correlatesofprotecti
TRANSCRIPT
Using Biomarkers in Vaccine Development and Evaluation
Biostat 578ALecture 10
Contributor: Steve Self
Immunological “Correlates of Protection”
• Key concept in vaccine development/evaluation– An immunologic measurement in response to
vaccination that is “correlated with protection”• Uses
– Guide for vaccine development– Bridging studies in vaccine production – Guide refinements of vaccine formulation– Basis for regulatory decisions– Guides for vaccination policy
• Precise meaning often confused- needs clarification and new terminology
Many Licensed Vaccines do not have a Known Correlate of Protection: List of FDA Licensed Vaccines (from FDA Website)
Product Name Trade Name Sponsor Immunological Correlate of Protection Known?
Anthrax Vaccine Adsorbed Biothrax BioPort Corp Partial, Antibodies
BCG (Bacille Calmette-Guérin) LiveTICE BCG Organon Teknika Corp No
BCG Live Mycobax Aventis Pasteur, Ltd
Diphtheria & Tetanus Toxoids Adsorbed No Trade Name Aventis Pasteur, Inc Yes, Antibodies
Diphtheria & Tetanus Toxoids Adsorbed No Trade Name Aventis Pasteur, Ltd
Diphtheria & Tetanus Toxoids & Acellular Pertussis Vaccine Adsorbed
Tripedia Aventis Pasteur, Inc
Diphtheria & Tetanus Toxoids & Acellular Pertussis Vaccine Adsorbed
Infanrix GlaxoSmithKline
Diphtheria & Tetanus Toxoids & Acellular Pertussis Vaccine Adsorbed
DAPTACEL Aventis Pasteur, Ltd
Diphtheria & Tetanus Toxoids & Acellular Pertussis Vaccine Adsorbed, Hepatitis B (recombinant) and Inactivated Poliovirus Vaccine Combined
Pediarix SmithKline Beecham Biologicals
Haemophilus b Conjugate Vaccine (Diphtheria CRM197 Protein Conjugate)
HibTITER Lederle Lab Div, American Cyanamid Co
Yes, Antibodies
Haemophilus b Conjugate Vaccine (Meningococcal Protein Conjugate)
PedvaxHIB Merck & Co, Inc
Haemophilus b Conjugate Vaccine (Tetanus Toxoid Conjugate)
ActHIB Aventis Pasteur, SA
Haemophilus b Conjugate Vaccine (Meningococcal Protein Conjugate) & Hepatitis B Vaccine (Recombinant)
Comvax Merck & Co, Inc
Many Licensed Vaccines do not have a Known Correlate of Protection: List of FDA Licensed Vaccines (from FDA Website)
Product Name Trade Name Sponsor Immunological Correlate of Protection Known?
Hepatitis A Vaccine, Inactivated Havrix GlaxoSmithKline No
Hepatitis A Vaccine, Inactivated VAQTA Merck & Co, Inc
Hepatitis A Inactivated and Hepatitis B (Recombinant) Vaccine
Twinrix GlaxoSmithKline
Hepatitis B Vaccine (Recombinant) Recombivax HB Merck & Co, Inc Partial, Antibodies
Hepatitis B Vaccine (Recombinant) Engerix-B GlaxoSmithKline
Influenza Virus Vaccine, Live, Intranasal FluMist MedImmune Vaccines, Inc Partial, Antibodies, CTLs suspected
Influenza Virus Vaccine, Trivalent, Types A and B Fluarix GlaxoSmithKline Biologicals
Influenza Virus Vaccine, Trivalent, Types A and B Fluvirin Evans Vaccines
Influenza Virus Vaccine, Trivalent, Types A and B Fluzone Aventis Pasteur, Inc
Japanese Encephalitis Virus Vaccine Inactivated JE-Vax Research Foundation for Microbial Diseases of Osaka University
No
Measles Virus Vaccine, Live Attenuvax Merck & Co, Inc Partial, Antibodies, CTLS and CD4s suspected
Measles and Mumps Virus Vaccine, Live M-M-Vax Merck & Co, Inc (not available) Partial, Antibodies
Measles, Mumps, and Rubella Virus Vaccine, Live M-M-R II Merck & Co, Inc
Measles, Mumps, Rubella and Varicella Virus Vaccine Live
ProQuad Merck & Co, Inc
Many Licensed Vaccines do not have a Known Correlate of Protection: List of FDA Licensed Vaccines (from FDA Website)
Product Name Trade Name Sponsor Immunological Correlate of Protection Known?
