Longitudinal AIDS Data Analysis - School of …ph7440/pubh7440/slide8.pdfLongitudinal AIDS Data Analysis Problem: Lengthy follow-up times required to evaluate efﬁcacy of a new treatment

Longitudinal AIDS Data Analysis

Problem: Lengthy follow-up times required to evaluateefficacy of a new treatment (e.g., survival time ofHIV-infected patients)

Chapter 8.1 Case Study: Analysis of AIDS Data – p. 1/22



Solution(?): Select an easily-measured biologicalmarker, known to be predictive of the clinical outcome,as a surrogate endpoint. In AIDS research, typically useCD4 count (number of lymphocytes/mm3 blood).





BUT several studies have cast doubt on this approach...





BUT several studies have cast doubt on this approach...

Example: Anglo-French “Concorde” trial showedimmediate AZT produces consistently higher CD4counts than deferred, but survival patterns in twogroups were nearly identical.


Our data: the ddI/ddC study467 persons randomized to didanosine (ddI) orzalcitabine (ddC)



HIV-infected patients with AIDS or two CD4 counts of300 or less, and who had failed or could not toleratezidovudine (AZT)

⇒ all are very ill





CD4 counts recorded at baseline, 2, 6, 12, and 18months (some missing)






covariates: age, sex, baseline AIDS Dx, baselineKarnofsky score, etc.






covariates: age, sex, baseline AIDS Dx, baselineKarnofsky score, etc.

outcome variables: clinical disease progression, death


Goal and Subplot of ddI/ddC study

Goal: Analyze the association among CD4 count,survival time, drug group, and AIDS diagnosis at studyentry (an indicator of disease progression status). Makerecommendations for clinical practice and use of CD4as surrogate marker for death in end-stage patients.


Goal and Subplot of ddI/ddC study

Goal: Analyze the association among CD4 count,survival time, drug group, and AIDS diagnosis at studyentry (an indicator of disease progression status). Makerecommendations for clinical practice and use of CD4as surrogate marker for death in end-stage patients.

Subplot: ddI granted preliminary license in USA basedprimarily on its ability to “boost” CD4 count at 2 months.ddC makers would like to show a similar boost and/orcomparable survival time (“equivalency trial").


Modeling of Longitudinal CD4 CountsWrite vector of CD4 counts for individual i asYi = (Yi1, . . . , Yisi

)T , and model

Yi = Xiα + Wiβi + ǫi ,

where



)T , and model


whereXi is a si × p design matrix



)T , and model


whereXi is a si × p design matrixα is a p × 1 vector of fixed effects



)T , and model


whereXi is a si × p design matrixα is a p × 1 vector of fixed effectsWi is a si × q design matrix (q < p), and



)T , and model


whereXi is a si × p design matrixα is a p × 1 vector of fixed effectsWi is a si × q design matrix (q < p), andβi is a q × 1 vector of subject-specific randomeffects, usually assumed iid N(0 , V)



)T , and model


whereXi is a si × p design matrixα is a p × 1 vector of fixed effectsWi is a si × q design matrix (q < p), andβi is a q × 1 vector of subject-specific randomeffects, usually assumed iid N(0 , V)

Wi has jth row wij = (1 , tij , (tij − 2)+) , toaccommodate CD4 response at two months.


Modeling of Longitudinal CD4 CountsWe account for the covariates by letting

Xi = (Wi | diWi | aiWi) , where




di = 1 if i received ddI; di = 0 if ddC




di = 1 if i received ddI; di = 0 if ddCai = 1 is i has an AIDS diagnosis at baseline; ai = 0if not





Likelihood:

n∏

i=1

Nsi(Yi|Xiα + Wiβi, σ

2Isi)

n∏

i=1

N3(βi|0,V) ,





Likelihood:

n∏

i=1


2Isi)

n∏

i=1

N3(βi|0,V) ,

Prior: N9(α|c,D) × IG(σ2|a, b) × IW (V|(ρR)−1, ρ)





Likelihood:

n∏

i=1


2Isi)

n∏

i=1

N3(βi|0,V) ,

Prior: N9(α|c,D) × IG(σ2|a, b) × IW (V|(ρR)−1, ρ)

⇒ easy full conditional distributions for Gibbs sampling!


