Empirical-likelihood-based semiparametric inference for the treatment effect in the two-sample problem with censoring

Summary

To compare two samples of censored data, we propose a unified semiparametric inference for the parameter of interest when the model for one sample is parametric and that for the other is nonparametric. The parameter of interest may represent, for example, a comparison of means, or survival probabilities. The confidence interval derived from the semiparametric inference, which is based on the empirical likelihood principle, improves its counterpart constructed from the common estimating equation. The empirical likelihood ratio is shown to be asymptotically chi-squared. Simulation experiments illustrate that the method based on the empirical likelihood substantially outperforms the method based on the estimating equation. A real dataset is analysed.

1 Introduction

Let X and Y be random variables representing some characteristic of two treatments, where X corresponds to the new treatment and Y to the control treatment. One may wish to compare the average treatment effects, so that Δ = E(X) − E(Y) is of interest. Alternatively, one may wish to consider the probability of one treatment being better than the other, which is equivalent to considering the survival competition probability, so that Δ = pr(X < Y) or Δ = pr(X > Y), or to compare the probability of death before a given time t₀ in survival analysis, so that Δ = pr(X < t₀) − pr(Y < t₀).When X and Y are both sampled from parametric families, standard statistical approaches such as maximum likelihood can be used (Brownie et al., 1986; Campbell & Ratnarkhi, 1993; Goddard & Hinberg, 1990; Hsieh & Turnbull, 1996). There are also many nonparametric approaches for comparing the two unknown continuous distributions F and G of X and Y , respectively, based on independent samples; see for example Gastwirth & Wang (1988), Hollander & Korwar (1982), and Li et al. (1996).

Sometimes enough is known about one treatment, usually the control treatment, for a parametric form to be assumed for the distribution G. However, this may not hold for F. This leads to a semiparametric two-sample model. Recently much attention has been paid to semiparametric inference. Li et al. (1999) proposed a semiparametric estimator for quantile comparison. They studied two-sample inference through the quantile comparison function G{F⁻¹(p)} assuming that G is known and F is unknown. A simpler case where Y is normally distributed has been studied by Hsieh & Turnbull (1996).

The contribution of this article is to develop a unified approach to inference about the parameter Δ, introduced in the first paragraph. Although a common estimating equation for Δ and the corresponding confidence interval are presented below, we focus on semiparametric inference about Δ by using empirical likelihood ratio methods; see for example Owen (1988, 1990, 2000), Qin (1994, 1999) and Qin & Lawless (1994). The empirical likelihood ratio has a limiting chi-squared distribution, leading to tests and confidence intervals ( Thomas and Grunkemeier, 1975) for a variety of problems, including linear models, generalised linear models and estimating equations. The empirical likelihood method has many advantages over competitors such as the normal-approximation-based method and the bootstrap method (Hall & La Scala, 1990). The appealing features of the empirical likelihood include improvement of the confidence region, increased accuracy of coverage through the use of auxiliary information, and easy implementation. We will derive the asymptotic distribution of the resulting empirical likelihood function in our context, explain how to establish the corresponding confidence interval, and illustrate the advantages numerically.

2 Estimation methods for a two-sample model with censored data

2.1 General framework

Assume that the variables $\{X_1^0,X_2^0,\cdots ,X_n^0\}$ and $\{Y_1^0,Y_2^0\cdots ,Y_m^0\}$ are randomly censored by two sequences of random variables {U₁, U₂, · · ·, U_n} and {V₁, V₂, · · ·, V_m}, respectively. The members of each sequence of random variables $\left\{X_i^0\right\}$, $\left\{Y_i^0\right\}$, {U_i} and {V_i} are independent and identically distributed with survival functions S(x), D(y), K(u), and Q(v), respectively, where S(x) and K(u) are two unknown functions. The distribution function G(y) = 1 − D(y) is of known form and depends on the unknown d-dimensional parameter θ, i.e., G(y) = G_θ(y). We assume that G_θ(y) has the density function g_θ(y). The observed sample are the pairs (X_i, δ_i), for i = 1, 2, · · · n, and the pairs (Y_j, η_j), for j = 1, · · ·, m, where $X_i=\min (X_i^0,U_i)$, ${\delta }_i=I(X_i^0\le U_i),i=1,2,\cdots ,n$ and $Y_j=\min (Y_j^0,V_j)$, ${\eta }_j=I(Y_j^0\le V_j),j=1,2,\cdots ,m$ in which I(·) denotes an indicator function. We assume that the two samples of pairs are independent.

The parameter of interest, Δ, may be a functional of the functions S(x), D(y), K(u) and Q(v). The estimation of the unknown functions S(x) and K(u) and the parameter θ is a secondary goal; these are considered as nuisance parameters. For any measurement Δ of the difference between the two samples, we assume that the information about θ, Δ and F(x) = 1 − S(x) is available in the known form of an unbiased estimation function, i.e.,

$$E_F\psi (X^0,{\theta }_0,\Delta )=0,$$ (2.1)

where θ₀ is the true value of θ and ψ, which may be a vector of functions, is assumed to be a real function for the sake of simplicity. As mentioned before plausible candidates for Δ are the following: (i) the difference in the means of the two censored samples, so that Δ = E(X⁰) − θ₀ and ψ(X⁰, θ₀, Δ) = X⁰ − θ₀ − Δ, where θ₀ = EY⁰; (ii) the difference of probabilities for the events X⁰ < t₀ and Y⁰ < t₀ for a given constant t₀, so that Δ = F(t₀) − G_θ0(t₀) and ψ(X⁰, θ₀, Δ) = I(X⁰ < t₀) − G_θ0(t₀) − Δ; or (iii) the probability of the event I(X⁰ > Y⁰), so that Δ = E{1 − G_θ0(X⁰)} and ψ(X⁰, θ₀, Δ) = 1 − G_θ0(X⁰) − Δ. Another candidate is (iv) a value on the receiver operating characteristic curve, so that $\Delta =1-F\left\{G_{{\theta }_0}^{-1}(1-p)\right\}$ and $\psi (X^0,{\theta }_0,\Delta )=1-I\{X^0\le G_{{\theta }_0}^{-1}(1-p)\}-\Delta$ for a given p ∈ (0, 1).

