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**|**HHS Author Manuscripts**|**PMC2601643

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Article sections

- Abstract
- 1. Introduction
- 2. The principal stratification framework
- 3. Augmented designs for estimation
- 4. Estimation
- 5. Simulation study
- 6. Discussion
- References

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Ann Appl Stat. Author manuscript; available in PMC 2008 December 12.

Published in final edited form as:

Ann Appl Stat. 2008 March; 2(1): 386–407.

doi: 10.1214/07-AOAS132PMCID: PMC2601643

NIHMSID: NIHMS68128

Fred Hutchinson Cancer Research Center, Fred Hutchinson Cancer Research Center, National Institute of Allergy and Infectious Diseases and Peking University

L. Qin, P. B. Gilbert, Vaccine and Infectious Disease Institute, Fred Hutchinson Cancer Research Center, Department of Biostatistics, University of Washington, 1100 Fairview Avenue NORTH, LE-400, SEATTLE, Washington 98109, USA, E-mail: gro.prahcs@niql, gro.prahcs@trebligp

D. Follmann, National Institute of Allergy and Infectious Diseases, 6700B Rockledge Drive MSC 7609, Bethesda, Maryland 20892, USA, E-mail: vog.hin.diain@nnamllofd

D. Li, School of Mathematical Science, Peking University, Beijing 100871, P.R. China, E-mail: nc.ude.ukp.htam@fdl

See other articles in PMC that cite the published article.

Assessing immune responses to study vaccines as surrogates of protection plays a central role in vaccine clinical trials. Motivated by three ongoing or pending HIV vaccine efficacy trials, we consider such surrogate endpoint assessment in a randomized placebo-controlled trial with case-cohort sampling of immune responses and a time to event endpoint. Based on the principal surrogate definition under the principal stratification framework proposed by Frangakis and Rubin [*Biometrics* **58** (2002) 21–29] and adapted by Gilbert and Hudgens (2006), we introduce estimands that measure the value of an immune response as a surrogate of protection in the context of the Cox proportional hazards model. The estimands are not identified because the immune response to vaccine is not measured in placebo recipients. We formulate the problem as a Cox model with missing covariates, and employ novel trial designs for predicting the missing immune responses and thereby identifying the estimands. The first design utilizes information from baseline predictors of the immune response, and bridges their relationship in the vaccine recipients to the placebo recipients. The second design provides a validation set for the unmeasured immune responses of uninfected placebo recipients by immunizing them with the study vaccine after trial closeout. A maximum estimated likelihood approach is proposed for estimation of the parameters. Simulated data examples are given to evaluate the proposed designs and study their properties.

The evaluation of vaccine efficacy in vaccine clinical trials is generally costly, either because it takes a long trial period for the clinical outcomes to be observed, or because the vaccine may only be partially effective. Therefore, identifying vaccine-induced immune responses as surrogate markers for the true study endpoint has spawned interest in vaccine research [Halloran (1998), Chan, Wang and Heyse (2003) and Gilbert et al. (2005)]. The potential surrogate would usually be measured shortly after administration of the study vaccine, and if it can be validated then the vaccine’s protective effect can be inferred from it. As knowledge builds on the immunological mechanism for protecting against disease by a pathogen, finding a good immunological surrogate is promising for iteratively guiding refinement of the vaccine formulation, and ultimately for providing a basis for regulatory decisions.

There is an extensive literature on the evaluation of surrogate endpoints for therapeutic development [e.g., Prentice (1989), Lin, Fleming and De Gruttola (1997), DeGruttola et al. (2002), Molenberghs et al. (2002) and Weir and Walley (2006)]. The assessment of an immunological surrogate focuses on contrasting the clinical outcome rate between vaccine recipients and placebo recipients, given the measured immune responses. Since immune response measurements are made post-randomization, this assessment is subject to selection bias [Frangakis and Rubin (2002) and Gilbert, Bosch and Hudgens (2003)]. To address this problem, Gilbert and Hudgens (2006) (henceforth GH) proposed to evaluate the value of a bio-marker as a surrogate endpoint by estimating the causal effect predictiveness (CEP) surface, which contrasts the clinical outcome rates between the vaccine recipients and placebo recipients within principal strata formed by joint values of the potential immune responses under assignment to vaccine or placebo. This work built on Frangakis and Rubin (2002)’s potential outcomes framework for evaluating principal surrogate endpoints. GH considered a binary clinical outcome and used a baseline predictor approach to predict the principal strata and estimate the CEP surface nonparametrically. We develop a similar method for a time-to-event clinical endpoint, which is most commonly used in vaccine clinical trials, and use the Cox proportional hazards model [Cox (1972)] to describe the relationship between the survival outcome and covariates including the potential surrogate. Our likelihood calculations utilize discrete failure time models, which are suitable for many vaccine trials because clinical endpoints are often assessed at pre-specified dates.

