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Evaluating Change in Hazard in Clinical Trials With Time-to-Event Safety Endpoints
Rafia Bhore, PhDStatistical Scientist, NovartisEmail: rafia.bhore@novartis.com
Midwest Biopharmaceutical Statistics WorkshopMuncie, IndianaMay 21, 2013
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop2
Outline
Motivation
Metrics of risk
Time-dependency of adverse events
Change-point methodology
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop3
Motivation
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop4
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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop5
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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop6
Metrics of risk
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop7
Metrics of Risk
1. Crude rates
2. Exposure-adjusted ratesa. Occurrences (events) per unit time of exposure (aka exposure-
adjusted event rate)b. Incidences (subjects) per unit time of exposure (aka exposure-
adjusted 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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop8
Type of metric Distribution Assumptions1. Crude rate Proportion (%) Binomial /
Beta-binomialAppropriate when risk is relatively constant, shorter duration of exposure, or rare
2. Exposure-adjusted incidence rate
Count per person-time
Poisson / Neg. Binomial
Appropriate when risk is relatively constant
3. Exposure-adjusted event rate
Count per person-time
Poisson / Neg. Binomial
Appropriate when risk is relatively constant
4. Cumulative rate
Based on time-to-event (%)
Parametric or Non-parametric
Risk can vary over time.
5. Hazard rate Based on time-to-event (count per person-time)
Parametric or Non-parametric
Risk can vary over time.
Different Metrics of RiskAn overview
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop9
Time-dependency of adverse events
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop10
Drug Exposure vs. Adverse Event RatesThree patterns of AEs – O’Neill, 1988
3000
2500
2000
1500
1000
500
00 1 2 3 4 5 6 7 8 9 10 11 12
MONTHS OF EXPOSURE
NUM
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OF
SUB
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1 2 3 4 5 6 7 8 9 10 11 12
MONTHS OF EXPOSURE
NU
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20
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864
CONSTANT
DELAYED
ACUTE
1 2 3 4 5 6 7 8 9 10 11 12
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00
MONTHS OF EXPOSURE
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INCREASING
DECREASING (ACUTE EVENTS)
(DELAYED EVENTS)
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop11
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 (Post-
herpetic neuralgia)
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop12
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, 2001New England Journal of Medicine – Lagakos, 2006
Study 1 Study 2
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Change-Point MethodologyA tool to test and estimate for change in risk
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop
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 CHANGE-POINT)
Change-point is defined as the time point at which an abrupt change occurs in the
risk/benefit due to a treatment
Definition of the Problem
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| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop16
Change-point models for hazard function
t0 ),exp()(t0 ),exp()(
0 ,)(
ttfttS
tth
Exponential Model Two-piece Piecewise Exponential
K-piece Piecewise Exponential
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th ,0,
)(2
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),[ tif
),[ tif
),[ tif),0[ tif
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jj-j
h (t)
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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop
Two-piece Piecewise Exponential Model
Test hypothesis of no change point, H0 ,vs. H1 of 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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point change One 0:vs.
point change No 0:
1
0
H
H
Estimation or Hypothesis Testing?Which comes first? (Chicken or Egg)
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop18
Log likelihood functions for exponential and 2-piece PWE
Maximum likelihood estimates of hazard rates, l’s, given t
Generalized to k (>2) change points (Bhore, Huque 2009)
Estimation of hazard ratesKnown change point
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| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop19
In real clinical data, change points are unknown
Consider log likelihood functions for 2-piece PWE
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].
Estimation of hazard ratesUnknown change point
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n
ii TTddL
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1986) (Yao and 0,e.g.? ),0[ ]?,[ interval restricted choose How to
,,,, where,;ˆ,ˆlogsupargˆ
)1(b
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a
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TTTTLba
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop20
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.
Gardner (2007) developed an efficient parametric bootstrap algorithm to estimate the confidence interval.
ˆ of functiona is )ˆ( and ˆˆlogˆ where,)ˆ(111
equation by the-1 level confidence the torelated is where
,)(logsup)(log:
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c
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cc
u ba
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop21
Simulated example of Change-Point
Change-point?
λ1 = 1
λ2 = 52.5
1.51
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop22
Estimation of change-pointSimulation 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)
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop23
Testing of Change PointsLikelihood Ratio Test (2-piece PWE)
)]ˆ([)ˆ(
log)ˆ()ˆ(
)ˆ(log)ˆ()ˆ(
:1991)(Loader statistic LRTRestricted
ˆ;ˆ,ˆlog0;ˆ,ˆlog2 :statistic LRT
or 0: vs.or 0:
212
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ii
i
i
i
LR
TTNTXN
XNTN
TXXl
LL
HH
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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop24
Goodness-of-fit: Selecting correct CP modelHammerstrom, 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
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop25
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
Simulation criteria for dataTrue underlying models for change-point
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop26
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=
| Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop
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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