Transcript Evaluating Correlates of Risk in HIV Vaccine Trials JoAnna Scott Biostat 578a – Vaccine Efficacy March 7, 2006

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.

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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?
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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

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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

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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.
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Case-Cohort Analysis (Q1)

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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.
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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.

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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.

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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.
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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
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Case-Cohort Design:
Time to Event


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Let Y j (ti )  1 if at risk
0 if not at risk
Then,

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
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

for person j at time ti.


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


Yi (ti )exp(zi (ti ) ) 
i
L( ) =  n
i1  Y (t )exp(z (t ) )
j i
j i
n
j1








)
1
if
N
(
t
)

N
(
t
i i
i i ; i.e., the censoring indicator.
where i 
0 if otherwise
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Case-Cohort Design:
Time to Event
Let r(x) be a fixed function such that r(0) = 1.
Then the partial likelihood is

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
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
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



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
Yi (ti ) r(zi (ti ) ) 
n
rii i
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
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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.

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Case-Cohort Design:
Time to Event
Then the pseudo-likelihood is
n
L( ) = 
i1
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rii
 r ji
jR(ti)

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
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
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Case-Cohort Design:
Time to Event
Now solving U( ) =   log L( )=0
yields the maximum pseudo-likelihood
estimate ˆ.

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Yi (ti ) r(zi (ti ) ) 
i.
L( ) = 
i1  Y j (ti ) r(z j (ti ) )
n
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jR(ti )
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Case-Cohort Design:
Time to Event
Hence,
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Yi (ti ) r(zi (ti ) )
l ( ) = log L( )  i log
i1
 Y j (ti) r(z j (ti) )
n
n



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
jR(t )
i




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
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
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

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=  i log Yi (ti ) + log r(zi (ti ) ) - log  Y j (ti ) r(z j (ti ) )
i1
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jR(t )
i
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Case-Cohort Design:
Time to Event
Therefore,
n
U( ) =   l ( ) =   i log
i1

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Yi (ti ) r(zi (ti ) )
 Y j (ti ) r(z j (ti ) )
jR(t )
i

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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
i
where r(u) =  u r(u).
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jR(t )
j i
j i
i
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Case-Cohort Design:
Time to Event
So asymptotic properties of ˆ 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.

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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.

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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
asymptotically equivalent to ˆ . Their
variance estimator turned out to be quite
algebraically complex.
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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.
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Case-Cohort Design:
Time to Event
They noticed that you could rewrite the
variance estimate in a computationally
easy manner.
 V
a  I 1 (1 )DT D

SC SC
where D is a subset of the matrix of dfbeta residuals
SC
that contains only those rows for the subcohort SC.
 = mn, the proportion of people in the subcohort.
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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
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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.

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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.

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Need to plan for in trial design.
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Q2: Surrogates of Protection
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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.
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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.

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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.

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Stepped Wedge Design

Cluster Randomized
Trial
 Multiple
time points
 Time of cross-over is
randomized
 Cross-over only
occurs in one direction
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Time
0 1
2
3
4
1
0 1
1
1
1
2
0 0
1
1
1
3
0 0
0
1
1
4
0 0
0
0
1
Cluster
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Stepped Wedge Design
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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.
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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.

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Research Questions (Q1)

Using a case-cohort analysis within a stepped
wedge trial requires some additional research.
 The
case-cohort 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?
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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.
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Research Questions (Q2)

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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
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Research Questions (Q2)

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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.
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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?
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Acknowledgements

Much thanks to Pamela Shaw, Jim
Hughes, and Peter Gilbert for all their help.
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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
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