In the principal stratification framework, the principal strata are subject to missingness as only the immune response to the actual treatment assignment (vaccine or placebo) is observed. This situation was described as the “fundamental challenge of causal inference” [Holland (1986)]. The unobserved immune response is missing for the subjects that receive the “opposite” assignment. We focus on a marginal estimand that conditions on the immune response to the vaccine. Consequently, the assessment of a surrogate in the Cox model framework can be cast as a problem of estimation with a missing covariate. Although methods for estimating the Cox model with missing covariates have been extensively studied [e.g., Lin and Ying (1993), Robins, Rotnitzky and Zhao (1994), Zhou and Pepe (1995), Paik and Tsai (1997), Chen and Little (1999), Herring and Ibrahim (2001), Chen (2002) and Little and Rubin (2002)], their application to the proposed surrogate assessment are not direct, as the missing data are entirely in the placebo group. Techniques are called for to predict the “missing” immune responses in the placebo recipients, or a random sample of them. Therefore, we extend the innovative designs proposed by Follmann (2006) for a binary endpoint to the Cox model setting.

Follmann (2006) proposed two novel components to vaccine trials: baseline irrelevant predictor (BIP), and closeout placebo vaccination (CPV), which enable inference about the vaccine-specific immune responses of placebo recipients. BIP utilizes association between the response of interest and another baseline immune response thought to be irrelevant to infection in the vaccinated subjects. CPV involves vaccinating uninfected placebo recipients after study completion. To match ongoing and pending HIV vaccine trials, we extend these strategies to accommodate a time to event clinical endpoint and sampling of immune responses via a case-cohort design [e.g., Prentice (1986), Borgan et al. (2000), Scheike and Martinussen (2004) and Kulich and Lin (2004)]. We focus on a sampling design that uses data from all infected subjects and a random subcohort of uninfected subjects for whom the immune response to the vaccine is measured (termed “immunogenicity subcohort,” *IC*). The methods also apply for other sampling designs, such as failure status-independent case-cohort sampling. We also consider measuring the BIP on some subjects outside the *IC*, which can help improve efficiency.

Under the BIP design placebo subjects cannot be selected into the *IC*; similarly, infected placebo subjects cannot enter *IC* in the CPV design. Such null selection probabilities violate a key assumption for most semiparametric approaches to handling missing covariates in Cox regression, including all that are based on partial likelihood. Accordingly, we employ a full-likelihood based estimation procedure based on DFT models. For continuous failure time data, we also consider an approximate semiparametric algorithm for the estimation of the BIP-alone design by extending the EM algorithm of Chen (2002).

The proposed methods will be applied to analyze three U.S. National Institutes of Health-sponsored HIV vaccine efficacy trials. These trials randomize HIV negative high risk volunteers to vaccine or placebo in a 1:1 ratio, and follow participants until a fixed number of HIV infection events. The first two trials (named STEP 502 [Mehrotra, Li and Gilbert (2006)] and HVTN 503) are ongoing in the Americas and South Africa, respectively, and evaluate Merck’s Adenovirus serotype 5 (Ad5) vector vaccine in approximately 3000 subjects. The third trial (named PAVE-100), co-sponsored by the U.S. Military HIV Research Program, the International AIDS Vaccine Initiative, and the Centers for Disease Control and Prevention, is being planned. The current PAVE-100 design will randomize approximately 8500 volunteers from 13 countries in the Americas, East Africa, and Southern Africa to placebo or the Vaccine Research Center’s prime-boost vaccine regimen (DNA prime:Ad5 vector boost). The trials plan to analyze approximately 100, 120 and 280 HIV infection events, respectively. A secondary objective of each trial is to evaluate the magnitude of CD8^{+} T cell response levels, as measured by the ELISpot assay from blood samples drawn after Ad5 immunization, as a surrogate for HIV infection. The neutralizing antibody titer to Ad5 is measured at baseline for all participants. Because it is inversely correlated with the CD8^{+} T cell responses [Catanzaro et al. (2006)], it potentially may be used as a BIP.

To develop our approach for assessing surrogate endpoints in vaccine trials, we present the general framework, assumptions, and definition of the estimands in Section 2, design considerations in Section 3, and an estimation procedure in Section 4. In Section 5 we evaluate the approach with simulated trials designed to match the aforementioned HIV trials. A discussion follows in Section 6.

In this section we introduce the principal stratification framework based on potential outcomes and principal stratification [Frangakis and Rubin (2002) and Rubin (2005)].

Let *n* denote the total number of subjects in the vaccine trial. For subject *i* (*i* = 1, …, *n*), let *V _{i}* denote the observed treatment indicator,

Suppose that {*V _{i}*,

A1. Stable unit treatment value assumption (SUTVA).

