Inferences on Relative Failure Rates in Stratified Mark-Specific Proportional Hazards Models with Missing Marks, with Application to Human Immunodeficiency Virus Vaccine Efficacy Trials (2024)

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Journal of the Royal Statistical Society Series C: Applied Statistics, Volume 64, Issue 1, January 2015, Pages 49–73, https://doi.org/10.1111/rssc.12067

Article history

Summary

1. Introduction

The primary objective of a preventive human immunodeficiency virus (HIV) vaccine efficacy trial is to assess vaccine efficacy VE to prevent HIV infection, where typically VE is defined as 1 minus the hazard ratio (vaccine/placebo) of HIV infection diagnosis. However, the great genetic variability of HIV poses a central challenge to developing a highly efficacious vaccine (Fauci et al., 2008). The trial population is exposed to many HIV genotypes but the vaccine contains only a few, and the vaccine is less likely to protect against HIVs with greater genetic distance from the sequences inside the vaccine (Gilbert et al., 1999). The trial has objectives to assess whether and how the vaccine impacts the infection rate with any HIV genotype and whether and how the vaccine effect varies by HIV genotype; assessment of this objective has been named ‘sieve analysis’ (Gilbert et al., 1998). Gilbert et al. (2008), Sun et al. (2009) and Sun and Gilbert (2012) developed sieve analysis methods using the competing risks failure time framework (Prentice et al., 1978), which attach a continuous ‘mark’ variable to HIV-infected subjects that measures the genetic distance of an infecting HIV sequence to a sequence inside the vaccine. The goal of the sieve analysis methods is evaluation of mark-specific vaccine efficacy, which here is defined as 1 minus the mark-specific hazard ratio (vaccine/placebo) of infection. Beyond HIV, the methods apply generally to any preventative vaccine efficacy trial for which the pathogen targeted by the vaccine is genetically diverse, which includes influenza, malaria, tuberculosis, dengue, streptococcus pneumoniae, human papilloma virus and hepatitis C virus.

Gilbert et al. (2008) and Sun et al. (2009) assumed no missing mark data in infected subjects, whereas Sun and Gilbert (2012) allowed marks missing at random. In practice there are missing marks; for example in the Vax004 trial 32 of 368 infected subjects had no HIV sequence data (Gilbert et al., 2008), owing to drop-out or to inability of the HIV sequencing technology to measure the infecting HIV sequence, and in the ‘Step’ trial 22 of 88 infected subjects had no HIV sequence data (Rolland et al., 2011). Whereas it is of scientific interest to evaluate a mark defined on the basis of the earliest available HIV sequence, a mark of particular scientific interest is defined on the basis of an HIV sequence measured near the time of acquisition, which is missing in a much larger fraction of infected subjects owing to the periodic (typically 6-monthly) diagnostic tests for HIV infection. Specifically, HIV sequences are measured from the earliest available post-infection blood sample, and a ‘near acquisition’ or ‘early’ sample may be defined as a sample that has been documented to be sufficiently near acquisition. In the Step trial, only 23 of the 66 infected subjects with sequence data had an early mark measured, defined as sampling within 3 weeks. Sun and Gilbert (2012) have provided details on the HIV testing algorithm that is used to define an early mark.

Sun and Gilbert (2012) is the only reference on sieve analysis that accommodates missing continuous marks. It develops two valid estimation approaches based on the stratified mark-specific proportional hazards model. The first uses inverse probability weighting (IPW) of the complete-case estimator, which leverages auxiliary predictors of whether the mark is observed, whereas the second, adapting Robins et al. (1994), augments the IPW complete-case estimator with auxiliary predictors of the missing marks. Sun and Gilbert (2012) restricted attention to estimation methods, and this paper is a sequel that develops corresponding inferential or hypothesis testing methods based on the augmented IPW estimator. An important new component of this work compared with the previous work is to centre it on the sieve analysis of the RV144 Thai trial, which recently delivered the landmark result that a prime boost HIV vaccine appeared to provide partial protection against HIV infection (estimated VE =31%; 95% confidence interval 1–51%; p = 0.04; Rerks-Ngarm et al. (2009)). This result has stimulated intense interest in sieve analysis, for two reasons. First, there is controversy about whether the vaccine is really partially working versus a false positive result (Gilbert et al., 2011), and the sieve analysis of HIV sequences can help to resolve this question. In particular, if evidence is found that the vaccine efficacy declines with genetic distance, and the distance is defined on the basis of known parts of HIV that contain putatively protective antibody epitopes, then an interpretation of real vaccine efficacy is supported. Secondly, the HIV vaccine field is grappling with how to modify the tested vaccine to increase its potential vaccine efficacy for the next efficacy trial, and understanding the relationship between vaccine efficacy and the genetic distance provides direct guidance on which HIV sequences to put inside the next generation vaccines.

