Nonparametric inference for assessing treatment efficacy in randomized clinical trials with a time-to-event outcome and all-or-none compliance

Abstract

To evaluate the biological efficacy of a treatment in a randomized clinical trial, one needs to compare patients in the treatment arm who actually received treatment with the subgroup of patients in the control arm who would have received treatment had they been randomized into the treatment arm. In practice, subgroup membership in the control arm is usually unobservable. This paper develops a nonparametric inference procedure to compare subgroup probabilities with right-censored time-to-event data and unobservable subgroup membership in the control arm. We also present a procedure to estimate the onset and duration of treatment effect. The performance of our method is evaluated by simulation. An illustration is given using a randomized clinical trial for melanoma.

1. Introduction

In randomized clinical trials, subjects often fail to comply with their assigned treatment regimen. The intent-to-treat analysis is a standard method for primary analysis of randomized trials that compares the treatment and control based on the initial randomization regardless of whether the subjects actually received their assigned treatment or not. However, in the presence of non-compliance, the intent-to-treat analysis estimates the programmatic effectiveness, not the biological efficacy of the treatment (Sommer & Zeger, 1991; Hollis & Campbell, 1999; Lachin, 2000).

Consider a randomized clinical trial with all-or-none compliance in the treatment arm, but no noncompliance in the control arm, where all-or-none compliance means that subjects either fully comply or fully do not comply with their assigned treatment regimen. To evaluate the treatment efficacy, one needs to compare the compliers in the treatment arm with the latent subgroup of patients in the control arm who would have received the treatment if they were offered the treatment. However, the membership of the latent compliance subgroup in the control arm is unobservable. For example, in the Eastern Cooperative Oncology Group trial E9288 (Kemeny et al., 2002), patients with liver metastases of colorectal cancer were randomized to receive surgical resection alone or surgical resection followed by chemotherapy. All patients received surgical resection, but only a portion of those randomized to the chemotherapy arm actually received the postoperative chemotherapy for reasons related to survival. To assess the biological efficacy of chemotherapy, one should compare patients in the chemotherapy arm who have actually received chemotherapy with those in the no-chemotherapy arm who would have received chemotherapy had they been assigned to receive postoperative chemotherapy. However, the latent chemotherapy-compliance status is not observable in the no-chemotherapy arm. Another example is Zelen’s (1979, 1990) single-consent design, in which subjects randomized to the control arm must receive the standard care, and those to the treatment arm can choose either the experimental treatment or the standard care.

This problem also arises from subgroup analysis when the subgroup status is identified by a diagnostic test that is performed in the treatment arm but not in the control arm. One example is the Multicenter Selective Lymphadenectomy Trial I (Morton et al., 2006). In this trial, newly diagnosed melanoma patients were randomized to a sentinel-node biopsy arm or a nodal observation arm. All patients underwent wide excision of the primary melanoma. In the biopsy arm, sentinel-node biopsy was performed on patients to identify the presence of sentinel-node metastases. An immediate complete lymphadenectomy was then performed for patients whose sentinel-node biopsies were positive for metastases. For all other patients in the trial, delayed complete lymphadenectomy was performed only when nodal recurrences became clinically detectable. An interesting question is whether the trial provides sufficient evidence that immediate complete lymphadenectomy improves survival relative to delayed complete lymphadenectomy in patients with sentinel-node metastases. The design of this trial does not facilitate a direct comparison since the sentinel-node status is unknown in the nodal observation arm.

There is a large literature on treatment noncompliance. For instance, Sommer & Zeger (1991) studied the problem with a dichotomous outcome and all-or-none compliance. Their approach has been extended to handle ordered categorical compliance (Goethgebeur & Molenberghs, 1996), all-or-none compliance with contamination (Cuzick et al., 1997) and clustered binary outcomes (Albert, 2002). Several methods are available for a time-to-event outcome with right-censored data. Robins & Tsiatis (1991) proposed a structural failure time model based on potential outcomes. Loeys & Goetghebeur (2003), Loeys et al. (2005) and Cuzick et al. (2007) proposed methods based on proportional hazards models. Frangakis & Rubin (1999) studied a nonparametric procedure to evaluate treatment efficacy in a randomized trial with missing outcomes and all-or-none noncompliance and discussed an extension to right-censored time-to-event data. Loeys & Goetghebeur (2003) proposed a nonparametric plug-in estimator of the survival function for compliers in the control arm, but its large-sample properties have not been studied.

The purpose of this paper is to develop a nonparametric inference method to assess treatment efficacy in randomized clinical trials with a right-censored time-to-event outcome and all-or-none compliance. We derive large sample properties of the estimated subgroup survival probabilities, which lead to an asymptotic inference procedure to compare a survival probability for compliers in the treatment arm with that for the latent compliers in the control arm. We also provide a procedure to determine the onset and duration of the treatment effect.

2. Methods

2.1. Notation and assumptions

Suppose that n₂ subjects are randomized to the treatment arm and n₁ subjects to the control arm. Let n = n₁ + n₂. In the treatment arm, one observes n₂ independent and identically distributed triplets (X_2i, δ_2i, g_2i) (i = 1, …, n₂), where for the ith subject the observation time X_2i = min(T_2i, C_2i) is the minimum of a nonnegative continuous survival time T_2i and a censoring time C_2i, δ_2i = I (T_2i ⩽ C_2i) is a failure indicator, and g_2i is a binary compliance indicator, 1 for compliance. In the control arm, one observes n₁ independent and identically distributed pairs (X_1i, δ_1i) (i = 1, …, n₁), where for subject i, X_1i = min(T_1i, C_1i), δ_1i = I (T_1i ⩽ C_1i) and the binary compliance indicator g_1i is not observed. Table 1 defines the subgroup survival functions by study arms and compliance status.

Table 1.