A2. Ignorable treatment assignments. Conditional on *W _{i}*,

Assumption A1 guarantees the “consistency” property (i.e., the observed outcomes for a subject assigned *V* equals his potential outcomes if assigned *V*) and that the potential outcomes of one subject are not impacted by the treatment assignments of other subjects. A2 holds for randomized, blinded trials.

Under the above assumptions, we define two vaccine efficacy estimands: 1. Conditional on joint potential outcomes (joint VE)

$$\begin{array}{l}VE({s}_{1},{s}_{0})\\ \equiv 1-\frac{Pr(T(1)={t}_{k}\mid T(1)\ge {t}_{k-1},S(1)={s}_{1},S(0)={s}_{0},R(1)=1,R(0)=1)}{Pr(T(0)={t}_{k}\mid T(0)\ge {t}_{k-1},S(1)={s}_{1},S(0)={s}_{0},R(1)=1,R(0)=1)}.\end{array}$$

2. Conditional on marginal potential outcome (marginal VE)

$$\begin{array}{l}VE({s}_{1})\equiv 1-\frac{Pr(T(1)={t}_{k}\mid T(1)\ge {t}_{k-1},S(1)={s}_{1},R(1)=1)}{Pr(T(0)={t}_{k}\mid T(0)\ge {t}_{k-1},S(1)={s}_{1},R(1)=1},\\ k=2,\dots ,K.\end{array}$$

The estimand *VE*(*s*_{1}, *s*_{0}) conditions on membership in the basic principal stratum {*S*(1) = *s*_{1}, *S*(0) = *s*_{0}, *R*(1) = *R*(0) = 1}, and *VE*(*s*_{1}) conditions on membership in a union of basic principal strata [Frangakis and Rubin (2002)]. The estimands condition on *R _{i}*(1) =

To help identify the estimands, only subjects with *R _{i}*(

A3. Equal drop-out and risk up to time *t*_{1}: *R _{i}*(1) = 1

A3 implies that subjects observed to be at risk at *t*_{1} will have *R _{i}*(1) =

In addition to A1–A3, identifiability of *VE*(*s*_{1}, *s*_{0}) requires a way to predict *S _{i}*(1) for subjects with

We propose a Cox model for the discrete cumulative hazard function Λ(*t*),

$$\begin{array}{l}d\mathrm{\Lambda}\left({t}_{k};V,S(1)={s}_{1},R(1)=1,W\right)=exp({Z}^{\prime}\mathit{\beta})d{\mathrm{\Lambda}}_{0}({t}_{k}),\\ k=2,\dots ,K,\end{array}$$

(1)

with *Z* = {*V*, *S*(1), *V S*(1), *W*′}′, ** β**= {

$$VE({s}_{1})=1-\frac{d\mathrm{\Lambda}({t}_{k};V=1,S(1)={s}_{1},R(1)=1)}{d\mathrm{\Lambda}({t}_{k};V=0,S(1)={s}_{1},R(1)=1)},\phantom{\rule{0.38889em}{0ex}}k=2,\dots ,K.$$

The discrete hazards always condition on {*R*(1) = 1} and, henceforth, we assume this implicitly. For subjects with a particular baseline covariate *w*, a similar estimand *VE*(*s*_{1}|*w*) can be formed by conditioning on *W* = *w* in the hazards.

The population estimand *VE*(*s*_{1}) contrasts the rate of the clinical event for subjects with *S*(1) = *s*_{1} under assignment to vaccine versus under assignment to placebo. Supposing *S*(1) is bounded below at value zero which indicates a negative immune response, we define *S* to be a *predictive surrogate* if *VE*(0) = 0 and *VE*(*s*_{1}) > 0 for all *s*_{1} > *C* for some constant *C* ≥ 0. These conditions reflect population level necessity and sufficiency of the immune response to achieve positive vaccine efficacy.

Under A1–A3 and the Cox model (1), the estimand equals

$$VE({s}_{1})=1-exp({\beta}_{1}+{s}_{1}{\beta}_{3}).$$

(2)

In equation (2) a negative value of *β*_{3} indicates that a higher immune response to vaccine predicts greater vaccine efficacy. On the other hand, *β*_{3} = 0 implies *VE*(*s*_{1}) is constant in *s*_{1} so that the marker does not predict vaccine efficacy. Therefore, testing *H*_{0} : *β*_{3} = 0 versus *H*_{1} : *β*_{3} < 0 assesses sufficiency. A value *β*_{1} = 0 indicates necessity, and both *β*_{1} = 0 and *β*_{3} < 0 indicate the marker is a predictive surrogate. The magnitude of *β*_{3} indicates the quality of the predictive surrogate with *β*_{3} = 0 suggesting no surrogate value [*VE*(*s*_{1}) is constant in *s*_{1}] and larger |*β*_{3}| suggesting greater surrogate value (greater predictiveness).