This paper is organized as follows. Notation, assumptions and the stratified mark-specific proportional hazards model are introduced in Section 2. Background on the estimation procedures that are needed for the testing procedures are described in Section 3. The testing procedures are developed, and asymptotic properties described, in Section 4. The finite sample performances of the tests are evaluated via simulations in Section 5. The application to the Thai trial is given in Section 6, and the asymptotic results and their proofs are placed in Appendix A.

The programs that were used to analyse the data can be obtained from

https://academic.oup.com/jrsssc/issue/

2. Model and missing mark data

2.1. Stratified mark-specific proportional hazards model

Let T be the failure time, V a continuous mark variable with bounded support [0,1] and Z(t) a possibly time-dependent p-dimensional covariate. The mark V is observable only when T is observed. Suppose that the conditional mark-specific hazard function at time t given the covariate history Z(s), for s≤t, depends on the current value Z(t) only. We consider the stratified mark-specific proportional hazards model

where $λk{t,v|z(t)}$ is the conditional mark-specific hazard function given covariate z(t) for an individual in the kth stratum, $λ0k(·,v)=λk{t,v|z(t)=0}$ is the unspecified baseline hazard function for the kth stratum, β(v) is the p-dimensional unknown regression coefficient function of v and K is the number of strata. Model (1) allows different baseline functions for different strata and flexibly allows for arbitrary mark-specific infection hazards over time in the placebo group. In practice, different key subgroups (e.g. men and women in the Thai trial) are assigned different baseline mark-specific hazards of HIV infection.

Arranging $β(v)=(β1(v),β2T(v))T$⁠, so that β₁(v) is the coefficient for vaccination status and β₂(v) for other covariates, the covariate- and stratum-adjusted mark-specific vaccine efficacy VE(v)=1− exp {β₁(v)}. Sun et al. (2009) developed some statistical procedures for model (1) under K = 1 based on observations of the random variables (X,Z(·),V) for δ = 1 and (X,Z(·)) for δ = 0, where X = min{T,C}, δ = I(T≤C) and C is a censoring random variable. Sun and Gilbert (2012) developed estimation procedures for model (1) allowing V to be missing for some subjects with δ = 1; these methods incorporate auxiliary covariates and/or auxiliary mark variables that inform about the probability V that is observed and about the distribution of V. This paper develops parallel hypothesis testing procedures for assessing VE(v). As summarized in Section 1, the two objectives are to assess whether the vaccine efficacy ever deviates from 0 (i.e. test VE(v)=0) and to assess whether the vaccine efficacy changes with the mark (i.e. test VE(v)=VE).

2.2. Missing data assumptions

Let R be the indicator of whether all possible data are observed for a subject; R = 1 if either δ = 0 (right censored) or if δ = 1 and V is observed; and R = 0 otherwise. Auxiliary variables A may be helpful for predicting missing marks. Since the mark can only be missing for failures, supplemental information is potentially useful only for failures, for predicting missingness and for informing about the distribution of missing marks. For example, if V is defined on the basis of the early virus, then V^*, the auxiliary mark information, may include sequences of later sampled viruses and can be considered a subset of A. In general, A could include multiple viral sequences per infected subject at multiple time points, giving information on intrasubject HIV evolution. The relationship between A and V can be modelled to help to predict V (see Section 5 for a simulated example).

