In many situations it is desirable or necessary to administer a set of tests to several different groups, and to ask if the results obtained in the different groups may be regarded as being essentially the same in some sense. In the case of two variables (one dependent and one independent) one may, for instance, ask if the errors of estimate and the regression lines may be regarded as being the same for the populations from which the different groups are drawn. For this case, the present article considers tests for three hypotheses regarding the populations from which the different groups are drawn: (a) HA, the hypothesis that all standard errors of estimate are equal; (b) HB, the hypothesis that all regression lines are parallel, (assuming HA); and (c) HC, the hypothesis that the regression lines are identical, (assuming HB). Test criteria for these three hypotheses and their sampling theory for large samples are presented. The results are extended to the case of several independent variables. An illustrative problem is presented for two groups, two independent and one dependent variable.

Evidence is cited to show that specificity, or lack of equivalence, in the comparable forms of tests has a tendency to lower the value of reliability coefficients but has no tendency to lower the value of observed trait coefficients. This implies that the greater the lack of equivalence, the higher will be coefficients corrected for attenuation. Errors of measurement are supposed to reduce the magnitude of observed trait coefficients. Since specificity does not lower the correlation between two tests and since the split-half and equivalent-form reliability coefficients treat specificity as error, it follows that these two coefficients cannot legitimately be used in Spearman's correction-for-attenuation formula.

In econometric literature a parameter in a theoretical model has been called identifiable if it can be uniquely determined in terms of the joint probability distribution of the observed variables. In this paper the identifiability of parameters in four different factor analysis models is considered. The last of these four models corresponds to Thurstone's factor analysis. In Sections 7 and 11, the possibility of a statistical testing of the models is discussed. Section 10 deals with the problem of actually determining the parameter r (the number of common factors) in terms of the probability distribution of the observed variables.

A formula for estimating real scores on a multiple-choice test from a knowledge of raw scores is derived. This formula does not involve the assumption of a binomial distribution of real scores as does the Calandra formula. Other important formulas derived show: the variance of real scores in terms of the variance of raw scores and the correlation between real scores and raw scores. If the variance of real scores (or of raw scores also) is binomial, the regression of real scores on raw scores is linear; but, otherwise the regression is curvilinear. Yet the linear estimating formula is a close approximation to the curvilinear relationship. Factors affecting the regression of real scores on raw scores and the correlation coefficient are: (1) the number of choices per question; (2) the number of questions answered; (3) the ratio of the average group raw score to the variance of raw scores.

By using the concepts of antimetry and n-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. It is shown that the number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called an n-clique, is introduced. n-cliques are shown to have a very close relationship to the existence of cliques in an artificial structure defined on the same set of elements, thus permitting the determination of n-cliques by means of the same simple matrix procedures used to obtain the clique structures. The presence of two or more m-cliques, where m is the number of elements in the group, is proved to mean an almost complete splitting of the group.

Samples are often obtained under circumstances which make it less likely to draw individuals from some parts of a population than from others. In this paper a method is presented which makes it possible, under certain assumptions, to correct for the resulting bias and thus reconstruct the means, standard deviations, and correlation coefficients of the original population.