Evaluating Correlates of Risk in HIV Vaccine Trials

Evaluating Correlates of Risk in HIV Vaccine Trials JoAnna Scott Biostat 578a – Vaccine Efficacy March 7, 2006 Research Questions Within an HIV vaccine trial, there is much interest in determining how a vaccine protects against infection and how long that vaccine will be able to protect against infection before requiring boosting.  I am interested in trying to determine how to answer these questions.  March 7, 2006 2 Research Questions 1. 2. 3. What immune responses seem to be associated with protection from HIV infection? Is immune response a surrogate of protection? What is the durability of the vaccine efficacy? 3 March 7, 2006 Correlates of Risk (Q1) Question 1: What immune responses are associated with the rate of infection?  This question attempts to answer which immune responses are correlates of risk (COR).  Correlates of risk: individual level predictors of risk  March 7, 2006 4 Correlates of Risk (Q1)  How to find CORS?  Traditionally, examine the immune responses of individuals who recover naturally from the disease.  Not possible as of yet with HIV infection.  Estimate from  vaccine trials. Need a different type of analysis that would require immune response measures collected on more people than in a standard trial design. March 7, 2006 5 Case-Cohort Analysis (Q1)     How to find CORS? Use a case-cohort analysis. In April of 1986, Ross Prentice published a paper in Biometrika introducing the case-cohort design (1). This innovative design uses a sub-sampling technique in survival data for estimating the relative risk of disease in a cohort study without collecting data from the entire cohort. 6 March 7, 2006 Case-Cohort Background This type of study was originally designed to allow efficient analysis of studies where the population size was too large to collect detailed data on all the participants, e.g. large survey studies.  These types of studies have large sample sizes which makes it too expensive and time consuming to analyze all data on all subjects.  March 7, 2006 7 Case-Cohort Background Prentice proposed randomly selecting a subcohort from the original sample at entry and only analyzing data on members of the subcohort and all cases.  Raw data is collected on all subjects, but the data would only be analyzed on cases and subcohort members.  March 7, 2006 8 Case-Cohort Background  For example, blood samples would be collected over time for all study participants and frozen for storage. Then, the biochemical analysis for specific covariates would only be performed on participants in the randomly selected subcohort or subjects that developed the disease of interest. 9 March 7, 2006 Case-Cohort Design: Time to Event  We are primarily interested in estimating the relative risk (hazard ratio) in the case of time to event data. Recall the partial likelihood function for k failure times, k exp(zi(ti) ) L( ) =  i1  exp(z j (ti) ) jRi March 7, 2006 10 Case-Cohort Design: Time to Event Let Y j (ti)  1 if at risk 0 if not at risk Then, n                    for person j at time ti.             Yi(ti)exp(zi(ti) ) i L( ) =  n i1 Y (t )exp(z (t ) ) j i j i j1  where i  1 if Ni(ti)  Ni(ti ) ; i.e., the censoring indicator. 0 if otherwise          March 7, 2006 11 Case-Cohort Design: Time to Event Let r(x) be a fixed function such that r(0) = 1. Then the partial likelihood is Yi(ti) r(zi(ti) ) i n rii i L( ) =  n  n  i1 Y (t ) r(z (t ) ) i1  r j i j i ji j1 j1 n                                                 where r ji  Y j(ti) r(z j(ti) )  assuming independent failure times and censoring for the full cohort data March 7, 2006 12 Case-Cohort Design: Time to Event Suppose a random subcohort C of size m is selected from the entire cohort and that {Ni , Yi} are available for all cohort members.  However, covariate histories are only available for the members of C and for subjects that fail.  March 7, 2006 13 Case-Cohort Design: Time to Event Then the pseudo-likelihood is n  ) =  L( i1                  jR(ti)  r ji rii                  i  This pseudo-likelihood differs from the partial likelihood in that the denominator is summing over subjects at risk in R(ti) rather than subjects at risk in the entire cohort 14 March 7, 2006 Case-Cohort Design: Time to Event  Now solving U( ) =   log L( )=0 yields the maximum pseudo-likelihood ˆ estimate . n ( ) =  L            i1 jR(ti ) Yi(ti) r(zi(ti) ) i .  