Estimation of Stratified Mark-Specific Proportional Hazards Models with Missing Marks

Estimation and Hypothesis Testing of Strain-Specific Vaccine Efficacy With Missing Strain Types With Application to a COVID-19 Vaccine Trial

Based on data from a randomized, controlled vaccine efficacy trial, this article develops statistical methods for assessing vaccine efficacy (VE) to prevent COVID-19 infections by a discrete set of genetic strains of SARS-CoV-2. Strain-specific VE adjusting for possibly time-varying covariates is estimated using augmented inverse probability weighting to address missing viral genotypes under a competing risks model that allows separate baseline hazards for different risk groups. Hypothesis tests are developed to assess whether the vaccine provides at least a specified level of VE against some viral genotypes and whether VE varies across genotypes. Asymptotic properties providing analytic inferences are derived and finite-sample properties of the estimators and hypothesis tests are studied through simulations. This research is motivated by the fact that previous analyses of COVID-19 vaccine efficacy did not account for missing genotypes, which can cause severe bias and efficiency loss. The theoretical properties and simulations demonstrate superior performance of the new methods. Application to the Moderna COVE trial identifies several SARS-CoV-2 genotype features with differential vaccine efficacy across genotypes, including lineage (Reference, Epsilon, Gamma, Zeta), indicators of residue match vs. mismatch to the vaccine-strain residue at Spike amino acid positions (identifying signatures of differential VE), and a weighted Hamming distance to the vaccine strain. The results show VE decreases against genotypes more distant from the vaccine strain, highlighting the need to update COVID-19 vaccine strains.

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan-Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.

Dynamic Classification of Plasmodium vivax Malaria Recurrence: An Application of Classifying Unknown Cause of Failure in Competing Risks

A standard competing risks set-up requires both time to event and cause of failure to be fully observable for all subjects. However, in application, the cause of failure may not always be observable, thus impeding the risk assessment. In some extreme cases, none of the causes of failure is observable. In the case of a recurrent episode of Plasmodium vivax malaria following treatment, the patient may have suffered a relapse from a previous infection or acquired a new infection from a mosquito bite. In this case, the time to relapse cannot be modeled when a competing risk, a new infection, is present. The efficacy of a treatment for preventing relapse from a previous infection may be underestimated when the true cause of infection cannot be classified. In this paper, we developed a novel method for classifying the latent cause of failure under a competing risks set-up, which uses not only time to event information but also transition likelihoods between covariates at the baseline and at the time of event occurrence. Our classifier shows superior performance under various scenarios in simulation experiments. The method was applied to Plasmodium vivax infection data to classify recurrent infections of malaria.

The mechanistic analysis of founder virus data in challenge models

Repeated low-dose challenge studies provide valuable information when evaluating candidate vaccines since they resemble the typical exposure of natural transmission and inform on the number of exposures prior to infection. Traditionally, the number of challenges to infection has been used as the outcome. This work uses the number of infecting viruses, or founder viruses at the time of infection, to more efficiently characterize a vaccine's mechanism of action. The vaccine mechanisms of action we consider are a Null mechanism (the vaccine offers no protection), a Leaky mechanism in which the number of founder viruses is reduced by some factor in vaccinated subjects, the All-or-None mechanism in which the vaccine randomly provides either complete protection or no protection in vaccinated subjects, and a Combination mechanism with both Leaky and All-or-None components. We consider two discrete marked survival models where the number of founder viruses follows a Poisson distribution with either a fixed mean parameter (Poisson model), or a random mean parameter that follows a Gamma distribution (negative binomial model). We estimate the models using maximum likelihood and derive likelihood ratio testing procedures that are accurate for small samples with boundary parameters. We illustrate the performance of these methodologies with a data example of simian immunodeficiency virus on nonhuman primates and a simulation study.