We assume that C is conditionally independent of (T,V) given Z(·) and the stratum. We also assume that V is missing at random (Rubin, 1976), i.e., given δ = 1 and W=(T,Z(T),A), the probability that V is missing depends only on the observed W, not on the value of V; this assumption is expressed as

Let $πk(Q)=P(R=1|Q)$ where Q=(δ,W). Then π_k(Q)=δr_k(W)+1−δ. The missingness at random (MAR) assumption (2) also implies that V is independent of R given Q:

Define $rk(w)=P(R=1|δ=1,W=w)$ and $ρk(v,w)=P(V⩽v|δ=1,W=w)$⁠. The stratum-specific definitions of r_k(w) and ρ_k(v,w) allow the models of the probability of complete-case and of the mark distribution to differ across strata.

Let τ be the end of the follow-up period, and $nk$ be the number of subjects in the kth stratum; the total sample size is $n=Σk=1Knk$⁠. Let ${Xki,Zki(·),δki,Rki,Vki,Aki;i=1,…,nk}$ be independent and identically distributed replicates of {X,Z(·),δ, R,V,A} from the kth stratum. The observed data are ${Oki;i=1,…,nk,k=1,…,K}$⁠, where O_ki={X_ki,Z_ki(·),R_ki, R_kiV_ki,A_ki} for δ_ki=1 and O_ki={X_ki,Z_ki(·),R_ki=1} for δ_ki=0. We assume that the O_ki are independent for all subjects.

2.3. Hypotheses to test

We develop procedures for testing the following two sets of hypotheses. Let [a,b]⊂(0,1). The first set of hypotheses is

The second set of hypotheses is

The null hypothesis H₁₀ implies that the vaccine affords no protection (nor increased risk) against any HIV genotype. The ordered alternative H_1m indicates that the vaccine provides protection for at least some of the HIV genotypes, whereas H_1a indicates that the vaccine provides protection and/or increased risk for some HIV genotypes. The null hypothesis H₂₀ implies that there is no difference in vaccine protection against different HIV genotypes. The ordered alternative H_2m indicates that vaccine efficacy decreases with v and H_2a indicates that the vaccine efficacy changes with v. With β₁(v) the first component of β(v), the first set of hypotheses is equivalent to H₁₀:β₁(v)=0 for v ∈ [a,b] versusH_1a:β₁(v)≠0 for some v or H_1m:β₁(v)≤0 with strict inequality for some v. The second set of hypotheses is equivalent to H₂₀:β₁(v) does not depend on v ∈ [a,b] versusH_2a:β₁(v) depends on v or H_2m:β₁(v) increases as v increases. We develop testing procedures for detecting departures from H₁₀ in the direction of H_1m and H_1a and for detecting departures from H₂₀ in the direction of H_2m and H_2a. The procedures are developed on the basis of the augmented IPW complete-case estimator that was developed by Sun and Gilbert (2012).

3. Estimation procedure with missing marks

The augmented IPW (AIPW) estimator for model (1) is obtained in two stages. First the IPW complete-case estimator is derived and second the AIPW estimator is obtained, which improves efficiency by accounting for information in the conditional distribution of V given the auxiliaries.

Let r_k(W_ki, ψ_k) be the parametric model for the probability of the complete case, with r_k(W_ki) defined in equation (2), where W_ki=(T_ki,Z_ki(T_ki),A_ki) and ψ_k is a q-dimensional parameter. For example, we can assume the logistic model with $logit{rk(Wki,ψk)}=ψkTWki$ for those with δ_ki=1, where W_ki=(T_ki,Z_ki(T_ki),A_ki). By equation (2), the maximum likelihood estimator $ψ^=(ψ^1,…,ψ^K)$ of ψ=(ψ₁,…,ψ_K) is obtained by maximizing the observed data likelihood

