After I had started to put some of my thoughts into a first draft for this talk, I selected “The Error” as a title. This selection gave me a great deal of flexibility since, no matter how I went wrong, I was still well within the limits of the title. After additional deliberation, I had more than a slight feeling that the title was more appropriate for a potential best-seller in the field of popular fiction than it was for a potential presidential address to this society. I have, on occasion, thought that I could write–in fact, I started my academic life in a School of Journalism. At least at the academic level, the journalistic style of writing rather quickly suppressed my interest therein as a means of making a livelihood.

Errors, random and otherwise, are often the heart of some rather undercover plots and subplots in best-sellers. But just as in the case of novels, in experimental designs errors also play a dominant role–unfortunately the experimental units, which are the players, often do not understand their roles as set forth in the design script. The latter is usually in the form of a mathematical model. A question might be raised in some instances as to whether it is more difficult to teach the experimental units what the model calls for, or to teach the designer what types of designs are potentially appropriate for expressing the behavior of experimental units, under specified experimental conditions.

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.

The effects of N and communality on the variability of zero and nonzero factor loadings were assessed using a Monte Carlo approach. It was found that increasing N or communality resulted in decreased sampling error of individual factor loadings, but for zero loadings N was found to have the greatest influence. It was also found that distributions of factor loadings become relatively elongated as communality increases.

Let θ > 0 be a measure of the average step size of a stochastic process {pn(θ) }n=1(∞). Conditions are given under which pn(θ) is approximately normally distributed when n is large and θ is small. This result is applied to a number of learning models where θ is a learning rate parameter and pn(θ) is the probability that the subject makes a certain response on the nth experimental trial. Both linear and stimulus sampling models are considered.

The paper presents a general framework for the dependent factor method in which judgmental as well as analytic criteria may be employed. The procedure involves a semi-orthogonal transformation of an oblique solution comprising a number of reference factors and a number of experimental factors (composites). It determines analytically the residual factors in the experimental field keeping the reference field constant. The method is shown to be a generalization of the multiple factor procedure [Thurstone, 1947] in so far as it depends on the use of generalized inverse in deriving partial structure from total pattern. It is also shown to provide an example of the previously empty category (Case III) of the Harris-Kaiser generalization [1964]. A convenient computational procedure is provided. It is based on an extension of Aitken's [1937] method of pivotal condensation of a triple-product matrix to the evaluation of a matrix of the form H − VA−1U' (for a nonsingular A).

Let A1, A2, ..., An be any n objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (Ai, Ai), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculating m coordinates {xia} for each Ai for a semi-metric d of preassigned form dij = d(|xi1 -xj1|, |xi2 -xj2|, ..., |xim - xjm|). The dimensionality m is sought to be as small as possible, yet satisfy the monotonicity condition that dij <dkl whenever the observed data indicate that Ai is “closer” to Aj than Ak is to Al. Minkowski and Euclidean spaces are special metric examples of d. A general coefficient of monotonicity μ is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing μ by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between “hard-squeeze” and “soft-squeeze” tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.