Parametric Modeling of Time-to-Event Data

With Possibly Non-Proportional Hazards

(Abridged version for distribution)

Keaven Anderson, Ph.D.

Merck Research LaboratoriesFor presentation at

28th Spring Symposium

New Jersey Chapter of the

American Statistical Association

May 31, 2007

2

Alternate title

An Alternative to the Cox Model for Clinical Trials with Preventive

Interventions

3

Acknowledgements

Shingles Prevention Study William Wang Ivan Chan

Human Papillomavirus Studies Lisa Lupinacci Eliav Barr

AFCAPS/TEXCAPS Robert Tipping

4

Objectives/Overview

Introduce parametric time-to-event model incorporating non-proportional hazards

Examples where model may be useful Disease prevention drugs and vaccines

Examples of models fit Published examples from the Framingham Heart

Study and uses applied New example: Shingles Prevention Study

Discuss software and next steps

5

Model Advantages/Target Audience

Advantages of parametric model Simple prediction of event rates by covariate

values Ability to model multi-state failure models

adjusting for covariates in a parsimonious fashion– QTWIST methods: see Cole et al (1994)

Incorporates proportional- and non-proportional-hazards models

– In simplest model, the scale parameter is a function of location

– This yields a powerful, 1 df likelihood ratio test for non-proportional hazards

Target audience Statisticians Health Economists: modeling of risk/benefit Epidemiologists: modeling of disease process

What (is it?)

7

Time-to-Event Data Time to event denoted by random variable T

Distribution of T governed for 0 < t < by Cumulative distribution function F(t) Cumulative hazard function (t) = -ln(1-F(t)) Hazard rate (t)= d/dt (t)

Proportional hazards (Cox) model Unknown underlying hazard rate (t) unrestricted Covariate vector X Unknown parameter vector Model: (t;X)=exp(’X) (t)

8

Accelerated Failure Time (AFT) Model

Covariate vector X Unknown parameter vector Location parameter = X’ Unknown, fixed dispersion parameter

Model (cdf for ln(T)): Underlying parametric cumulative distribution G(t)

is specified (e.g., double exponential, normal) G(t;X)=G((t-)/), -<t<

G=Double exponential F=parametric proportional hazards model for T with Weibull distribution

9

Varying Location and DispersionAccelerated Failure Time (VLDAFT) Model

Same setup as standard AFT, except that now dispersion is modeled by:

a linear function of location: ln() = 0+ 1 ,

or

a linear function of covariates: ln() = ’Y – for some covariate vector Y (may be same as X)– and parameter vector

References Nelson (1984) J of Testing and Evaluation Anderson (1991) Biometrics

10

Graph of log cumulative hazard function versus log time

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2

0

0.05

0.1

0.1 5

0.2

0.2 5

0.3

0.3 5

0.4

0.4 5

1 2

Standard AFT VLDAFT

Proportional hazards Converging or diverging hazards

When/Why: Examples

Examples:Clinical Trial Data from

Vaccine and Drug Studies

Example: Lipid Lowering with

Simvastatin and Lovastatin

14

0.00

1.00

Scandinavian Simvastatin Survival Study (4S)

Secondary prevention 4444 patients Cholesterol: 272 ± 23 mg/dL Simvastatin 20 mg/d

40 mg/d in 37% LDL-C reduced 38% Survival and events

30% decreased death rate 34% decreased CHD events

Subsequent secondary prevention trials

Reprinted from The Lancet, Vol. 344, Scandinavian Simvastatin Survival Study Group, 1383-1389, copyright 1994, with permission from Elsevier.

Proportion Alive

Years Since Randomization

0 1 2 3 4 5 6

0.95

0.90

0.85

0.80

Simvastatin

Placebo

Log rank: p=0.0003

Slide source: lipidsonline.org

15

No. at risk:

Lovastatin N = 3,304 N = 3,270 N = 3,228 N = 3,184 N = 3,134 N = 1,688

Placebo N = 3,301 N = 3,251 N = 3,211 N = 3,159 N = 3,092 N = 1,644

Air Force/Texas Coronary Atherosclerosis Prevention Study (AFCAPS/TexCAPS)Primary End Point: First Acute Major Coronary Event

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0 1 2 3 4 5 5+

Cum

ula

tive inci

dence

Placebo

Lovastatin

37% risk reduction(P < 0.001)