Let K(x) be a kernel function with support [−1,1] and let h = h_n be a bandwidth. Let N_ki(t,v)=I(X_ki≤t,δ_ki=1,V_ki≤v) and Y_ki(t)=I(X_ki≥t). Let Q_ki=(δ_ki,W_ki) and π_k(Q_ki, ψ_k)=δ_kir_k(W_ki, ψ_k)+1−δ_ki. The first-stage estimator is the IPW estimator $β^ipw(v)$⁠, which solves the estimating equation for β$Uipw(v,β,ψ^)=0$⁠, where

Here

and

for j = 0,1, where z^⊗0=1 and z^⊗1=z for any $z∈Rp$⁠. The score function (5) can be viewed as an extension of the score function that is used for the cause-specific Cox model (Prentice et al., 1978) for a particular failure cause J = j, for which the counting process counts only events of type j. It borrows strength from observations having marks in the neighbourhood of v. The kernel function is designed to give greater weight to observations with marks near v than those further away.

The baseline function λ_0k(t,v) can be estimated by $λ^0kipw(t,v)$⁠, obtained by smoothing the increments of the following estimator of the doubly cumulative baseline function $Λ0k(t,v)=∫0t∫0vλ0k(s,u)dsdu$⁠:

For example, one can use the following kernel smoothing:

where $Kh1(1)(x)=K(1)(x/h1)/h1$ and $Kh2(2)(x)=K(2)(x/h2)/h2$⁠, with K⁽¹⁾(·) and K⁽²⁾(·) the kernel functions and $h1$ and $h2$ the bandwidths.

Following Robins et al. (1994), Sun and Gilbert (2012) proposed a more efficient procedure for estimating equation (1) by incorporating the knowledge of ρ_k(w,v) in the estimation procedure. Let w=(t,z,a) and $gk(a|t,v,z)=P(Aki=a|Tki=t,Vki=v,Zki=z,δki=1)$⁠. Then

If no auxiliary variables are available or if A_ki is conditionally independent of V_ki given (T_ki,Z_ki, δ_ki), then

In this case, ρ_k(w,v) can be estimated by

where $λ^kipw(t,u|z)=λ^0kipw(t,u)exp{β^ipw(u)Tz}$⁠. When the auxiliary marks A_ki are correlated with V_ki conditional on T_ki,Z_ki and δ_ki=1, the conditional distribution ρ_k(w,v) involves the function $gk(a|t,u,z)$⁠, for which a parametric or semiparametric model may be developed to describe the dependence between A_ki and V_ki. Let $g^k(a|t,u,z)$ be an estimator of $gk(a|t,u,z)$ with a convergence rate of at least (nh)^−1/2. Then ρ_k(w,v) can be estimated by

Let $Nkix(t)=I(Xki⩽t,δki=1)$ and $Nkiv(v)=I(Vki⩽v)$⁠. The AIPW estimating equation for β is $Uaug{v,β,ψ^,ρ^(·)}=0$⁠, where

and

and

for j = 0,1. The AIPW estimator of β(v) solves equation (10) and is denoted by $β^aug(v)$⁠. The estimator of the cumulative function $B(v)=∫0vβ(u)du$ is given by $B^aug(v)=∫0vβ^aug(u)du$⁠. Note that there is no $ψ^k$ in $Z¯k(t,β)$⁠; this is a difference between the IPW and AIPW estimators.

To implement the estimation procedures in practice, one can use arbitrary auxiliaries for estimating $ψ^k$⁠; these auxiliaries may include covariates and marks at multiple time points pre infection and post infection respectively. In contrast, although in principle arbitrary auxiliaries may also be used for the terms $g^k(a|t,u,z)$ in equation (9), owing to the curse of dimensionality the method is expected to perform best in practice with a univariate auxiliary, where semiparametric or fully parametric models for $gk(a|t,u,z)$ would be required to include multivariate auxiliaries.

Sun and Gilbert (2012) proved that the estimators $β^ipw(t,v)$ and $β^aug(t,v)$ are consistent and that $β^aug(v)$ is more efficient than $β^ipw(v)$⁠. In the next section, we develop some hypothesis testing procedures for assessing mark-specific vaccine efficacy based on $B^aug(v)$⁠.