Notation of survival functions by study arms and compliance status

	Treatment	Control
Compliers (gi = 1)	S2(t) = pr(T2i > t | g2i = 1)	S1(t) = pr(T1i > t | g1i = 1)
Noncompliers (gi = 0)	G2(t) = pr(T2i > t | g2i = 0)	G1(t) = pr(T1i > t | g1i = 0)
Overall	—	H1(t) = pr(T1i > t)

Let Λ₁(t), ${\mathrm{\Lambda }}_2^S\left(t\right)$ and ${\mathrm{\Lambda }}_2^G\left(t\right)$ denote the cumulative hazard functions of H₁(t), S₂(t) and G₂(t), respectively. In addition, let D₁(t) = pr(C_1i > t), $D_2^S\left(t\right)=\text{pr}(C_{2i}>t|g_{2i}=1)$ and $D_2^G\left(t\right)=\text{pr}(C_{2i}>t|g_{2i}=0)$.

The following assumptions are made throughout the paper.

Assumption 1. The compliance proportion is independent of randomization, i.e., pr(g_1i =1) = pr(g_2i = 1) ≡ p.

Assumption 2. The survival function of noncompliers is independent of randomization, i.e., G₁(t) = G₂(t), for all t.

Assumption 3. In the control arm, the censoring time is independent of the survival time. In the treatment arm, the censoring time is independent of the survival time conditional on the compliance status.

2.2. The estimator

The overall survival function for the control arm is H₁(t) = pr(g_1i = 1)S₁(t) + pr(g_1i = 0)G₁(t). This, together with Assumptions 1 and 2, implies that

$$S_1(t)=\{H_1(t)-(1-p)G_2(t)\}/p,$$

which leads to the plug-in estimator (Loeys & Goetghebeur, 2003)

$${\widehat{S}}_1(t)=\{{\widehat{H}}_1(t)-(1-\widehat{p}){\widehat{G}}_2(t)\}/\widehat{p}.$$ (1)

Here Ĥ₁(t) = Π_{X1i:n1 ⩽ t}{1 – 1/(n₁ – i + 1)}^δ_1i:n1 is the Kaplan & Meier (1958) estimator of H₁(t), where 0 ⩽ X_11:n1 ⩽ ⋯ ⩽ X_1n1:n1 are the ordered observation times in the control arm and δ_1i:n1 is the failure indicator for X_1i:n1, Ĝ₂(t) is the Kaplan–Meier estimator of G₂(t) based on the noncompliers in the treatment arm and $\widehat{p}={\sum }_{i=1}^{n_2}{g}_{2i}/n_2$.

Let Ŝ₂(t) be the Kaplan–Meier estimator of S₂(t) based on the compliers (g_2i = 1) in the treatment arm. We estimate S₂(t) – S₁(t), a measure of treatment efficacy, by Ŝ₂(t) – Ŝ₁(t).

Remark 1. Frangakis & Rubin (1999, § 5) derived a different estimator of S₁(t) that is consistent for S₁(t) under independent censoring conditional on compliance status in both arms. Because compliance status is unobserved in the control arm, the conditional independent censoring assumption in the control arm is untestable. Furthermore, the plug-in estimator defined in (1) may not be consistent under the assumptions of Frangakis & Rubin (1999) since the conditional independent censoring assumption in the control arm renders the Kaplan–Meier estimator Ĥ₁(t) generally inconsistent for H₁(t). We make different assumptions in Assumption 3 that the censoring time is independent of the survival time conditional on the compliance status in the treatment arm, but unconditionally independent in the control arm. Thus, under Assumption 3, Ĥ₁(t) is consistent for H₁(t) and Ĝ₂(t) is consistent for G₂(t). Consequently, the plug-in estimator Ŝ₁(t) is consistent for S₁(t).

2.3. Asymptotic properties

The counting process and martingale theory are used to study the asymptotic properties of Ŝ₁(t) and Ŝ₂(t) – Ŝ₁(t). For the control arm, define the at-risk processes

$$Y_{1i}(t)=I(X_{1i}⩾t),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Y}_{1(t)}=\sum _{i=1}^{n_1}{Y}_{1i}(t),$$

and counting processes

$$N_{1i}(t)=I(X_{1i}⩽t,{\delta }_{1i}=1),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{N}_1(t)=\sum _{i=1}^{n_1}{N}_{1i}(t).$$

Then $M_{1i}(t)=N_{1i}(t)-{\int }_0^t{Y}_{1i}(u)\hspace{0.17em}\text{d}{\mathrm{\Lambda }}_1(u)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=1,\dots ,n_1)$, are orthogonal locally integrable martingales with respect to the filtration 𝒡₁(t) = σ [{N_1i (u), Y_1i (u)}, i = 1, …, n₁ : 0 ⩽ u ⩽ t], the failure and censoring histories up to time t.

Similarly, let Y_2i (t) and N_2i (t) denote the corresponding at-risk process and counting process for subject i in the treatment arm. For the compliance subgroup in the treatment arm, define $Y_2^S(t)={\sum }_{i=1}^{n_2}\hspace{0.17em}{g}_{2i}\hspace{0.17em}{Y}_{2i}(t)$, $N_2^S(t)={\sum }_{i=1}^{n_2}\hspace{0.17em}{g}_{2i}\hspace{0.17em}{N}_{2i}(t)$ and $M_2^S(t)={\sum }_{i=1}^{n_2}\hspace{0.17em}{g}_{2i}\hspace{0.17em}{M}_{2i}^S(t)$, where $M_{2i}^S(t)=N_{2i}(t)-{\int }_0^t\hspace{0.17em}{Y}_{2i}(u)\text{d}{\mathrm{\Lambda }}_2^S(u)$. For the noncompliance subgroup in the treatment arm, define $Y_2^G(t)={\sum }_{i=1}^{n_2}(1-g_{2i})Y_{2i}(t)$, $N_2^G(t)={\sum }_{i=1}^{n_2}(1-g_{2i})N_{2i}(t)$ and $M_2^G(t)={\sum }_{i=1}^{n_2}(1-g_{2i})M_{2i}^G(t)$, where $M_{2i}^G(t)=N_{2i}(t)-{\int }_0^t{Y}_{2i}(u)\text{d}{\mathrm{\Lambda }}_2^G(u)$.

