Analyses of multinomial mixture distributions: new tests for stochastic models of cognition and action

Mixture distributions are formed from a weighted linear combination of 2 or more underlying basis distributions [g(x) = sigma j alpha j fj(x); sigma alpha j = 1]. They arise frequently in stochastic models of perception, cognition, and action in which a finite number of discrete internal states are entered probabilistically over a series of trials. This article reviews various distributional properties that have been examined to test for the presence of mixture distributions. A new multinomial maximum likelihood mixture (MMLM) analysis is discussed for estimating the mixing probabilities alpha j and the basis distributions fj(x) of a hypothesized mixture distribution. The analysis also generates a maximum likelihood goodness-of-fit statistic for testing various mixture hypotheses. Stochastic computer simulations characterize the statistical power of such tests under representative conditions. Two empirical studies of mental processes hypothesized to involve mixture distributions are summarized to illustrate applications of the MMLM analysis.