Years of follow-up

Downs JR et al. JAMA 1998;279:1615–1622Copyright ©1998, American Medical Association. Slide source: lipidsonline.org

16

Air Force/Texas Coronary Atherosclerosis Prevention Study (AFCAPS/TexCAPS)Event Rates by Baseline HDL-C Tertile

0

2

4

6

8

10

12

14

16

34 35–39 40

Lovastatin

Placebo

Even

t ra

te p

er

1,0

00

pati

ent-

years

at

risk

HDL-C (mg/dL)

Downs JR et al. JAMA 1998;279:1615–1622

-44%risk reduction

-45%risk reduction

-15%risk reduction

Slide source: lipidsonline.org

Example: Prevention of Fractures

18

Black D et al. N Engl J Med 2007;356:1809-1822

Incidence of Fractures during the 3-Year Study Period

Example: Prevention of Pre-cancerous Cervical

Lesions with GARDASIL®

Thanks to Lisa Lupinacci, Eliav Barr

for February, 2007 ACIP Slides

20

Modeling questions for GARDASIL®

Population Impact

Events detected by semi-annual examinations Should events be modeled as occurring in the

interval between exams?

Possible improving effectiveness over time? Vaccine only prevents infection; it does not cure

infection Early events may be largely associated with HPV

infections prevalent at baseline Later events may be more likely to be associated

with new infections

Can modeling with differing dispersion by treatment adequately fit apparent non-proportional hazards?

Example: EpidemiologyFramingham Heart Study

22

Framingham Equation for CHD

Anderson et al, Circulation, 1991 12 years of follow-up for coronary heart disease

incidence in a population free of disease at baseline

Risk factors in covariate vector X: age, gender, SBP (systolic pressure), total

cholesterol, HDL cholesterol, cigarette smoking, diabetes, ECG LVH

Location model: = X’

Scale model: ln() = 0+ 1 1 = 0 implies proportional hazards 1 > 0 implies diverging hazards 1 < 0 implies converging hazards

23

24

Example: Prevention of Herpes Zoster (HZ)

with ZOSTAVAX ®

Thanks to Bill Wang, Ivan Chan

for access to MRL’s

FDA Advisory Committee slides

26

Herpes Zoster (HZ) Epidemiology

HZ is a consequence of reactivation of varicella-zoster virus years after development of varicella (chickenpox)

An estimated 1 million cases of HZ per year in the US 50,000 to 60,000 hospitalizations

– 12,000 to 19,000 with primary diagnosis of HZ 70 to 80% of those hospitalized for HZ are

immunocompetent

Lifetime risk of developing HZ ~30% Among people who reach the age of 85 years,

up to ~50% will have developed one or more episodes of HZ

Risk factors for HZ: age, immunosuppression

27

Typical HZ Eruption

Courtesy of Dr. Kenneth Schmader, Duke University and Durham VA Medical Centers.

28

Epidemiology of HZ/PHNOccurrence by Age

Ra

te p

er

10

00 p

er a

nnu

m.

0 10 20 30 40 50 60 70 80+

11

10

9

8

7

6

5

4

3

2

1

0

HZ per1000 per annum.

PHN per1000 per annum.

Age (years)

Hope-Simpson, J. Royal College Pract. (1975).

29

Shingles Prevention Study (Oxman et al., NEJM 2005)

N = 38,546 subjects ≥60 years of age randomized 1:1 to receive ZOSTAVAX® or placebo

Single dose of vaccine with potency ranging from 18,700 to 60,000 PFU (median 24,600 PFU)

Average of 3.1 years of HZ surveillance and ≥6-month follow-up of HZ pain after HZ rash onset

Conducted by Dept. of Veteran Affairs (VA) in collaboration with the National Institutes of Health (NIH) and Merck & Co., Inc.