Let

$${\widehat{\mathrm{\Lambda }}}_1(t)={\int }_0^t\frac{\text{d}{N}_1(u)}{Y_1(u)},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{\mathrm{\Lambda }}}_2^S(t)={\int }_0^t\frac{\text{d}{N}_2^S(u)}{Y_2^S(u)},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{\mathrm{\Lambda }}}_2^G(t)={\int }_0^t\frac{\text{d}{N}_2^G(u)}{Y_2^G(u)}$$

be the Nelson–Aalen estimators of Λ₁(t), ${\mathrm{\Lambda }}_2^S(t)$ and ${\mathrm{\Lambda }}_2^G(t)$.

The following lemma is needed to derive the joint distribution of Ĥ₁(t), Ŝ₂(t), Ĝ₂(t) and p̂.

Lemma 1. Assume that Λ₁(τ) < ∞, ${\mathrm{\Lambda }}_2^S(\tau )<\infty$, ${\mathrm{\Lambda }}_2^G(\tau )<\infty$, D₁(τ) > 0, $D_2^S(\tau )>0$ and $D_2^G(\tau )>0$, where τ is usually the time when data collection ends. Assume n₁/n → ρ₁ and n₂/n → ρ₂ for 0 < ρ₁, ρ₂ < 1 as n → ∞. Then, for any t in (0, τ],

$$n^{1/2}\hspace{0.17em}\left(\begin{array}{c}{\widehat{H}}_1(t)-H_1(t)\\ {\widehat{S}}_2(t)-S_2(t)\\ {\widehat{G}}_2(t)-G_2(t)\\ \widehat{p}-p\end{array}\right)\to N\hspace{0.17em}\left(\left(\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right),\mathrm{\Sigma }(t)\right)$$

in distribution, as n → ∞, where the variance–covariance matrix is

$$\mathrm{\Sigma }(t)=\left(\begin{array}{cccc}{H_1}^2(t){\rho }_1^{-1}{\sigma }_{11}(t)& 0& 0& 0\\ 0& {S_2}^2(t){\rho }_2^{-1}{\sigma }_{22}(t)& 0& -S_2(t){\rho }_2^{-1}{\sigma }_{24}(t)\\ 0& 0& {G_2}^2(t){\rho }_2^{-1}{\sigma }_{33}(t)& -G_2(t){\rho }_2^{-1}{\sigma }_{34}(t)\\ 0& -S_2(t){\rho }_2^{-1}{\sigma }_{24}(t)& -G_2(t){\rho }_2^{-1}{\sigma }_{34}(t)& {\rho }_2^{-1}p(1-p)\end{array}\right),$$

with

$$\begin{array}{l}{\sigma }_{11}(t)=E\hspace{0.17em}\left[{\left\{{\int }_0^t\frac{\text{d}{M}_{1i}(u)}{H_1(u-)D_1(u-)}\right\}}^2\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sigma }_{22}(t)=E\hspace{0.17em}\left[{\left\{{\int }_0^t\frac{\text{d}{M}_{2i}^S(u)g_{2i}}{p{S}_2(u-)D_2^S(u-)}\right\}}^2\right],\\ {\sigma }_{33}(t)=E\hspace{0.17em}\left[{\left\{{\int }_0^t\frac{\text{d}{M}_{2i}^G(u)(1-g_{2i})}{(1-p)G_2(u-)D_2^G(u-)}\right\}}^2\right],\\ {\sigma }_{24}(t)=E\hspace{0.17em}\left\{(g_{2i}-p){\int }_0^t\frac{\text{d}{M}_{2i}^S(u)g_{2i}}{p{S}_2(u-)D_2^S(u-)}\right\},\\ {\sigma }_{34}(t)=E\hspace{0.17em}\left\{(g_{2i}-p){\int }_0^t\frac{\text{d}{M}_{2i}^G(u)(1-g_{2i})}{(1-p)G_2(u-)D_2^G(u-)}\right\}.\end{array}$$

Furthermore, σ₁₁(t), σ₂₂(t), σ₃₃(t), σ₂₄(t) and σ₃₄(t) can be consistently estimated by

$$\begin{array}{l}{\widehat{\sigma }}_{11}(t)=n_1\sum _{i=1}^{n_1}{\left\{{\int }_0^t\frac{\text{d}{\widehat{M}}_{1i}(u)}{Y_1(u)}\right\}}^2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{\sigma }}_{22}(t)=n_2\sum _{i=1}^{n_2}\hspace{0.17em}{\left\{{\int }_0^t\frac{\text{d}{\widehat{M}}_{2i}^S(u)g_{2i}}{Y_2^S(u)}\right\}}^2,\\ {\widehat{\sigma }}_{33}(t)=n_2\sum _{i=1}^{n_2}\hspace{0.17em}{\left\{{\int }_0^t\frac{\text{d}{\widehat{M}}_{2i}^G(u)(1-g_{2i})}{Y_2^G(u)}\right\}}^2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{\sigma }}_{24}(t)=\sum _{i=1}^{n_2}(g_{2i}-\widehat{p}){\int }_0^t\frac{\text{d}{\widehat{M}}_{2i}^S(u)g_{2i}}{Y_2^S(u)},\\ {\widehat{\sigma }}_{34}(t)=\sum _{i=1}^{n_2}(g_{2i}-\widehat{p}){\int }_0^t\frac{\text{d}{\widehat{M}}_{2i}^G(u)(1-g_{2i})}{Y_2^G(u)},\end{array}$$

where ${\widehat{M}}_{1i}(t)=N_{1i}(t)-{\int }_0^t{Y}_{1i}(u)d{\widehat{\mathrm{\Lambda }}}_1(u)$, ${\widehat{M}}_{2i}^S(t)=N_{2i}(t)-{\int }_0^t{Y}_{2i}(u)\text{d}{\widehat{\mathrm{\Lambda }}}_2^S(u)$ and ${\widehat{M}}_{2i}^G(t)=N_{2i}(t)-{\int }_0^t{Y}_{2i}(u)\text{d}{\widehat{\mathrm{\Lambda }}}_2^G(u)$.

The joint limiting distribution of Ŝ₁(t) and Ŝ₂(t) is stated below.

