Chester W. Harris

Oblique factor analytic solutions by orthogonal transformations

A general framework for obtaining all possible factor analytic solutions, orthogonal and oblique, for a given common factor space is developed in detail. Interestingly, and seemingly paradoxically, any one of these solutions may be obtained by orthogonal transformations of selected matrices; thus an oblique solution may be determined by orthogonal transformations. Within the possible oblique solutions, two distinct categories of solutions emerge, a special case of the simpler of which apparently provides a definitive solution to the problem of independent, but correlated, clusters. Possible further specializations of the general approach to specific problems are discussed.

Some rao-guttman relationships

An examination of the determinantal equation associated with Raos canonical factors suggests that Guttmans best lower bound for the number of common factors corresponds to the number of positive canonical correlations when squared multiple correlations are used as the initial estimates of communality. When these initial communality estimates are used, solving Raos determinantal equation (at the first stage) permits expressing several matrices as functions of factors that differ only in the scale of their columns; these matrices include the correlation matrix with units in the diagonal, the correlation matrix with squared multiple correlations as communality estimates, Guttmans image covariance matrix, and Guttmans anti-image covariance matrix. Further, the factor scores associated with these factors can be shown to be either identical or simply related by a scale change. Implications for practice are discussed, and a computing scheme which would lead to an exhaustive analysis of the data with several optional outputs is outlined.

A Factor Analytic Interpretation Strategy.

THE purpose of this paper is to illustrate the use of a strategy for determining the common factors in a set of data. C. Harris (1967) suggested using several different computing algorithms for the initial solution, obtaining derived solutions, both orthogonal and oblique, comparing the results, and regarding as the important substantive findings those factors that are robust with respect to method. This paper illustrates a way of comparing the results. The factor results used for this illustration of a factor analytic interpretation strategy are the reanalyses, by seven different solutions, of the data from nine of the Guilford studies as reported by C. Harris (1967). The initial component and factor methods used

Some Recent Developments in Factor Analysis

THE major purpose of this paper is to describe some recent developments in factor analysis that seem to me to be worthy of attention on the part of those who see factor analysis as one useful tool in their research. In order to describe these developments in a meaningful context, I shall cover some old ground that may be quite familiar to you; for this I ask your indulgence. My descriptions will be rather general, without the attention to technical details that would be required in order to satisfy many of my colleagues. I do not know whether this will disappoint or please you. This paper is divided into five sections: Component Analysis, Image Analysis, Scale-Free Factor Analysis, A Note on the Rotation Problem, and Extensions to Multiple-Modes. The first three topics were selected to emphasize distinctions among models of factor analysis, in the hope that such distinctions will help provide research workers with better bases for choosing among possible models. It will become clear that not all the methods or procedures that you may associate with factor analysis are treated in these first three topics; by omitting them I am implying that although they may be interesting or important methods in terms of the history of factor analysis, they appear to have no substantial place today. The excellent book by Harry Harman called Modern Factor Analysis (Harman, 1960) is in my view an extremely important historical document, but much of it is quite out of date as a guide to present practice even though its copyright date is 1960.

Some Problems in the Description of Change

THE purpose of this discussion is to explore several possibilities for treating change over time in a set of attributes. For the most part, we shall assume that the &dquo;same&dquo; set of persons has been mea~ured, in some fashion, on two separate occasions, and that this change over time is to be described. Such change may be called &dquo;growth&dquo; or &dquo;development&dquo; if one wishes; we shall use the fairly neutral term. If only one attribute has been measured on each of the two occasions, we shall call this a univariate problem. If more than one attribute has been measured on the two occasions, we shall call the problem multivariate. For the univariate problem, crude gain, standard score gain, or residual gain might be examined. Problems such as the reliability of difference scores arise here. In addition to the work of DuBois (1957), papers by Lord (1956, 1958) and McNemar (195

A Factor Analysis of Selected Senate Roll Calls, 80Th Congress

) are relevant to the univariate case. Our interest is in the multivariate case. We begin by making

Relationships between two systems of factor analysis

Since it is not possible for us to list the full record of every member of Congress on all issues, we have selected what we consider to be some of the most significant roll calls during the first session of the 8oth Congress. The selection was made on the basis of issues in which the League has been active, with the exception of a few issues of general interest on which the organization took no position.!

Effects of an Auditory Perceptual Remediation Program on Reading Performance

Considering only population values, it is shown that the complete set of factors of a correlation matrix with units in the diagonal cells may be transformed into the factors derived by factoring these correlations with communalities in the diagonal cells. When the correlations are regarded as observed values, the common factors derived as a transformation of the complete set of factors of the correlation matrix with units in the diagonal cells satisfy Lawleys requirement for a maximum likelihood solution and are a first approximation to Raos canonical factors.

Separation of data as a principle in factor analysis

Four different commercially produced programs were employed to determine their relative effectiveness with a population of school children who had failed an auditory perception test. Replications of the study at two different grade levels consistently support the hypothesis that the program to remediate auditory perceptual development successfully achieved its purpose, but these results also consistently failed to support the hypothesis that increasing competency in auditory perception had a significantly different impact on the childs typical reading and spelling behavior than more traditional types of reading instruction.

Relationship of Selected Variables to Ability to Handle a Bowling Ball

Two systems of factor analysis—factoring correlations with units in the diagonal cells and factoring correlations with communalities in the diagonal cells—are considered in relation to the commonly used statistical procedure of separating a set of data (scores) into two or more parts. It is shown that both systems of factor analysis imply the separation of the observed data into two orthogonal parts. The matrices used to achieve the separation differ for the two systems of factor analysis.