Meningococcal Polysaccharide (Serogroups A, C, Y and W-135) Diphtheria Toxoid Conjugate Vaccine
Menactra Aventis Pasteur, Inc Yes for some serotypes, Antibodies, no for other serotypes
Meningococcal Polysaccharide Vaccine, Groups A, C, Y and W-135 Combined
Menomune-A/C/Y/W-135
Aventis Pasteur, Inc
Mumps Virus Vaccine Live Mumpsvax Merck & Co, Inc Partial, Antibodies
Pneumococcal Vaccine, Polyvalent Pneumovax 23 Merck & Co, Inc Partial, Serotype-Specific Antibodies
Pneumococcal 7-valent Conjugate Vaccine (Diphtheria CRM197 Protein)
Prevnar Lederle Lab Div, American Cyanamid Co
Poliovirus Vaccine Inactivated (Human Diploid Cell) Poliovax Aventis Pasteur, Ltd (not available)
No
Poliovirus Vaccine Inactivated (Monkey Kidney Cell) IPOL Aventis Pasteur, SA
Rabies Vaccine Imovax Aventis Pasteur, SA Yes, Antibodies
Rabies Vaccine RabAvert Chiron Behring GmbH & Co
Rabies Vaccine Adsorbed No Trade Name BioPort Corp1 (not available)
Rubella Virus Vaccine Live Meruvax II Merck & Co, Inc No
Smallpox Vaccine, Dried, Calf Lymph Type Dryvax Wyeth Laboratories, Inc(available only thru CDC or DoD programs)
Partial, Antibodies
Many Licensed Vaccines do not have a Known Correlate of Protection: List of FDA Licensed Vaccines (from FDA Website)
Product Name Trade Name Sponsor Immunological Correlate of Protection Known?
Tetanus & Diphtheria Toxoids Adsorbed for Adult Use No Trade Name Massachusetts Public Health Biologic Lab
Yes, Antibodies
Tetanus & Diphtheria Toxoids Adsorbed for Adult Use DECAVAC Aventis Pasteur, Inc
Tetanus & Diphtheria Toxoids Adsorbed for Adult Use No Trade Name Aventis Pasteur, Ltd(not available)
Tetanus Toxoid No Trade Name Aventis Pasteur, Inc
Tetanus Toxoid Adsorbed No Trade Name Massachusetts Public Health Biologic Lab
Tetanus Toxoid Adsorbed No Trade Name Aventis Pasteur, Inc
Tetanus Toxoid, Reduced Diphtheria Toxoid and Acellular Pertussis Vaccine, Adsorbed
Adacel Aventis Pasteur, Ltd No for Acellular Pertussis
Tetanus Toxoid, Reduced Diphtheria Toxoid and Acellular Pertussis Vaccine, Adsorbed
Boostrix GlaxoSmithKline Biologicals
Typhoid Vaccine Live Oral Ty21a Vivotif Berna Biotech, Ltd No
Typhoid Vi Polysaccharide Vaccine TYPHIM Vi Aventis Pasteur, SA
Varicella Virus Vaccine Live Varivax Merck & Co, Inc No
Yellow Fever Vaccine YF-Vax Aventis Pasteur, Inc No
Summary of Licensed Vaccines and Correlates of Protection
• The immune responses responsible for protection of most licensed vaccines are unknown– Correlates known: 5 vaccine types– Correlates partially known: 7 vaccine types– Correlates unknown: 9 vaccine types
• Only antibody responses have been identified as correlates of protection
• For many licensed vaccines T cell responses are suspected to play a role in protection, but T cells have not yet been documented as correlates of protection
Utility of Biomarkers: Prediction
• Correlates are useful only to the extent that they build bridges… predicting effects in a new setting based on effects observed in another setting
• Different types and sizes of bridges:– Across vaccine lots, across different vaccine
formulations, across human populations, across viral populations, across species
• One correlate can be useful in building one type of bridge but not another
• Propose using the term predictor of protection (POP) to clarify and specify two essential elements:– What measurement(s) are used as basis for
prediction?– What target for prediction?