Theorem 1. Assume that the assumptions of Lemma 1 hold. At a given time-point t in (0, τ],

$$n^{1/2}\hspace{0.17em}\left(\begin{array}{c}{\widehat{S}}_1(t)-S_1(t)\\ {\widehat{S}}_2(t)-S_2(t)\end{array}\right)\to N\left\{\left(\begin{array}{c}0\\ 0\end{array}\right),\hspace{0.17em}\left(\begin{array}{cc}{\nu }_{11}(t)& {\nu }_{12}(t)\\ {\nu }_{12}(t)& {\nu }_{22}(t)\end{array}\right)\right\}$$

in distribution, as n → ∞, n₁/n → ρ₁ and n₂/n → ρ₂, where

$$\begin{array}{ll}{\nu }_{11}(t)=\hfill & \frac{H_1^2(t)}{p^2}{\rho }_1^{-1}{\sigma }_{11}(t)+\frac{{(1-p)}^2{G_2}^2(t)}{p^2}{\rho }_2^{-1}{\sigma }_{33}(t)+\frac{{\{H_1(t)-G_2(t)\}}^2}{p^4}{\rho }_2^{-1}{\sigma }_{44}\hfill \\ \hfill & -2\frac{(1-p)G_2(t)\{H_1(t)-G_2(t)\}}{p^3}{\rho }_2^{-1}{\sigma }_{34}(t),\hfill \end{array}$$ (2)

$${\nu }_{22}(t)={S_2}^2(t){\rho }_2^{-1}{\sigma }_{22}(t),$$ (3)

$${\nu }_{12}(t)=\frac{S_2(t)\{H_1(t)-G_2(t)\}}{p^2}{\rho }_2^{-1}{\sigma }_{24}(t).$$ (4)

The proofs of Lemma 1 and Theorem 1 are provided in the Appendix.

2.4. Pointwise confidence intervals for survival probabilities

Let ν̂₁₁(t), ν̂₂₂(t) and ν̂₁₂(t) be the consistent estimates of ν₁₁(t), ν₂₂(t) and ν₁₂(t) obtained by replacing the theoretical quantities in (2)–(4) with their consistent estimators. It follows from Theorem 1 that the 100(1 – α)% confidence intervals for S₁(t) and S₂(t) at a given t ∈ [0, τ] are given by

$${\widehat{S}}_1(t)\pm z_{1-\alpha /2}\hspace{0.17em}{\left\{\frac{{\widehat{\nu }}_{11}(t)}{n}\right\}}^{1/2},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{S}}_2(t)\pm z_{1-\alpha /2}\hspace{0.17em}{\left\{\frac{{\widehat{\nu }}_{22}(t)}{n}\right\}}^{1/2},$$

respectively, where z_1–α/2 is the upper α/2 quantile of the standard normal distribution.

Furthermore, a 100(1 – α)% confidence interval for S₂(t) – S₁(t) is given by

$$\{{\widehat{S}}_2(t)-{\widehat{S}}_1(t)\}\pm z_{1-\alpha /2}{n}^{-1/2}\widehat{\sigma }(t),$$

where σ̂²(t) = ν̂₁₁(t) + ν̂₂₂(t) – 2ν̂₁₂(t).

2.5. Onset and duration of treatment effect

In practice, one is often interested in the estimation of the time interval in which S₂(t) exceeds S₁(t). Below we provide a procedure to determine the onset and duration of the time interval with a confidence level 1 – α. The procedure is based on ideas from Berger & Boos (1999).

Step 1. Construct a one-sided, α/2-level test of H_0t : S₂(t) – S₁(t) = 0 versus H_at : S₂(t) – S₁(t) > 0 for a given t ∈ [0, τ]. Define Z_t = {Ŝ₂(t) – Ŝ₁(t)}/{n^−1/2σ̂(t)}. We reject the null hypothesis H_0t if Z_t > z_1–α/2.

Step 2. Choose starting value t_s ∈ [0, τ]. If Z_ts accepts H_0ts, no confidence statement is made. If Z_ts rejects H_0ts, we test sequentially downward and upward from t_s. Let L be the last t ⩽t_s for which H_0t is rejected, and U be the last t ⩾ t_s for which H_0t is rejected. Then [L, U] is the largest interval containing t_s for which Z_t rejects H_0t for all t ∈ [L, U].

Theorem 2. Let [L, U] be defined by the above algorithm. Then,

$$\text{pr}\{S_2(t)-S_1(t)>0,\hspace{0.17em}t\hspace{0.17em}\in \hspace{0.17em}[L,\hspace{0.17em}U]\}⩾1-\alpha ,$$

as n → ∞, n₁/n → ρ₁ and n₂/n → ρ₂ for some constants 0 < ρ₁, ρ₂ < 1.

As remarked by Berger & Boos (1999), the starting value t_s should be chosen before the study. Any starting point satisfying L ⩽ t_s ⩽ U will lead to the same interval [L, U]. The choice of t_s requires some prior information about the time frame during which S₂(t) is likely to be higher than S₁(t). If prior knowledge is unavailable, one can repeat the procedure at k different starting points and use level α/(2k) for each one-sided test to achieve the overall confidence level 1 – α.

3. Simulation studies

We carried out some simulations to investigate the finite sample performance of our method for different combinations of sample size, censoring rate and compliance proportion.

In the first simulation, the survival times were generated from Weibull distributions S₁(t) = exp(−t^0.8), S₂(t) = exp(−t^0.75/2) and G₁(t) = G₂(t) = exp(−t^0.7/4). The censoring times were generated from an exponential distribution with hazard rate λ chosen to give a prespecified overall censoring rate. For each simulation, 1000 Monte Carlo samples were generated. Table 2 reports the empirical mean of the plug-in estimate Ŝ₁(t), the empirical standard error of Ŝ₁(t), the empirical mean of estimated standard error of Ŝ₁(t) and the achieved coverage probability of the 95% confidence interval for S₁(t). For the moderate sample size n₁ = n₂ = 100, the plug-in estimator Ŝ₁(t) and its standard error have very small biases and the confidence interval has very small coverage probability error in almost all cases. The only exception is that when the censoring rate is 70%, the coverage rate, 90.9%, is notably lower than the nominal level 95% in the right tail, S(t) = 0.25. Hence, one should interpret the analysis results for the right tail with caution in the presence of heavy censoring. Furthermore, as the sample sizes increase to n₁ = n₂ = 500, the performance of our method is satisfactory in all cases considered.

