Note on functional measurement and data analysis

The guiding idea of functional measurement is that measurement theory and substantive theory form an organic unity. Psychological scales are inherent in the statement of quantitative psychological laws, and these laws themselves form the base and frame for psychological measurement. Valid scales thus depend on empirically valid laws. But establishing empirical validity of any law requires appropriate data analysis. Several statistical problems are discussed with respect to simple algebraic laws. To illustrate the necessity for proper tests of goodness of fit for algebraic models, five sets of experimental data are reanalyzed. In each case, the factorial plot and the analysis of variance showed that the data were nonadditive. Nevertheless, an additive model was fit to the data. The correlations between the data and the predictions from the additive model were extremely high, ranging from .964 to .9997. The corresponding observed-predicted scatterplots also gave little sign of the deviations from additivity. These correlation-scatterplot analyses conceal and obscure what the factorial plot and the analysis of variance reveal and make clear. Other topics discussed are accepting and rejecting the null hypothesis, the use of nonmetric smoothing for parameter estimation, and problems of stimulus-response-model generality. An extension of functional measurement is suggested for a practicable error theory for nonmetric analysis.     

Note on the factor structure of the Bern Sex-Role Inventory

The scores of BSRI, a questionnaire measuring psychological androgyny, i.e. the extent to which a person possess traits that are traditionally considered feminine and traits that are traditionally considered masculine, were subjected to a factor analysis. The BSRI was administered to 100 women and 100 men, between 20 and 60 years of age. The analysis gave two factors for women, interpreted as a Femininity factor and a Masculinity factor, and three factors for men, interpreted as a Femininity factor and two Masculinity factors. A shorter version of the BSRI is also suggested. These results support the notion that femininity and masculinity are best considered two independent dimensions.     

Note on the computation of biserial correlations in item analysis

A rapid method is described for machine computation of biserial correlations in item analysis with several criteria. This method has been found to yield biserial correlations from punched IBM cards at the rate of about 41 per hour.     

Note on use of extreme criterion groups in item discrimination analysis

The purpose of this note is to reconsider the Kelley-Cureton definition of optimal extreme groups for estimating item-criterion correlations. Optimal tail per cents are derived, using the criterion of minimum sampling variance of the tetrachoric correlation coefficient, and the findings are related to earlier work of Mosteller. It is shown that upper and lower 27 per cent groups yield the most precise estimate of the tetrachoric coefficient only when the population correlation is close to zero. When the population value is .4, extreme 20 per cent groups provide estimates with the smallest sampling error variance. It is further shown, however, that 27 per cent extremes yield highly efficient estimates. Thus no change is recommended in traditional item analysis procedures.     

Note on Number

This article presents a high-resolution copy of C. G. Jung’s Note on Number, transcribed and translated into English, with some notes by Roy Freeman.     

Alpha factor analysis

A distinction is made between statistical inference and psychometric inference in factor analysis. After reviewing Rao's canonical factor analysis (CFA), a fundamental statistical method of factoring, a new method of factor analysis based upon the psychometric concept of generalizability is described. This new procedure (alpha factor analysis, AFA) determines factors which have maximum generalizability in the Kuder-Richardson, or alpha, sense. The two methods, CFA and AFA, each have the important property of giving the same factors regardless of the units of measurement of the observable variables. In determining factors, the principal distinction between the two methods is that CFA operates in the metric of the unique parts of the observable variables while AFA operates in the metric of the common (communality) parts.On the other hand, the two methods are substantially different as to how they establish the number of factors. CFA answers this crucial question with a statistical test of significance while AFA retains only those alpha factors with positive generalizability. This difference is discussed at some length. A brief outline of a computer program for AFA is described and an example of the application of AFA is given.The first version of this paper was prepared while the senior author was a U. S. Public Health Service Fellow at the Center for Advanced Study in the Behavioral Sciences and while the junior author was Director of Research of the Palo Alto Public Schools.     

Longitudinal factor analysis

A model is presented for factor analysing scores on a set of psychological tests administered as both pre- and postmeasures in a study of change. The model assumes that the same factors underlie the tests on each occasion, but that factor scores as well as factor loadings may change between occasions. Factors are defined to be orthogonal between as well as within occasions. A two-stage least squares procedure for fitting the model is described, and generally provides a unique rotation solution for the factors on each occasion.We thank M. W. Browne, R. P. McDonald, R. Pruzek, and the Managing Editor for their constructive comments.     

More on EM for ML factor analysis

We address several issues that are raised by Bentler and Tanaka's [1983] discussion of Rubin and Thayer [1982]. Our conclusions are: standard methods do not completely monitor the possible existence of multiple local maxima; summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate, even if there exists a unique local maximum; EM and LISREL can be viewed as complementary, albeit not entirely adequate, tools for factor analysis.This work was partially supported by the Program Statistics Research Project at Educational Testing Service.     

Some mathematical notes on three-mode factor analysis

The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.     

A Note on Harmony

In the proof-theoretic semantics approach to meaning, harmony, requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony, requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony: imposes the existence of certain transformations of derivations, known as reduction and expansion. We propose a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules.     

Note on guessing probabilities

A hypothesis of stationary response probabilities (SRP), or that subjects equate the unconditional probability of each response to its frequency of occurrence in the task, is shown to be equivalent to the proposal of Atkinson et al. (1965) that guessing may occur only among unlearned responses. This hypothesis is contrasted with the usual assumption of stationary guessing probabilities (SGP). It is shown that assuming SGP when SRP is true results in a biased estimate of learning which first increases and then decreases with the actual degree of learning, and is a decreasing function of list length. This can lead to improper inferences, including erroneous rejection of an all-or-none model and spurious evidence of increasing difficulty in long lists.