For the case of complete data, Qin (1997) used an empirical likelihood ratio statistic to test the hypothesis that the two populations have the same mean and constructed a confidence interval for the mean difference of two populations in a semiparametric model. This case and Li et al.’s (1999) work on the vertical quantile comparison function for two-sample tests can be viewed as special cases of ψ(·, ·, ·).

2.2 Estimation of Δ based on the common estimating equation

A naive and direct method of estimating Δ is to solve equation (2.1 ) with F and θ replaced by estimates, which may be obtained by maximum likelihood estimation.

The likelihood function based on the samples {(X_i, δ_i), i = 1, 2, · · ·, n} and {(Y_j, η_j), j = 1, 2, · · ·, m} is

$$L(F,\theta )=\prod _{i=1}^n{\{F\left(X_i\right)-F(X_i-)\}}^{{\delta }_i}{\{1-F\left(X_i\right)\}}^{1-{\delta }_i}\prod _{j=1}^m{g}_{\theta }^{{\eta }_j}\left(Y_j\right){\{1-G_{\theta }\left(Y_j\right)\}}^{1-{\eta }_j},$$

provided Q(v) does not depend on θ. We assume that θ ∈ Θ ⊂ R^d. It can be shown that L(F, θ) attains its maximum value at the point $(\widehat{F},{\widehat{\theta }}_{\mathrm{MLE}})$, where ${\widehat{\theta }}_{\mathrm{MLE}}$ is the maximum likelihood estimator based on the sample (Y_j, η_j), and $\widehat{F}$ is the Kaplan-Meier estimator:

$$\widehat{F}\left(t\right)=1-\prod _{u\le t}\left\{1-\frac{dN\left(u\right)}{Y\left(u\right)}\right\},$$

where N(u) = Σ I(X_i ≤ u, δ_i = 1) and Y(u) = Σ I(X_i ≥ u). The estimator of Δ, denoted by $\widehat{\Delta }$, is defined as the solution of the estimating equation ${\int }_0^{{\tau }_1}\psi (x,{\widehat{\theta }}_{\mathrm{MLE}},\Delta )d\widehat{F}\left(x\right)=0$, which is equivalent to

$$\sum _{i=1}^n\frac{{\delta }_i}{\widehat{K}\left(X_i\right)}\psi (X_i,{\widehat{\theta }}_{\mathrm{MLE}},\Delta )=0,$$ (2.2)

where $\widehat{K}$ is the Kaplan-Meier estimator of K, i.e.,

$$\widehat{K}\left(t\right)=\prod _{u\le t}\left\{1-\frac{d{N}^c\left(u\right)}{Y\left(u\right)}\right\},$$

in which N^c(u) = Σ I(X_i ≤ u, δ_i = 0), and τ₁ = sup{t : S(t)K(t) > 0}.

We assume that m/n → ζ > 0 as n, m → ∞. The asymptotic normality of the estimator $\widehat{\Delta }$ is given in the following theorem.

THEOREM 1

Suppose that Assumptions 1-4 in the Appendix hold. Then

$$\frac{1}{\sqrt{n}}\sum _{i=1}^n{W}_{ni}\psi (X_i,{\widehat{\theta }}_{\mathrm{MLE}},\Delta )\to N(0,{\sigma }_{\psi }^2),$$

in distribution, where $W_{ni}={\delta }_i∕\widehat{K}\left(X_i\right)$, ${\sigma }_{\psi }^2={\sigma }^2(\theta ,\Delta )+{\beta }_0^{\mathrm{T}}(\theta ,\Delta ){\Sigma }^{-1}{\beta }_0(\theta ,\Delta )$, Σ = ζ⁻¹I(θ) with I(θ) given in Assumption 2, ${\sigma }^2(\theta ,\Delta )=\mathrm{var}\left\{\psi (X,\theta ,\Delta )\right\}+E{\int }_0^{{\tau }_1}{\{\psi (X,\theta ,\Delta )-G(\theta ,u)\}}^{2}I(X\ge u){\lambda }^c\left(u\right)∕K\left(u\right)du$, G(θ, u) = S⁻¹(u)E{ψ(X, θ, Δ)I(X ≥ u)}, ${\beta }_0(\theta ,\Delta )={\int }_0^{{\tau }_1}{\stackrel{.}{\psi }}_{\theta }(u,\theta ,\Delta )dF\left(u\right)$, and ${\stackrel{.}{\psi }}_{\theta }(u,\theta ,\Delta )=\partial \psi (x,\theta ,\Delta )∕\partial \theta$, λ^c(t) is the hazard function of U. Furthermore, if Assumption 5 holds, then

$$\sqrt{n}(\widehat{\Delta }-\Delta )\to N(0,{\sigma }_{\Delta }^2),$$

in distribution, where ${\sigma }_{\Delta }^2={\sigma }_{\psi }^2{\gamma }^{-2}$, and $\gamma (\theta ,\Delta )=E\left\{{\stackrel{.}{\psi }}_{\Delta }(X,\theta ,\Delta )\right\}$, in which ${\stackrel{.}{\psi }}_{\Delta }(x,\theta ,\Delta )=\partial \psi (x,\theta ,\Delta )∕\partial \Delta$.

