This talk revisits previous work of Kalbfleisch and Lawless (JASA, 1985) on the analysis of panel data under a Markov assumption. Within this framework, each of random sample of n subjects is observed at times t0 < . . . < tk < tk+1 < . . . < tm, 0 < k ≤ m − 1, m > 2. The authors performed the analysis taking account the number of i → j transitions nij,k in [tk, tk+1) and the respective transition probabilities pij(tk, tk+1 | zk), where zk = z(tk) is the value of covariates z at tk.
In contrast to their approach, our analysis/estimation takes into account the i → j transition intensities qij(tk | zk) for given covariates zk, the number of transitions, nij,k, and the amount of time τi,k each subject spent in state i on [tk, tk+1), possibly subject to right censoring for both nij,k and τi,k. Some distributional identities of the time-inhomogeneous Markov chains are derived to develop the likelihood function of continuously observed n random sample (paths) of the Markov chains. Also for generating the sample paths for numerical study and testing the model.
Under piecewise constant intensity, explicit forms of score function and observed information matrix of regression coefficients of covariates are obtained in terms of matrix tensor representations. In particular, the information matrix is of block diagonal and positive definite regardless the statistics of sample paths and the values of covariates. These appealing features provide robust estimation of the coefficients. Importantly, they are useful in simplifying derivation of the standard errors of ML estimators (MLEs) and in showing the consistency and large sample properties of the MLEs. A numerical study confirms the accuracy of ML estimation. The results generalize the above paper and Albert (Ann. Math. Stat., 1962).
This talk is based on a joint work with Professor Halina Frydman.