The immune response to the study vaccine, *S*(1), cannot be measured in placebo recipients, but it may be inferred when utilizing either the BIP or CPV designs (see Figure 1).

Illustration of an HIV vaccine trial design under the BIP and CPV strategies. Under BIP or BIP + CPV, baseline measurements of W and B are obtained from all (or a random sample of) the study participants prior to the randomization at time 0. The study **...**

Assume a baseline covariate *B* is available that does not affect (i.e., is “irrelevant” for) clinical risk after accounting for the immune response *S*(1) and first-phase covariates *W* :

A4. *d* Λ (*t _{k}*;

Assumptions A1–A3 imply that the relationship between *S*(1) and *B* is the same regardless of treatment assignment

$$[{S}_{i}(1)\mid {V}_{i}=1,\phantom{\rule{0.16667em}{0ex}}{B}_{i},\phantom{\rule{0.16667em}{0ex}}{R}_{i}(1)=1]\stackrel{d}{=}[{S}_{i}(1)\mid {V}_{i}=0,\phantom{\rule{0.16667em}{0ex}}{B}_{i},\phantom{\rule{0.16667em}{0ex}}{R}_{i}(1)=1].$$

(3)

Therefore, *S _{i}*(1) can be predicted or imputed for placebo subjects based on

In case-cohort designs, good baseline predictors need to be highly correlated with the biomarker *S*(1), and preferably include first-phase (measured on everyone) inexpensive covariates to achieve efficiency gains.

This design entails vaccinating uninfected placebo subjects after the study closeout, and measuring their immune response
${S}_{i}^{c}(1)$. The closeout measurement
${S}_{i}^{c}(1)$ is made at a visit *t*_{1} time units after vaccination, to match the measurement schedule in the vaccine trial. We need to make an additional assumption to bridge the marker values *S _{i}* (1) and
${S}_{i}^{c}(1)$. Let
${S}_{i}^{\text{true}}(1)$ be the true immune response at time

A5. *Time constancy of*
${S}_{i}^{\text{true}}(1)$: For uninfected placebo recipients,
${S}_{i}(1)={S}_{i}^{\text{true}}(1)+{e}_{i1}\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}{S}_{i}^{c}(1)={S}_{i}^{\text{true}}(1)+{e}_{i2}$, where *e _{i}*

This assumption implies that the true immune response is unchanged from time *t*_{1} to study closeout plus *t*_{1}, and the measurement errors have the same distribution. Thus, *S _{i}*(1) and
${S}_{i}^{c}(1)$ are exchangeable and one can be used in lieu of the other. To be concrete, suppose only one shot is given, the trial is three years, and

Under A5, the distribution of [*S _{i}*(1)|

The BIP and CPV designs can be combined by imputing *S _{i}*(1) with
${S}_{i}^{c}(1)$ for all uninfected placebo recipients with
${S}_{i}^{c}(1)$ measured, and predicting

Estimation of the estimand is challenged by the amount of missing *S*(1)’s. We focus on the maximum estimated likelihood (MEL) estimation procedure that applies to all three designs. We then briefly outline an approximate EM-type algorithm for estimation with the BIP-alone design.

We present below the estimation procedure for the BIP + CPV design, which includes estimation under the BIP- or CPV-alone designs as special cases.

Let *IC _{V}* denote the immunogenicity cohort that contributes second-phase data

$$\begin{array}{l}logL(\mathit{\beta},{\mathbf{\lambda}}_{0})=\sum _{i\in I{C}_{V}}log{L}_{1}({O}_{i})+\sum _{i\in I{C}_{P}}log{L}_{2}({O}_{i})+\sum _{i\in \overline{IC},IB}log{L}_{3}({O}_{i})\\ +\sum _{i\in \overline{IC},\overline{IB}}log{L}_{4}({O}_{i}),\end{array}$$

(4)

