Testing Markovianity in the Three-state Progressive Model via future-past Association

Mar Rodríguez-Girondo and Jacobo de Uña Álvarez have a paper currently available as a Universidade de Vigo Discussion paper in Statistics and Operations Research. They develop a test for the Markov property in a progressive three-state model subject to continuous observation up to right-censoring. The test is based on calculating Kendall's Tau at each time point t, which involves calculating the difference between the concordance and discordance probabilities for two pairs of (Z,T) where Z=sojourn in state 1, T=time to entry in state 3, given both subjects are in state 2 at time t, i.e. Z <= t < T. If the Markov property holds we expect Kendall's Tau to stay at around zero. For a non-Markov process tau would vary with time away from 0. An estimator for Tau is developed and a bootstrap resampling algorithm is proposed to estimate a p-value for tau at a fixed time point t, based on independently sampling Z and T from their empirical distributions. A trace of p-values at a grid of time points can then be produced.

Unlike the Cox-proportional hazards approach where a single statistic is produced, here the p-value varies depending on the choice of t chosen. A superior power to the Cox-PH approach was obtained in simulations but only for a good choice of t. An omnibus statistic based on some weighted integral of the absolute value (or square) of tau over the observation range would be useful.

The paper only deals with the progressive 3-state case. It's unclear how the method would be extended to incorporate complicated past history in a more general multi-state model. However, there is scope to extend it to testing whether a particular state within a model has semi-Markov dependencies. In its current form, while the test is an interesting concept, using a simple Cox-PH seems a much more attractive prospect in practice. Update: This paper is now published in Biometrical Journal.