Table 2.

Performance of Ŝ₁(t), estimated standard error and confidence interval

p	CR	S1(t)	n1 = n2 = 100				n1 = n2 = 500
			Ŝ1(t)	SE	$\widehat{SE}$	ACP	Ŝ1(t)	SE	$\widehat{SE}$	ACP
0.7	0.3	0.50	0.50	0.08	0.08	94.2	0.50	0.04	0.04	94.8
		0.25	0.25	0.09	0.09	94.7	0.25	0.04	0.04	94.9
	0.5	0.50	0.50	0.08	0.08	94.8	0.50	0.04	0.04	95.2
		0.25	0.25	0.10	0.10	94.6	0.25	0.04	0.04	95.6
	0.7	0.50	0.50	0.10	0.09	94.5	0.50	0.04	0.04	94.9
		0.25	0.24	0.15	0.13	90.9	0.25	0.06	0.06	95.1
0.5	0.3	0.50	0.50	0.12	0.11	94.5	0.50	0.05	0.05	96.0
		0.25	0.24	0.13	0.13	94.6	0.25	0.06	0.06	95.4
	0.5	0.50	0.50	0.12	0.12	94.1	0.50	0.05	0.05	95.9
		0.25	0.24	0.15	0.14	94.6	0.25	0.06	0.06	95.3
	0.7	0.50	0.50	0.14	0.13	94.3	0.50	0.06	0.06	95.4
		0.25	0.24	0.19	0.18	94.6	0.25	0.08	0.08	94.9
0.3	0.3	0.50	0.49	0.20	0.19	95.7	0.50	0.08	0.08	95.9
		0.25	0.24	0.23	0.23	96.1	0.25	0.10	0.10	95.8
	0.5	0.50	0.49	0.21	0.20	95.6	0.50	0.08	0.09	95.4
		0.25	0.23	0.25	0.24	95.3	0.25	0.10	0.11	95.4
	0.7	0.50	0.49	0.22	0.21	95.9	0.50	0.09	0.09	96.0
		0.25	0.23	0.30	0.29	95.7	0.25	0.13	0.13	96.2

Ŝ₁(t), the empirical mean of estimated survival probability; SE, the empirical standard error of Ŝ₁(t); $\widehat{SE}$, the empirical mean of estimated standard error of Ŝ₁(t); ACP, the achieved coverage probability of the 95% confidence interval for S₁(t); p, the compliance proportion; CR, the censoring rate.

A similar simulation was conducted under conditions that emulate the Multicenter Selective Lymphadenectomy Trial I, in which the overall censoring rate is 87%, the observed proportion of node-positive patients is 15% and the sample sizes are n₁ = 500 and n₂ = 764. In this simulation, the survival times were generated from Weibull distributions S₁(t) = exp(−t^0.87/48.3), S₂(t) = exp(−t^0.87/89.8) and G₁(t) = G₂(t) = exp(−t^0.79/117.2). The results are summarized in Table 3. It is seen that similar to the previous simulation, the plug-in estimate Ŝ₁(t) and its standard error estimate show small bias and the confidence interval has small coverage probability error except in the right tail.

Table 3.

Simulation designed to emulate the Multicenter Selective Lymphadenectomy Trial I

n1	n2	p	CR	S1(t)	Ŝ1(t)	SE	$\widehat{SE}$	ACP
500	764	0.15	0.87	0.50	0.51	0.25	0.25	95.30
				0.25	0.22	0.63	0.55	91.70

p, the sample size; CR, the censoring rate; Ŝ₁(t), the empirical mean of estimated survival probability; SE, the empirical standard error of Ŝ₁(t); $\widehat{SE}$, the empirical mean of estimated standard error of Ŝ₁ (t); ACP, the achieved coverage probability of the 95% confidence interval for S₁(t).

We conducted more simulations under different scenarios that include different survival distributions and a null case where S₁ = S₂. The performance of Ŝ₂(t) – Ŝ₁(t) was also investigated. The results were similar to those for Ŝ₁(t) and thus omitted. We also carried out a small sensitivity analysis of Assumption 2 and did not observe serious estimation bias and coverage error under moderate censoring when Assumption 2 is slightly violated.

4. An example

We applied our method to analyse the Multicenter Selective Lymphadenectomy Trial I described in § 1. Between January 1994 and March 2002, 769 patients were randomized to the sentinel-node biopsy arm and 500 patients to the nodal observation arm. Our analysis excludes five patients in the biopsy arm whose sentinel-node status was unavailable. Immediate complete lymphadenectomy was performed on 122 of 764 patients in the biopsy arm whose biopsies were positive for sentinel-node metastases. For other patients, delayed complete lymphadenectomy was performed upon clinically observable nodal relapse.

To apply our method, it is important that Assumption 2 in § 2.1 is reasonable for this trial. Because removing the sentinel node among the node-negative patients is not expected to change their survival experience, it is reasonable to assume that G₁(t) = G₂(t), where G₁(t) and G₂(t) represent the survival probabilities of node-negative patients in the observation arm and the biopsy arm, respectively.

We applied our method to estimate the benefit of immediate versus delayed complete lymphadenectomy on patients with sentinel-node metastases with respect to melanoma-specific survival, i.e., survival until death due to melanoma. Let S₁(t) and S₂(t) be the survival probabilities for patients with sentinel-node metastases in the observation arm and the biopsy arm, respectively. Table 4 reports the estimated survival probabilities Ŝ₁(t) and Ŝ₂(t), the estimated survival difference Ŝ₂(t) – Ŝ₁(t) and their estimated standard errors at some given time-points. It is seen from Table 4 that the 95% confidence interval for S₂(t) – S₁(t) is (0.02, 0.42) at 2.5 years, which implies that the 2.5-year melanoma-specific survival probability for patients with sentinel-node metastases in the biopsy arm is significantly higher than that of the observation arm at 0.05 significance level. Furthermore, using the procedure in § 2.5, we found that immediate complete lymphadenectomy significantly improves melanoma-specific survival relative to delayed complete lymphadenectomy for patients with sentinel-node metastases over the time interval [2.05, 2.97] years at 95% confidence level.

