Long Subject Presentation Sojourn Models for Ordinal Data B Frame 9/23/2009 Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Background Info This SSP is to cover the details of a model developed for zero heavy ordinal data. This model is scheduled to be presented at the October 2009 DIA/FDA/Pharma workshop on Modeling and Simulation, check their website for a .ppt download. This work is also accepted for publication in JPP with major revision (which I did), but I have no idea when it might be in print. Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation The Published Model Actually was three models, one for each of dizziness, drowsiness, and dropout. Here we will focus on a model like the one used for drowsiness. Using fake data of course. Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation The Basic Problem 80% of Subjects 0,0,0 … 0,0,0 0,0,1,1,1,2,2,2,2,2,2,1,1 The Rest So what we see here is many subjects with no change from baseline, a zero AE score, and some that have a Markovian pattern. Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Lets Jump Right In! Take a look at control stream c400.txt. This is a simple Markovian cummulative logit model for our fake data. The sojourn aspect and a John Draper random effect transform are hinted at but not invoked, yet. Wolverine Pharmacometrics Corporation

Running this control on T1SIM.TXT ETABAR IS THE ARITHMETIC MEAN OF THE ETA-ESTIMATES, AND THE P-VALUE IS GIVEN FOR THE NULL HYPOTHESIS THAT THE TRUE MEAN IS 0. ETABAR: 0.16E+01 SE: 0.10E+00 P VAL.: 0.37E-52 OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ******** ETA1 + 4.80E+01 The numbers look bad but the pictures are even worse, next slide??? Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation No sense computing skewness or kurtosis here! Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Nomenclature Data: 0 0 0 0 0 0 0 11 1 1 1 …. 2 2 2 2 2 3 3 3 1 1 1 1 0 0 0 Day 1 2 3 4 … TAE … t TAE = Time of first non-zero AE t = last day of study Wolverine Pharmacometrics Corporation

Likelihood for the data Before TAE (if it ever occurs) : 1- h(j) where j is the day number and h is the hazard. At TAE : h(j=TAE)Probability of transition from 0 (on day TAE-1) to 1, 2, or 3 on day TAE After TAE : Probability of transition between previous days score and the current days score. This is the standard Markovian ordinal regression probability as used in c400.txt Wolverine Pharmacometrics Corporation

So what are the pieces / parts? Review time to event modeling with attention to the one change point Weibull hazard for discrete time. Review truncated distributions. Present coding for NONMEM VI implementation. Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Time to Event Material Let X be an observed event time and C be a censoring time. In our example here the time of the first occurrence of a non-zero AE, TAE would be X and C would be the last day of observation if no non zero AE were observed, here t, the last day of the study. Additionally, let  =1 if X is observed and 0 otherwise. Letting Y = min (X,S), the likelihood for a subjects data assuming a density f, and survival function S is… This is pretty standard representation for right censored data, see Klein and Moeshberger for a nice intro. Wolverine Pharmacometrics Corporation

Functions of Interest for Continuous Time Processes Given a density function f(x), we have The survival function Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation The hazard function h(x) = f(x)/S(x) = -d(ln(S(x)))/dx The cummulative hazard function = H(x) Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation And Finally Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Weibull Example Density : Survival Function: Hazard function: The home work will explore these relationships. Here lambda is scale and alpha is shape Cummulative Hazard: Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation For Discrete Time… If X is discrete valued, i.e. can only assume values xj, j = 1,2, …. Explain what the j and j-1 subscripts do… Wolverine Pharmacometrics Corporation

One Change Point Weibull Consider a two part log Weibull cumulative hazard 1 if t change point =tc Where : c = 0 of t > change point = tc And (1, 1) and (2, 2) are the Weibull parameters for times  and > the change point, respectively Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Forcing the two log cumulative hazards to be equal at the change point..  Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation Truncation Our AE scores are 0, 1, 2, or 3 and we will model these with a cumulative logit model, but at TAE the only possible transitions are to 1, 2, or 3. For our model to be a probability model it must integrate or sum to unity. If we sum up these probabilities we get the probability that AE1, so if we divide each of them by this quantity and then preform the summation the probabilities do add up to unity. Wolverine Pharmacometrics Corporation

NONMEM Implementation of Sojourn Part The data contains a flag (IND1) that is always zero except at TAE ID DV TIME OLDT IND1 1 0 1 0 0 1 0 2 1 0 1 1 3 2 1 1 1 4 3 0 Wolverine Pharmacometrics Corporation

A simple sojourn model c401.txt /nmdata100indest.csv ;TIME TO FIRST AE STRUCTURAL PART ;WEIBULL SHAPE PARAMETER ALPH1=THETA(1) ;HILL COEFFICIENT GAM1=THETA(6) ;MEDIAN TAE USE SIGMOID EMAX MODEL MED1=THETA(3)*(1-TDOS**GAM1*THETA(4)/(THETA(5)**GAM1+TDOS**GAM1)) IF (MED1.LE.0) THEN PRINT MED1 EXIT 1 100 ENDIF ; CHANGE OF VARIABLE (ALPHA, MEDIAN) -> (ALPHA, LAMBDA) LAM1=0.693/MED1**ALPH1 H1ZT1=LAM1*TIME**ALPH1 ; COMPUTE H(T) H1ZT0=LAM1*OLDT**ALPH1; COMPUTE H(T_) I1HAZ=H1ZT1-H1ZT0 ; COMPUTE INTEGRAL OF h(t) from T_ to T Q1=EXP(-I1HAZ) ; 1-h(T) Explain what T and T_ are Wolverine Pharmacometrics Corporation

The likelihood prior to TAE IF (NEWIND.NE.2) THEN FLG1=0 ENDIF IF (IND1.EQ.1) THEN FLG1=1 ;LIKELIHOOD FOR DATA PRIOR TO TRANSITION OUT OF AE=0 STATE IF (FLG1.EQ.0) THEN L=Q1 ;CONTRIBUTION HERE IS 1-h(T) Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation The likelihood at TAE IF (IND1A.EQ.1) THEN ;Probabilities for TRANSITION TO Y=1, Y=2, Y=3 when it is FIRST transition PA1=1-P11 ;CORRECT LIKELIHOOD SINCE ZERO CANNOT OCCUR HERE FUDG1=P11 PB1=(P11-P12)/FUDG1 PC1=(P12-P13)/FUDG1 PD1=P13/FUDG1 ENDIF ;LIKELIHOOD FOR DATA AT TRANSITION OUT OF AE=0 STATE IF (FLG1.EQ.1.AND.IND1A.EQ.1.AND.DV.EQ.1) THEN L=(1-Q1)*PB1 IF (FLG1.EQ.1.AND.IND1A.EQ.1.AND.DV.EQ.2) THEN L=(1-Q1)*PC1 IF (FLG1.EQ.1.AND.IND1A.EQ.1.AND.DV.EQ.3) THEN L=(1-Q1)*PD1 Wolverine Pharmacometrics Corporation