One application of Theorem 1 is to construct the confidence interval for Δ, for which we need to estimate ${\sigma }_{\Delta }^2$. This can be done by using a plug-in method. Let

$${\widehat{\sigma }}_1^2({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })=\frac{1}{n}\sum _{i=1}^n\frac{{\delta }_i{\psi }^2(X_i,{\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })}{\widehat{K}\left(X_i\right)}+\frac{1}{n}{\int }_0^{\Lambda }\frac{d{N}^c\left(u\right)}{{\widehat{K}}^2\left(u\right)}\{{\widehat{G}}_1({\widehat{\theta }}_{\mathrm{MLE}},u)-{\widehat{G}}_2({\widehat{\theta }}_{\mathrm{MLE}},u)\},$$

where Λ is the maximum of the uncensored observations,

$${\widehat{G}}_1(\theta ,u)=\frac{1}{n\widehat{S}\left(u\right)}\sum _{i=1}^n\frac{{\delta }_i{\psi }^2(X_i,\theta ,\widehat{\Delta })I(X_i\ge u)}{\widehat{K}\left(X_i\right)},$$ (2.3)

$${\widehat{G}}_2(\theta ,u)={\left\{\frac{1}{n\widehat{S}\left(u\right)}\sum _{i=1}^n\frac{{\delta }_i\psi (X_i,\theta ,\widehat{\Delta })I(X_i\ge u)}{\widehat{K}\left(X_i\right)}\right\}}^2,$$ (2.4)

$${\widehat{\beta }}_0({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })=\frac{1}{n}\sum _{i=1}^n{W}_{ni}{\stackrel{.}{\psi }}_{\theta }(X_i,{\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta }),\widehat{\gamma }({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })=\frac{1}{n}\sum _{i=1}^n{W}_{ni}{\stackrel{.}{\psi }}_{\Delta }(X_i,{\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta }).$$ (2.5)

Replace σ²(θ, Δ), β₀(θ, Δ), γ(θ, Δ) and Σ in ${\sigma }_{\Delta }^2$ by their estimators ${\widehat{\sigma }}_1^2({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })$, ${\widehat{\beta }}_0({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })$, $\widehat{\gamma }({\widehat{\theta }}_{\mathrm{MLE}},\widehat{\Delta })$ and $\widehat{\Sigma }=-\zeta ∕n{\partial }^2{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)∕\partial {\theta }^2$, to obtain an estimator ${\widehat{\sigma }}_{\Delta }^2$ of ${\sigma }_{\Delta }^2$. It is easy to prove that the estimator is consistent. As a consequence, the approximate 100(1 − α)% confidence interval of Δ is $(\widehat{\Delta }-z_{1-\alpha ∕2}\sqrt{{\widehat{\sigma }}_{\Delta }^2},\widehat{\Delta }+z_{1-\alpha ∕2}\sqrt{{\widehat{\sigma }}_{\Delta }^2})$, where z_1−α/2 denotes the 1 −α/2 quantile of the standard normal distribution.

2.3 Empirical likelihood inference

In this section, we apply the empirical likelihood principle to obtain a confidence interval for Δ. To deal with the censoring we propose an adjusted empirical likelihood function and prove that the resulting log likelihood ratio is still asymptotically chi-squared under mild conditions.

Let θ₀ be the true value of parameter θ. Note that the auxiliary information depends only on the unknown distributions F and K and the parameter θ₀, since E {δψ(X, θ₀, Δ)/K(X−)} = 0. If the function K(·) were known, then we could infer about Δ based on the observation sample {δ_i/K(X_i−), i = 1, 2, · · ·, n} and the sample {(Y_j, η_j), j = 1, 2, · · ·, m}. Let F_p be the distribution function that assigns probabilities p_i at points δ_i/K(X_i−). Then the adjusted empirical likelihood is $L_{\mathrm{adj}}(\theta ,\Delta )={\prod }_{i=1}^n{p}_i{\prod }_{j=1}^m{g}_{\theta }^{{\eta }_j}\left(Y_j\right){\{1-G_{\theta }\left(Y_j\right)\}}^{1-{\eta }_j}$, where ${\sum }_{i=1}^n{p}_i=1$ and p_i ≥ 0 for each i, and the auxiliary information (2.1) is expressed as

$$\sum _{i=1}^n\frac{{\delta }_i}{K(X_i-)}\psi (X_i,\theta ,\Delta )=0.$$ (2.6)

Again, we replace K(·) with its Kaplan-Meier estimator $\widehat{K}(\cdot )$, and our adjusted empirical likelihood, evaluated at Δ and θ, is defined by L_adj(θ, Δ) subject to the constraints

$$\sum _{i=1}^n{p}_i=1,p_i\ge 0,\text{for}i=1,2,\cdots ,n,$$ (2.7)

$$\sum _{i=1}^n\frac{p_i{\delta }_i}{\widehat{K}(X_i-)}\psi (X_i,\theta ,\Delta )=0.$$ (2.8)