Table 4.

Melanoma-specific survival probabilities, and standard errors in parentheses, for patients with sentinel-node metastases

	t (years)
	1	2	2.5	3
Delayed complete lymphadenectomy Ŝ1(t)	0.90 (0.04)	0.75 (0.08)	0.65 (0.09)	0.64 (0.10)
Immediate complete lymphadenectomy Ŝ2(t)	0.97 (0.02)	0.88 (0.03)	0.86 (0.03)	0.83 (0.04)
Difference Ŝ2(t) – Ŝ1(t)	0.06 (0.04)	0.13 (0.08)	0.22 (0.10)	0.20 (0.11)

5. Discussion

Most existing methods for assessing treatment efficacy compare either mean survival times (Robins & Tsiatis, 1991) or hazard rates (Loeys & Goetghebeur, 2003; Loeys et al., 2005; Cuzick et al., 2007) between study arms using parametric or semiparametric models. The difference in subgroup survival probabilities provides a useful alternative measure of treatment efficacy. Our method is fully nonparametric. If the proportional hazards assumption between S₁(t) and S₂(t) holds, our nonparametric method may not be as efficient as a proportional hazards model-based method. On the other hand, our method is more robust when the hazards of S₁(t) and S₂(t) are not proportional.

As illustrated in our simulations, caution is needed when drawing inference in the right tail of a survival distribution for moderate sample size, especially under heavy censoring and extremely unbalanced compliance proportions.

The focus of this paper is on comparison of subgroup survival probabilities at a fixed time-point based on the plug-in estimator Ŝ₁(t). As pointed out by Loeys & Goetghebeur (2003), Ŝ₁(t) is not a proper estimate of the entire survival curve because it is not monotonically non-increasing. One possible solution is to apply isotonic regression to the plug-in estimator to obtain a proper survival function. Properties of the resulting estimator, however, are difficult to study. An alternative approach is to consider nonparametric maximum likelihood estimation of S₁.

Appendix. Proofs

To prove the main results, we need the following lemma.

Lemma A1. Under the assumptions of Lemma 1, we have ${\text{sup}}_{t\in [0,\tau ]}\hspace{0.17em}|Y_2^S(t)/n_2-y_2^S(t)|\to 0$, ${\text{sup}}_{t\in [0,\tau ]}\hspace{0.17em}|Y_2^G(t)/n_2-y_2^G(t)|\to 0$ and sup_{t ∈ [0,τ]} |Y₁(t)/n₁ – y₁(t)| → 0, in probability, where $y_2^S(t)=p{S}_2(t-)D_2^S(t-)$, $y_2^G(t)=(1-p)G_2(t-)D_2^G(t-)$ and y₁(t) = H₁(t−) D₁(t−).

Proof. For each i = 1, …, n₂, Y_2i = I (X_2i ⩾ t) is a left-continuous nonincreasing random function. We assume that S₂(t) and D₂(t) are continuous functions. At a given time-point t ∈ [0, τ],

$$E\{g_{2i}{Y}_{2i}(t)\}=\text{pr}(g_{2i}=1)\text{pr}\{Y_{2i}(t)=1|g_{2i}=1\}=p{S}_2(t-)D_2^S(t-).$$

By the weak law of large numbers, at each given time-point t,

$$\left|\frac{{\sum }_{i=1}^{n_2}{g}_{2i}{Y}_{2i}(t)}{n_2}-p{S}_2(t-)D_2^S(t-)\right|\to 0$$

in probability, as n₂ → ∞. By the dominated convergence theorem,

$$E\{g_{2i}\hspace{0.17em}{Y}_{2i}(t+)\}=E\hspace{0.17em}\left\{\underset{k\to \infty }{\text{lim}}{g}_{2i}\hspace{0.17em}{Y}_{2i}\hspace{0.17em}\left(t+\frac{1}{k}\right)\right\}=\underset{k\to \infty }{\text{lim}}E\hspace{0.17em}\left\{g_{2i}\hspace{0.17em}{Y}_{2i}\hspace{0.17em}\left(t+\frac{1}{k}\right)\right\}=p{S}_2(t)D_2^S(t).$$

Thus, as $n_2\to \infty |\{{\sum }_{i=1}^{n_2}\hspace{0.17em}{g}_{2i}\hspace{0.17em}{Y}_{2i}(t+)\}/n_2-p{S}_2(t)D_2^S(t)|\to 0$ in probability. Using arguments similar to those in the proof of Chung (2001, Theorem 5.5.1), it can be shown that ${\text{sup}}_{t\in [0,\tau ]}\hspace{0.17em}|Y_2^S(t)/n_2-y_2^S(t)|\to 0$ in probability, as n₂ → ∞, where $y_2^S(t)=p{S}_2(t-)D_2^S(t-)$.

The other results can be proved along the same lines.

Proof of Lemma 1. It is well known that Λ̂₁(t), ${\widehat{\mathrm{\Lambda }}}_2^S(t)$ and ${\widehat{\mathrm{\Lambda }}}_2^G(t)$ are consistent estimators of Λ₁(t), ${\mathrm{\Lambda }}_2^S(t)$ and ${\mathrm{\Lambda }}_2^G(t)$, respectively (Andersen et al., 1993), and that p̂ is a consistent estimator of p. It is also clear that Λ̂₁(t) is independent of ${\widehat{\mathrm{\Lambda }}}_2^S(t)$, ${\widehat{\mathrm{\Lambda }}}_2^G(t)$ and p̂. Furthermore, ${\mathrm{\Lambda }}_2^S(t)$ and ${\widehat{\mathrm{\Lambda }}}_2^G(t)$ are independent. We have

$$n_1^{1/2}\{{\widehat{\mathrm{\Lambda }}}_1(t)-{\mathrm{\Lambda }}_1(t)\}=n_1^{-1/2}\hspace{0.17em}{\int }_0^t\frac{\text{d}{M}_1(u)}{y_1(u)}+n_1^{-1/2}\hspace{0.17em}{\int }_0^t\hspace{0.17em}\left\{\frac{1}{Y_1(u)/n_1}-\frac{1}{y_1(u)}\right\}\hspace{0.17em}\text{d}{M}_1(u).$$ (A1)

