Evaluating Change in Hazard in
Clinical Trials With Time-to-Event
Safety Endpoints
Rafia Bhore, PhD
Statistical Scientist, Novartis
Email: rafia.bhore@novartis.com
Midwest Biopharmaceutical Statistics Workshop
Muncie, Indiana
May 21, 2013
Outline
 Motivation
 Metrics of risk
 Time-dependency of adverse events
 Change-point methodology
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Motivation
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US FDA Regulations
FDA regulations created from these laws
 Federal Food and Drug Cosmetic (FD&C) Act (1938)
• submit evidence of safety to the FDA
 Kefauver-Harris Amendments (1962)
• Strengthened rules for drug safety
• In addition to safety, effectiveness of drug needs to be demonstrated
 Food and Drug Administration Amendments Act (FDAAA) (2007)
• Enhanced authority on monitoring safety
 FDA Safety and Innovation Act (FDASIA) (2012)
• Better adapt to truly global supply chain (Chinese and Indian drug suppliers)
Safety – an older/consistent regulatory requirement
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Why quantitative methods for evaluation of safety?
 Safety evaluation required by regulators
 Extensive collection of safety data
• E.g., extensive safety data collected in new application
(NDA/BLA/PMA) packages comprising several clinical trials
• Abundance of descriptive safety analyses
 Surprises in post-hoc review of safety data
• Descriptive analyses not adequate. No planned inferential analyses.
 Top reason why new applications for
drugs/biologics/devices go to FDA Advisory Panels
 Understand risk of “major” events
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Metrics of risk
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Metrics of Risk
1. Crude rates
2. Exposure-adjusted rates
a. Occurrences (events) per unit time of exposure (aka exposureadjusted event rate)
b. Incidences (subjects) per unit time of exposure (aka exposureadjusted incidence rate)
3. Cumulative rates
- Life table method or Kaplan-Meier method
4. Hazard rates and functions
- Instantaneous measure of risk
- Similar to cumulative rates
- constant, decreasing, or increasing
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Different Metrics of Risk
An overview
1. Crude rate
Type of metric
Distribution
Assumptions
Proportion (%)
Binomial /
Appropriate when risk
Beta-binomial is relatively constant,
shorter duration of
exposure, or rare
2. ExposureCount per personadjusted
time
incidence rate
Poisson /
Appropriate when risk
Neg. Binomial is relatively constant
3. Exposureadjusted
event rate
Count per persontime
Poisson /
Neg. Binomial
4. Cumulative
rate
Based on time-toevent (%)
Parametric or
Risk can vary over
Non-parametric time.
5. Hazard rate
Based on time-toevent (count per
person-time)
Parametric or
Risk can vary over
Non-parametric time.
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Appropriate when risk
is relatively constant
Time-dependency of adverse events
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Drug Exposure vs. Adverse Event Rates
3000
NUMBER OF EVENTS
NUMBER OF SUBJECTS
Three patterns of AEs – O’Neill, 1988
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
7
8
9
24
22
20
18
16
14
12
10
8
6
4
2
0
10 11 12
ACUTE
CONSTANT
DELAYED
1
4
5
6
7
8
9
10 11 12
0.07
INCREASING
0.05
0.04
0.03
0.02
0.01
(Risk per unit time)
0.06
*HAZARD RATE
CUMULATIVE
3
MONTHS OF EXPOSURE
MONTHS OF EXPOSURE
ADVERSE EVENT RATE
2
(DELAYED EVENTS)
CONSTANT
DECREASING (ACUTE EVENTS)
0
0
1
2
3
4
5
6
7
8
9 10 11 12
MONTHS OF EXPOSURE
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MONTHS FROM INITIAL DRUG EXPOSURE
Time-to-event Endpoints
Time-to-event endpoint is a measure of time for an event
from start of treatment until time that event occurs
• Safety Outcomes
- Invasive breast cancer in Women’s Health Study
- CV Thrombotic Events in a large clinical trial
- Safety Signals detected through biochemical markers,
• Change in grade of Liver Function Tests
• Abnormalities in serum creatinine and phosphorus
• Abnormal elevations in other lab tests
• Efficacy Outcomes
- Time-to-Relapse, Overall survival (SCLC), Cessation of Pain (Postherpetic neuralgia)
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Increased risk of Invasive Breast Cancer?
Women’s Health Initiative Study on Estrogen Plus Progestin (JAMA 2002)
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Increased risk of Cardiovascular Thrombotic events?
FDA Advisory Committee Meeting – Li, 2001
New England Journal of Medicine – Lagakos, 2006
Study 1
Study 2
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Change-Point Methodology
A tool to test and estimate for change in risk
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Definition of the Problem
 Risk abruptly changes over time
 Define risk using time-to-event outcome
 Is there a change in hazard?
 Is this statistically significant?
 What is the estimated time of change? (aka CHANGEPOINT)
Change-point is defined as the time point at which
an abrupt change occurs in the risk/benefit
due to a treatment
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Change-point models for hazard function
 Let (Ti , i) be the observed data (time & censoring variable) with
hazard function h(t) and survival function S(t)
 Assume hazard is constant piecewise in k intervals of time
 Total of k hazard rates l1,..., lk and (k-1) change points t1,...,tk-1
Exponential Model
h (t )   ,
0t
S (t )  exp( t ), 0  t
f (t )   exp( t ), 0  t
Two-piece
Piecewise Exponential
 1 ,
h (t )  
2 ,
K-piece
Piecewise Exponential
1 if t  [0, 1 )
 if t  [ , )
1 2
 2
0  t  