• Need typology for empirical basis of prediction
“Surrogates of Protection” (SOPs) vsCorrelates of Risk (CORs)
• Correlates of risk:– Individual-level predictors of risk– Estimable from cohort, nested case-control or nested case-cohort)
studies of different types of individuals – CORs among vaccinees– CORs among non-vaccinees
Natural history studies (general high-risk cohorts, highly exposed seronegative cohorts)
Control groups in randomized vaccine trials• Surrogates of protection:
– Individual- or group-level predictors of vaccine efficacy (i.e., individual- or group-level surrogate endpoints)
• An immune response identified to be a COR may be studied further to see if it is also a SOP and/or a POP
How Find a COR?
• Examine immune responses of individuals who recover naturally from disease
– Traditional approach to vaccine development– Immune responses preferentially present in those who
recover are CORs– In HIV, very few individuals naturally recover
The Center for HIV/AIDS Vaccine Immunology (CHAVI) is initiating a large study of Highly Exposed Seronegatives to identify CORs
• Animal challenge models– Challenge animals with a pathogen– Just prior to challenge, measure the immune response to
vaccination– Compare immune response levels in protected and
unprotected animals The Gates Foundation may be funding large monkey
challenge studies to facilitate “discovery” of CORs
Direct Assessment of a POP by Meta-Analysis
• N pairs of immunologic and clinical endpoint assessments among vaccinees and non-vaccinees– Pairs chosen to reflect specific target of prediction– Examples
1. Predict efficacy of vaccine to new viral strain: N strain-specific assessments of immunogenicity and efficacy
2. Predict efficacy of new vaccine formulation: N vaccine efficacy trials of “comparable vaccines but with different formulations”
– Plot of vaccinee/non-vaccinee contrast in endpoint rates (VE) vs contrast in immunologic response
Prediction for target based on observed immunologic response Prediction error read directly from scatter in plot
– Data intensive approach; often infeasible
Schematic Example 1. Plot of Estimated VEs(s) versus Mean Difference in Antibody Titers to Strain s [10 strains s]; Large Phase III Trial
This result would support that strain-specific antibody titer is a fairly reliable POP for predicting vaccine efficacy against new viral strains
Indirect Assessment of POPs:From CORs to SOPs to POPs
• Data for direct assessment of POPs are rarely available but CORs can often be identified (e.g., Vax004)
• Two indirect strategies for assessing a COR as a SOP/POP – Prentice (1989) criterion for a “statistical surrogate” endpoint:
COR to SOP: Can an individual-level regression model for risk be identified that is 1) consistent across vaccinated and unvaccinated individuals and 2) fully explains differences in risk between vaccinees and non-vaccinees?
SOP to POP: Can an individual-level regression model with the properties described above be used as the basis for prediction of protective effects in novel settings?
– Frangakis and Rubin (2002) criterion for a “principal surrogate” endpoint: COR to SOP: Do causal vaccine effects on the immune response
predict causal vaccine effects on risk? [addressed further in Lecture 12]
SOP to POP: Can the estimated “causal effect predictiveness” of the immune response be used as the basis for prediction of protective effects in novel settings?
