Case-Cohort Approach to Assessing Immunological Correlates of Risk, With Application to Vax004. Biostat 578A: Lecture 11

Transcription

1 Case-Cohort Approach to Assessing Immunological Correlates of Risk, With Application to Vax4 Biostat 578A: Lecture A manuscript pertinent to this talk is posted on the course webpage (JIDimmunearticle5pdf) Case-Cohort Designs for HIV Vaccine Trials p/

2 Outline Design of Vax4 for assessing immunological correlates of risk Methods: Case-cohort sampling design Cox proportional hazards model Application to Vax4 Case-Cohort Designs for HIV Vaccine Trials p/

3 Assessing Antibodies as Correlates of Risk in Vax4 Secondary objective: Assess if various in vitro measurements of antibody levels in vaccinees correlate with HIV infection rate 8 antibody assays that measure binding/neutralization of the MN or GNE8 HIV strains ELISAs to measure antibody binding: gp, V, V3, CD4 blocking Functional assay: Neutralization of MN HIV- Case-Cohort Designs for HIV Vaccine Trials p3/

4 Assessing Antibodies as Correlates of Risk in Vax4 Sampling design Specimens collected: Month,, 6,, 8, 4, 3, 36 (troughs) Month 5, 5, 65, 5, 85, 45, 35 (peaks) Specimens assayed: Random subcohort" of 5% of all vaccinees (n=74, all time points) n=63/ in subcohort uninfected/infected All infected vaccinees (n=39, last sample prior to infection) Case-Cohort Designs for HIV Vaccine Trials p4/

5 The Cox Model with The Case-Cohort Sampling Design Cox proportional hazards model λ(t Z) = λ (t)exp { β T Z(t) } λ(t Z) = conditional failure hazard given covariate history until time t β = unknown vector-valued parameter λ (t) = λ(t ) = unspecified baseline hazard function Z are expensive" covariates only measured on failures and subjects in the subcohort Case-Cohort Designs for HIV Vaccine Trials p5/

6 Notation and Set-Up (Matches Kulich and Lin, 4, JASA) T = failure time (eg, time to HIV infection diagnosis) C = censoring time X = min(t,c), = I(T C) N(t) = I(X t, = ) Y (t) = I(X t) Cases are subjects with = Controls are subjects with = Case-Cohort Designs for HIV Vaccine Trials p6/

7 Notation and Set-Up (Matches Kulich and Lin, 4, JASA) Consider a cohort of n subjects, who are stratified by a variable V with K categories ε = indicator of whether a subject is selected into the subcohort α k = Pr(ε = V = k), where α k > (X ki, ki,z ki (t), t τ,v ki,ε ki ) observed for all subcohort subjects At least (X ki, ki,z ki (X ki )) observed for all cases Case-Cohort Designs for HIV Vaccine Trials p7/

8 Estimation of β With full data, β would be estimated by the MPLE, defined as the root of the score function U F (β) = n τ {Z i (t) Z F (t,β)}dn i (t), () i= where Z F (t,β) = S () F (t,β)/s() F (t,β); S () n F (t,β) = n Z i (t)exp { β T Z i (t) } Y i (t) i= S () n F (t,β) = n exp { β T Z i (t) } Y i (t) i= Case-Cohort Designs for HIV Vaccine Trials p8/

9 Estimation of β Due to missing data () cannot be calculated under the case-cohort design Many modified estimators have been proposed, all of which replace Z F (t,β) with an approximation Z C (t,β), so are roots of U C (β) = K k= n k τ {Z ki (t) Z C (t,β)}dn ki (t) i= The double indices k, i reflect the stratification Case-Cohort Designs for HIV Vaccine Trials p9/

10 Estimation of β The case-cohort at-risk average is defined as where Z C (t,β) S () C (t,β)/s() C (t,β), S () K n k C (t,β) = n ρ ki (t)z ki (t)exp { β T Z ki (t) } Y ki (t) k= i= S () K n k C (t,β) = n ρ ki (t)exp { β T Z ki (t) } Y ki (t) k= i= Case-Cohort Designs for HIV Vaccine Trials p/

11 Estimation of β The potentially time-varying weight ρ ki (t) is set to zero for subjects with incomplete data, eliminating them from the estimation Cases and subjects in the subcohort have ρ ki (t) > Usually ρ ki (t) is set as the inverse estimated sampling probability (Using the same idea as the Weighted GEE methods of Robins, Rotnitzky, and Zhao, 994, 995) Different case-cohort estimators are formed by different choices of weights ρ ki (t) Two classess of estimators (N and D), described next Case-Cohort Designs for HIV Vaccine Trials p/