where

$$\begin{array}{l}{L}_{1}({O}_{i})=\prod _{j=2}^{{M}_{i}-1}{(1-{\lambda}_{0j})}^{exp\{{V}_{i}{\beta}_{1}+{S}_{i}(1){\beta}_{2}+{V}_{i}{S}_{i}(1){\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}{R}_{i}({V}_{i})}\\ \times {\{1-{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+{S}_{i}(1){\beta}_{2}+{V}_{i}{S}_{i}(1){\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}\}}^{{\delta}_{i}{R}_{i}({V}_{i})}\\ \times {{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+{S}_{i}(1){\beta}_{2}+{V}_{i}{S}_{i}(1){\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}}^{(1-{\delta}_{i}){R}_{i}({V}_{i})},\\ {L}_{2}({O}_{i})=\prod _{j=2}^{{M}_{i}}{(1-{\lambda}_{0j})}^{exp\{{V}_{i}{\beta}_{1}+{S}_{i}^{c}{\beta}_{2}+{V}_{i}{S}_{i}^{c}{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}{R}_{i}({V}_{i})},\\ {L}_{3}({O}_{i})=\int \prod _{j=2}^{{M}_{i}-1}{(1-{\lambda}_{0j})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}{R}_{i}({V}_{i})}\\ \times {\{1-{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}\}}^{{\delta}_{i}{R}_{i}({V}_{i})}\\ \times {{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}}^{(1-{\delta}_{i}){R}_{i}({V}_{i})}d\phantom{\rule{0.16667em}{0ex}}P(s\mid {B}_{i},{W}_{i}),\\ {L}_{4}({O}_{i})=\int \prod _{j=2}^{{M}_{i}-1}{(1-{\lambda}_{0j})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}{R}_{i}({V}_{i})}\\ \times {\{1-{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}\}}^{{\delta}_{i}{R}_{i}({V}_{i})}\\ \times {{(1-{\lambda}_{0,{M}_{i}})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}}^{(1-{\delta}_{i}){R}_{i}({V}_{i})}d\phantom{\rule{0.16667em}{0ex}}P(s\mid {W}_{i}).\end{array}$$

Here **λ**_{0} = {λ_{02}, …, λ_{0}* _{K}* }

In the Cox model formulation, the estimand *VE*(*s*_{1}) depends only on ** β** while the parameters in the conditional c.d.f.’s

For a categorical *W*, *P* (*s*|*w*) and *P* (*s*|*b*, *w*) can be estimated nonparametrically. However, if *W* is continuous, then nonparametric estimation will require smoothing and much larger sample sizes are needed for tractable computation. Therefore, if *W* is continuous or multi-component, parametric assumptions on the conditional c.d.f.’s will usually be needed to achieve stable estimation in practice. An advantage of the MEL approach is that it can straightforwardly accommodate any approach to estimating the nuisance parameters *P* (*s*|*w*) and *P* (*s*|*b*, *w*). In the MEL approach we first estimate these distributions consistently using data from the vaccine recipients, and then construct the estimated likelihood *L*(** β**,

We outline three key steps in the evaluation of the log-likelihood (4) in the absence of the first-phase covariates *W* :

*1. Estimation of p*(*s*) *and p*(*s*|*b*). Let *p*(*s*), *p*(*b*), and *p*(*s*, *b*) be marginal and joint p.d.f.s (or p.m.f.s for discrete variables) for *S*(1) and *B*. Because vaccine recipients in the *IC _{V}* provide nonrandom samples of

$$\begin{array}{l}p(s)={f}_{11}(s){p}_{11}+{f}_{10}(s){p}_{10},\\ p(b)={f}_{11}(b){p}_{11}+{f}_{10}(b){p}_{10},\\ p(s,b)={f}_{11}(s,b){p}_{11}+{f}_{10}(s,b){p}_{10},\end{array}$$

(5)

where, for *h* = 1, 0, *f*_{1}* _{h}*(·) is the conditional p.d.f. or p.m.f. of

We sketch the estimation for two special cases where (A) (*S*(1), *B*) are categorical and (B) (*S*(1), *B*) are bivariate normally distributed.

(A) If *S*(1) and *B* have discrete values with *J* and *L* categories, respectively, then *f*_{1}* _{h}*(

$$\begin{array}{l}{\widehat{f}}_{1h}({s}_{j})=\frac{{\sum}_{i\in I{C}_{V}}I({S}_{i}(1)={s}_{j},{\delta}_{i}=h)}{{\sum}_{i\in I{C}_{V}}I({\delta}_{i}=h)},\\ \widehat{p}\left(S(1)={s}_{j}\mid {b}_{l}\right)=\frac{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}{\delta}_{i}I({S}_{i}(1)={s}_{j})}{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}{\delta}_{i}}{\widehat{p}}_{11}\\ +\frac{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}(1-{\delta}_{i})I({S}_{i}(1)={s}_{j})}{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}(1-{\delta}_{i})}{\widehat{p}}_{10}.\end{array}$$

(B) If (*S*(1), *B*) are jointly normally distributed, then *p*(*s*) and *p*(*s*|*b*) are both normal densities and thus can be estimated using estimates of the first and second moments from expressions in (5).

Evaluating the likelihood (4) involves integrations over *s*, which are briefly described in the Appendix.