Without the restrictions (2.7) and (2.8), the adjusted empirical likelihood has the maximum value $n^{-n}{\prod }_{j=1}^m{g}_{{\widehat{\theta }}_{\mathrm{MLE}}}^{{\eta }_j}\left(Y_j\right){\{1-G_{{\widehat{\theta }}_{\mathrm{MLE}}}\left(Y_j\right)\}}^{1-{\eta }_j}$, where ${\widehat{\theta }}_{\mathrm{MLE}}$ is the maximum likelihood estimate of θ based on the sample {(Y_j, η_j), j = 1, 2, · · ·, m}, as in §2.2. Our adjusted semiparametric empirical likelihood ratio statistic is

$$R\left(\Delta \right)=\underset{\mathcal{P},\theta }{\sup }\prod _{i=1}^n\left(n{p}_i\right)\prod _{j=1}^m{g}_{\theta }^{{\eta }_j}\left(Y_j\right){\{1-G_{\theta }\left(Y_j\right)\}}^{1-{\eta }_j}{\left[\prod _{j=1}^m{g}_{{\widehat{\theta }}_{\mathrm{MLE}}}^{{\eta }_j}\left(Y_j\right){\{1-G_{{\widehat{\theta }}_{\mathrm{MLE}}}\left(Y_j\right)\}}^{1-{\eta }_j}\right]}^{-1},$$

where $\mathcal{P}$ denotes the region of possible p_i subject to the constraints (2.7) and (2.8). We first investigate R(Δ) for a given θ, which we denote by R(θ, Δ). The empirical log likelihood ratio of R(θ, Δ) is described as $\mathcal{L}(\theta ,\Delta )={\sup }_{\mathcal{P}}{\sum }_{i=1}^n\log \left(n{p}_i\right)+{\ell }_g\left(\theta \right)-{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)$, where ${\ell }_g\left(\theta \right)={\sum }_{j=1}^m\log \left[g_{\theta }^{{\eta }_j}\left(Y_j\right){\{1-G_{\theta }\left(Y_j\right)\}}^{1-{\eta }_j}\right]$. Provided that ${\sum }_{i=1}n{\psi }^2(X_i,\theta ,\Delta )W_{ni}^2>0$ and the origin is inside the convex hull of {ψ(X_i, θ, Δ)W_ni, i = 1, 2, · · · n}, where $W_{ni}={\delta }_i∕\widehat{K}(X_i-)$, there exits a unique maximum value of $\mathcal{L}(\theta ,\Delta )$ or R(θ, Δ) (Qin & Lawless, 1994). By the Lagrange multiplier method, we conclude that $\mathcal{L}(\theta ,\Delta )$ attains its maximum value at ${\widehat{p}}_i=n^{-1}{\{1+\lambda \left(\theta \right)W_{ni}\psi (X_i,\theta ,\Delta )\}}^{-1}$, for i = 1, 2, · · ·, n. Alternatively, the estimated empirical log likelihood ratio is

$$\mathcal{L}(\theta ,\Delta )=\sum _{i=1}^n\log \{1+\lambda \left(\theta \right)W_{ni}\psi (X_i,\theta ,\Delta )\}+{\ell }_g\left(\theta \right)-{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right),$$ (2.9)

and the adjusted empirical log likelihood is defined by

$${\ell }_{\mathrm{EL}}(\theta ,\Delta )=\sum _{i=1}^n\log \{1+\lambda \left(\theta \right)W_{ni}\psi (X_i,\theta ,\Delta )\}+{\ell }_g\left(\theta \right),$$ (2.10)

where λ(θ) is determined by the equation:

$$\frac{1}{n}\sum _{i=1}^n\frac{\psi (X_i,\theta ,\Delta )W_{ni}}{1+\lambda \left(\theta \right)W_{ni}\psi (X_i,\theta ,\Delta )}=0.$$ (2.11)

In the following, we use ∥ · ∥ to denote the Euclidean norm.

THEOREM 2

Suppose that Assumptions 1-4 in the Appendix hold. Then (i) ℓ_EL(θ, Δ) attains its maximum value at some point ${\widehat{\theta }}_{\mathrm{EL}}$ in the interior of the ball ∥θ − θ₀∥ < n^−ϱ (1/3 < ϱ < 1/2) such that ${\widehat{\theta }}_{\mathrm{EL}}$ and $\widehat{\lambda }=\lambda \left({\widehat{\theta }}_{\mathrm{EL}}\right)$ satisfy $\partial {\ell }_{\mathrm{EL}}({\widehat{\theta }}_{\mathrm{EL}},\Delta )∕\partial \theta =0$ and (2.11); and (ii) the maximiser ${\widehat{\theta }}_{\mathrm{EL}}$ of R(θ, Δ) is strongly consistent and such that $2\rho (\theta ,\Delta )\mathcal{L}({\widehat{\theta }}_{\mathrm{EL}},\Delta )\to {\chi }_{\left(1\right)}^2$, in distribution, where

$$\rho (\theta ,\Delta )=\frac{{\sigma }_1^2(\theta ,\Delta )+{\beta }_0^{\mathrm{T}}(\theta ,\Delta ){\Sigma }^{-1}{\beta }_0(\theta ,\Delta )}{{\sigma }_0^2(\theta ,\Delta )+{\beta }_0^{\mathrm{T}}(\theta ,\Delta ){\Sigma }^{-1}{\beta }_0(\theta ,\Delta )},$$

in which Σ = ζ⁻¹I(θ) with I(θ) given in Theorem 1, and ${\sigma }_0^2(\theta ,\Delta )={\int }_0^{{\tau }_1}{\psi }^2(u,\theta ,\Delta )∕K\left(u\right)dF\left(u\right)$.