A Hybrid Approach for the Stratified Mark-Specific Proportional Hazards Model with Missing Covariates and Missing Marks, with Application to Vaccine Efficacy Trials

Deployment of the recently licensed CYD-TDV dengue vaccine requires understanding of how the risk of dengue disease in vaccine recipients depends jointly on a host biomarker measured after vaccination (neutralization titer - NAb) and on a "mark" feature of the dengue disease failure event (the amino acid sequence distance of the dengue virus to the dengue sequence represented in the vaccine). The CYD14 phase 3 trial of CYD-TDV measured NAb via case-cohort sampling and the mark in dengue disease failure events, with about a third missing marks. We addressed the question of interest by developing inferential procedures for the stratified mark-specific proportional hazards model with missing covariates and missing marks. Two hybrid approaches are investigated that leverage both augmented inverse probability weighting and nearest neighborhood hot deck multiple imputation. The two approaches differ in how the imputed marks are pooled in estimation. Our investigation shows that NNHD imputation can lead to biased estimation without properly selected neighborhood. Simulations show that the developed hybrid methods perform well with unbiased NNHD imputations from proper neighborhood selection. The new methods applied to CYD14 show that NAb is strongly inversely associated with risk of dengue disease in vaccine recipients, more strongly against dengue viruses with shorter distances.

Analysis of the time-varying Cox model for the cause-specific hazard functions with missing causes

This paper studies the Cox model with time-varying coefficients for cause-specific hazard functions when the causes of failure are subject to missingness. Inverse probability weighted and augmented inverse probability weighted estimators are investigated. The latter is considered as a two-stage estimator by directly utilizing the inverse probability weighted estimator and through modeling available auxiliary variables to improve efficiency. The asymptotic properties of the two estimators are investigated. Hypothesis testing procedures are developed to test the null hypotheses that the covariate effects are zero and that the covariate effects are constant. We conduct simulation studies to examine the finite sample properties of the proposed estimation and hypothesis testing procedures under various settings of the auxiliary variables and the percentages of the failure causes that are missing. These simulation results demonstrate that the augmented inverse probability weighted estimators are more efficient than the inverse probability weighted estimators and that the proposed testing procedures have the expected satisfactory results in sizes and powers. The proposed methods are illustrated using the Mashi clinical trial data for investigating the effect of randomization to formula-feeding versus breastfeeding plus extended infant zidovudine prophylaxis on death due to mother-to-child HIV transmission in Botswana.

Combining Viral Genetics and Statistical Modeling to Improve HIV-1 Time-of-infection Estimation towards Enhanced Vaccine Efficacy Assessment

Knowledge of the time of HIV-1 infection and the multiplicity of viruses that establish HIV-1 infection is crucial for the in-depth analysis of clinical prevention efficacy trial outcomes. Better estimation methods would improve the ability to characterize immunological and genetic sequence correlates of efficacy within preventive efficacy trials of HIV-1 vaccines and monoclonal antibodies. We developed new methods for infection timing and multiplicity estimation using maximum likelihood estimators that shift and scale (calibrate) estimates by fitting true infection times and founder virus multiplicities to a linear regression model with independent variables defined by data on HIV-1 sequences, viral load, diagnostics, and sequence alignment statistics. Using Poisson models of measured mutation counts and phylogenetic trees, we analyzed longitudinal HIV-1 sequence data together with diagnostic and viral load data from the RV217 and CAPRISA 002 acute HIV-1 infection cohort studies. We used leave-one-out cross validation to evaluate the prediction error of these calibrated estimators versus that of existing estimators and found that both infection time and founder multiplicity can be estimated with improved accuracy and precision by calibration. Calibration considerably improved all estimators of time since HIV-1 infection, in terms of reducing bias to near zero and reducing root mean squared error (RMSE) to 5-10 days for sequences collected 1-2 months after infection. The calibration of multiplicity assessments yielded strong improvements with accurate predictions (ROC-AUC above 0.85) in all cases. These results have not yet been validated on external data, and the best-fitting models are likely to be less robust than simpler models to variation in sequencing conditions. For all evaluated models, these results demonstrate the value of calibration for improved estimation of founder multiplicity and of time since HIV-1 infection.

