Parametric Modeling of Time-to-Event Data
With Possibly Non-Proportional Hazards
(Abridged version for distribution)
Keaven Anderson, Ph.D.
Merck Research Laboratories
For presentation at
28th Spring Symposium
New Jersey Chapter of the
American Statistical Association
May 31, 2007
Alternate title
An Alternative to the Cox Model for
Clinical Trials with Preventive
Interventions
2
Acknowledgements
 Shingles Prevention Study
 William Wang
 Ivan Chan
 Human Papillomavirus Studies
 Lisa Lupinacci
 Eliav Barr
 AFCAPS/TEXCAPS
 Robert Tipping
3
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
4
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-proportionalhazards models
– In simplest model, the scale parameter is a function of
location
– This yields a powerful, 1 df likelihood ratio test for nonproportional hazards
 Target audience
 Statisticians
 Health Economists: modeling of risk/benefit
 Epidemiologists: modeling of disease process
5
What (is it?)
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)
7
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
8
Varying Location and Dispersion
Accelerated 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
9
Graph of log cumulative hazard function
versus log time
VLDAFT
Standard AFT
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
Proportional hazards
Converging or
diverging hazards
10
When/Why: Examples
Examples:
Clinical Trial Data from
Vaccine and Drug Studies
Example:
Lipid Lowering with
Simvastatin and Lovastatin
Scandinavian Simvastatin Survival Study (4S)
Proportion
Secondary prevention
Alive
1.00
4444 patients
Cholesterol: 272 ± 23 mg/dL
Simvastatin 20 mg/d
0.95
 40 mg/d in 37%
 LDL-C reduced 38%
 Survival and events
0.90
 30% decreased death rate
 34% decreased CHD events
 Subsequent secondary prevention trials 0.85




Simvastatin
Placebo
0.80
Slide source: lipidsonline.org
0.00
Log rank: p=0.0003
0
1
2
3
4
5
6
Years Since Randomization
Reprinted from The Lancet, Vol. 344, Scandinavian Simvastatin Survival Study
Group, 1383-1389, copyright 1994, with permission from Elsevier.
14
Air Force/Texas Coronary Atherosclerosis Prevention
Study (AFCAPS/TexCAPS)
Primary End Point: First Acute Major Coronary Event
0.07
Cumulative incidence
0.06
0.05
37% risk reduction
(P < 0.001)
Placebo
0.04
0.03
Lovastatin
0.02
0.01
0.00
No. at risk:
0
1
2
4
3
Years of follow-up
5+
5
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
Downs JR et al. JAMA 1998;279:1615–1622
Copyright ©1998, American Medical Association.
Slide source: lipidsonline.org
15
Air Force/Texas Coronary Atherosclerosis Prevention
Study (AFCAPS/TexCAPS)
Event Rates by Baseline HDL-C Tertile
Event rate per 1,000
patient-years at risk
16
-45%
risk reduction
14
-44%
risk reduction
12
Lovastatin
Placebo
-15%
risk reduction
10
8
6
4
2
0
 34
35–39
HDL-C (mg/dL)
Downs JR et al. JAMA 1998;279:1615–1622
 40
Slide source: lipidsonline.org
16
Example:
Prevention of Fractures
Incidence of Fractures during the 3-Year Study Period
Black D et al. N Engl J Med 2007;356:1809-1822
18
Example:
Prevention of Pre-cancerous Cervical
Lesions with GARDASIL®
Thanks to Lisa Lupinacci, Eliav Barr
for February, 2007 ACIP Slides
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?
20
Example: Epidemiology
Framingham Heart Study
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
22
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
Herpes Zoster (HZ) Epidemiology
 HZ is a consequence of reactivation of varicellazoster 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
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Typical HZ Eruption
Courtesy of Dr. Kenneth Schmader, Duke University and Durham VA Medical Centers.
27
Epidemiology of HZ/PHN
Rate per 1000 per annum.
Occurrence by Age
11
10
9
8
7
6
5
4
3
2
1
0
HZ per
1000 per annum.
PHN per
1000 per annum.
0
10
20
30
40
50
Age (years)
60
70
80+
Hope-Simpson, J. Royal College Pract. (1975).
28
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 ≥6month 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.
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Shingles Prevention Study
Subjects Enrolled
38,546
ZOSTAVAX™
19,270
Censored Before End of Study
793 (4.1%) Died
57 (0.3%) Withdrew
61 (0.3%) Lost to follow-up
Completed Study
18,359 (95.3%)
Placebo
19,276
Censored Before End of Study
792 (4.1%) Died
75 (0.4%) Withdrew
52 (0.2%) Lost to follow-up
Completed Study
18,357 (95.2%)
Average duration of HZ surveillance, 3.1 years (range, up to 4.9 years)
30
Shingles Prevention Study: Population
Gender:
Male
Female
Age (in years):
Mean
Range
Race:
Black
Hispanic
White
Other
ZOSTAVAX ®
Placebo
11,403 (59.2%)
7867 (40.8%)
11,357 (58.9%)
7919 (41.1%)
69.4
60 to 99
69.4
59 to 94
395 (2.0%)
265 (1.4%)
18,393 (95.4%)
214 (1.1%)
420 (2.2%)
248 (1.3%)
18,381 (95.4%)
223 (1.2%)
31
ZOSTAVAX® Efficacy: HZ Incidence
Percent of Subjects With HZ
Estimate of the Cumulative Incidence of HZ Over Time
by Vaccination Group (MITT Population)
6
Placebo (n=642)
p<0.001
5
4
3
2
ZOSTAVAX (n=315)
1
0
0
1
2
3
4
Time Since the Start of Follow-Up (in Years)
Number of subjects at risk
ZOSTAVAX 19254
Placebo
19247
18994
18915
18626
18422
9942
9806
1906
1856
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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 KaplanMeier estimates reasonably well?
33
Shingles Prevention Study
HZ Model Form

 ln( t )    
Pr{T  t}  exp   exp 


 


   0   1  Age   2  Vaccine   3  Age  Vaccine
ln    0  1 Vaccine
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How?
Existing Software
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!
36
Conclusions/Recommendations
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
38
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
39
Contact
 Keaven_anderson@merck.com
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