*2. Maximization and implementation*. The estimated log-likelihood log *L*(** β**,

In this subsection we present an estimation approach that uses regression calibration to impute the missing *S _{i}*(1)s for subjects with a BIP

Because the missingness of *S*(1) does not depend on unobserved *S*(1), and we assume the censoring distribution does not depend on *S*(1), the log-likelihood for the BIP-alone design can be expressed up to a constant factor as

$$\begin{array}{l}l(\mathit{\beta},\mathit{\alpha},{\mathrm{\Lambda}}_{0})\\ =\sum _{i\in IC}\{{\delta}_{i}({Z}_{i}^{\prime}\mathit{\beta})-{\mathrm{\Lambda}}_{0}({X}_{i})exp({Z}_{i}^{\prime}\mathit{\beta})\}\\ +\sum _{i\in \overline{IC},IB}log\left\{\int exp\{{\delta}_{i}({Z}_{i}^{\prime}\mathit{\beta})-{\mathrm{\Lambda}}_{0}({X}_{i})exp({Z}_{i}^{\prime}\mathit{\beta})\}d\phantom{\rule{0.16667em}{0ex}}P(s\mid {V}_{i},{W}_{i},{B}_{i})\right\}\\ +\sum _{i\in \overline{IC},\overline{IB}}log\left\{\int exp\{{\delta}_{i}({Z}_{i}^{\prime}\mathit{\beta})-{\mathrm{\Lambda}}_{0}({X}_{i})exp({Z}_{i}^{\prime}\mathit{\beta})\}d\phantom{\rule{0.16667em}{0ex}}P(s\mid {V}_{i},{W}_{i})\right\}\\ +{\delta}_{i}log(d{\mathrm{\Lambda}}_{0}({X}_{i})),\end{array}$$

where *X _{i}* denotes the observed failure time, Λ

The log-likelihood score equations can be solved via an iterative EM algorithm [Chen and Little (1999), Herring and Ibrahim (2001), Chen (2002)]. For computational efficiency, we propose to modify the double-semiparametric EM-algorithm of Chen (2002) to incorporate the auxiliary covariate *B* as a predictor of the missing *S*(1). Given equation (3) and the relationship *S _{i}*(1) =

- Calibration-step: Prediction of unobserved
*S*(1)s by_{i}*Ŝ*(1) = E_{i}*^*(*S*(1)|*B*)._{i} - E-step: Given parameter values at the
*m*th iteration (*β*^{(}^{m}^{)}, ${\mathrm{\Lambda}}_{0}^{(m)}(X)$,*α*^{(}^{m}^{)}, ${p}_{\mathit{klj}}^{(m)}$,*θ*^{(}^{m}^{)}), for*p*denote the probability mass of the observed distinct values of_{klj}*S*(1) at discrete levels of*V*=*ν*and_{k}*W*=_{d}*w*(_{l}*W*=*W*_{d}*W*where_{c}*W*and_{d}*W*denote the categorical and continuous covariates in_{c}*W*, resp.), and**α**^{(}^{m}^{)}denote the parameters in the distribution*P*(*W*|_{c}*S*(1),*V*,*W*,_{d}*X*,*δ*). Calculate conditional expectations under*P*(*S*(1)|*V*,*W*,_{d}*X*,*δ*). - M-step: Update (
, Λ*β*_{0}(*X*),,*α**p*,_{klj}*θ*) by solving the corresponding score equations. - Repeat the E-step and M-step above until convergence.

The advantage of the ACEM algorithm is that it can account for continuous failure times and is computationally fast; however, since it uses regression calibration, it performs well only for the rare event situation with a highly predictive BIP. Prevention trials, which usually have a low event rate, are an application area.

We conducted a simulation study to evaluate the performance of the proposed strategies for estimating the estimand *VE*(*s*_{1}) and thereby assessing a predictive surrogate in the Cox model setting. To simulate the real scenarios, we roughly follow the design of the three HIV vaccine efficacy trials described in the introduction. We suppose a total sample size of 5000, with 2500 subjects per arm. The treatment indicator *V* = 1 if assigned vaccine and *V* = 0 if assigned placebo. Under the case-cohort sampling, the immunogenicity subcohort (*IC*) consists of all infected vaccine recipients and a random sample of uninfected vaccine recipients, which include a combination of 25% or 50% of uninfected vaccine recipients. We considered one auxiliary covariate *B* as the BIP for the potential immunological surrogate *S*(1). The variables *S*(1) and *B* were generated from a bivariate normal distribution with mean zero and variance 0.4 for each component [reflecting the variance of the ELISPOT assay used to measure *S*(1) = CD8^{+} T cell response], and correlation *ρ* = 0.6 or 0.9. For the BIP-alone and BIP + CPV designs, we assume that *B* was measured from all individuals in the *IC* and from 50% or 37.5% of those not in the *IC*, as a precision factor. In the BIP-alone approach, *S*(1) was treated as missing for all placebo recipients, while for the BIP + CPV and CPV-alone approaches, we assume 25% or 50% uninfected placebo recipients got the CPV measurement *S ^{c}*(1). Infection times were generated from the continuous-time Cox model λ (

We first conducted estimation through the MEL algorithm for discrete failure times using all three designs. For the BIP-alone design, a second simulation was conducted to compare the performance of the MEL approach for grouped failure times, versus that of the ACEM algorithm assuming continuous failure times were observed in a rare event setting. To evaluate efficiencies for the parameter estimates, estimates from the Cox model using the full simulated data were obtained as an unattainable “gold standard.”

