Evaluating Change in Hazard in Clinical Trials With Time-to-Event Safety Endpoints

Evaluating Change in Hazard in Clinical Trials With Time-to-Event Safety Endpoints

Rafia Bhore, PhDStatistical Scientist, NovartisEmail: [email protected]

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


Motivation


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


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


Metrics of risk


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


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


Time-dependency of adverse events


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

BER

OF

SUB

JEC

TS

1 2 3 4 5 6 7 8 9 10 11 12

MONTHS OF EXPOSURE

NU

MB

ER O

F EV

ENTS

20

2422201816141210

864

CONSTANT

DELAYED

ACUTE

1 2 3 4 5 6 7 8 9 10 11 12

0.060.050.040.030.020.01

00

MONTHS OF EXPOSURE

0.07

AD

VER

SE E

VEN

T R

ATE

CU

MU

LATI

VE

CONSTANT

MONTHS FROM INITIAL DRUG EXPOSURE

*HAZ

ARD

RAT

E(R

isk

per u

nit t

ime)

INCREASING

DECREASING (ACUTE EVENTS)

(DELAYED EVENTS)


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)


Increased risk of Invasive Breast Cancer?Women’s Health Initiative Study on Estrogen Plus Progestin (JAMA 2002)


Increased risk of Cardiovascular Thrombotic events?FDA Advisory Committee Meeting – Li, 2001New England Journal of Medicine – Lagakos, 2006

Study 1 Study 2


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

15


Change-point models for hazard function

t0 ),exp()(t0 ),exp()(

0 ,)(

ttfttS

tth

Exponential Model Two-piece Piecewise Exponential

K-piece Piecewise Exponential

tt

th ,0,

)(2

1

),[ tif

),[ tif

),[ tif),0[ tif

1-kk

1

212

11

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


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

17

point change One 0:vs.

point change No 0:

1

0

H

H

Estimation or Hypothesis Testing?Which comes first? (Chicken or Egg)


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

n

ii

n

ii

n

ii

n

ii

u

TTddL

TdTL

12

11221121

11

)(loglog);,(log

loglog)(log

n

ii

n

ii T

d

T

d

1

22

1

11

ˆ ,ˆ


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

n

ii

n

ii TTddL

12

11221121 )(loglog);,(log

1986) (Yao and 0,e.g.? ),0[ ]?,[ interval restricted choose How to

,,,, where,;ˆ,ˆlogsupargˆ

)1(b

b

)1()1()1()1(21

na

a

nn

T

TTTTLba


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:

21 veve

c

cuLtLtI

cc

u ba


Simulated example of Change-Point

Change-point?

λ1 = 1

λ2 = 52.5

1.51


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)


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

211210

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


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


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


True model: 2-piece Piecewise Exponential (N=150)Pairwise comparison of models

2=


True model: 2-piece Piecewise Exponential (N=40)Pairwise comparison of models

2=


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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Evaluating Change in Hazard in Clinical Trials With Time-to-Event Safety Endpoints

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