12 N-estimators The subcohort is considered a sample from all study subjects regardless of failure status The whole covariate history Z(t) is used for all subcohort subjects For cases not in the subcohort, only Z(T i ) (the covariate at the failure time) is used Prentice (986, Biometrika): ρ i (t) = ε i /α for t < T i and ρ i (T i ) = /α Self and Prentice (988, Ann Stat): ρ i (t) = ε i /α for all t Case-Cohort Designs for HIV Vaccine Trials p/

13 N-estimators General stratified N-estimator ρ ki (t) = ε i / α k (t) for t < T ki and ρ ki (T ki ) = α k (t) is a possibly time-varying estimator of α k α k is known by design, but nonetheless estimating α k provides greater efficiency for estimating β (Robins, Rotnitzky, Zhao,994) A time-varying weight can be obtained by calculating the fraction of the sampled subjects among those at risk at a given time point (Barlow, 994; Borgan et al,, Estimator I) Case-Cohort Designs for HIV Vaccine Trials p3/

14 D-estimators Weight cases by throughout their entire at-risk period D-estimators treat cases and controls completely separately α k apply to controls only, so that α k should be estimated using data only from controls Conditional on failure status, the D-estimator case-cohort design is similar to that of the case-control design whether or not the subcohort sampling is done retrospectively Case-Cohort Designs for HIV Vaccine Trials p4/

15 General D-estimator D-estimators ρ ki (t) = ki + ( ki )ε ki / α k (t) Borgan et al (, Estimator II) obtained by setting α k (t) = n n ε ki ( ki )Y ki (t)/ ( ki )Y ki (t), i i ie, the proportion of the sampled controls among those who remain at risk at time t Under Computing", the course webpage includes R code for Borgan s Estimator II with a time-independent expensive covariate of interest (contributed by Michal Kulich) Case-Cohort Designs for HIV Vaccine Trials p5/

16 Main Distinctions between N- and D- Estimators D-estimators require data on the complete covariate histories of cases N-estimators only require data at the failure time for cases For Vax4, the immune response in cases was only measured at the visit prior to infection, so N-estimators are valid while D-estimators are not valid Case-Cohort Designs for HIV Vaccine Trials p6/

17 Main Distinctions between N- and D- Estimators For N-estimators, the sampling design is specified in advance, whereas for D-estimators, it can be specified after the trial (retrospectively) D-estimators more flexible Case-Cohort Designs for HIV Vaccine Trials p7/

18 Example: Application to Vax4 Randomly selected subject-specific antibody profiles GNE8 CD4 Blocking MN CD4 Blocking Percent blocking Percent blocking GNE8 V MN V Optical density 3 Optical density GNE8 V MN V3 Optical density 3 Optical density GNE8/MN gp MN Neutralization Optical density 3 Log titer Months Since Entry Case-Cohort Designs for HIV Vaccine Trials p8/

19 Peak Antibody Levels of Vaccinees (Solid/dotted = Uninfected/infected) Optical density Optical density Optical density Percent blocking GNE8 CD4 Blocking MN CD4 Blocking Percent blocking GNE8 V MN V Optical density GNE8 V3 MN V3 Optical density GNE8/MN gp MN Neutralization Log titer Months Since Entry Case-Cohort Designs for HIV Vaccine Trials p9/

20 Tests for Different Antibody Levels, Uninfected vs Infected Vaccinees Wei-Johnson (985, Biometrika) tests linearly combine Wilcoxon statistics across the 7 time-points Overall/aggregate tests of whether peak antibody levels differ between infected (n=39) and uninfected (n=63) vaccinees Antibody Variable Wei-Johnson p-value MN CD4 74 GNE8 CD4 45 MN V 3 GNE8 V 8 MN V3 GNE8 V3 3 MN/GNE8 gp 39 MN Neutralization 6 Case-Cohort Designs for HIV Vaccine Trials p/

21 Results of Case-Cohort Cox Model Analysis Fit Prentice (986) case-cohort Cox model, using α = 74/3598 = 484 Antibody HR of HIV infection by Ab Quartile P-value for P-value for variable Q Q Q3 Q4 difference trend MN CD GNE8 CD4 Binding MN V GNE8 V MN V GNE8 V MN/GNE8 gp MN Neutralization Case-Cohort Designs for HIV Vaccine Trials p/

22 Interpretation of Vax4 Results MN CD4 blocking, GNE8 CD4 blocking, GNE8 V, GNE8 V3, MN Neutralization responses inversely correlated with HIV infection rate Estimated V E S negative for low responses, zero for medium responses, positive for high responses Two possible explanations High antibody levels cause protection and low antibody levels cause increased susceptibility [Causation Hypothesis] Antibody levels mark individuals by their intrinsic risk of infection [Association Hypothesis] New methods needed to discriminate these Addressed by Dean Follmann, covered in Lecture Case-Cohort Designs for HIV Vaccine Trials p/