Prior SelectionWe would prefer a vague (low-information) prior, butmust take care with variance components, since if bothhave improper priors, the posterior will be improper!Possible solutions:



Add constraints to reduce parameter count, or



Add constraints to reduce parameter count, orUse vague but proper priors in a hierarchicallycentered parametrization (reduce correlations)




Our rule of thumb: Take ρ = n/20 and R = E(V) =

Diag((r1/8)2, (r2/8)2, (r3/8)2), where ri is total range ofplausible parameter values across individuals






⇒ ±2 prior standard deviations covers half the plausiblerange (since R is roughly the prior mean of V)






⇒ ±2 prior standard deviations covers half the plausiblerange (since R is roughly the prior mean of V)

Other priors can be vague except for “placeholder” α4

(drug intercept): insist it be close to 0 since patients arerandomized to drug


Exploratory plots of CD4 count

020

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CD

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020

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d) square root CD4 count over time, ddC group

Sample sizes show increasing missingness over time –ddI: (230, 182, 153, 102, 22); ddC: (236, 186, 157, 123, 14)


MCMC convergence monitoring plots

iteration

gran

d in

t

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68

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G&R: 1.01 , 1.03 ; lag 1 acf = 0.823

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iterationV

inv_

12

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Note: horizonal axis is iteration number; vertical axes areα1, α2, α3, α4, α5, α6, β8,1, β8,2, β8,3, σ, (V−1)11, and (V−1)12.


Fixed effect (α) priors and posteriors

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AID

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Priors (dashed lines) and estimated posteriors (solid lines)for the α parameters; note no “learning” for α4 (placeholder).


Point and interval estimatesmode 95% interval

Baseline intercept α1 9.938 9.319 10.733slope α2 –0.041 –0.285 0.204change in slope α3 –0.166 –0.450 0.118

Drug intercept α4 0.004 –0.190 0.198slope α5 0.309 0.074 0.580change in slope α6 –0.348 –0.671 –0.074

AIDS Dx intercept α7 –4.295 –5.087 –3.609slope α8 –0.322 –0.588 –0.056change in slope α9 0.351 0.056 0.711

ddI trajectories significantly different both before andafter the changepoint


Point and interval estimatesmode 95% interval

Baseline intercept α1 9.938 9.319 10.733slope α2 –0.041 –0.285 0.204change in slope α3 –0.166 –0.450 0.118

Drug intercept α4 0.004 –0.190 0.198slope α5 0.309 0.074 0.580change in slope α6 –0.348 –0.671 –0.074

AIDS Dx intercept α7 –4.295 –5.087 –3.609slope α8 –0.322 –0.588 –0.056change in slope α9 0.351 0.056 0.711

ddI trajectories significantly different both before andafter the changepoint

AIDS Dx main effects also all significantChapter 8.1 Case Study: Analysis of AIDS Data – p. 10/22

Fitted population model by drug and Dx

months after enrollment

squa

re r

oot C

D4

coun

t

02

46

810

0 2 6 12 18

No AIDS Dx at baselineddIddC

AIDS Dx at baseline

Obtained by setting βi = ǫi = 0, α = posterior mode




squa

re r

oot C

D4

coun

t

02

46

810

0 2 6 12 18


AIDS Dx at baseline


On average, only AIDS-free ddI patients get a “boost”...




squa

re r

oot C

D4

coun

t

02

46

810

0 2 6 12 18


AIDS Dx at baseline


On average, only AIDS-free ddI patients get a “boost”...