Some Examples using the “Prentice Criterion” Framework
• From CORs to SOPs:– Influenza vaccine: Strain-specific Ab titer and risk of
clinical infection– rgp120 HIV-1 vaccine (Vax004): Binding Ab titers and
risk of infection
• From SOPs to POPs:– Influenza vaccine: Strain-specific Ab titer and strain-
specific VEs
1943 Influenza Vaccine Field Trial(Salk, Menke, and Francis)
• Study subjects– 1,776 men in 3651st Service Unit of ASTP at the
University of Michigan)– Age 18-47– Housed (mainly) in dormitories and fraternities– Dined in 3 mess halls– Common daily activities
1943 Influenza Vaccine Field Trial(Salk, Menke, and Francis)
• Treatment– Trivalent vaccine w/ components Weiss Strain A,
PR8 Strain A, Lee Strain B– Placebo control– Treatment assignment and delivery:
Men arranged alphabeticallyAlternate individuals inoculated with 1 ml of
vaccine/placebo subcutaneously Subjects blinded to assignmentAll inoculations completed over 7 day period
(Oct 25-Nov 2)
• Follow-up and serologic assessments– Blood for serology at vaccination, + 2 weeks and
at end of study for sample of participants– Every 10th vaccinee and every 5th placebo
recipient included in sample (approx 10% and 20% of study cohort, respectively)
– 35 participants lost to follow-up (19 controls, 16 vaccinees) for retention rate of 98%
1943 Influenza Vaccine Field Trial(Salk, Menke, and Francis)
1943 Influenza Vaccine Field Trial
• Clinical Endpoints– Daily “sick call”, clinic and hospital-based
surveillance– Multiple throat washes for viral culture– Blood samples
Results
• Weiss Strain A– Case incidence
Controls: 8.45 / 100Vaccinees: 2.25 / 100
– Estimated VEs = 73% • PR8 Strain A
– Case incidenceControls: 8.22 / 100Vaccinees: 2.25 / 100
– Estimated VEs = 73%
Strain-specific Ab Titer:COR? Also a SOP?
• COR models– Estimate relationship between Ab titer and risk
within control group (COR among non-vaccinees)– Estimate relationship between Ab titer and risk
within vaccine group (COR among vaccinees)– Assess consistency between two COR models
• Ab titer as SOP?– Compute predicted efficacy based on
Observed effect of vaccination on Ab titerCOR model among non-vaccinees (w/
extrapolation)Observed risk in control group
– Compare predicted VEs with observed VEs
3 4 5 6 7 8 9
05
10
15
20
25
log(Ab titer)
Ca
se In
cid
en
ce/1
00
Weiss strain Type A: Control Gp
3 4 5 6 7 8 9
51
01
52
02
53
0
log(Ab titer)
Pe
rce
nt
ControlVaccine
Expected Risk
Observed Risk
Estimated Incidence as a Function of Log Antibody Titer (from logistic regression)
Logistic Regression Models:Estimated Coefficients (SE)
Control Gp Only Control and Vaccine Gps Model 1 Model 2 Model 3 Model 4 Intercept 1.80 (0.54) -2.38 (0.12) 1.62 (0.45) 1.80 (0.54)log(Titer) -1.03 (0.14) - -0.98 (0.12) -1.03 (0.14)Tmt - -1.39 (0.25) 0.33 (0.32) -0.43 (1.28)Tmt*log(Titer) - - - 0.16 (0.25)
Weiss Strain A
3 4 5 6 7 8 9
05
10
15
20
25
log(Ab titer)
Ca
se In
cid
en
ce/1
00 Control-O
Control-EVaccine-OVaccine-E
Weiss strain Type A
3 4 5 6 7 8 9
51
01
52
02
53
0
log(Ab titer)
Pe
rce
nt
ControlVaccine
Model-Fit is good, based on Observed and Expected Incidence
Estimated and Predicted VEs:Weiss Strain A
• Direct estimates of VEs (w/o use of Ab titer)– Est-VEsCrude = 73%