Assuming Λ₁(τ) < ∞, $n_1^{-1/2}\hspace{0.17em}{M}_1$ converges weakly to a zero-mean Gaussian process by the martingale central limit theorem (Andersen et al., 1993). It follows from Lin & Ying (2001, Lemma A.1) and our Lemma A1 that $n_1^{-1/2}\hspace{0.17em}{\int }_0^t[1/\{Y_1(u)/n_1\}-1/y_1(u)]\hspace{0.17em}\text{d}{M}_1(u)\to 0$ in probability uniformly in t ∈ [0, τ]. This, together with (A1), implies that

$$n^{1/2}\{{\widehat{\mathrm{\Lambda }}}_1(t)-{\mathrm{\Lambda }}_1(t)\}=n_1^{-1/2}{\rho }_1^{-1/2}\sum _{i=1}^{n_1}{\int }_0^t\frac{\text{d}{M}_{1i}(u)}{y_1(u)}+o_p(1)$$ (A2)

uniformly in t ∈ [0, τ]. Similarly, we can show that as n → ∞ and n₂/n → ρ₂,

$$n^{1/2}\{{\widehat{\mathrm{\Lambda }}}_2^S(t)-{\mathrm{\Lambda }}_2^S(t)\}=n_2^{-1/2}{\rho }_2^{-1/2}\sum _{i=1}^{n_1}{\int }_0^t\frac{\text{d}{M}_{2i}^S(u)g_{2i}}{y_2^S(u)}+o_p(1)$$ (A3)

and

$$n^{1/2}\{{\widehat{\mathrm{\Lambda }}}_2^G(t)-{\mathrm{\Lambda }}_2^G(t)\}=n_2^{-1/2}{\rho }_2^{-1/2}\sum _{i=1}^{n_2}{\int }_0^t\frac{\text{d}{M}_{2i}^G(u)(1-g_{2i})}{y_2^G(u)}+o_p(1)$$ (A4)

uniformly in t ∈ [0, τ].

Let 𝒟 [0, τ]³ be the metric space consisting of {f₁(t), f₂(t), f₃(t)}, where f_k : [0, τ] → R for k = 1, 2, 3 are right-continuous functions with left limits. The metric of 𝒟[0, τ]³ is defined as d(f, g) = max_{t ∈ [0,τ]}{‖f_k (t) – g_k(t) ‖: 1 ⩽ k ⩽ 3} for f, g ∈ 𝒟[0, τ]³. It is easy to see from equations (A2)–(A4) that the stochastic process n^1/2[{Λ̂₁(t) – Λ₁(t)}, $\{{\widehat{\mathrm{\Lambda }}}_2^S(t)-{\mathrm{\Lambda }}_2^S(t)\}$, $\{{\widehat{\mathrm{\Lambda }}}_2^G(t)-{\mathrm{\Lambda }}_2^G(t)\}$, (p̂ – p)] in 𝒟[0, τ]³ × R is asymptotically equivalent to a sum of independent and identically distributed random vectors. By the multivariate central limit theorem, its finite-dimensional distributions converge asymptotically to zero-mean multivariate normal distributions. Moreover, because the elements, n^1/2{Λ̂₁(t) – Λ₁(t)}, n^1/2 $\{{\widehat{\mathrm{\Lambda }}}_2^S(t)-{\mathrm{\Lambda }}_2^S(t)\}$ and $n^{1/2}\{{\widehat{\mathrm{\Lambda }}}_2^G(t)-{\mathrm{\Lambda }}_2^G(t)\}$, are square-integrable martingales with respect to their marginal filtrations, their tightness follows from the proof of Pollard (1984, Theorem VIII.13). Hence, $n^{1/2}[\{{\widehat{\mathrm{\Lambda }}}_1(t)-{\mathrm{\Lambda }}_1(t)\},\hspace{0.17em}\{{\widehat{\mathrm{\Lambda }}}_2^S(t)-{\mathrm{\Lambda }}_2^S(t)\},\{{\widehat{\mathrm{\Lambda }}}_2^G(t)-{\mathrm{\Lambda }}_2^G(t)\},\hspace{0.17em}(\widehat{p}-p)]$ converges weakly to a zero-mean Gaussian stochastic process {𝒲₁(t), 𝒲₂(t), 𝒲₃(t), 𝒲₄} in 𝒟[0, τ]³ × R with the variance-covariance functions between 𝒲₁(t₁), 𝒲₂(t₂), 𝒲₃(t₃) and 𝒲₄ given by

$$\left(\begin{array}{cccc}{\rho }_1^{-1}{\sigma }_{11}(t_1)& 0& 0& 0\\ 0& {\rho }_2^{-1}{\sigma }_{22}(t_2)& 0& {\rho }_2^{-1}{\sigma }_{24}(t_2)\\ 0& 0& {\rho }_2^{-1}{\sigma }_{33}(t_3)& {\rho }_2^{-1}{\sigma }_{34}(t_3)\\ 0& {\rho }_2^{-1}{\sigma }_{24}(t_2)& {\rho }_2^{-1}{\sigma }_{34}(t_3)& {\rho }_2^{-1}p(1-p)\end{array}\right).$$

