_{k=1}^J{𝒮}_{kn}(D_{kn},K_{kn},m_k))}^{\otimes Jd}$ and ${\Vert {\eta }_{jk}^{*}-{\eta }_{jk}\Vert }_{{\mathrm{\Phi }}_k}=O(n^{-{\nu }_k})\text{for}j=1,\dots ,Jd\text{and}1\le k\le J\}$ . It can be similarly argued that the ε-bracketing number associated with L₂(P)-norm is bounded by $C{(1/\epsilon )}^{Jd}{(1/\epsilon )}^{C{\sum }_{k=1}^J{q}_{kn}}{(1/\epsilon )}^{C{\sum }_{k=1}^J{q}_{kn}}$ , which leads to ℒ₄ being Donsker. Furthermore, for any r(Y; ζ, η) ∈ ℒ₄, Pr² → 0 as n → ∞. Hence I_3,n = o_p(n^−1/2) by van der Vaart & Wellner (1996, Corollary 2.3.12). By Cauchy-Schwarz inequality and assumptions A1–A4, it can be easily shown that $\Vert I_{4,n}\Vert \le C{\sum }_{j=1}^{Jd}{\sum }_{k=1}^{J}d({\hat{\mathit{\zeta }}}_{kn},{\mathbf{\zeta }}_{k0}){\Vert {\eta }_{jk}^{*}-{\eta }_{n,jk}^{*}\Vert }_{{\mathrm{\Phi }}_k}=O_p(n^{-{\text{min}}_k[\text{min}\{(p_k+1){\nu }_k,(1+{\nu }_k)/2\}]})=o_p(n^{-1/2})$ , where ζ̂_kn and ζ_k0 are respectively the collections of the components of ζ̂_n and ζ₀ associated with cause k. So (B1) holds. (B2) holds by similarly verifying that the class ${ℒ}_5(\delta )=\{l_{\mathit{\theta }}^{*}(\mathbf{Y};\mathit{\zeta })-l_{\mathit{\theta }}^{*}(\mathbf{Y};{\mathit{\zeta }}_0):\mathit{\zeta }\in {\mathit{𝒯}}_n\text{and}d(\mathit{\zeta },{\mathit{\zeta }}_0)\le \delta \}$ is P-Donsker and for any r(Y; ζ) ∈ ℒ₅(δ), Pr² → 0 as δ → 0. (B3) can be easily established using Taylor expansion and the convergence rate derived in Theorem 1. Hence the proof is complete.