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

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en

ce/1

00

Weiss strain Type A: Control Gp

3 4 5 6 7 8 9

51

01

52

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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

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log(Ab titer)

Ca

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Control-EVaccine-OVaccine-E

Weiss strain Type A

3 4 5 6 7 8 9

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log(Ab titer)

Pe

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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

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nce/

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Control-OControl-EVaccine-OVaccine-E

Weiss strain Type A

3 4 5 6 7

02

46

810

log(Ab titer)

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Control-OControl-EVaccine-OVaccine-E

PR8 strain Type A

3 4 5 6 7 8 9

510

1520

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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

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01

00

Predicted VEs

Ob

serv

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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