An experimenter threw individually 219 different dice of four different brands and recorded even and odd outcomes for one block of 20,000 trials for each die—4,380,000 throws in all. The resulting data on runs offer a basis for comparing the observed properties of such a physical randomizing process with theory and with simulations based on pseudo-random numbers and RAND Corporation random numbers. Although generally the results are close to those forecast by theory, some notable exceptions raise questions about the surprise value that should be associated with occurrences two standard deviations from the mean. These data suggest that the usual significance level may well actually be running from 7 to 15 percent instead of the theoretical 5 percent.

The data base is the largest of its kind. A set generated by one brand of dice contains 2,000,000 bits and is the first handmade empirical data of such size to fail to show a significant departure from ideal theory in either location or scale.

A simplified proof of a lemma by Ledermann [1938], which lies at the core of the factor indeterminacy issue, is presented. It leads to a representation of an orthogonal matrix T, relating equivalent factor solutions, which is different from Ledermann's [1938] and Guttman's [1955]. T is used to evaluate bounds on the average correlation between equivalent sets of uncorrelated factors. It is found that the minimum average correlation is independent of the data.

A limitation of the Tukey HSD procedure for multiple comparison has been the requirement of equal number of observations for each group. Three approximation techniques have been suggested when the group sizes are unequal. Each of these techniques was empirically analyzed to determine its effect on Type I error. Two of the considered variables, average group size and differences in group size, caused differing actual probabilities of Type I error. One of the three techniques (Kramer's) consistently provided actual probabilities in close agreement with corresponding nominal probabilities.

In the method of multiple paired comparisons the dominance of object j over object k is observed upon p attributes. The present paper develops a covariance analysis for these paired comparisons in terms of a linear model which includes scale, bias, and interaction effects, along with s covariants upon which the comparisons are presumably dependent. The covariance model gives rise to adjusted parameter estimates and hypothesis tests for the residual pairwise layout from which the effects of the s covariants have been removed. These estimation and testing procedures are illustrated with an analysis of political judgment data, and their relevance to the general problem of residual scaling is discussed.

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squares Q obtained from the least-squares monotone regression of yi on xi. Treating Q as a function of the yi, we prove that the gradient ▽Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient of d = Q1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that the yi and hence indirectly Q are functions of other variables.

The method of deriving the second derivatives of the goodness-of-fit functions of maximum likelihood and least-squares confirmatory factor analysis is discussed. The full set of second derivatives is reported.

A simple Bayesian solution for the joint estimation of several parallel profile differentials for multivariate normal populations is obtained. This generalizes a previous result for a single profile differential.