A Markov chain with transition probabilities pij(θ) that are functions of a parameter vector θ is defined. One of t input values is delivered to the chain on each trial. Under the hypothesis H0 the parameter vector is independent of the input; under the hypothesis H1 the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H0 against H1, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to “within-subject effects” for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.