But ddC patients “catch up” by the end of the period!Chapter 8.1 Case Study: Analysis of AIDS Data – p. 11/22

Changepoint vs. linear decay model

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q-q plots indicate a reasonable degree of normality inthese cross validation residuals, rij = yij − E(Yij|y(ij))


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q-q plots indicate a reasonable degree of normality inthese cross validation residuals, rij = yij − E(Yij|y(ij))∑

ij |rij | = 66.37 and 70.82, respectively ⇒ similar fits!Chapter 8.1 Case Study: Analysis of AIDS Data – p. 12/22

Residual and CPO comparison

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⇒ CD4 counts adequately explained by the linear model!Chapter 8.1 Case Study: Analysis of AIDS Data – p. 13/22

CD4 “Boost” at Two MonthsDirect way of capturing the chance of a CD4 response:Define Ri = 1 if Yi2 − Yi1 ≥ 0, and Ri = 0 otherwise, fori = 1, . . . ,m = 367.



Fit the probit regression model:

pi ≡ P (Ri = 1|γ) = Φ(γ0 + γ1di + γ2ai) ,

since calculations easy under a flat prior on γ




pi ≡ P (Ri = 1|γ) = Φ(γ0 + γ1di + γ2ai) ,


Results:mode 95% LL 95% UL

γ0 (intercept) .120 –.135 .378γ1 (treatment) .226 –.040 .485γ2 (AIDS Dx) –.339 –.610 –.068




pi ≡ P (Ri = 1|γ) = Φ(γ0 + γ1di + γ2ai) ,


Results:mode 95% LL 95% UL

γ0 (intercept) .120 –.135 .378γ1 (treatment) .226 –.040 .485γ2 (AIDS Dx) –.339 –.610 –.068

⇒ Boost more likely for patients without an AIDS Dx and(to a lesser extent) taking ddI


CD4 “Boost” at Two MonthsWe can use the fitted probit model to transform our γ(g)

Gibbs samples to the probability-of-response scale for atypical patient in each drug-diagnosis group




Results:ddI ddC

AIDS diagnosis at baseline .502 .413No AIDS diagnosis at baseline .637 .550




Results:ddI ddC


Posterior median probability of a response is almost.09 larger for ddI




Results:ddI ddC


Posterior median probability of a response is almost.09 larger for ddIAIDS-negative patients have a .135 larger chance ofresponding than an AIDS-positive patient




Results:ddI ddC


Posterior median probability of a response is almost.09 larger for ddIAIDS-negative patients have a .135 larger chance ofresponding than an AIDS-positive patient

But for even the best response group (AIDS-free ddIpatients), there is a substantial estimated probability ofnot experiencing a boost


Survival AnalysisDoes the CD4 “boost” translate into an improvement insurvival?



To check, fit a proportional hazards model:

h(t|z,β) = h0(t) exp(z′β)

where we employ the covariates





where we employ the covariatesz0 = 1 for all patients





where we employ the covariatesz0 = 1 for all patientsz1 = 1 for ddI patients with a CD4 response, and 0otherwise





where we employ the covariatesz0 = 1 for all patientsz1 = 1 for ddI patients with a CD4 response, and 0otherwisez2 = 1 for ddC patients without a CD4 response, and0 otherwise





where we employ the covariatesz0 = 1 for all patientsz1 = 1 for ddI patients with a CD4 response, and 0otherwisez2 = 1 for ddC patients without a CD4 response, and0 otherwisez3 = 1 for ddC patients with a CD4 response, and 0otherwise


Survival AnalysisLoglikelihood (Cox and Oakes, 1984):

log L(β) =∑

i∈U

log h(ti|zi,β) +m∑

i=1

log S(ti|zi,β) ,

where



log L(β) =∑

i∈U


i=1

log S(ti|zi,β) ,

whereh denotes the hazard function



log L(β) =∑

i∈U


i=1

log S(ti|zi,β) ,

whereh denotes the hazard functionS denotes the survival function



log L(β) =∑

i∈U


i=1

log S(ti|zi,β) ,

whereh denotes the hazard functionS denotes the survival functionU the collection of uncensored failure times(observed deaths), and



log L(β) =∑

i∈U


i=1

log S(ti|zi,β) ,

whereh denotes the hazard functionS denotes the survival functionU the collection of uncensored failure times(observed deaths), andzi = (z0i, z1i, z2i, z3i)

′



log L(β) =∑

i∈U


i=1

log S(ti|zi,β) ,

whereh denotes the hazard functionS denotes the survival functionU the collection of uncensored failure times(observed deaths), andzi = (z0i, z1i, z2i, z3i)

′

Our parametrization uses nonresponding ddI patientsas a reference group; β1, β2, and β3 capture the effect ofbeing in one of the other 3 drug-response groups.