Recall that Ĥ₁(t) = Π_u⩽t{1–dΛ̂₁(u)}, ${\widehat{S}}_2(t)={\prod }_{u⩽t}\{1-\text{d}{\widehat{\mathrm{\Lambda }}}_2^S(u)\}$ and ${\widehat{G}}_2(t)={\prod }_{u⩽t}\{1-\text{d}{\widehat{\mathrm{\Lambda }}}_2^G(u)\}$, respectively. It follows from the functional delta method (Andersen et al., 1993) that the joint stochastic process n^1/2[{Ĥ₁(t) – H₁(t)}, {Ŝ₂(t) – S₂(t)}, {Ĝ₂(t) – G₂(t)}, (p̂ – p)] converges weakly to a zero-mean Gaussian process { ${𝒲}_1^{*}(t)$, ${𝒲}_2^{*}(t)$, ${𝒲}_3^{*}(t)$, ${𝒲}_4^{*}$} with the variance-covariance function between ${𝒲}_1^{*}(t_1)$, ${𝒲}_2^{*}(t_2)$, ${𝒲}_3^{*}(t_3)$ and ${𝒲}_4^{*}$ given by

$$\left(\begin{array}{cccc}{H_1}^2(t_1){\rho }_1^{-1}{\sigma }_{11}(t_1)& 0& 0& 0\\ 0& {S_2}^2(t_2){\rho }_2^{-1}{\sigma }_{22}(t_2)& 0& -S_2(t_2){\rho }_2^{-1}{\sigma }_{24}(t_2)\\ 0& 0& {G_2}^2(t_3){\rho }_2^{-1}{\sigma }_{33}(t_3)& -G_2(t_3){\rho }_2^{-1}{\sigma }_{34}(t_3)\\ 0& -S_2(t_2){\rho }_2^{-1}{\sigma }_{24}(t_4)& -G_2(t_3){\rho }_2^{-1}{\sigma }_{34}(t_3)& {\rho }_2^{-1}p(1-p)\end{array}\right).$$

Therefore, for any t ∈ [0, τ], n^1/2[{Ĥ₁(t) – H₁(t)}, {Ŝ₂(t) – S₂(t)}, {Ĝ₂(t) – G₂(t)}, (p̂ – p)] converges to a multivariate normal distribution with mean zero and variance-covariance matrix Σ(t).

To prove the consistency of σ̂₁₁(t), we have

$$\begin{array}{ll}{\widehat{\sigma }}_{11}(t)=\hfill & \frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}{\left[{\int }_0^t\left\{\frac{1}{Y_1(u)/n_1}-\frac{1}{y_1(u)}\right\}\hspace{0.17em}\text{d}{\widehat{M}}_{1i}(u)\right]}^2\hfill \\ \hfill & +\frac{2}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}\left[{\int }_0^t\hspace{0.17em}\left\{\frac{1}{Y_1(u)/n_1}-\frac{1}{y_1(u)}\right\}\hspace{0.17em}\text{d}{\widehat{M}}_{1i}(u)\hspace{0.17em}{\int }_0^t\frac{\text{d}{\widehat{M}}_{1i}(u)}{y_1(u)}\right]+\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}{\left\{{\int }_0^t\frac{\text{d}{\widehat{M}}_{1i}(u)}{y_1(u)}\right\}}^2.\hfill \end{array}$$

By the uniform consistency of Λ̂₁(t) and Y₁(t)/n₁ and the fact that |Y_1i (t)| ⩽ 1, we have

$$\begin{array}{r}\hfill \underset{t\in [0,\tau ]}{\text{sup}}\left|\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}{\left[{\int }_0^t\hspace{0.17em}\left\{\frac{1}{Y_1(u)/n_1}-\frac{1}{y_1(u)}\right\}\hspace{0.17em}\text{d}{\widehat{M}}_{1i}(u)\right]}^2\right|\to 0,\\ \hfill \underset{t\in [0,\tau ]}{\text{sup}}\left|\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}\left[{\int }_0^t\left\{\frac{1}{Y_1(u)/n_1}-\frac{1}{y_1(u)}\right\}\hspace{0.17em}d{\widehat{M}}_{1i}(u)\hspace{0.17em}{\int }_0^t\frac{\text{d}{\widehat{M}}_{1i}(u)}{y_1(u)}\right]\right|\to 0,\\ \hfill \underset{t\in [0,\tau ]}{\text{sup}}\hspace{0.17em}\left|\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}{\left\{{\int }_0^t\frac{\text{d}{\widehat{M}}_{1i}(u)}{y_1(u)}\right\}}^2-\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}\left\{{\int }_0^t\frac{\text{d}{M}_{1i}(u)}{y_1(u)}\right\}\right|\to 0,\end{array}$$

in probability. Thus, ${\widehat{\sigma }}_{11}(t)=n_1^{-1}{\sum }_{i=1}^{n_1}{\{{\int }_0^t\text{d}{M}_{1i}(u)/y_1(u)\}}^2+o_p(1)$ uniformly in t ∈ [0, τ]. Furthermore, it follows from the uniform law of large numbers (Newey & McFadden, 1994, Lemma 2.4) that

$$\underset{t\in [0,\tau ]}{\text{sup}}\left|\frac{1}{n_1}\sum _{i=1}^{n_1}\hspace{0.17em}{\left\{{\int }_0^t\frac{\text{d}{M}_{1i}(u)}{y_1(u)}\right\}}^2-E\left[{\left\{{\int }_0^t\frac{\text{d}{M}_{1i}(u)}{y_1(u)}\right\}}^2\right]\right|\to 0$$

in probability. Therefore, ${\widehat{\sigma }}_{11}(t)\to E[{\{{\int }_0^t\text{d}{M}_{1i}(u)/y_1(u)\}}^2]\equiv {\sigma }_{11}(t)$ in probability uniformly in t ∈ [0, τ].

The uniform consistency of σ̂₂₂(t), σ̂₃₃(t), σ̂₂₄(t) and σ̂₃₄(t) can be proved similarly. The consistency of p̂(1 – p̂) follows directly from the weak law of large numbers.

Proof of Theorem 1. It follows from Lemma 1 and the delta method that at a given time-point t ∈ [0, τ ], n^1/2[{Ŝ₁(t) – S₁(t)}, {Ŝ₂(t) – S₂(t)}] is asymptotically normally distributed with mean zero and variance-covariance matrix f Σ(t) f′, where

$$f=\left(\begin{array}{cccc}1/p& 0& -(1-p)/p& -\{H_1(t)-G_2(t)\}/p^2\\ 0& 1& 0& 0\end{array}\right).$$

Applying Slutsky’s theorem, we have ν̂₁₁(t) → ν₁₁(t), ν̂₂₂(t) → ν₂₂(t) and ν̂₁₂(t) → ν₁₂(t) in probability as n → ∞, n₁/n → ρ₁ and n₂/n → ρ₂.

Proof of Theorem 2. The theorem can be proved using essentially the same arguments given in the appendix of Berger & Boos (1999). Thus, we omit the details.