Were it not for the term ρ(θ, Δ), the confidence interval could be easily derived without estimating variance as in traditional empirical likelihood (Owen, 2000, p.23). With censorship, the situation becomes complex, and we need to estimate ρ(θ, Δ). As in §2.2, we replace σ²(θ, Δ), β₀(θ, Δ) and Σ in ρ(θ, Δ) by their estimators ${\widehat{\sigma }}^2({\widehat{\theta }}_{\mathrm{EL}},\widehat{\Delta })$ and ${\widehat{\beta }}_0({\widehat{\theta }}_{\mathrm{EL}},\widehat{\Delta })$, which are defined in a similar way to (2.3)-(2.5), and $\widehat{\Sigma }=-(\zeta ∕n){\partial }^2{\ell }_g\left({\widehat{\theta }}_{\mathrm{EL}}\right)∕\partial {\theta }^2$. The quantity ${\sigma }_0^2(\theta ,\Delta )$ is estimated by ${\widehat{\sigma }}_0^2({\widehat{\theta }}_{\mathrm{EL}},\widehat{\Delta })=n^{-1}{\sum }_{i=1}^n{\left\{W_{ni}\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\widehat{\Delta })\right\}}^2$, where $\widehat{\Delta }$ is a consistent estimator of Δ. The resulting $\widehat{\rho }({\widehat{\theta }}_{\mathrm{EL}},\widehat{\Delta })$ is a consistent estimator of ρ(θ, Δ). The confidence interval of Δ based on empirical likelihood can then be established. Another approach is not to estimate Δ in the adjusted factor ρ(θ, Δ). A confidence interval can then be constructed on the basis of the quantity $2\rho ({\widehat{\theta }}_{\mathrm{EL}},\Delta )\mathcal{L}(\widehat{\theta },\Delta )$.

Corollary 1

Suppose that the Assumptions of Theorem 2 hold. Then ${\lim }_{n\to \infty }\mathrm{pr}(\Delta \in I_{\alpha })=1-\alpha$, where $I_{\alpha }=\{\Delta :2\rho ({\widehat{\theta }}_{\mathrm{EL}},\Delta )\mathcal{L}(\widehat{\theta },\Delta )\le c_{\alpha }\}$, and $\mathrm{pr}({\chi }_{\left(1\right)}^2\le c_{\alpha })=1-\alpha$.

Theorem 2 showed that the maximum point is close to the true value, and the adjusted empirical log likelihood ratio is distributed approximately as chi-squared at the maximum point. One way of obtaining this maximum point is to consider ∂ℓ_EL(θ, Δ)/∂θ = 0, which results in the score equation of semiparametric empirical likelihood, i.e.,

$$\frac{1}{n}\lambda \left(\theta \right)\sum _{i=1}^n\frac{{\stackrel{.}{\psi }}_{\theta }(X_i,\theta ,\Delta )W_{ni}}{1+\lambda \left(\theta \right)\psi (X_i,\theta ,\Delta )W_{ni}}=\frac{1}{m}\sum _{j=1}^m\frac{\partial \log \left[g_{\theta }^{{\eta }_j}\left(Y_j\right){\{1-G_{\theta }\left(Y_j\right)\}}^{{\eta }_j}\right]}{\partial \theta }.$$ (2.12)

There may be multiple solutions of (2.12) but, if (2.12) has a unique root, this root is ${\widehat{\theta }}_{\mathrm{EL}}$.

An obvious result holds when we use ${\widehat{\theta }}_{\mathrm{MLE}}$ in (2.9), as stated in the following theorem.

THEOREM 3

Suppose that Assumptions 1-4 hold. Then $2r(\theta ,\Delta )\mathcal{L}({\widehat{\theta }}_{\mathrm{MLE}},\Delta )\to {\chi }_{\left(1\right)}^2$, in distribution, where $r(\theta ,\Delta )={\sigma }_1^2(\theta ,\Delta )∕{\sigma }_0^2(\theta ,\Delta )$.

Theorem 3 can be proved in a similar way to Theorem 2 by using Lemma A.3 and Taylor expansion. We omit the details.

Theorem 3 indicates that the empirical likelihood function based on the estimator ${\widehat{\theta }}_{\mathrm{MLE}}$ is still asymptotically chi-squared, but its asymptotic variance is bigger than that of ${\widehat{\theta }}_{\mathrm{EL}}$ (see Lemma A.5). A confidence interval is available, based on the result of Theorem 3.

3 Simulation Study

In our simulation experiments, we consider only one-dimensional θ for the sake of simplicity. We mainly compare the lengths and coverages of confidence intervals based on ${\widehat{\theta }}_{\mathrm{EL}}$ and the estimating equation. Note that closed-form solutions of (2.11) and (2.12) rarely exist, and we use the Newton-Raphson algorithm for implementation. Codes in S-plus are developed which are modified from El.s, a function written by A.B.Owen.

Example 1

Let X⁰ and Y⁰ be independent Un(0, 6) and Un(0, 10) random variables. The censored random variables U and V are generated by the uniform distributions Un(0, c_x) and Un(0, c_y), respectively. The parameters c_x and c_y are used to control the rate of censored data; the larger the c_x and c_y, the lighter the censorship. The four choices used for (c_x, c_y) are a combination of c_x = 12.5 and c_x = 15 with c_y = 15 and c_y = 20. We are concerned with semiparametric inference for Δ = E(X) − θ₀ = −2. The simulation study is conducted as follows. For each of the 4 censoring configurations, sample sizes (n, m) = (30, 30), (50, 50) and (100, 100) are considered. For each censoring configuration and sample size, 2000 independent sets of data are generated. The confidence interval is based on the 95% level, and Table 1 summarises the results. The confidence interval for Δ based on the adjusted empirical likelihood is almost symmetric about the true value. Coverages based on the empirical likelihood method are consistently close to the nominal level, and the coverages of the normal approximation are all larger than that of the empirical likelihood method and the nominal level. The lengths of the confidence intervals are inversely proportional to square root of the sample size and censoring degree of sample X, and the confidence interval based on the empirical likelihood method is consistently and substantially shorter than its normal counterpart.

Table 1.