4. Testing of mark-specific vaccine efficacy

The covariate-adjusted vaccine efficacy VE(v) is defined through the first component of β(v). Let $B1(v)$ be the first component of the cumulative coefficient function B(v). The hypothesis tests concerning VE(v) are constructed on the basis of the first component $B^1aug(v)$ of the AIPW estimator $B^aug(v)$⁠. The cumulative estimator $B^aug(v)$ has more stable large sample behaviour and a faster convergence rate than $β^aug(v)$⁠.

Let $WB(v)=n1/2{B^aug(v)−B^aug(a)}−n1/2{B(v)−B(a)}$ for v ∈ [a,b]. In Appendix A we show that W_B(v), v ∈ [a,b], converges weakly to a p-dimensional mean 0 Gaussian process with continuous sample paths on v ∈ [a,b]. Further, the distribution of W_B(v), for v ∈ [a,b], can be approximated by using the Gaussian multipliers resampling method (of Lin et al. (1993)) based on

where ${ξki,i=1,…,nk,k=1,…,K}$ are independent and identically distributed standard normal random variables and $H^ki(v)$ is defined in expression (22) in Appendix A. Let $WB1(v)$ and $WB1*(v)$ be the first component of W_B(v) and $WB*(v)$ respectively. With the Gaussian multipliers method, the variance $var{B^1aug(v)−B^1aug(a)}$ can be consistently estimated by $var^{B^1aug(v)−B^1aug(a)}=n−1var*{WB1*(v)}$⁠, where $var*{WB1*(v)}$ is the first component on the diagonal of the covariance given in expression (23) in Appendix A.

4.1. Testing the null hypothesis H₁₀

Consider the test process $Q(1)(v)=n1/2{B^1aug(v)−B^1aug(a)}$⁠, v ∈ [a,b]. Then $Q(1)(v)=WB1(v)+n1/2{B1(v)−B1(a)}$⁠, v ∈ [a,b]. Under H₁₀, $B1(v)−B1(a)=0$ for v ∈ [a,b], which motivates the following test statistics for testing H₁₀:

The test statistics $Ta1(1)$ and $Ta2(1)$ capture general departures H_1a, whereas the test statistics $Tm1(1)$ and $Tm2(1)$ are sensitive to the monotone departures H_1m. It is easy to derive that all the test statistics $Ta1(1)$⁠, $Ta2(1)$⁠, $Tm1(1)$ and $Tm2(1)$ are consistent against their respective alternative hypotheses, and Appendix A derives their limiting distributions under H₁₀.

Under H₁₀, the distribution of Q⁽¹⁾(v), v ∈ [a,b], can be approximated by the conditional distribution of $WB1*$⁠, v ∈ [a,b], given the observed data sequence. Hence, the distributions of $Ta1(1)$⁠, $Ta2(1)$⁠, $Tm1(1)$ and $Tm2(1)$ under H₁₀ can be approximated by the conditional distributions of $Ta1*(1)=supv∈[a,b]|WB1*(v)|$⁠, $Ta2*(1)=∫abWB1*(v)2dvar*{WB1*(v)}$⁠, $Tm1*(1)=infv∈[a,b]WB1*(v)$ and $Tm2*(1)=∫abWB1*(v)dvar*{WB1*(v)}$⁠, given the observed data sequence, respectively. The critical values $ca1(1)$ and $ca2(1)$ of the test statistics $Ta1(1)$ and $Ta2(1)$ can be approximated by the (1−α)-quantile of $Ta1*(1)$ and $Ta2*(1)$⁠, which can be obtained by repeatedly generating a large number, say 500, of independent sets of normal samples ${ξki,i=1,…,nk,k=1,…,K}$ while holding the observed data sequence fixed. Similarly, the critical values $cm1(1)$ and $cm2(1)$ of the test statistics $Tm1(1)$ and $Tm2(1)$ can be approximated by the α-quantile of $Tm1*(1)$ and $Tm2*(1)$⁠, which again can be obtained by repeatedly generating independent sets of normal samples ${ξki,i=1,…,nk,k=1,…,K}$⁠. At the significance level α, the tests based on $Ta1(1)$ and $Ta2(1)$ reject H₁₀ in favour of H_1a if $Ta1(1)>ca1(1)$ and $Ta2(1)>ca2(1)$ respectively, and the tests based on $Tm1(1)$ and $Tm2(1)$ reject H₁₀ in favour of H_1m if $Tm1(1)<cm1(1)$ and $Tm2(1)<cm2(1)$ respectively.