Figure 2 plots the true *VE*(*s*_{1}) curve for different true parameters (*β*_{1}, *β*_{3}) in model (2). It shows that when *β*_{3} = −0.7, *VE*(0) = 0 and *VE*(*s*_{1}) > 0 for *s*_{1} > 0, indicating that the immune response variable is a predictive surrogate.

Illustration of the estimand VE(s_{1}) as a function of the standardized potential surrogate *S*(1) over the range of observable values with different values for *β*_{3}.

Table 1 presents simulation results for the MEL approach in different settings. It can be seen that the method has excellent performance. There are generally small biases, small variances of the estimates and good power of the test of *H*_{0} : *β*_{3} = 0 for surrogate value. As more CPV or auxiliary BIP information is available, both the accuracy and precision of the estimates improve. The efficiency of the BIP-involved designs increases as the correlation between the BIP and *S*(1) increases. The CPV-alone design is less efficient because none of the infected placebo subjects have *S ^{c}*(1) measured. Figure 3 displays the relative efficiencies of the parameter estimators from the three designs with missing

Relative efficiencies of parameter estimators. For designs with BIP, “Large Missing” and “Medium Missing” patterns indicate the IC size of 25% or 50% with additional 25% or 37.5% BIP data, respectively. For the design with **...**

Table 2 lists results from both the MEL approach and the ACEM algorithm under the BIP-alone design and the medium missing case (the *IC* size of 50% with additional 37.5% first phase BIP data). It demonstrates that the performance of the ACEM method is very sensitive to the prediction accuracy of the baseline predictor. When the BIP is a fairly inaccurate predictor of *S*(1) (*ρ* = 0.6), the ACEM method produces large biases and does not control the type-I error rate; while if the calibration is reliable (*ρ* = 0.9), then the ACEM algorithm can generally estimate well. The MEL estimation outperforms the ACEM in most settings, with a slight loss of efficiency due to the grouping of the survival times.

We have proposed a framework for assessing an immunological predictive surrogate in a vaccine trial with a time to event endpoint and case-cohort sampling of the immunological biomarker. While we have focused on the methods development for vaccine trials, the proposed principles are applicable for evaluating predictive surrogate endpoints in other biomedical applications.

We have discussed study designs and estimation procedures, and provided simulation results to demonstrate their validity and applicability under assumptions. We plan to apply the BIP-alone design to the three ongoing or pending HIV vaccine efficacy trials. As demonstrated by the simulation study, if good baseline irrelevant predictors exist, then a predictive surrogate can be evaluated effectively. The CPV-alone design is also a useful tool for the assessment that is complimentary to the BIP-alone design. If resources permit, the BIP + CPV design merits consideration because it improves accuracy and efficiency compared to the BIP-alone design if baseline predictors are not closely correlated with the potential predictive surrogate, or if A4 appears to be violated (i.e., the BIP affects clinical risk after controlling for the potential surrogate and first-phase baseline covariates).

For simplicity, we assumed equal drop-out and risk for each subject under assignment to vaccine or placebo over the time interval [0, *t*_{1}] (assumption A3), and restricted the analysis to subjects at risk at the time the immune response is measured, *t*_{1}. To include all randomized subjects, A3 can be relaxed by postulating that the future immune response that will be measured at time *t*_{1} impacts the risk of infection over [0, *t*_{1}]. With the DFT Cox model (1), the likelihood contribution of a subject with early infection during [0, *t*_{1}] can be obtained as
$\int \{1-{(1-{\lambda}_{01})}^{exp\{{V}_{i}{\beta}_{1}+s{\beta}_{2}+{V}_{i}s{\beta}_{3}+{W}_{i}^{\prime}{\mathit{\beta}}_{4}\}}\}d\phantom{\rule{0.16667em}{0ex}}P\phantom{\rule{0.16667em}{0ex}}(s)$, where *P* (·) is the marginal (*P* (*s*)) or conditional distribution of *S*(1) (*P* (*s*|*B _{i}*)) if the BIP

