Report

Evaluating Change in Hazard in Clinical Trials With Time-to-Event Safety Endpoints Rafia Bhore, PhD Statistical Scientist, Novartis Email: [email protected] Midwest Biopharmaceutical Statistics Workshop Muncie, Indiana May 21, 2013 Outline Motivation Metrics of risk Time-dependency of adverse events Change-point methodology 2 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Motivation 3 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 4 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 5 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Metrics of risk 6 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 7 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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. 8 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Appropriate when risk is relatively constant Time-dependency of adverse events 9 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 10 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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) 11 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Increased risk of Invasive Breast Cancer? Women’s Health Initiative Study on Estrogen Plus Progestin (JAMA 2002) 12 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Increased risk of Cardiovascular Thrombotic events? FDA Advisory Committee Meeting – Li, 2001 New England Journal of Medicine – Lagakos, 2006 Study 1 Study 2 13 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Change-Point Methodology A tool to test and estimate for change in risk 14 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 15 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 ) , 0t 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 , ) 16 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 17 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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) 18 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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) 19 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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. 20 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop Simulated example of Change-Point λ2 = 5 Change-point? λ1 = 1 2.5 1.5 1 21 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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) 22 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 23 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 24 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop 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 25 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop True model: 2-piece Piecewise Exponential (N=150) Pairwise comparison of models 2= 26 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop True model: 2-piece Piecewise Exponential (N=40) Pairwise comparison of models 2= 27 | 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) 28 | Change in Hazard | Rafia Bhore | 21 May 2013 | Midwest Biopharmaceutical Statistics Workshop