• Predicted VEs – Based on [Risk | Ab, Controls] plus [Ab | Vaccine]– Pred-VEs = 82%
• “Prentice Criterion” for a surrogate endpoint – Vaccine effect on surrogate completely explains
effect on clinical endpoint– Log(Ab titer) satisfies criterion as a surrogate of
protection
3 4 5 6 7 8 9
05
1015
2025
log(Ab titer)
Cas
e In
cide
nce/
100
Control-OControl-EVaccine-OVaccine-E
Weiss strain Type A
3 4 5 6 7
02
46
810
log(Ab titer)
Cas
e In
cide
nce/
100
Control-OControl-EVaccine-OVaccine-E
PR8 strain Type A
3 4 5 6 7 8 9
510
1520
2530
log(Ab titer)
Per
cent
ControlVaccine
Weiss strain Type A
3 4 5 6 7
010
2030
40
log(Ab titer)
Per
cent
ControlVaccine
PR8 strain Type A
Estimated Incidence as a Function of Log Antibody Titer, Weiss & PR8 Strains A
Logistic Regression Models:Estimated Coefficients (SE)
Control Gp Only Control and Vaccine Gps Model 1 Model 2 Model 3 Model 4 Intercept -1.37 (0.59) -2.41 (0.12) -1.27 (0.53) -1.37 (0.59)log(Titer) -0.27 (0.15) - -0.29 (0.14) -0.27 (0.15)Tmt - -1.36 (0.26) -0.89 (0.34) -0.22 (1.79)Tmt*log(Titer) - - - -0.13 (0.34)
PR8 Strain A
Estimated and Predicted VE:PR8 Strain A
• Direct estimate of VEs (w/o use of Ab titer)– Est-VEsCrude = 73%
• Predicted VE – Based on [Risk | Ab, Controls] plus [Ab | Vaccine]– Pred-VEs = 33%
• “Prentice Criterion” for a surrogate endpoint – Log(Ab titer) does not satisfy criterion as a
surrogate of protection– Only ½ of overall protective effect is predicted
from effect on Ab titer
Discussion
• Protection from PR8 Strain A only partly described by PR8 Ab titer
• A (Prentice) surrogate of protection will have:– The same association between immune
response and risk in vaccinees and in non-vaccinees
– Consistency of the within-group association and the between-group association (VEs)
Control
Vaccine
Weiss Strain A
Risk
Ab Titer
Control
Vaccine
PR8 Strain A
Risk
Ab Titer
Explained by COR model
Not explained by COR model
Discussion
• Protection from PR8 Strain A only partly described by PR8 Ab titer– A possible explanation is that antibodies
are protective, but the measurements reflect something else besides protective responses (i.e., measurement error) Measurement error attenuates within-
group association Q. How to accommodate measurement
errors in assessment of COR as a SOP?
Control
Vaccine
PR8 Strain A
Risk
Ab Titer
De-attenuated COR models to accommodate measurement error; Adjusted model consistent w/ SOP
Discussion
• Protection from PR8 Strain A only partly described by PR8 Ab titer– Another possible explanation is that there
are other protective immune responses that were not measuredE.g., cell-mediated immune responses
– Another possible explanation is that PR8 Strain A has different protective determinants than Weiss Strain A
POP for Strain-specific VEs:Direct Assessment
• Strain-specific Ab titer as a POP for emerging viral strains?
• Basis of prediction from SMF study– N = 2 (2 pairs of strain-specific Ab responses
and estimated VEs)– Plot observed strain-specific VEs vs
mean Ab titer (Vaccine vs Control)Predicted VE based on Ab titer distributions
(Vaccine vs Control) and COR model among non-vaccinees
P-VE for emergent viral strain
Prediction interval of efficacyfor new viral strain??