30

Subjects Enrolled38,546

Placebo19,276

ZOSTAVAX™19,270

Completed Study18,357 (95.2%)

Completed Study18,359 (95.3%)

Censored Before End of Study793 (4.1%) Died 57 (0.3%) Withdrew 61 (0.3%) Lost to follow-up

Censored Before End of Study792 (4.1%) Died 75 (0.4%) Withdrew 52 (0.2%) Lost to follow-up

Average duration of HZ surveillance, 3.1 years (range, up to 4.9 years)

Shingles Prevention Study

31

Shingles Prevention Study: Population

Gender:MaleFemale

Age (in years):MeanRange

Race:BlackHispanicWhiteOther

Placebo

11,357 (58.9%)7919 (41.1%)

69.459 to 94

420 (2.2%)248 (1.3%)

18,381 (95.4%)223 (1.2%)

ZOSTAVAX ®

11,403 (59.2%)7867 (40.8%)

69.460 to 99

395 (2.0%)265 (1.4%)

18,393 (95.4%)214 (1.1%)

32

0 1 2 3 4

Time Since the Start of Follow-Up (in Years)

0

1

2

3

4

5

6P

erce

nt o

f Sub

ject

s W

ith H

Z

ZOSTAVAX (n=315)

Placebo (n=642)p<0.001

ZOSTAVAX 19254 18994 18626 9942 1906

Placebo 19247 18915 18422 9806 1856v211p4ACMkm_t2hz_mittv4 Oct. 1, 2005

Number of subjects at risk

ZOSTAVAX® Efficacy: HZ IncidenceEstimate of the Cumulative Incidence of HZ Over Time

by Vaccination Group (MITT Population)

33

Shingles Prevention Study

Modeling issues summary Vaccine efficacy decreases with age Vaccine efficacy decreases with time since

vaccination– Proportional hazards assumption violated

Hazard of developing HZ is relatively smooth

Question: Can a VLDAFT model fit? Does dispersion varying with vaccine address

proportional hazards issue? Any age-vaccine interactions required? Do HZ incidence prediction curves fit match

Kaplan-Meier estimates reasonably well?

34

Shingles Prevention Study HZ Model Form

VaccineAgeVaccineAge 3210

)ln(expexp}Pr{

ttT

Vaccine 10ln

How?Existing Software

36

Software

I have used a C program for the analyses presented here Currently undocumented Could link to R or rewrite in R and provide

documentation

SAS macro developed at Boston University (R. D’Agostino)

Lack of software availability is a drawback!

Conclusions/Recommendations

38

Conclusions/Recommendations Proportional hazards may not be suitable for many

situations where there may be a delayed or waning effect Non-proportional parametric models appear suitable for

many preventive treatments; examples here were Lipid lowering Vaccines Prevention of fractures

Parametric survival models can provide simple equations to: Predict future outcomes for patients Model cost-benefit (especially if Markov modeling

undertaken) Graphically describe benefit over time

Immediate plans: as a summer intern project, analyze Merck datasets to further evaluate value of methods in a pharmaceutical setting

39

References Anderson KM, A nonproportional hazards Weibull accelerated failure time

model. Biometrics, 1991;47:281-288. Anderson KM, Wilson PWF, Odell PM, Kannel WB, An updated coronary risk

profile. A statement for health professionals. Circulation 1991;83:356-362 Anderson KM, Odell PM, Wilson PWF and Kannel WB, Cardiovascular

disease risk profiles. American Heart Journal, 1990;121:293-8 Cole BF, Gelber RD, Anderson KM. Parametric approaches to quality-

adjusted survival analysis. Biometrics 1994;50:621-631 Black D et al. Once-yearly zoledronic acid for treatment of

postmenopausal osteoporosis. N Engl J Med 2007;356:1809-1822 Downs JR et al. Primary prevention of acute coronary events with

lovastatin in men and women with average cholesterol levels. Result of AFCAPS/TEXCAPS. JAMA 1998;279:1615–1622

The FUTURE II Study Group. Quadrivalent vaccine against human papillomavirus to prevent high-grade cervical lesions. N Engl J Med 2007;356:1915-27

Odell PM, Anderson KM, Kannel WB. New models for predicting cardiovascular events. J Clin Epidemiol 1994;47:582-592

Oxman, MN et al A vaccine to prevent herpes zoster and postherpetic neuralgia in older adults N Engl J Med 2005;352:2271-2284

Scandinavian Simvastatin Survival Study Group. Randomised trial of cholesterol lowering in 4444 patients with coronary heart disease: the Scandinavian Simvastatin Survival Study (4S). The Lancet 1994;344:1383-1389

40

Contact

Keaven_anderson@merck.com