Probability Models, Estimation, and Classification for Multivariate Dichotomous Populations

Many scientific investigations involve observations on a vector Z of dichotomous variables Z$_i$, Z$_i$ = 0, 1, i = 1, $\cdots$, I, from populations or subpopulations $\Pi^{(m)}$, m = 1, $\cdots$, M. Two examples are given, one on attitudes towards science of low and high I.Q. groups of high-school students and one on physiological responses of low and normal Apgar score groups of infants. A probability model is developed for $p^{(m)}(z) = P(Z = z | \Pi^{(m)})$ in the form p$^{(m)}$(z) = f(z)[1 + h$_S$(a$^{(m)}$, z)], m = 1, $\cdots$, M, f(z) $\geq$ 0, z $\epsilon \delta,$ where $\delta$ is the set of 2$^I$ discrete points in the sample space. The model depends on a set S of orthogonal polynomials $\phi_\gamma$(z) on $\delta, h_S(a^{(m)}, z) = \Sigma_{\gamma\epsilon S}a^{(m)}_\gamma \phi_\gamma (z), a^{(m)}$ being a distinguishing vector of parameters for $\Pi^{(m)}$. For special S, analogy of the model with that for a 2$^I$-factorial is noted. Benefits arise when I is large and N, the total sample size, is small relative to 2$^I$. Estimation of parameters f(z), z $\epsilon \delta$, and a$^{(m)}, m = 1, \cdots, M$, subject to appropriate constraints is considered. Application to classification procedures is discussed and illustrated for the examples.