0 20 40 60 80 100
02
04
06
08
01
00
Predicted VEs
Ob
serv
ed
VE
s
Assessing ability to predict VEs across viral strains
Weiss PR8
Problems with Prentice Framework
• COR models in non-vaccinees may not be estimable– If the COR is “response to vaccine” then cohort study
relating COR to risk in non-vaccinees is impossible– If no variation in putative COR among non-vaccinees
• In these cases the causal inference approach (based on Frangakis and Rubin) may be more useful
• Statistical surrogates (satisfying the Prentice criteria for a surrogate endpoint) are based on net effects, not causal effects, implying this criterion may mislead– See Frangakis and Rubin (2002)
Introduction to Causal Inference Approach from CORs to CSOPs (Expanded on in Lecture 12)
• In the causal inference paradigm, causal vaccine efficacy is based on comparing risk within the same individual if he/she were assigned vaccine versus if assigned control
• A difference within the same individual is directly attributable to vaccine, and thus is a causal effect
• A CSOP, i.e., a “Causal Surrogate of Protection”, is defined in this framework (defined below)
Causal Inference Approach from CORs to CSOPs
• VEcausal = 1 – Pr[Y(1) = 1]/Pr[Y(0)=1] – Y(1) = indicator of outcome if assigned vaccine– Y(0) = indicator of outcome if assigned placebo
• Interpretation of VEcausal: Percent reduction in risk for a subject assigned vaccine versus assigned control
• In randomized, blinded trial, VEcausal can be estimated by comparing event rates in vaccine and control groups
Causal Inference Approach: From CORs to CSOPs
• Approach to assessing whether a COR is a CSOP: Study how causal vaccine efficacy varies over groups defined by fixed values of both the immune response if assigned vaccine, X(1), and the immune response if assigned control, X(0)
• VEcausal(x1,x0) = 1- Pr[Y(1)=1|X(1)=x1,X(0)=x0] Pr[Y(0)=1|X(1)=x1,X(0)=x0]
– Compares risk for the same individual who would have immune responses x1 under vaccine and x0 under control
Simplification of Causal Vaccine Efficacy Parameter
• For many immunological measurements, X(0) is constant (e.g., ~0) for all subjects, because placebo does not induce responses– Causal VE can be rewritten as VEcausal(x1,x0=c) = VEcausal(x1)
= 1-Pr[Y(1)=1|X(1)=x1]/Pr[Y(0)=1|X(1)=x1]
Simplified interpretation: Percent reduction in risk for a vaccinated individual with response x1 compared to if he/she had not been vaccinated
– E.g., VEcausal(x1=high response) = 0.5: an individual with high immune response to vaccine has halved risk compared to if he/she had not been vaccinated
Interpretation of VEcausal(x1)
– VEcausal(0)=0 implies the immune response is causally necessary as defined by Frangakis and Rubin (FR) (2002): the vaccine can only have efficacy in a person if it stimulates x1 > 0
– VEcausal(x1) increasing with x1 implies a higher immune response to vaccine directly causes lower risk- implies a COR is a CSOP
– Motivates terminology “Causal Surrogate of Protection” (CSOP)The slope of increase of VEcausal(x1) with x1 measures the
strength of the causal correlation of x1 with protectionThis slope is a measure of the associative effect in the
terminology of FR– VEcausal(x1) constant in x1 implies that this immune response
has no causal effect on risk, i.e., x1 is a COR but not a CSOP
Interpretation of VEcausal(x1)
• Note that there must be some protection in order for a COR to be a CSOP– VEcausal = 0 and no enhancement of risk at any
immune response level implies VEcausal(x1) = 0 for all x1- not a CSOP
• “Causal surrogate of protection” is only meaningful when there is some protection (VEcausal > 0)!
