1. Long Subject Presentation Sojourn Models for Ordinal Data
B Frame
9/23/2009
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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.
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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.
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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.
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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.
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6. 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: E+01
SE: E+00
P VAL.: E-52
OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ********
ETA1
E+01
The numbers look bad but the pictures are even worse, next slide???
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No sense computing skewness or kurtosis here!
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Nomenclature
Data: …
Day … TAE … t
TAE = Time of first non-zero AE
t = last day of study
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9. 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
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10. 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.
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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.
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12. Functions of Interest for Continuous Time Processes
Given a density function f(x), we have
The survival function
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The hazard function h(x)
= f(x)/S(x) = -d(ln(S(x)))/dx
The cummulative hazard function = H(x)
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And Finally
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Weibull Example
Density :
Survival Function:
Hazard function:
The home work will explore these relationships. Here lambda is scale and alpha is shape
Cummulative Hazard:
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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…
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17. 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
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Forcing the two log cumulative hazards to be equal at the change point.. 
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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.
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20. NONMEM Implementation of Sojourn Part
The data contains a flag (IND1) that is always zero except at TAE
ID DV TIME OLDT IND1
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21. 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-H1ZT ; COMPUTE INTEGRAL OF h(t) from T_ to T
Q1=EXP(-I1HAZ) ; 1-h(T)
Explain what T and T_ are
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22. 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)
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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
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