For randomized placebo-controlled efficacy trials of an HIV vaccine evaluated in HIV-uninfected volunteers, a primary objective is to evaluate the effect of vaccination on the incidence of HIV infection. Another objective, which was secondary in 2 trials of antibody-based vaccines (Flynn and others, 2005
; Pitisittithum and others, 2006
) and co-primary in 2 trials of T cell–based vaccines (Buchbinder and others, 2008
), is to evaluate the effect of vaccination on HIV viral load measured after HIV infection. Two causal approaches of interest for assessing the latter objective are intent-to-treat (ITT) methods that assess the burden of illness (a composite end point that combines the infection end point and the postinfection end point) (Chang and others, 1994
) and conditional methods that assess the postinfection end point in the principal stratum of subjects who would be HIV-infected under either treatment assignment (the so-called “always infected” stratum). As discussed by Gilbert and others (2003)
[henceforth GBH] and others, the 2 approaches address different substantive questions. For example, in registrational/licensure trials, the ITT approach may be most appropriate for the primary analysis and the conditional approach would be used in secondary analyses of the mechanistic vaccine effect on the postinfection outcome, whereas in preregistrational test-of-concept trials, the conditional approach might be used for the primary analysis (Mehrotra and others, 2006
Here, we develop methodology for the conditional approach to evaluate the causal vaccine effect on the postinfection end point. For concreteness, we refer to the first postrandomization event as infection and the outcome of interest measured after this event as viral load. However, the method has general application to evaluating causal treatment effects on outcomes measured after a postrandomization event, including studies of quality of life (Rubin, 2000
), prostate cancer severity (Shepherd and others, 2008
), kidney disease, and cancer screening (Joffe and others, 2007
Because the set of trial participants who are in the always infected stratum is unknown, causal treatment effects for this group are not identified from randomized trial data. Two approaches for addressing the nonidentifiability have been to derive sharp bounds for causal effects (Jemiai and Rotnitzky, 2003; Hudgens and others, 2003
; Zhang and Rubin, 2003
) and to estimate causal effects under an additional set of identifiability assumptions that include models describing the nature and degree of possible selection bias, with a sensitivity analysis to explore how the inferences vary over a range of the selection models (GBH; Hayden and others, 2005
; Hudgens and Halloran, 2006
; Jemiai and Rotnitzky, 2005
(henceforth JR); Jemiai and others, 2007
; Shepherd, Gilbert, Jemiai and others, 2006
; Shepherd and others, 2007
). All these methods assume that viral load is observed for all infected subjects. Therefore, the validity of their inferences depends on a missing completely at random assumption (MCAR). While the MCAR assumption is often untenable, trials may collect sufficient data on participant characteristics to make a missing at random (MAR) assumption plausible. This article extends the approach of GBH and JR to accommodate MAR missingness of the viral load end point.
To illustrate the new method, we focus on the first efficacy trial (Flynn and others, 2005
) and define the end point Yp
to be the pre-antiretroviral therapy (pre-ART) log10
viral load at the month 12 visit post HIV infection diagnosis. Only subjects who have not started ART by the month 12 visit contribute a value Yp
. We exclude viral load values measured after ART initiation because ART strongly effects viral load levels (Gilbert and others, 2003
). Inferences about Yp
apply to a population where ART is not prescribed during the first 12 months after infection diagnosis.
Of the 368 subjects who became HIV-infected during the trial, 121 had Yp observed, 138 had missing data because they initiated ART prior to the month 12 study visit, and 109 had missing data because they dropped out prior to initiating ART and prior to the visit. shows the VaxGen data on Yp = Y3 in relationship to 2 key covariates: Y1 is early pre-ART square root CD4 cell count, and Y2 is early pre-ART log10 viral load; these variables average all pre-ART values measured within 2 months of infection diagnosis. Viral load early and at 12 months were positively correlated (), with Spearman rank correlation 0.68 (0.55) for the vaccine (placebo) group. There is evidence of a negative correlation between early CD4 cell count and viral load at 12 months () for the vaccine group (Spearman correlation = − 0.33) but not for the placebo group (Spearman correlation = 0.16).
Fig. 1. VaxGen trial data: jittered pre-antiretroviral (pre-ART) viral loads at the month 12 postinfection diagnosis visit (Y3) ((a) and (b)); pre-ART viral loads at the month 12 postinfection diagnosis visit versus square root CD4 cell counts early after infection (more ...)
In addition, lower levels of Y1
and higher levels of Y2
were highly predictive of ART initiation (P
0.001 in a multivariate Cox model; Gilbert and others, 2005
), showing that MCAR was badly violated. MAR may be plausible, however, because physicians base decisions to prescribe ART on the monitoring of CD4 cell counts, viral loads, HIV-related clinical events, and comorbidities (Hammer and others, 2008
). In fact, MAR missingness due to ART initiation may approximately hold “systematically” in settings where ART is offered to all infected participants when their biomarkers are observed to cross prespecified thresholds or they present with prespecified symptomatic illnesses.
Two popular approaches to making valid inferences under MAR are inverse probability weighted (IPW)–based methods (e.g. Robins and others, 1995
) and likelihood-based methods (e.g. Little and Rubin, 2002
). For some HIV vaccine trials, the former methods are expected to be relatively inefficient and possibly unstable because the estimated weights for some subjects are expected to be near zero. Specifically, for a subject with Yp
observed, the weight in the denominator of the estimating equation equals the estimated probability that the subject did not drop out by the month 12 visit multiplied by the estimated probability that the subject did not start ART by the visit conditional on not dropping out. These estimates can be computed as fitted values in regression models based on the subject's covariates. The latter conditional probability may be near zero for subjects whose CD4 cell counts drop below the prespecified threshold at which ART initiation is recommended (currently the recommended threshold is between 200 and 500 cells/mm3
depending on the country). In fact, for ethical reasons any efficacy trial will offer ART to all infected participants who meet treatment criteria, such that the more successful the treatment coverage the closer some estimated weights in the denominator may be to zero. Therefore, for some HIV vaccine trials, IPW methods are expected to provide relatively imprecise inferences.
While likelihood methods are less subject to the instability problem, they are susceptible to misspecification of the model relating Yp
to covariates. To partially ameliorate this problem, we use the robust likelihood–based method of Little and An (2004)
(henceforth LA) that is based on penalized splines of the propensity score. Our approach only handles monotone missing data (i.e. dropout); an alternative approach would handle nonmonotone missingness using parametric multiple imputation and Monte Carlo Markov chain, for example, by extending the method of Mogg and Mehrotra (2007)
for analyzing viral load data to address postrandomization selection bias. The article is organized as follows. Section 2 describes the causal estimand of interest and identifiability assumptions. Section 3 shows how to combine the methodologies of GBH/JR and LA into a procedure for consistently estimating the causal estimand under an MAR assumption. Section 4 evaluates the new method in a simulation study. Section 5 applies the method to the first HIV vaccine efficacy trial, and Section 6 offers concluding remarks.