Exchangeability in multivariate Markov chain models

Time-homogeneous Markov chain models with state space [0, 1]k are useful in analysis of binary follow-up data on k individuals that interact. The number of parameters increases exponentially with k so more restrictive models are imperative for statistical inference. The hypothesis that the matrix of transition probabilities is invariant under permutation of individuals is discussed. It is shown that if individuals are exchangeable, then the process counting the number of individuals occupying a given state is a Markov chain. This reduction of data is sufficient if either at most a single individual may change state between two consecutive time points or if a state is absorbing. Similar results are obtained for exchangeability within two subgroups. Inference in the multivariate process reduces to a univariate problem if individuals are independent given the group's previous response. It is shown how conditional independence could be tested assuming exchangeability. The different hypotheses re examined in an analysis of the occurrence of bacteria in milk samples of Danish dairy cattle.