Survival AnalysisBegin with a parametric baseline hazard, say Weibull:

h0(t) = ρtρ−1



h0(t) = ρtρ−1

Result: extremely high posterior correlation between ρand β0! So fix ρ = 1 (return to constant baseline hazard,i.e., an exponential survival model).



h0(t) = ρtρ−1


Resulting posterior quantiles:

median 95% LL 95% ULβ0 (baseline) –7.00 –7.34 –6.67β1 (ddI resp) –.07 –.54 .38β2 (ddC nonresp) .06 –.39 .53β2 (ddC resp) –.53 –1.10 .02



h0(t) = ρtρ−1


Resulting posterior quantiles:

median 95% LL 95% ULβ0 (baseline) –7.00 –7.34 –6.67β1 (ddI resp) –.07 –.54 .38β2 (ddC nonresp) .06 –.39 .53β2 (ddC resp) –.53 –1.10 .02

⇒ only the ddC responders seem different!Chapter 8.1 Case Study: Analysis of AIDS Data – p. 18/22

Survival Analysis

Typical nice feature of parametric MCMC analysis:Our posterior γ(g) samples may be easily transformedto investigate:


Survival Analysis


the survival function at time t:

S(t|z,β) = exp{−t exp(z′β)} ,

or


Survival Analysis


the survival function at time t:

S(t|z,β) = exp{−t exp(z′β)} ,

orthe median survival time:

θ(z,β) = (log 2) exp(−z′β)

(set S(t|z,β) = 1/2 and solve for t)


EstimatedS(t) and median survival

days since randomization

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⇒ Clinically significant improvement in survival only forddC patients experiencing a boost!


Conclusions

ddC is less successful than ddI in producing a CD4boost in patients with advanced HIV infection, BUT...


Conclusions


This superior CD4 performance does not seem totranslate into improved survival.


Conclusions



That is, CD4 is prognostic, but not a surrogate endpoint.


Conclusions




Recommendations


Conclusions




RecommendationsRethink the practice of licensing drugs basedprimarily on an increase in CD4 count.


Conclusions




RecommendationsRethink the practice of licensing drugs basedprimarily on an increase in CD4 count.Review the use of these drugs with end-stagepatients (low efficacy, unpleasant side effects)


Conclusions




RecommendationsRethink the practice of licensing drugs basedprimarily on an increase in CD4 count.Review the use of these drugs with end-stagepatients (low efficacy, unpleasant side effects)Reconsider placebo trials?!?


Medical journal references:

“Main paper”: ABRAMS, D.I., GOLDMAN, A.I., ET AL. (1994).Comparative trial of didanosine and zalcitabine inpatients with human immunodeficiency virus infectionwho are intolerant of or have failed zidovudine therapy.New England Journal of Medicine, 330, 657–662.


Medical journal references:

“Main paper”: ABRAMS, D.I., GOLDMAN, A.I., ET AL. (1994).Comparative trial of didanosine and zalcitabine inpatients with human immunodeficiency virus infectionwho are intolerant of or have failed zidovudine therapy.New England Journal of Medicine, 330, 657–662.

Bayesian followup paper: GOLDMAN, A.I., CARLIN, B.P.,CRANE, L.R., LAUNER, C., KORVICK, J.A., DEYTON, L., AND

ABRAMS, D.I. (1996). Response of CD4 lymphocytes andclinical consequences of treatment using ddI or ddC inpatients with advanced HIV infection. Journal ofAcquired Immune Deficiency Syndromes and HumanRetrovirology, 11, 161–169.


Documents

Longitudinal AIDS Data Analysis - School of …ph7440/pubh7440/slide8.pdfLongitudinal AIDS Data Analysis Problem: Lengthy follow-up times required to evaluate efﬁcacy of a new treatment