Hypothesis tests for stratified mark-specific proportional hazards models with missing covariates, with application to HIV vaccine efficacy trials

This article develops hypothesis testing procedures for the stratified mark-specific proportional hazards model with missing covariates where the baseline functions may vary with strata. The mark-specific proportional hazards model has been studied to evaluate mark-specific relative risks where the mark is the genetic distance of an infecting HIV sequence to an HIV sequence represented inside the vaccine. This research is motivated by analyzing the RV144 phase 3 HIV vaccine efficacy trial, to understand associations of immune response biomarkers on the mark-specific hazard of HIV infection, where the biomarkers are sampled via a two-phase sampling nested case-control design. We test whether the mark-specific relative risks are unity and how they change with the mark. The developed procedures enable assessment of whether risk of HIV infection with HIV variants close or far from the vaccine sequence are modified by immune responses induced by the HIV vaccine; this question is interesting because vaccine protection occurs through immune responses directed at specific HIV sequences. The test statistics are constructed based on augmented inverse probability weighted complete-case estimators. The asymptotic properties and finite-sample performances of the testing procedures are investigated, demonstrating double-robustness and effectiveness of the predictive auxiliaries to recover efficiency. The finite-sample performance of the proposed tests are examined through a comprehensive simulation study. The methods are applied to the RV144 trial.

Estimation of Stratified Mark-Specific Proportional Hazards Models under Two-Phase Sampling with Application to HIV Vaccine Efficacy Trials

An objective of preventive HIV vaccine efficacy trials is to understand how vaccine-induced immune responses to specific protein sequences of HIV-1 associate with subsequent infection with specific sequences of HIV, where the immune response biomarkers are measured in vaccine recipients via a two-phase sampling design. Motivated by this objective, we investigate the stratified mark-specific proportional hazards model under two-phase biomarker sampling, where the mark is the genetic distance of an infecting HIV-1 sequence to an HIV-1 sequence represented inside the vaccine. Estimation and inference procedures based on inverse probability weighting of complete-cases and on augmented inverse probability weighting of complete-cases are developed. Asymptotic properties of the estimators are derived and their finite-sample performances are examined in simulation studies. The methods are shown to have satisfactory performance, and are applied to the RV144 vaccine trial to assess whether immune response correlates of HIV-1 infection are stronger for HIV-1 infecting sequences similar to the vaccine than for sequences distant from the vaccine. Augmented inverse probability weighting; auxiliary variables; case-cohort design; censored failure time; competing risks; HIV vaccine efficacy trial.

Mark-specific additive hazards regression with continuous marks

For survival data, mark variables are only observed at uncensored failure times, and it is of interest to investigate whether there is any relationship between the failure time and the mark variable. The additive hazards model, focusing on hazard differences rather than hazard ratios, has been widely used in practice. In this article, we propose a mark-specific additive hazards model in which both the regression coefficient functions and the baseline hazard function depend nonparametrically on a continuous mark. An estimating equation approach is developed to estimate the regression functions, and the asymptotic properties of the resulting estimators are established. In addition, some formal hypothesis tests are constructed for various hypotheses concerning the mark-specific treatment effects. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a data set from the first HIV vaccine efficacy trial is provided.

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1-11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.