Y j(ti) r(z j(ti) )            March 7, 2006 15 Case-Cohort Design: Time to Event Hence, n ( ) = log L( )   log  l i i1                          jR(ti) Yi(ti) r(zi(ti) )  Y j(ti) r(z j(ti) )                          =  i log Yi(ti) + log r(zi(ti) ) - log  Y j (ti) r(z j (ti) ) i1 jR(ti) n March 7, 2006 16 Case-Cohort Design: Time to Event Therefore, n    ) =   l ( ) =    log U( i i1                Y j (ti) z j (ti) r (z j(ti) ) zi(ti) r (zi(ti) ) =  i { -  } r(z j (ti) ) i1 jR(t )  Y (t ) r(z (t ) ) n jR(ti) Yi(ti) r(zi(ti) )  Y j(ti) r(z j(ti) )                where r(u) =  u r(u). i jR(t ) i j i j i March 7, 2006 17 Case-Cohort Design: Time to Event ˆ So asymptotic properties of  will derive  from those of the score statistic U( ).  Note that one cannot just use theory developed for survival analysis to work through the asymptotic properties of this function. To use that theory we would need to include everyone who was at risk and who died at each time point.  March 7, 2006 18 Case-Cohort Design: Time to Event In the case-cohort pseudo-likelihood, we are not including everyone at risk for each time point. We are adding cases into our dataset later that were not necessarily in earlier risk sets.  This means that the risk sets are not nested.  March 7, 2006 19 Case-Cohort Design: Time to Event  In 1988, Self and Prentice (3) developed the asymptotic theory behind the relative risk estimate for the case-cohort design. They produced an estimate that was ˆ asymptotically equivalent to  . Their variance estimator turned out to be quite algebraically complex. March 7, 2006 20 Case-Cohort Design: Time to Event  In 1998, Therneau and Li (4) published a technical report describing how to obtain these parameter and variance estimates using any proportional hazards regression program that support an offset command, dfbeta residuals, and the (start, stop] notation to describe risk intervals, i.e. the S-plus coxph command and the SAS phreg command. 21 March 7, 2006 Case-Cohort Design: Time to Event They noticed that you could rewrite the variance estimate in a computationally easy manner.  V  I 1(1 )DT D a SC SC  where DSC is a subset of the matrix of dfbeta residuals that contains only those rows for the subcohort SC.  = mn, the proportion of people in the subcohort. March 7, 2006 22 Case-Cohort Design: Time to Event  In S-plus, simply use the proportional hazard model to get the parameter estimate. Then fix-up the variance using the previous formula. An example follows: dfbeta <- resid(fit, type=‘dfbeta’) d2<-dfbeta[data$subcohort==1] fit$naive.var <- fit$var fit$var <- fit$var + (1-alpha)*t(d2)%*%d2  March 7, 2006 23 Case-Cohort Design This case-cohort design that Prentice developed has been modified by several different people resulting in many estimators available for use in finding CORs in an HIV vaccine trial.  Kulich and Lin (6) has divided the estimators into two categories, Nestimators and D-estimators.  March 7, 2006 24 Q1: Correlates of Risk  The Borgan estimator II (a D-estimator) appears to be the estimator that I would choose for answering question 1.  Advantage: Tends to be more efficient, allows prospective and retrospective sampling, can account for time dependency of covariates.  Disadvantage: To get unbiased estimates, you need the entire covariate history of the cases.  Need to plan for in trial design. 25 March 7, 2006 Q2: Surrogates of Protection    Q2: What immune responses are surrogates of protection? Surrogates of Protection: individual or group level predictors of vaccine efficacy (i.e. surrogate endpoints). Essentially, we are interested in determining if causal vaccine effects on immune response predict causal vaccine effects on risk. 26 March 7, 2006 Q2: Surrogates of Protection Traditionally, use a meta-analysis to compare hazard ratios of immune response on HIV infection rates across several different trials.  However, multiple trials on individual vaccines typically don’t happen.  