Nonlinear Factor Analysis

ACKNOWLEDGMENTS The present monograph, except for minor revisions, was presented as a doctoral dissertation at the University of New England in 1963. I wish to thank my supervisor, Dr. J. A. Keats of the University of Queensland, for his encouragement, advice, and criticism in the course of this work. I am particularly indebted to Dr. E. J. Burr, of the Department of Mathematics, University of New England. His advice and criticism at stages in the work has helped me to avoid a number of lapses from formality and rigor in the mathematics to be presented. Any errors that remain are, of course, my responsibility. NOTE ON FORMAT AND NOTATION Sections, equations, tables and figures, are numbered on Peano's decimal system. Thus, (4.8.2) means the second equation of section 8 of Chapter and similarly for tables and figures. The sections of Chapter 5 correspond to some of the sections of Chapter 4, and are given section numbers identical with those of the sections in Chapter 4 on which they are based. An attempt has been made to conform to the preferred conventions in mathematical notation. In general, capital Roman letters (in italics) denote matrices, lower case Roman letters (in italics) without subscripts denote vectors, and Greek letters denote scalars. The few exceptions should be clear from the context. (For example, a script R (fit) is used for a covariance matrix, and the corresponding correlation matrix is the ordinary R.) A script E (5) is employed for the expectation operator. A prime after a matrix or vector denotes transposition. Glossary o] major symbols (with equation of first appearance). ~b, unspecified latent "characterization." g(~b), density function of ~b. h(y I ¢~), conditional density function of y given hi(yi [ ~), conditional density function of any yi given ~b. p~ ] ~b, conditional probability that a dichotomous y~ = 1, given if_ N(') m %/~-~ ~ exp-dx, the normal ogive function. ]~(~) ~ ~(y; ] ~I,), a prescribed function. p~., probability of a "positive" response. q~, probability of a "negative" response. vector of r linearly independent functions in t statistically independent variates x~.. a vector of r ortho-normal functions in the x~. B, an r X n matrix, "factor loadings." v =-[v~ ... vr] = zF'C-~, a vector of "component variates." w ~ [w~ ... wr] = hL, a vector of "true parts" of the "component variates." d =-[d, ... d~] = eUF~C-~, a …