Fundamental Problem of Causal Inference Approach
– In controls, X(1) is not measured- it is the immune response he/she would have had had he/she been vaccinated
– To estimate VEcausal(x1) a technique is needed for predicting the X(1)’s of controls
– Approaches suggested by Dean Follmann (Covered in Lecture 12)Exploit correlations of X(1) with subject-specific
characteristics measured in both vaccinees and controls Immunological measurements Immune response to a non-HIV vaccine or blank-
vectorCloseout vaccination of uninfected control subjects
Assume the (unmeasured) X(1) during the trial equals the immune response Xc measured after the trial
Causal Inference Approach
• This approach most useful when:– The range of immune responses in controls is
very narrow [e.g., X(0) ~ zero for the VaxGen trials], which simplifies VEcausal(x1) to vary only in x1
– Limited variability of X(0) in controls makes difficult assessing whether a COR is a SOP within the Prentice framework
Causal Inference Approach: VaxGen Illustration [U.S. Trial]
• ? is the risk for a placebo recipient with Qk quartile antibody response that he/she would have had had he/she been vaccinated
Q1 Q2 Q3 Q4
Vaccine 0.18 0.10 0.10 0.08
Placebo ? ? ? ?
Risk of Infection by Antibody Quartile
Causal Inference Approach: VaxGen Illustration
• Idea: Control/adjust for the antibody response if assigned vaccine– Decreasing relative risks (vaccine/placebo)
with increasing antibody levels implies a CSOP- some causal effect
– Constant relative risks (vaccine/placebo) with increasing antibody levels implies not a CSOP- no causal effect
VaxGen Illustration: Example 1 [COR is a CSOP]
• A CSOP- a higher vaccine-induced antibody response directly causes a lower risk of infection (relative risks 1, 0.56, 0.56, 0.44)
Q1 Q2 Q3 Q4
Vaccine 0.18 0.10 0.10 0.08
Placebo 0.18 0.18 0.18 0.18
VaxGen Illustration: Example 2 [COR Not a CSOP]
• Not a CSOP- the level of vaccine-induced antibody response does not causally effect the risk of infection (relative risks 0.5, 0.5, 0.5, 0.5)
Q1 Q2 Q3 Q4
Vaccine 0.18 0.10 0.10 0.08
Placebo 0.36 0.20 0.20 0.16
VaxGen Illustration
– Estimates for Example 1:VEcausal(Q1) = 1 – 0.18/0.18 = 0VEcausal(Q2) = 1 – 0.10/0.18 = 0.44VEcausal(Q3) = 1 – 0.10/0.18 = 0.44VEcausal(Q4) = 1 – 0.08/0.18 = 0.56
VEcausal(x1) increasing in antibody quartile implies a CSOP
– Estimates for Example 2:VEcausal(Q1) = 1 – 0.18/0.36 = 0.5VEcausal(Q2) = 1 – 0.10/0.20 = 0.5VEcausal(Q3) = 1 – 0.10/0.20 = 0.5VEcausal(Q4) = 1 – 0.08/0.16 = 0.5
VEcausal(x1) constant in antibody quartile implies not a CSOP
Illustration with 1943 Influenza Trial [Much Variation in X(0)]
• Imputation of X(1) (= log ab titer) for controls– Assume any two control subjects with log ab
titers X1(0) < X2(0) have X1(1) < X2(1); i.e., a higher response for a control subject implies a higher response had he/she received vaccine
– This equipercentile assumption is X(1) = Fv-1(Fc(X(0)))Fv = empirical distribution of log ab titer in
vaccine groupFc = empirical distribution of log ab titer in
control group– This assumption allows construction of a
complete dataset of {X(1),X(0)} for all trial participants
Imputed X(1)’s corresponding to the observed x0’s in controls
exp(x0) observed in controls
Imputed exp(X(1))
16 128
32 256
64 512
128 1024
256 2048
512 4096
1024 8192
Imputed X(1)’s corresponding to the observed X0’s in controls
• The imputation scheme yields a simple relationship– Imputed X(1) = log(8) + x0
• For vaccinees with lowest observed X(1)=log(32), X(0) is unknown– For these subjects impute X(0)=log(16)
