1. Goodness of Fit of a Joint Model for Event Time and Nonignorable Missing Longitudinal Quality of Life Data – A Study by Sneh Gulati* *with Jean-Francois Dupuy and Mounir Mesbah

2. HISTORY  Project is a result of the sabbatical spent at University of South Brittany  In survival studies two variables of interest: terminal event and a covariate (possibly time dependent)

3. Dupuy and Mesbah (2002) modeled above for unobserved covariates We propose here a test statistic to validate their model Still in progress

4. BREAKDOWN OF TALK  I) Preliminary Results  II) Missing Observations  III) Dupuy’s Model  IV) Goodness of fit for Dupuy’s Model

5. Preliminaries  The survival time (or duration time to some terminal event) T is often modeled by the Cox Regression Model : ( t | Z) =  (t)exp{ T Z)}   (t) is baseline hazard rate, Z is the vector of covariates. Survival times are censored on right – one observes X = min (T, C) and  = I { T C}

6. Types of Covariates  External – Not directly involved with the failure mechanism  Internal – Generated by the individual under study – observed only as long as the individual survives

7. Solution to the full model  Parameter vector obtained  0 by maximizing the following:

8. Estimate of the cumulative hazard function:

9. Goodness of Fit:  Graphical Methods  Chi-Squared Type Tests  Lin’s Method of Weights

10. Problem – Missing Data  Missing Covariates – often due to drop out

11.  Let denote the history of the covariate upto time t:  Let T be the time to some event. Then the hazard of T at time t is ((t)| )dt = lim dt →0 Pr( t < T < t + dt)| )

12. CLASSIFICATION OF THE DROP-OUT PROCESS  Completely Random Dropout – Drop-out process is independent of both observed and non observed measurements.  Random Drop-out – Drop-out process is independent of unobserved measurements, but depends on the observed measurements.  Nonignorable Drop-out – Drop-out process depends on unobserved measurements.

13. Approaches  Use only the complete observations  Replace missing values with sample mean.  Estimate missing values with consistent estimators so that the likelihood is maximized (IMPUTATION)

14. Previous Notable Work for Nonignorable Dropout  Diggle and Kenward (1994)  Little (1995), Hogan and Laird (1997) - Essentially one integrates out the unobserved covariates  Martinussen (1999) – uses EM algorithm

15. Work of Dupuy and Mesbah  Subjects measured at discrete time intervals  Terminal Event – Disease Progression  Patients can dropout and covariate can be unobserved at dropout

16. The Model  n subjects observed at fixed times t j, t 0 = 0 <... < t j-1 < t j <... <  0 <  0  t = t j – t j-1 <  1 <   Let Z = internal covariate and Z i (t) = value of Z at time t for the i th individual  Z i, j denote the response for the ith subject on (t j, t j+1 ].

17. Hazard Rate  = (t)exp( T w(t)) where w(t) = (z(t – t), z(t)) T  = (    ) T or  = (    ) T

18. Assumptions  1) The covariate vector Z is assumed to have uniformly bounded continuous density f(z,   2) The censoring time C has continuous distribution function G C (u)  3) The censoring distribution is assumed to be independent of the unobserved covariate, and of the parameters ,  and .

19. Likelihood L(  ) = Let us call the above model Equation (1)

20. Solution  Method of Sieves: Replace original parameter space  of the parameters () by an approximating space  n, called the sieve.

21. Instead of the hazard function,  one considers;  n,i =  n (T (i) ) T (i), i = 1, 2,..., p(n), where T (1)  T (2) ...  T (p(n) are the order statistics corresponding to the distinct dropout times T 1  T 2 ...  T p(n) Hence the approximating sieve is  n = {  = ( , ,  n ):   R p,   R 2,  n, 1   n, 2 ...   n, p(n) }.

22. One maximizes the psuedo-likelihood function: Ln(  ) =

23. here L (i) (  ) =

24. THE MLE The MLE Obtained via the EM algorithm is identifiable and asymptotically normally distributed

25. Goodness-of-fit for Dupuy’s Model  Issue of Model Checking Important  PROBLEM – MISSING DATA  Could use DOUBLE SAMPLING or IMPUTATION

26. SOLUTION: Validate model in Equation (1) – Marginal Model  Done by Using the Weights Method of Lin (1991)

27. Development of the Test Statistic  Using a random weight function, WG(.) define a class of weighted pseudo-likelihood functions given by  WL n () = Call the above equation (2)

28. where WL (i) (  ) =

29. Define the maximizer of equation (2) as: The test statistic is a function of

30. Asymptotic Results For  Under the model in Equation (1), the vector converges in distribution to a bivariate normal distribution with zero mean and a covariance matrix

32. If model in Equation (1) is correct, the weighted and the nonweighted MLE’s should be close to each other:

33. Under the model in Eqn (1), the vector converges in distribution to a bivariate normal distribution with zero mean and a covariance matrix D W =

34. Note that

35. Proof still in progress  Proposed method: Show that score function for weighted likelihood and unweighted likelihood are asymptotically joint normal.  Use counting process techniques and martingale theory.

36. THE TEST STATISTIC Under the model in Equation (1), the above statistic will have a chi-square distribution with 2 degrees of freedom.