Likelihood Ratio-Based Tests for Longitudinal Safety Data October - PowerPoint PPT Presentation

Likelihood Ratio-Based Tests for Longitudinal Safety Data October 24, 2014 Ram Tiwari and Lan Huang Office of Biostatistics, CDER, FDA 1

Disclaimer The views expressed by the speakers of this talk are their own and do not necessarily represent those of FDA. 2

Useful references 1. Lan Huang, Jyoti Zalkikar, and Ram Tiwari. A likelihood based method for signal detection with application to FDA’s drug safety data. Journal of the American Statistical Association (JASA), 106 (496), 1230-1241, 2011 2. Lan Huang, Jyoti Zalkikar, Ram Tiwari. Likelihood ratio tests for longitudinal drug safety data. Statistics in Medicine, 33(14), 2408-2424, 2014 3. Lan Huang, Ted Gou, Jyoti Zalkikar, Ram Tiwari. A review of statistical methods for safety surveillance. Therapeutic Innovation & Regulatory Science, 48 (1), 98-108, 2014 3

Outline • Basic LRT method for large post-market safety database • OB in-house tool development and illustration • Longitudinal LRT method for data with exposure information • Discussion 4

Background • Large databases: AERS, Vigibase, MAUDE • Data mining methods (frequentist, Bayesian, OMOP/IMEDS) • Objective of the safety exploration – Signal detection in large safety database – Clinical trials database – Passive/active 5

I x J Safety Data-Matrix for AERS Database Drugs … … J 1 j Row total … … n 1 J 1 n 11 n 1j n 1. … … … n 2 J 2 n 21 n 2. … … … … … … … AEs … … … i n ij n i J n i. … … … … … … … … … … I n I 1 n I j n IJ IJ … … n . J Col. n .1 n .J n.. total 6

2x2 Table of nij Other drugs Drug j Subtracted n i . AE i n ij Other AEs Subtracted Subtracted Subtracted Subtracted n.. n j . • Fix a drug, say Drug j, and construct a 2X2 table for each AE: If there are, say, 16,000 AEs, then there are 16,000 such 2x2 tables • Most of the frequentist’s methods and some Bayesian methods work with 2X2 tables 7

Statistical model and hypothesis ~ ( ) n Pois n p . ij i i    ~ (( .. ) ) n n Pois n n q . . j ij i i   0 : * H p q p for all AEs, i i i  : H p q for all at least one AE, i a i i RR=p/q, AE i vs. other AEs  max / max LR L L 0 a 8

Likelihood Ratio Test (LRT) statistic    n  n n     . ij j ij n n n       . ij j ij        ˆ ˆ     n n n ( , ) L p q   . .. . i i a , LR   ij ˆ n ( *)   L p . j n   0   . j       n   .. • Test Statistic is MaxLR=max of LR_ij (i=1 to I) over i=1,…,I AEs. • LogLR and MaxLogLR can be used for faster computation 9

Re-parametrization and adjustment  n n n     . ij j ij      n n n       ij . j ij  n   n n  n n n       ij . j ij ij . j ij LR          ij n n E n E         .   .  j j . ( ) ij j ij n n n . .. . i i     n n .. .. n n n   . . . i j j E n . ij i n n .. .. To adjust for a covariate (such as age or gender)(stratified analysis), we simply calculate the age-adjusted or gender adjusted expected cases. We first calculate the E_ijk, k=1,2 (by gender), then we combine them together. k n      . j k k [ ] E E n . ij ij i k n k k .. 10

Theory behind the multinomial simulation for the null data Assume that the marginal totals n1., …, nI., are fixed. Under H0, assume that n1j, …, nIj are ind distributed as ~ ( ). n Poisson n p 1 1. j ....  ~ ( ), 0, . n Poisson n p p unknown Ij 1. Then, n n 1. . I ( ,..., ) | ~ ( ,( ,..., )). n n n mult n 1 . . j Ij j j .. .. n n 11

Hypothesis testing • The distribution of MaxLR under H0 is not analytically tractable, we use Monte Carlo method to obtain the empirical dist. • Cases can be generated using multinomial distribution (n.j , (n1./n..), (n2./n..), …., (nI./n..)) assuming homogeneous reporting rate. 12

P-value calculation • Calculate MaxLR from observed data (one) • Calculate MaxLRs from the 9,999 simulated null data. • Threshold at alpha=0.05 is 95 percentile of the 10,000 (=1+9999) MaxLRs. • Reject H0 if obs MaxLR> threshold. • Compare the observed MLR and the ones from simulation ---  p-value = P(MLR> obs MaxLR)= Max # of times simulated MaxLR> obs MaxLR /10000. • Gatekeeping step-down process (1 st , 2 nd , 3 rd , …) 13

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OB in-house tool for LRT method 15