Mark-specific hazard ratio model with missing multivariate marks

An objective of randomized placebo-controlled preventive HIV vaccine efficacy (VE) trials is to assess the relationship between vaccine effects to prevent HIV acquisition and continuous genetic distances of the exposing HIVs to multiple HIV strains represented in the vaccine. The set of genetic distances, only observed in failures, is collectively termed the 'mark.' The objective has motivated a recent study of a multivariate mark-specific hazard ratio model in the competing risks failure time analysis framework. Marks of interest, however, are commonly subject to substantial missingness, largely due to rapid post-acquisition viral evolution. In this article, we investigate the mark-specific hazard ratio model with missing multivariate marks and develop two inferential procedures based on (i) inverse probability weighting (IPW) of the complete cases, and (ii) augmentation of the IPW estimating functions by leveraging auxiliary data predictive of the mark. Asymptotic properties and finite-sample performance of the inferential procedures are presented. This research also provides general inferential methods for semiparametric density ratio/biased sampling models with missing data. We apply the developed procedures to data from the HVTN 502 'Step' HIV VE trial.

Goodness-of-fit test of the stratified mark-specific proportional hazards model with continuous mark

Motivated by the need to assess HIV vaccine efficacy, previous studies proposed an extension of the discrete competing risks proportional hazards model, in which the cause of failure is replaced by a continuous mark only observed at the failure time. However the model assumptions may fail in several ways, and no diagnostic testing procedure for this situation has been proposed. A goodness-of-fit test procedure for the stratified mark-specific proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazards depends nonparametrically on both time and the mark is proposed. The test statistics are constructed based on the weighted cumulative mark-specific martingale residuals. The critical values of the proposed test statistics are approximated using the Gaussian multiplier method. The performance of the proposed tests are examined extensively in simulations for a variety of the models under the null hypothesis and under different types of alternative models. An analysis of the 'Step' HIV vaccine efficacy trial using the proposed method is presented. The analysis suggests that the HIV vaccine candidate may increase susceptibility to HIV acquisition.

Comprehensive sieve analysis of breakthrough HIV-1 sequences in the RV144 vaccine efficacy trial

The RV144 clinical trial showed the partial efficacy of a vaccine regimen with an estimated vaccine efficacy (VE) of 31% for protecting low-risk Thai volunteers against acquisition of HIV-1. The impact of vaccine-induced immune responses can be investigated through sieve analysis of HIV-1 breakthrough infections (infected vaccine and placebo recipients). A V1/V2-targeted comparison of the genomes of HIV-1 breakthrough viruses identified two V2 amino acid sites that differed between the vaccine and placebo groups. Here we extended the V1/V2 analysis to the entire HIV-1 genome using an array of methods based on individual sites, k-mers and genes/proteins. We identified 56 amino acid sites or "signatures" and 119 k-mers that differed between the vaccine and placebo groups. Of those, 19 sites and 38 k-mers were located in the regions comprising the RV144 vaccine (Env-gp120, Gag, and Pro). The nine signature sites in Env-gp120 were significantly enriched for known antibody-associated sites (p = 0.0021). In particular, site 317 in the third variable loop (V3) overlapped with a hotspot of antibody recognition, and sites 369 and 424 were linked to CD4 binding site neutralization. The identified signature sites significantly covaried with other sites across the genome (mean = 32.1) more than did non-signature sites (mean = 0.9) (p < 0.0001), suggesting functional and/or structural relevance of the signature sites. Since signature sites were not preferentially restricted to the vaccine immunogens and because most of the associations were insignificant following correction for multiple testing, we predict that few of the genetic differences are strongly linked to the RV144 vaccine-induced immune pressure. In addition to presenting results of the first complete-genome analysis of the breakthrough infections in the RV144 trial, this work describes a set of statistical methods and tools applicable to analysis of breakthrough infection genomes in general vaccine efficacy trials for diverse pathogens.