March 7, 2006 27 Q3: Durability of VE To examine the durability of vaccine efficacy, we will need to be able to examine VE over time.  The methods I am interested in using to look at the question is a Stepped Wedge Design (7) trial.  March 7, 2006 28 Stepped Wedge Design  Cluster Randomized Trial  Multiple Time Cluster 0 1 0 1 0 0 2 1 1 3 1 1 4 1 1 time points  Time of cross-over is randomized  Cross-over only occurs in one direction 1 2 3 4 0 0 0 0 0 0 1 0 1 1 March 7, 2006 29 Stepped Wedge Design      Allows for a model based analysis Has reduced sensitivity to between-cluster variation Controls for temporal trends Can be useful in a Phase III trial to support licensure and after licensure in Phase IV trials. Allows for measuring durability of vaccine efficacy over long term follow-up. 30 March 7, 2006 Research Questions The most efficient way to look at these questions is with one trial.  I.e., Performing a case-cohort analysis within a stepped wedge trial.  This would need to be accounted for during the trial design.  March 7, 2006 31 Research Questions (Q1)  Using a case-cohort analysis within a stepped wedge trial requires some additional research. analysis would result in a log-hazard ratio for each the clusters. Hence we would need to perform a test of homogeneity.  What test would you use?  If the test result was that the log-hazard ratios were homogeneous, how would you combine them into one log hazard ratio?  The case-cohort March 7, 2006 32 Research Questions (Q2)  Using a stepped wedge design is a novel way to design a mock meta-analysis.  We need to perform a meta-analysis to determine if immune response is a causal surrogate of protection.  First to determine causation, we would like to create a random intervention to get a contrast in a potential surrogate. March 7, 2006 33 Research Questions (Q2)   So basically we want to randomly introduce different levels of immune response. One way this could be done is to have a high and low dose of vaccine, which should introduce a high and low level of immune response in the participants.  For example, use a 3 arm trial: high and low dose and a placebo arm.  Set the 3 arm trial within a stepped wedge trial 34 March 7, 2006 Research Questions (Q2)   Can treat the different clusters comparisons from a stepped wedge design trial as a metaanalysis. For example, suppose that there are 9 clusters, 3 for each arm, then comparing all low dose hazard ratios to each other, all high dose, and all placebo to come up with what simulates 3 trials to compare in a metaanalysis. 35 March 7, 2006 Research Questions (Q3)  The stepped wedge design at present does not account for time effects across the steps, which is necessary to be able to measure durability of the vaccine efficacy.  How could the method be updated to account for these time effects? March 7, 2006 36 Acknowledgements  Much thanks to Pamela Shaw, Jim Hughes, and Peter Gilbert for all their help. March 7, 2006 37 References 1. 2. 3. 4. 5. 6. 7. Prentice R.L., “A case-cohort design for epidemiologic cohort studies and disease prevention trials” Biometrika: 73(1), 1986 pgs 1-11. Prentice R.L, Pyke R., “Logistic disease incidence models and case-control studies”. Biometrika: 66(3), 1979 pgs 403-11. Self S.G., Prentice R.L., “Asymptotic distribution theory and efficiency results for case-cohort studies”. The Ann. of Stat.: 16(1), 1988 pgs 64-81. Therneau T.M., Li H., “Computing the Cox Model for Case Cohort Designs”. Technical Report Series Section of Biostatistics: 62, June 1998. Gilbert P.B, et al., “Correlation between Immunologic Responses to a Recombinant Glycoprotein 120 Vaccine and Incidence of HIV-1 Infection in a Phase 3 HIV-1 Preventive Vaccine Trial”. Journ. Infect. Dis.: 191 March 1, 2005 pgs. 666-77. Kulich, M. and Lin, D.Y., “Improving the Efficiency of Relative Risk Estimation in Case-Cohort Studies”. JASA: 99(467), Sept. 2004. pgs. 832 – 844. Hussey M. and Hughes J. “Cluster Randomized Crossover Designs: Design and Analysis of the Stepped Wedge Design”, Biostat 578 Website -http://faculty.washington.edu/peterg/Vaccine2006/articles/HusseyHughes.pdf March 7, 2006 38