A solution is presented to the problem of determining a proper correction for spuriousness in correlation coefficients. The general case developed is the estimate of correlation between u and v, both being linear functions of the same set of variables. Special formulae relate to overlapping scales correlations, part-whole correlations, and item-total test correlations.

In this paper a method is developed for analyzing data resulting from the use of a modification of the usual paired comparisons procedure which allows for ratings of size of difference. The scaling model is essentially an extension of the Thurstonian successive intervals model. The method is applied to scaling of a set of nine “immoral” actions and the results agree rather well with those from conventional successive intervals scaling of the same stimuli.

Significance tests are developed for evaluating the absolute invariance of the dominance and composition scales presented in a previous paper [Bechtel, 1967]. These tests derive from the multivariate normality of the estimated scale values—this multinormality stemming from that of the observations upon which these estimated scale parameters are based. The scales compared by means of the present techniques will typically be constructed under distinct experimental conditions, which may take the concrete form of treatments, occasions, individuals, or groups. Thus the present tests possess a wide range of applicability including various experimental designs in which scale configuration is the dependent variable. Illustrative data are used in statistically comparing the scale structures of two different individuals.

The misclassification process is represented by a stochastic matrix containing the probabilities that an individual who belongs in one cell is counted as belonging to another (or perhaps the same) cell. These probabilities are supposed known. If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged. It is shown how to calculate loss of power in this case and also how to calculate the change in size of the test if the errors are not independent. A modified test criterion is suggested when errors are not independent and a numerical example is included.

A Monte Carlo approach is employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings. A number of sample correlation matrices were generated from a specified population, factored, and transformed to a least-squares fit to the population values. Influences of factor strength, communality and loading size are discussed in relation to the statistics summarizing the results of the above procedures. Influences producing biased estimators of the population values are also discussed.

The basic concepts of nonlinear factor analysis are introduced and some extensions of the general theory are developed. An elementary account of the class of multiple-factor polynomial models is presented, using more elementary algebraic methods than have been employed in earlier accounts of this theory. Working formulas are developed for the multiple-factor polynomial model without product terms.

Some empirical results are presented.

A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that such improper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.

Latent partition analysis has been formulated to study the relationships between two or more partitions of the same set of items. The major structural hypothesis is that a latent partition underlies the manifest partitions; that is, it is assumed that each item belongs to a latent category and that the manifest categories are derived by dividing and combining the latent categories. We have found that by examining manifest categories it is possible to reconstruct information about the latent partition and about its relation to the manifest partitions.

Usual system: take as scale value the normal deviate corresponding to the area p″ = (p − p0)/(1 − p0), where p is the proportion of correct answers and p0 is the chance proportion. If p ≦ p0, the item difficulty is unscalable. Proposed system: take as scale value the normal deviate corresponding to the area p′ = px, with x so chosen that p′ = .5 when p is half-way between p0 and 1. Table gives D (=500 + 100z) for p = .00(.01)1.00 and for a (= number of alternatives) 2,3,4,5,6,7, ∞.

For the tests in which the score on an item is not restricted to 0 and 1, but is any number on a continuous scale, a procedure for estimating an examinee's true score is given. For the case of 0, 1 item scoring this problem was considered by Lord [1959]. Following Lord, the least squares estimation procedure is used and the regression coefficient is obtained, which is compared with the generalized KR(20) and KR(21) formulas. Also, results are discussed using analysis of variance models.

Estimates of test size (probability of Type I error) were obtained for several specific repeated measures designs. Estimates were presented for configurations where the underlying covariance matrices exhibited varying degrees of heterogeneity. Conventional variance ratios were employed as basic statistics in order to produce estimates of size for a conventional test, an ∊-adjusted test, and ∊-adjusted test and a conservative test. Indices for empirical distributions of two estimators of ∊j, a measure of covariance heterogeneity, were also provided.

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.