We have said that psychometric methods involving algorithms are completely objective—at least they are if the algorithm is in the form of a program for a digital computer. These objective procedures need Monte Carlo and other computer runs to determine their properties, but so do many equation-oriented techniques. The objective algorithms are flexible but not flaccid. They offer a way of dealing with complexities that formerly seemed beyond our grasp. As the computer revolution continues in psychometrics, we can expect objective algorithmic methods to become the rule rather than the exception.

A previous article was concerned with simultaneous linear prediction. There one was given a set of predictor tests or items and one predicted a set of predictands (also tests or items, or perhaps criteria.) We proposed a simultaneous prediction which was a certain weighted sum of the predictors. In the present article the constraint that the prediction be a weighted sum is relaxed. We seek a general function of the predictors which will maximize the quantity chosen for measuring prediction efficiency. This quantity is the same as the one used in linear prediction and we justify this approach by showing it is the appropriate one when there is only one predictand. In order to solve the problem we restrict consideration to a vector of predictors having only a finite number of possible values, i.e., it possesses discrete probability distribution weights. This can be applied in the case of dichotomous items for instance. It may also be used in continuous distributions as an approximation, by first dividing the original range of values into a finite number of intervals. Then one attributes to the interval the weight corresponding to the probability mass it underlies in the original distribution.

A model for the measurement of the discrepancy between two scores is presented and discussed as a paradigm for the study of growth or experimentally produced change. The model assumes two tests or measures differing in complexity, and it analyzes the true difference between the test scores into a component that is entirely dependent on the first or base-line test and a second component that is entirely independent of it. Equations for estimating both components are given and these are compared with other measurement efforts with similar goals.

In many decision problems the decision-maker must ask a series of questions or select cues relevant to but prior to the decision. There are many strategies which would prescribe an order for asking such questions or selecting such cues. It is the purpose of the paper to describe a manner in which the strategy or strategies of the decision-maker can be analyzed. From this model can be derived limits corresponding to the possible use of a particular strategy, and the effective or observed use of the strategy or strategies by the decision-maker.

Certain aspects of point estimation are treated for two kinds of exponential latency processes. The first unitary process is represented by a simple exponential density in which the rate parameter may be viewed as an unknown constant or as a random variable. If a second, slower exponential process is grafted onto the first, there results a postulated two-component latency between stimulus and response. Moments estimators are derived for the two parameters of this latter density, and the relevance of the second parameter to decision time is emphasized.

A general method is presented for estimating variance components when the experimental design has one random way of classification and a possibly unbalanced fixed classification. The procedure operates on a sample covariance matrix in which the fixed classes play the role of variables and the random classes correspond to observations. Cases are considered which assume (i) homogeneous and (ii) nonhomogeneous error variance, and (iii) arbitrary scale factors in the measurements and homogeneous error variance. The results include maximum-likelihood estimations of the variance components and scale factors, likelihood-ratio tests of the goodness-of-fit of the model assumed for the design, and large-sample variances and covariances of the estimates. Applications to mental test data are presented. In these applications the subjects constitute the random dimension of the design, and a classification of the mental tests according to objective features of format or content constitute the fixed dimensions.

Kaiser has given the varimax criterion for the solution of the rotation problem in factor analysis as well as a practical computational procedure for maximizing this criterion. In the present paper, the maximization condition is shown as a matrix equation involving only the unknown orthogonal rotation matrix. This matrix equation can be solved iteratively as a sequence of symmetric eigenproblems.

Certain properties and consequences of factor indeterminancy are deduced from an orthogonal transformation derived by Ledermann. The first deduction is the matrix of cosines between maximally distinct factor axes which represent the same factor solution. This result is used to investigate the indeterminancy of factor extensions.

The relationship between the factor pattern, F, derived from fallible (containing measurement error) observations on variables and the factor pattern, F*, derived from infallible observations on variables is investigated. A widely believed relationship between F and F*, viz.,F* = AF where A is a diagonal matrix containing the inverses of the square roots of the reliabilities of the variables, is shown to be false for several factor analytic techniques. Under suitable assumptions, it is shown that for Kaiser and Caffrey's “alpha factor analysis”F* and F are related by F* = AF. Empirical examples for which the corresponding elements of F* and AF are equal to two decimal places are presented. The implications of the equality of F* and AF for alpha factor analysis are discussed.

In the course of developing the minres method of factor analysis the troublesome situation of communalities greater than one arose. This problem—referred to as the generalized Heywood case—is resolved in this paper by means of a process of minimizing the sum of squares of off-diagonal residuals. The resulting solution is superior to the otherwise very efficient original minres method without requiring additional computing time.

This paper presents the results from an investigation of the true probability distributions of the range of rank totals. A procedure for generating an approximation to the true distributions is also given. A comparison of the results of this approximation with an extensive criterion of generated true and sample distributions, and with other approximations is indicated. Accurate estimates of the critical ranges necessary to reach significance at three commonly used alpha levels, where the number of judges and items is less than or equal to sixteen, are presented in tabular form.