A4 is a strong untestable assumption. Because we assume *B* and *S*(1) are correlated, A4 implies that the phase one covariates *W* capture all the causes of *S*(1) and the clinical endpoint [in the sense of Pearl (2000)]. Furthermore, it may be difficult to find a baseline covariate *B* that is known to not affect clinical risk after accounting for *S*(1). We suggest three potentially useful *B*’s for vaccine trials. First, a study that vaccinated 75 individuals simultaneously with hepatitis A and B vaccines showed a linear correlation of 0.85 among A- and B-specific antibody titers [Czeschinski, Binding and Witting (2000)]. Given there is little cross-reactivity among the hepatitis A and B proteins, *B* = hepatitis A titer may be an excellent baseline predictor for *S*(1) = hepatitis B titer that satisfies A4. For HIV vaccine trials, two available scalar *B*’s may plausibly satisfy A4. First, Follmann (2006) considered as *B* the antibody titer to a rabies glycoprotein vaccine. Because rabies is not acquired sexually, it is plausible that anti-rabies antibodies are independent of risk of HIV infection given *S*(1). Second, in the ongoing HIV vaccine efficacy trials, a current leading candidate *B* is the titer of antibodies that neutralize the Adenovirus serotype 5 vector that carries the HIV genes in the vaccine. This *B* has been shown to inversely correlate with the *S*(1) of primary interest (T cell response levels measured by ELISpot) [Catanzaro et al. (2006)], and since Adenovirus 5 is a respiratory infection virus, A4 may plausibly hold.

In general, though, it is desirable to relax A4, and fortunately this can be done by including *B* as a component of *W* in the Cox model (1) and estimating its coefficient (as suggested by the Associate Editor). This extra coefficient for *B* is identified by the data from vaccine recipients with *B* measured. Based on the argument given by Follmann (2006) and Gilbert and Hudgens (2006) for the setting of the BIP-alone design and a dichotomous clinical endpoint, we conjecture that the estimand *VE*(*s*_{1}) will be identified from the observed data as long as at least one of the interaction terms of *B* with *V* or *W* with *V* is omitted from the Cox model.

Our approach specified a Cox regression with *S _{i}*(1) and

We have presented estimated likelihood based methods to accommodate missing data in case-cohort designs, as well as a regression calibration based double-semiparametric EM algorithm that has reasonable performance when the regression calibration is reliable and the event is rare. This approximate algorithm enjoys the convenience of regression calibration to incorporate auxiliary information, and has faster and easier implementation for the continuous failure time model. Alternative estimation methods such as multiple imputation may also be useful, provided the posterior distribution can be properly specified. In addition, a full likelihood approach that maximized over (** β**,

The authors thank Huayun Chen for providing his fortran code for the EM algorithm of Chen (2002), and the Editor, Associate Editor and Referees for helpful comments.

For discrete *S*(1) and *B*, the integrations can be replaced by finite summations. When *S*(1) is continuous, the integrations can be made easier by positing parametric models. Assume *S*(1) ~ N(*μ*(·), *σ* (·)^{2}), where *μ*(·), *σ* (·)^{2} represent the first two moments of *p*(*s*) or *p*(*s*|*b*). Then for a given function *g*(*s*) *of s*, *∫g*(*s*)*p*(*·*) *ds* = *∫g* (*μ*(*·*) + *σ* (*·*)*u*)(*u*) *du*, where *p*(·) denotes *p*(*s*) or *p*(*s*|*b*) and (*u*) is the standard normal density function. Because the integrand *g*(*s*) in (4) is a smooth function of *s*, numerical methods such as Gaussian quadrature can be applied to evaluate the integration. Based on our experience, only a small number (around 15) of evaluations is needed to get stable quadrature results.

When *B* has discrete values *b _{l}*,

$$\begin{array}{l}\int g\phantom{\rule{0.16667em}{0ex}}(s)p\phantom{\rule{0.16667em}{0ex}}(s)\phantom{\rule{0.16667em}{0ex}}ds\approx {p}_{11}\frac{1}{{\sum}_{i\in I{C}_{V}}{\delta}_{i}}\sum _{i\in I{C}_{V}}{\delta}_{i}g({S}_{i}(1))\\ +{p}_{10}\frac{1}{{\sum}_{i\in I{C}_{V}}(1-{\delta}_{i})}\sum _{i\in I{C}_{V}}(1-{\delta}_{i})g\phantom{\rule{0.16667em}{0ex}}({S}_{i}(1)),\\ \int g\phantom{\rule{0.16667em}{0ex}}(s)p\phantom{\rule{0.16667em}{0ex}}(s\mid {b}_{l})\phantom{\rule{0.16667em}{0ex}}ds\approx {p}_{11}\frac{1}{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}{\delta}_{i}}\sum _{i\in I{C}_{V},{B}_{i}={b}_{l}}{\delta}_{i}g\phantom{\rule{0.16667em}{0ex}}({S}_{i}(1))\\ +{p}_{10}\frac{1}{{\sum}_{i\in I{C}_{V},{B}_{i}={b}_{l}}(1-{\delta}_{i})}\sum _{i\in I{C}_{V},{B}_{i}={b}_{l}}(1-{\delta}_{i})g\phantom{\rule{0.16667em}{0ex}}({S}_{i}(1)).\end{array}$$

^{1}Supported by US NIH-NIAID Grant 2 RO1 AI054165-04 and NIH Grant R37 AI29168.

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