4.2. Testing the null hypothesis H₂₀

Let

Then

where

is a transformation of F₁(·). We note that $Γ(·,B1)=0$ under H₂₀ and $Γ(·,B1)≠0$ under the alternatives, motivating Q⁽²⁾(v) as the test process and the following test statistics for testing H₂₀:

where $a<a′<b$⁠. We choose $a′>a$ to avoid 0 in the denominator of Q⁽²⁾(v). In practice, one can choose $a′$ close to a to make use of available data and to ensure that the tests are consistent.

By the asymptotic results shown in Appendix A and the continuous mapping theorem, under H₂₀ the distribution of Q⁽²⁾(v), v ∈ [a,b], can be approximated by the conditional distribution of $Γ(v,WB1*)$⁠, v ∈ [a,b], given the observed data sequence. Hence, the distributions of $Ta1(2)$⁠, $Ta2(2)$⁠, $Tm1(2)$ and $Tm2(2)$ under H₂₀ can be approximated by the conditional distributions of

and

given the observed data sequence, respectively. Similarly to Section 4.1, the respective critical values $ca1(2)$ and $ca2(2)$ of the test statistics $Ta1(2)$ and $Ta2(2)$ can be approximated by the (1−α)-quantiles of the conditional distributions of $Ta1*(2)$ and $Ta2*(2)$ obtained through repeatedly generating independent sets of normal samples ${ξki,i=1,…,nk,k=1,…,K}$ while holding the observed data sequence fixed. The critical values $cm1(2)$ and $cm2(2)$ for $Tm1(2)$ and $Tm2(2)$ can be approximated similarly. At the significance level α, the tests based on $Ta1(2)$ and $Ta2(2)$ reject H₂₀ in favour of H_2a if $Ta1(2)>ca1(2)$ and $Ta2(2)>ca2(2)$ respectively, and the tests based on $Tm1(2)$ and $Tm2(2)$ reject H₂₀ in favour of H_2m if $Tm1(2)<cm1(2)$ and $Tm2(2)<cm2(2)$ respectively.

The tests $Ta1(2)$ and $Ta2(2)$ capture general departures H_2a whereas the tests $Tm1(2)$ and $Tm2(2)$ are sensitive to the monotone departure H_2m. Note that the derivative $dΓ(v,B1)/dv=(v−a)−1{β1(v)−(v−a)−1B1(v)}⩾0$ under H_2m with strict inequality for at least some v ∈ [a,b]. This plus the fact that $Γ(v,B1)$ is non-decreasing with $Γ(b,B1)=0$ lead to the results that the tests based on $Tm1(2)$ and $Tm2(2)$ are consistent against H_2m and the tests based on $Ta1(2)$ and $Ta2(2)$ are consistent against H_2a. The proofs are given in the second paragraph following theorem 1 in Appendix A.

In Sections 4.1 and 4.2, we considered two types of test statistics, namely the integration-based test statistics and the supremum-based test statistics, for each pair of hypotheses. The former are generalizations of the Cramér–von Mises test statistic and involve integration of deviations over the whole range of the mark, whereas the latter are extensions of the classic Kolmogorov–Smirnov test statistic for testing the goodness of fit of a distribution function, and take the supremum of such deviations. As demonstrated in a comprehensive analysis of the relative powers of the classic Kolmogorov–Smirnov test and the Cramér–von Mises test by Stephens (1974), we expect that the two types of test statistic have different powers for different true alternative distributions. The integration-based test statistics are best suited for situations where the true alternative distribution deviates a little over the whole support of the mark and the supremum-based test statistics may have more power against situations where the true alternative has large deviations over a small section of the support. For example, for testing differential VE(v), H₂₀, the supremum-based tests will tend to be relatively more powerful if $VE^(v)$ is very high for a small range of marks near a and declines sharply to 0 and is constant at 0 for all other marks.