Analysis of HLA A*02 association with vaccine efficacy in the RV144 HIV-1 vaccine trial

The RV144 HIV-1 vaccine trial demonstrated partial efficacy of 31% against HIV-1 infection. Studies into possible correlates of protection found that antibodies specific to the V1 and V2 (V1/V2) region of envelope correlated inversely with infection risk and that viruses isolated from trial participants contained genetic signatures of vaccine-induced pressure in the V1/V2 region. We explored the hypothesis that the genetic signatures in V1 and V2 could be partly attributed to selection by vaccine-primed T cells. We performed a T-cell-based sieve analysis of breakthrough viruses in the RV144 trial and found evidence of predicted HLA binding escape that was greater in vaccine versus placebo recipients. The predicted escape depended on class I HLA A*02- and A*11-restricted epitopes in the MN strain rgp120 vaccine immunogen. Though we hypothesized that this was indicative of postacquisition selection pressure, we also found that vaccine efficacy (VE) was greater in A*02-positive (A*02(+)) participants than in A*02(-) participants (VE = 54% versus 3%, P = 0.05). Vaccine efficacy against viruses with a lysine residue at site 169, important to antibody binding and implicated in vaccine-induced immune pressure, was also greater in A*02(+) participants (VE = 74% versus 15%, P = 0.02). Additionally, a reanalysis of vaccine-induced immune responses that focused on those that were shown to correlate with infection risk suggested that the humoral responses may have differed in A*02(+) participants. These exploratory and hypothesis-generating analyses indicate there may be an association between a class I HLA allele and vaccine efficacy, highlighting the importance of considering HLA alleles and host immune genetics in HIV vaccine trials.

IMPORTANCE

The RV144 trial was the first to show efficacy against HIV-1 infection. Subsequently, much effort has been directed toward understanding the mechanisms of protection. Here, we conducted a T-cell-based sieve analysis, which compared the genetic sequences of viruses isolated from infected vaccine and placebo recipients. Though we hypothesized that the observed sieve effect indicated postacquisition T-cell selection, we also found that vaccine efficacy was greater for participants who expressed HLA A*02, an allele implicated in the sieve analysis. Though HLA alleles have been associated with disease progression and viral load in HIV-1 infection, these data are the first to suggest the association of a class I HLA allele and vaccine efficacy. While these statistical analyses do not provide mechanistic evidence of protection in RV144, they generate testable hypotheses for the HIV vaccine community and they highlight the importance of assessing the impact of host immune genetics in vaccine-induced immunity and protection. (This study has been registered at ClinicalTrials.gov under registration no. NCT00223080.).

Inferences on relative failure rates in stratified mark-specific proportional hazards models with missing marks, with application to HIV vaccine efficacy trials

This article develops hypothesis testing procedures for the stratified mark-specific proportional hazards model in the presence of missing marks. The motivating application is preventive HIV vaccine efficacy trials, where the mark is the genetic distance of an infecting HIV sequence to an HIV sequence represented inside the vaccine. The test statistics are constructed based on two-stage efficient estimators, which utilize auxiliary predictors of the missing marks. The asymptotic properties and finite-sample performances of the testing procedures are investigated, demonstrating double-robustness and effectiveness of the predictive auxiliaries to recover efficiency. The methods are applied to the RV144 vaccine trial.

Mark-specific proportional hazards model with multivariate continuous marks and its application to HIV vaccine efficacy trials

For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates (Prentice and others, 1978. The analysis of failure time in the presence of competing risks. Biometrics 34, 541-554). Inspired by previous research in HIV vaccine efficacy trials, the cause of failure is replaced by a continuous mark observed only in subjects who fail. This article studies an extension of this approach to allow a multivariate continuum of competing risks, to better account for the fact that the candidate HIV vaccines tested in efficacy trials have contained multiple HIV sequences, with a purpose to elicit multiple types of immune response that recognize and block different types of HIV viruses. We develop inference for the proportional hazards model in which the regression parameters depend parametrically on the marks, to avoid the curse of dimensionality, and the baseline hazard depends nonparametrically on both time and marks. Goodness-of-fit tests are constructed based on generalized weighted martingale residuals. The finite-sample performance of the proposed methods is examined through extensive simulations. The methods are applied to a vaccine efficacy trial to examine whether and how certain antigens represented inside the vaccine are relevant for protection or anti-protection against the exposing HIVs.