Example (Myocardial infarction) PT #Drug n.j PRR025 sB05 BCPNN025 EB05 LRT (>1) (>2) (>0) (>2) (p<0.05) Myocardial 1416 26,848 242 36 137 35 51 infarction #Drug Nij PRR025 LRT sB05 BCPNN025 EB05 N (Generic) (>1) (P<0.05) (>2) (>0) (>2) 1 Rosiglitazone 2231 2 Metformin And Rosiglitazone 322 3 Calcium Chloride And Glucose And 637 Magnesi 4 Clopidogrel 419 5 Rosuvastatin 398 6 Atorvastatin 506 7 Calcium Chloride And Icodextrin And 150 Magn 8 Ticagrelor 109 9 Glimepiride And Rosiglitazone 46 16 10 Glyceryl Trinitrate 175

LRT to longitudinal LRT (Motivation) LRT Longitudinal LRT Count data Count data with exposure information Large post-market observational Observational or clinical trial data safety data Drug signals for one AE Same Or AE signals for one drug Multiple AEs and drugs Same Fixed time analysis Same Analysis over time using cumulative Use alpha-spending for analysis count data without planned alpha over time control 17 covariate adjustment by stratification same

Longitudinal LRT method (sequential LRT) for active surveillance • General – Compare multiple AEs by drug – Compare two drugs for one AE of interest (1 st occurrence or without recurrence) – Compare multiple drugs for one AE of interest (may have recurrence or combined AE terms) • Control error rates and false discovery rate (FDR) 18

Define countable cases and drug exposure • Countable cases: AEs that occur during the exposure period (other definitions: AEs occur several days after the drug exposure) • Drug exposure – Event-time – Person-time – Exposure-time • Time – calendar time – time after drug exposure 19

Definition of event-time AE 1 AE 2 AE 3 AE 1 * ** s=1 P 1 i=1,js P 1 i=2,js P 2 i=1,js AE 1 * ** s=2 P 1 i=1,js AE 2 AE 1 s=3 * ** P 1 i=2,js P 1 i=1,js 20

Definition of person-time AE 1 s=1 * ** P i=1,js s=2 AE 1 * ** P i=1,js AE 1 ** s=3 * P i=1,js 21

Definition of exposure-time AE 1 AE 2 AE 3 AE 1 s=1 * ** P ds s=2 AE 1 * ** P ds AE 2 AE 1 * ** s=3 P ds 22

Working data structure (with event-time) Drugs Drugs … 1 2 3 14 … 1 2 3 14 Row total … … 1 n11 n1j n1J … … 1 P11 P1j P1J P1. … … … 2 n21 n2J … … … 2 P21 P2J P2. … … … … … … AEs … … … … … … … … … … i nij niJ … … … i Pij PiJ PiJ … … … … … … … … … … … … … … … I nI1 nIj nIJ … … I PI1 PIj PIJ PIJ … … Col n.1 n.j n.J … … Col P.1 P.j P.J P.J total total J=14 in the above table. At look k (k=1,…, K=5), there are two tables constructed from the individual level data. Pij is the event-time (unit here is day) for the AE i and drug j. We suppress k in the notation. 23

Exposure-based longitudinal LRT Methods using event-time  ~ ( ), n Poisson p P . ijk ijk i k    ( ) ~ ( ( )) n n Poisson q P P . .. . jk ijk ijk k i k    1 ,..., , 1 ,..., , 1 ,..., . i I j J K K H0: p i =q i over i=1,…I, if J (drug) is fixed. RR i =p i /q i ,=1 under H0, i=1,…, I, which is relative event-rate of ith AE vs. other AEs for fixed drug j. 24

The likelihood ratio is then  ˆ n ˆ n n ( ) ( ) . p ijk q jk ijk  , , ijk H ijk H LR a a ijk ˆ n ( ) . p jk , ijk H 0  n n n  n n n . ijk jk ijk ( ) ( ) . ijk jk ijk  .. P P k P  . . i k i k LR ijk n n . jk ( ) . jk P .. k   n n n n P    n n n . . . ijk jk ijk jk i k ( ) ( ) ; . ijk jk ijk E  ijk E n E P . .. ijk jk ijk k   Test max max , 1 ,..., LR LR i I jk i ijk statistic is 25

Working data structure (with person-time) AE of interest J=1 1 n11k 1 P11k drug I=2 n21k 2 P21k Col P..= Col n.1k= total total P11k+P21k n11k+n21k 26

Sequential LRT (with person-time)  ~ ( ), n Poisson p P 11 11 11 k k k  ( ) ~ ( ) n Poisson q P 21 21 21 k k k RR P 11 11 k k | ~ ( , ) n n Binomial n  11 . 1 . 1 k k k ( ) RR P P 11 11 21 k k H0: p 1 =q 2 , Ha: p1>q2 RR 1 =p 1 /q 2 , is relative risk of ith AE vs. the other AE for fixed drug j; or relative risk of ith drug vs. the other drug for fixed AE j. 27