[the lowest observed response in controls]• For Weiss Strain A, the dataset has the following
principal strata mass points (x1,x0) at which VEcausal(x1,x0) can be estimated (on log scale):
(32,16),(128,16),(256,32),(512,64),(1024,128),(2048,256),(4096,512),(8192,1024)
Estimation of VEcausal(x1,x0)
• Logistic regression model in vaccine group to estimate Pr(Y=1|X(1)=x1,X(0)=x0,Z=vaccine) at each point (x1,x0) specified earlier
• Logistic regression model in control group to estimate Pr(Y=1|X(1)=x1,X(0)=x0,Z=control) at each point (x1,x0)
• VEcausal(x1,x0) is estimated as one minus the ratio of these estimated probabilities
Interpretation
• Subjects with antibody titers (32,16) under (vaccine,control) have causal efficacy ~0.38
• Subjects with antibody titers (128,16) under (vaccine,control), with X(1) = X(0) + log(8), have causal efficacy ~0.75– Efficacy approximately constant across the
7 principal strata of individuals with non-low antibody titers
– Suggests a threshold of efficacy: antibody titers 128 confer ~75% protection
Interpretation, Continued
• Ability to assess Ab titer as a CSOP is limited because can only study VEcausal(x1,x0) over a narrow set of (x1,x0) values– Cannot assess FR dissociative effects,
because X(1) never equals X(0)– Limited ability to assess FR associative effects
Cannot assess the slope of VE(X(1),X(0)=c) with X(1) increasing for X(0) fixed at a constant level
Predicted VEcausal
• Can predict the overall vaccine efficacy for a population with a certain distribution of principal strata (x1,x0) by summing estimated stratum-specific VEcausal(x1,x0) estimates– E.g., internal to the Salk trial:Predicted VEcausal =
(x1,x0) {# subjects in PS(x1,x0) Est.VEcausal(x1,x0)} = 0.75
– Close to observed VEcausal = 0.73• Comparing Predicted VEcausal and Observed
VEcausal is one level of diagnostic for the imputation assumption
Discussion from the Example
• Causal estimation sensitive to imputation assumption– E.g., changing the assumption
X(1)=log(32) implies X(0)=log(16) to X(1)=log(32) implies X(0)=log(4) changes the estimated VEcausal for lowest titer responders from 0.38 to 0.73
• Only a small set of principal strata (x1,x0) exist with non-negligible probability– A strength- focus inference on the relevant/meaningful sub-
populations – A limitation- cannot assess how causal efficacy varies over
certain regions of the plane (x1,x0) • When have a solid basis for imputation, the causal approach
may be a useful complement to the Prentice approach when (X(1),X(0)) both substantially vary
Implication: Causal Approach Best Motivated when X(0) is Constant
• FR causal approach attractive when X(0)=c for all trial participants– The range of (X(1),X(0)) collapses from 2
dimensions to oneOften will be able to estimate
VEcausal(X(1),X(0)=c) over a meaningful range for X(1)
– Plots of Estimated VEcausal(X(1),X(0)=c) highly interpretable
– Straightforward to assess FR associative and disassociative effects
– Lighter imputation assumptions than when X(0) varies
From a “Causal Surrogate or of Protection” (CSOP) to a POP
• Consider the problem of predicting protection against a new viral strain
• Predicted strain-specific VEcausal can be computed based on: – The estimated S-S VEcausal(S-S X(1)) for S-S
X(1)’s spanning the observed range in vaccinees– The estimated distribution of S-S X(1)’s in
vaccinees• A plot of Observed S-S VEcausal versus Predicted S-
S VEcausal informs about the value of the CSOP as a POP
• This approach can be taken using data from a single (large) trial or across multiple trials