Example 1. Confidence intervals and the corresponding coverage based on the empirical likelihood (EL) and normal approximation (Norm) methods

Sample size				CI(95%)		coverage
n	m	cx	cy	EL	Norm	EL	Norm
30	30	12.5	15	(−2.686, −1.311)	(−3.153, −0.847)	94.46	97.39
			20	(−2.682, −1.311)	(−3.354, −0.646)	94.13	98.60
		15	15	(−2.652, −1.313)	(−3.106, −0.894)	94.66	97.72
			20	(−2.688, −1.347)	(−3.399, −0.601)	95.28	98.15
50	50	12.5	15	(−2.531, −1.457)	(−2.753, −1.247)	95.02	97.84
			20	(−2.534, −1.46)	(−2.904, −1.096)	95.68	98.54
		15	15	(−2.522, −1.469)	(−2.751, −1.249)	95.07	97.53
			20	(−2.530, −1.477)	(−2.926, −1.074)	94.9	98.34
100	100	12.5	15	(−2.386, −1.619)	(−2.442, −1.558)	94.35	96.60
			20	(−2.379, −1.613)	(−2.519, −1.481)	93.78	97.64
		15	15	(−2.374, −1.625)	(−2.433, −1.567)	94.45	97.35
			20	(−2.369, −1.619)	(−2.498, −1.502)	94.98	97.69

CI(95%), approximate 95% confidence interval.

Example 2

We use a scenario similar to that in Example 1, except that X⁰ and Y⁰ are exponential random variables with density functions 1/6exp(−x/6) and 1/10 exp(−x/10), respectively. Here, we consider Δ = pr(X⁰ > Y⁰). Consequently, θ₀ = 10, ψ(x, θ₀, Δ) = exp(−x/θ₀) − Δ, and Δ = 0.625. The simulation results are given in Table 2. The conclusions are the same as those in Example 1, in general. However, the confidence interval based on the empirical likelihood method is not symmetric about the true value in this example. These confidence intervals also depend more upon the censorship of X than did those in Example 1.

Table 2.

Example 2. Confidence intervals and the corresponding coverage based on the empirical likelihood (EL) and normal approximation (Norm) methods

Sample size				CI(95%)		coverage
n	m	cx	cy	EL	Norm	EL	Norm
30	30	12.5	15	(0.403, 0.777)	(0.245, 1.005)	96.90	100
			20	(0.411, 0.785)	(0.274, 0.976)	97.20	100
		15	15	(0.46, 0.792)	(0.254, 0.996)	98.05	100
			20	(0.455, 0.789)	(0.282, 0.968)	98.35	100
50	50	12.5	15	(0.460, 0.755)	(0.333, 0.917)	96.55	100
			20	(0.461, 0.758)	(0.355, 0.895)	97.40	100
		15	15	(0.506, 0.769)	(0.341, 0.909)	98.15	100
			20	(0.506, 0.768)	(0.363, 0.887)	98.35	100
100	100	12.5	15	(0.507, 0.723)	(0.420, 0.830)	96.05	100
			20	(0.508, 0.722)	(0.435, 0.815)	96.65	100
		15	15	(0.553, 0.742)	(0.426, 0.824)	98.10	100
			20	(0.551, 0.740)	(0.440, 0.810)	98.30	100

CI(95%), approximate 95% confidence interval.

4 Real Data Analysis

Mantel et al. (1977) reported a litter-matched study of the tumourigenesis of a drug. One rat was randomly selected from each of 50 litters and given the drug. Two rats from each litter were selected as controls and given a placebo; see Mantel et al. (1977) for the details of the study. We are interested in comparing the average survival times of the treatment and control groups. Let X and Y be the survival times of the treatment and control groups. We first calculate the Kaplan-Meier estimate of the survival distribution of Y. A Q-Q plot then shows that the exponential-distribution assumption for Y is appropriate. Consequently, the estimated average survival time of the control group is 476, and the estimate of the difference of the averages, Δ = E(X) − E(Y), of the survival times is −240. The 95% confidence intervals for Δ based on the normal approximation and our empirical likelihood method are (−676.5, 195.4) and (−258.8, −68.7), respectively. The conclusions drawn from these two confidence intervals are puzzling because the first interval suggests that there is no significant difference, but the second interval implies the contrary. Recalling the performance of the two methods in the simulation experiment, we prefer the conclusion based on the empirical likelihood method. For comparison, we also conducted a log-rank test on the dataset and obtained a p–value less than 10⁻³, which reinforces our conclusion based on the empirical likelihood method. Moreover, the 99% confidence interval (empirical likelihood) is (−304.7, −47.4), which indicates a significant difference at the 1% significance level.

5 Conclusion

The finite-sample behaviour of our method seems superior to that of the common estimating equation. Furthermore implementation of the normal-approximation method requires ψ(x, θ₀, Δ) to be continuously differentiable in Δ, whereas, this assumption is unnecessary for constructing the confidence interval in the empirical likelihood framework. In addition, in absence of censoring, the factors ρ(θ, Δ) and r(θ, Δ) equal 1, so that the confidence interval based on empirical likelihood does not require the calculation of any covariance or variance. The factors ρ(θ, Δ) and r(θ, Δ) can therefore be viewed as the cost of censoring.

If the function ψ(x, θ₀, Δ) in equation (2.1 ) is discontinuous at θ₀, then neither the normal-approximation-based method nor the empirical likelihood method is applicable even in the absence of censoring. In particular, this happens if ψ(x, θ₀, Δ) includes the term I{X⁰ < G_θ0(p)}. Hence, the current version is not applicable to the problems of two-sample quantile comparison, and inference for the ROC curve. In principle, the method proposed in this paper can be extended to cover these cases by incorporating smoothing techniques. Detailed investigation of these issues will be discussed in further research.