5. Simulation study

5.1. Numerical assessment of the tests under correctly specified models

We conduct a simulation study to evaluate the finite sample performance of the testing procedures proposed. The empirical sizes and powers of the test statistics are assessed for various models, sample sizes (500 and 800) and choices of bandwidths. The powers of the tests are evaluated in both situations where a correlated auxiliary variable is used and where it is absent.

We consider K = 1 stratum. Let Z_ki be the treatment indicator with P(Z_ki=1)=0.5. The (T_ki, V_ki) are generated from the following mark-specific proportional hazards model:

where α, β and γ are constants. Under model (12), λ₀(t,v)= exp (γv) and VE(v)=1− exp (α+βv). For α = 0 and β = 0, VE(v)=0, indicating no vaccine efficacy, and, for β = 0, VE(v)=VE, indicating mark invariant vaccine efficacy, whereas β>0 indicates VE(v) decreasing in v. We examine the hypothesis testing procedures for the following specific models M1–M5 respectively:

implying that VE(v)=0;

implying that VE(v) does not depend on v;

implying that VE(v) decreases;

implying that VE(v) decreases;

implying that VE(v) decreases.

We generate the censoring times from an exponential distribution, independent of (T,V), with censoring rates ranging from 20% to 30%. We take τ = 2.0. The complete-case indicator R_ki is generated with conditional probability $rk(Wki)=P(Rki=1|δki=1,Wki)$⁠, where

With ψ_k0=0.2 and ψ_k1=−0.2 about 50% of observed failures are missing marks.

Conditionally on (T_ki, Z_ki, V_ki), we assume that the auxiliary marks follow the model

for $i=1,…,nk,$k = 1,…,K, where V_ki are the possibly missing marks, U_ki is uniformly distributed on [0,1] independent of V_ki and θ>0 is an association parameter between A_ki and V_ki. The correlation coefficient ρ between A_ki and V_ki is 1 for θ = 0. Since A_ki is observed for all observed failure times, the AIPW estimator in this case is the full data estimator. The A_ki and V_ki are independent for θ = ∞, yielding ρ = 0. In addition, the θ-values of 0.8, 0.4 and 0.2 correspond to ρ = 0.78, 0.92, 0.98.

Under model (14), the conditional density of A_ki given (T_ki, Z_ki, V_ki) is

The likelihood function for θ is

It is easy to show that the maximum likelihood estimator equals

The density estimator $gk(a|t,v,z;θ^)$ is plugged into equation (9) to obtain $ρ^kipw(w,v)$⁠, which is used to construct the AIPW estimator of β in equation (10).

The performances of the test procedures proposed are evaluated through simulations for the models described in expressions (12), (13) and (14) under the settings M1–M5, where M1 is a setting under the null hypothesis H₁₀ and M2 is a setting under the null hypothesis H₂₀. We consider the situations where no auxiliary information is provided and where the correlation between the auxiliary mark and the mark of interest is ρ = 0.92 (under model (14) with θ = 0.4). Table 1 presents the empirical sizes and powers of the tests $Ta1(1)$⁠, $Ta2(1)$⁠, $Tm1(1)$ and $Tm2(1)$ for testing H₁₀ at the nominal level 0.05. Table 2 presents the empirical sizes and powers of the tests $Ta1(2)$⁠, $Ta2(2)$⁠, $Tm1(2)$ and $Tm2(2)$ for testing H₂₀ at the nominal level 0.05. The results are presented for n = 500 with $h1=0.1$ and $h=h2=0.15$⁠, 0.2, and for n = 800 with $h1=0.1$ and $h=h2=0.1$⁠, 0.15. We take a = 0, b = 1 and $a′=0.5$ for the tests. The Epanechnikov kernel $K(x)=0.75(1−x2)I(|x|⩽1)$ is used throughout the numerical analysis.