 h (t) 
t  
 j if t  [ j-1 , j )


k if t  [ k-1 , )
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Estimation or Hypothesis Testing?
Which comes first? (Chicken or Egg)
Two-piece Piecewise Exponential Model
 Test hypothesis of no change point, H0 ,vs. H1 of one
change point.
H0 :  0
No change point
vs.
H1 :   0
One change point
• We can expand statistical methods to more than one change-point
 Estimation (Point and 95% Confidence Interval/Region)
• Estimate where the change point(s) occurs
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Estimation of hazard rates
Known change point
 Log likelihood functions for exponential and 2-piece PWE
n
n
i 1
i 1
log L( )   log    Ti  d log     Ti
u
n
n
i 1
i 1
log L(1 , 2 ; )  d1 log 1  d 2 log 2   1  (Ti   )  2  Ti   

 Maximum likelihood estimates of hazard rates, l’s, given t
ˆ1 
d1
n
 Ti   
i 1
, ˆ2 
d2
n



T


 i
i 1
 Generalized to k (>2) change points (Bhore, Huque 2009)
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Estimation of hazard rates
Unknown change point
 In real clinical data, change points are unknown
 Consider log likelihood functions for 2-piece PWE
n
n
i 1
i 1
log L(1 , 2 ; )  d1 log 1  d 2 log 2   1  (Ti   )  2  Ti   

 Estimate t using a grid search that maximizes profile log
likelihood
• Substitute MLE of hazard rates into log L and maximize log L wrt t
over a restricted interval [ta, tb].


ˆ  arg sup log L ˆ1 , ˆ2 ; , where   T(1) , T(1) ,, T(n 1) , T( n 1) 
 a   b
How to choose restricted interval [ a , b ]?  [0, ) ?
e.g., a  0 and  b  T( n 1) (Yao 1986)
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Confidence region/interval for change-point, t
 An approximate confidence region for the change point, t,
was given by Loader (1991).
• Underlying likelihood function is not a smooth function of t. Hence
confidence region may be a union of disjoint intervals.


I  t : log L(t )  sup log L(u )  c ,
 a u  b


where c is related to the confidence level 1 -  
by the equation
1    1  e c 1  v (ˆ )e c , where ˆ  log ˆ1 ˆ2 and v (ˆ ) is a function of ˆ




 Gardner (2007) developed an efficient parametric
bootstrap algorithm to estimate the confidence interval.
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Simulated example of Change-Point
λ2 = 5
Change-point?
λ1 = 1
2.5
1.5
1
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Estimation of change-point
Simulation example
E.g. Result: Change in hazard is estimated to occur at
0.81 units of time (95% CI: 0.64 to 0.99 units of time)
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Testing of Change Points
Likelihood Ratio Test (2-piece PWE)
H 0 :   0 or 1  2  vs. H1 :   0 or 1  2 




2
LRT statistic :  LR
 2 log L ˆ, ˆ;0  log L ˆ1 , ˆ2 ;ˆ

Restricted LRT statistic (Loader 1991) :
 X (ˆ) Ti 
 N  X (ˆ)  Ti 
l (ˆ)  X (ˆ) log 
  N  X (ˆ)  log 

ˆ
ˆ
 N  (Ti   ) 
 N  [Ti  (Ti   )] 
 One would think that LRT statistic has χ2 distribution with two degrees
of freedom. Not true because of discontinuity at change-point
 See Bhore, Huque (2009), Gardner (2007) & Loader (1991) for details
on computing significance level
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Goodness-of-fit: Selecting correct CP model
Hammerstrom, Bhore, Huque (2006 JSM, 2007 ENAR)
Consider 6 time-to-event models
1. Exponential (constant hazard)
2. Two-piece PWE with decreasing hazard
3. Two-piece PWE with increasing hazard
4. Three-piece PWE with V shape
5. Three-piece PWE with upside down V shape
6. Weibull
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Simulation criteria for data
True underlying models for change-point
Sample size, N = 150 or 40 subjects
1. 2-piece Piecewise Exponential (15 models)
•
λ1 = 1
•
λ2 = 0.2, 0.5, 1, 2, 5
•
Change point,  = 30th, 50th, 70th percentile of λ1
2. 3-piece Piecewise Exponential (9 models)
•
Early:Mid:Late hazard rates = 0.25:1:0.3 or 2:1:2
•
Change point,  = 20th:50th, 20th:70th, or 50th:20th percentiles of
early and middle hazards
3. Weibull (25 models)
•
Shape = 0.25, 0.5, 1, 2, 5 and Scale = 0.5, 2, 3, 3.5, 4
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True model: 2-piece Piecewise Exponential (N=150)
Pairwise comparison of models
 2=
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True model: 2-piece Piecewise Exponential (N=40)
Pairwise comparison of models
 2=
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Concluding Remarks
 Uncontrolled or open-label Phase II/III clinical trials
provide a major source of long-term safety/efficacy data
for a single group.
• Crude incidence rates underestimate the incidence of delayed
events
• Visual check of Kaplan-Meier curves are not sufficient to detect
change in hazard
 Change-point methodology (new in application to clinical
trials) can be applied to test whether and estimate where
a change in hazard occurs.
• Piecewise exponential model is robust for modeling change in
hazard (Bhore and Huque 2009).
• Percentile bootstrap preferred for computing CIs (work not shown)
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