APPENDIX

Theory

Assumption 1

We assume that τ_S ≤ τ_K, where τ_S = sup{x : S(x) > 0} and τ_K = sup{x : K(x) > 0}, and

$${\int }_0^{{\tau }_1}\frac{{\psi }^2(u,\theta ,\Delta )}{K\left(u\right)}dF\left(u\right)<\infty .$$

Assumption 2

The functions ψ(x, θ, Δ) and ${\stackrel{.}{\psi }}_{\theta }(x,\theta ,\Delta )$ are continuous and bounded by some function M(x) in a neighbourhood of the true value θ₀ such that, ${\int }_0^{{\tau }_1}\frac{M\left(u\right)}{K^2\left(u\right)}dF\left(u\right)<\infty$, and $E_F{\stackrel{.}{\psi }}_{\theta }(X,\theta ,\Delta )\ne 0$.

Assumption 3

The density function g_θ(y) is three times differentiable with respect to θ on A = {y : g_θ(y) > 0}, which is assumed to be independent of θ. There exists a function M₁(y) such that, for any y ∈ A and θ in the neighbourhood of θ₀, E_θ|M₁(Y)| < ∞.

Assumption 4

The information matrix I(θ) with entries

$$I_{ij}=-{\int }_0^{{\tau }_2}\frac{{\partial }^2\log g_{\theta }\left(y\right)}{\partial {\theta }_i\partial {\theta }_j}\{1-Q\left(y\right)\}{g}_{\theta }\left(y\right)dy-{\int }_0^{{\tau }_2}\frac{{\partial }^2\log \{1-G_{\theta }\left(y\right)\}}{\partial {\theta }_i\partial {\theta }_j}\{1-G_{\theta }\left(y\right)\}dQ\left(y\right),$$

for i, j = 1, · · ·, d, which are continuous, is positive definite. Here τ₂ = sup{t : D(t)Q(t) > 0}.

Assumption 5

The function ψ(x, θ, Δ) is continuously differentiable in Δ, and $E\left\{{\stackrel{.}{\psi }}_{\Delta }(X,\theta ,\Delta )\right\}\ne 0$.

We next state preliminary lemmas. Their detailed proofs are referred to Zhou and Liang (2004).

Lemma A.1

Under Assumptions 1 and 2, we have that

$$\frac{1}{\sqrt{n}}\sum _{i=1}^n{W}_{ni}\psi (X_i,{\theta }_0,\Delta )\to N\{0,{\sigma }^2({\theta }_0,\Delta )\},$$

in distribution, where σ²(θ₀, Δ) is defined in Theorem 2.

Lemma A.2

Suppose that Assumptions 1, 2 and 5 hold. Then

$$\begin{array}{c}\frac{1}{n}\sum _{i=1}^n{\left\{\frac{\psi (X_i,\theta ,\Delta ){\delta }_i}{\widehat{K}(X_i-)}\right\}}^2={\sigma }_0^2(\theta ,\Delta )+o_P\left(1\right),\hfill \\ \frac{1}{n}\sum _{i=1}^n\frac{{\stackrel{.}{\psi }}_{\theta }(X_i,\theta ,\Delta ){\delta }_i}{\widehat{K}(X_i-)}={\beta }_0(\theta ,\Delta )+o_P\left(1\right),\text{and}\frac{1}{n}\sum _{i=1}^n\frac{{\delta }_i{\stackrel{.}{\psi }}_{\Delta }(X_i,\theta ,\Delta )}{\widehat{K}(X_i-)}=\gamma (\theta ,\Delta )+o_P\left(1\right).\hfill \end{array}$$

Lemma A.3

Suppose that Assumptions 1-4 hold. Then λ(θ) = O_P(n^−ϱ) uniformly on {θ : ∥θ − θ₀∥ ≤ cn^−ϱ}, where 1/3 < ϱ < 1/2, c is some constant, and

$$\lambda \left(\theta \right)=\frac{\sum _{i=1}^n\psi (X_i,\theta ,\Delta )W_{ni}}{\sum _{i=1}^n{\left\{\psi (X_i,\theta ,\Delta )W_{ni}\right\}}^2}+o_P\left(n^{-\varrho }\right)$$

uniformly on {θ : ∥θ − θ₀∥ ≤ cn^−ϱ}, where λ(θ) satisfies (2.11).

Lemma A.4

Assume that the conditions of Theorem 2 hold, and that ϱ is defined as in Lemma A.1 . Then R(θ, Δ) attains its local maximum value at ${\widehat{\theta }}_{\mathrm{EL}}$ such that $\parallel {\widehat{\theta }}_{\mathrm{EL}}-{\theta }_0\parallel =O\left(n^{-\varrho }\right)$, almost surely and ${\widehat{\theta }}_{\mathrm{EL}}$ is a root of (2.12).

Lemma A.5

Suppose that Assumptions 1-4 hold. Then $\sqrt{n}\left(\begin{array}{c}\hfill {\widehat{\theta }}_{\mathrm{EL}}-{\theta }_0\hfill \\ \hfill \widehat{\lambda }\hfill \end{array}\right)\to N(0,{\Sigma }_1)$, in distribution, where

$${\Sigma }_1=\left\{\begin{array}{cc}\hfill C_{\theta }{\sigma }^2{\Sigma }^{-1}{\beta }_0{\beta }_0^{\mathrm{T}}{\Sigma }^{-1}+{(\Sigma -{\beta }_0{\sigma }_0^2{\beta }_0^{\mathrm{T}})}^{-1}\Sigma {(\Sigma -{\beta }_0{\sigma }_0^2{\beta }_0^{\mathrm{T}})}^{-1}\hfill & \hfill \ast \hfill \\ \hfill C_{\theta }{\beta }_0^{\mathrm{T}}{(\Sigma -{\beta }_0{\sigma }_0^2{\beta }_0^{\mathrm{T}})}^{-1}-C_{\theta }^2{\sigma }^2{\beta }_0^{\mathrm{T}}{\Sigma }^{-1}\hfill & \hfill C_{\theta }^2{\sigma }^2+C_{\theta }^2{\beta }_0^{\mathrm{T}}{\sigma }_0^2{\beta }_0\hfill \end{array}\right\},$$

$C_{\theta }={({\sigma }_0^2+{\beta }_0^{\mathrm{T}}{\Sigma }^{-1}{\beta }_0)}^{-1}$, ${\sigma }_0^2={\sigma }_0^2(\theta ,\Delta )$ and ${\beta }_0^2={\beta }_0^2(\theta ,\Delta )$ as given in Theorem 2.

Proof of Theorem 1

By Assumption 5, we have

$$\frac{1}{\sqrt{n}}\sum _{i=1}^n\psi (X_i,{\widehat{\theta }}_{\mathrm{MLE}},\Delta )=\frac{1}{\sqrt{n}}\sum _{i=1}^n{W}_{ni}\psi (X_i,\theta ,\Delta )+\gamma (\theta ,\Delta )\sqrt{n}({\widehat{\theta }}_{\mathrm{MLE}}-\theta )+o_P\left(1\right).$$

Theorem 1 follows from Lemmas A.1 and A.2 .

Proof of Theorem 2

The result in (i) is a direct consequence of Lemma A.3 . We now prove result (ii). By using Lemma A.4 and Taylor expansion, we can show that

$$\begin{array}{cc}\hfill & 2\mathcal{L}({\widehat{\theta }}_{\mathrm{EL}},\Delta )=-2\widehat{\lambda }\sum _{i=1}^n\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}+{\widehat{\lambda }}^2\sum _{i=1}^n{\left\{\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}\right\}}^2\hfill \\ \hfill & -2\frac{\partial {\ell }_g^{\mathrm{T}}\left({\widehat{\theta }}_{\mathrm{MLE}}\right)}{\partial \theta }({\widehat{\theta }}_{\mathrm{EL}}-{\widehat{\theta }}_{\mathrm{MLE}})+{({\widehat{\theta }}_{\mathrm{EL}}-{\widehat{\theta }}_{\mathrm{MLE}})}^{\mathrm{T}}\frac{{\partial }^2{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)}{\partial \theta \partial {\theta }^{\mathrm{T}}}({\widehat{\theta }}_{\mathrm{EL}}-{\widehat{\theta }}_{\mathrm{MLE}})+o_P\left(1\right).\hfill \end{array}$$ (A.1)

Note that (2.11) implies that

$$\frac{1}{n}\sum _{i=1}^n\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}=\frac{\widehat{\lambda }}{n}\sum _{i=1}^n{\left\{\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}\right\}}^2+o_P\left(n^{-\varrho }\right);$$ (A.2)

see Lemma A.3 . Recall that ${\widehat{\theta }}_{\mathrm{MLE}}$ is the maximiser of ℓ_g(θ) and that $\partial {\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)∕\partial \theta =0$. This implies that

$$\frac{1}{n}\frac{\partial {\ell }_g^{\mathrm{T}}\left({\widehat{\theta }}_{\mathrm{EL}}\right)}{\partial \theta }=\frac{1}{n}\frac{{\partial }^2{\ell }_g^{\mathrm{T}}\left({\widehat{\theta }}_{\mathrm{MLE}}\right)}{\partial \theta \partial {\theta }^{\mathrm{T}}}({\widehat{\theta }}_{\mathrm{EL}}-{\widehat{\theta }}_{\mathrm{MLE}})+o_P\left(n^{-\varrho }\right).$$

On the other hand, (2.12) and Lemma A.2 yield that

$$\frac{1}{n}\frac{\partial {\ell }_g\left({\widehat{\theta }}_{\mathrm{EL}}\right)}{\partial \theta }=\frac{\widehat{\lambda }}{n}\sum _{i=1}^n\frac{{\stackrel{.}{\psi }}_{\theta }(X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}}{1+\widehat{\lambda }\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}}=\widehat{\lambda }{\beta }_0+o_P\left(1\right).$$

As a result,

$${\widehat{\theta }}_{\mathrm{EL}}-{\widehat{\theta }}_{\mathrm{MLE}}=\widehat{\lambda }{\left\{\frac{1}{n}\frac{{\partial }^2{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)}{\partial \theta \partial {\theta }^{\mathrm{T}}}\right\}}^{-1}{\beta }_0+o_P\left(1\right).$$ (A.3)

Noting the fact that $1∕n{\partial }^2{\ell }_g\left({\widehat{\theta }}_{\mathrm{MLE}}\right)∕\partial \theta \partial {\theta }^{\mathrm{T}}\to -{\zeta }^{-1}I\left(\theta \right)$, together with (A.1)-(A.3), we obtain that

$$2\mathcal{L}({\widehat{\theta }}_{\mathrm{EL}},\Delta )={\left(\sqrt{n}\widehat{\lambda }\right)}^2\left[\frac{1}{n}\sum _{i=1}^n{\left\{\psi (X_i,{\widehat{\theta }}_{\mathrm{EL}},\Delta )W_{ni}\right\}}^2+{\beta }_0^{\mathrm{T}}{\Sigma }^{-1}{\beta }_0\right]+o_P\left(1\right).$$

Using Lemmas A.2 and A.5 , we complete the proof of Theorem 2.