On rotating to smooth functions

Tucker has outlined an application of principal components analysis to a set of learning curves, for the purpose of identifying meaningful dimensions of individual differences in learning tasks. Since the principal components are defined in terms of a statistical criterion (maximum variance accounted for) rather than a substantive one, it is typically desirable to rotate the components to a more interpretable orientation. Simple structure is not a particularly appealing consideration for such a rotation; it is more reasonable to believe that any meaningful factor should form a (locally) smooth curve when the component loadings are plotted against trial number. Accordingly, this paper develops a procedure for transforming an arbitrary set of component reference curves to a new set which are mutually orthogonal and, subject to orthogonality, are as smooth as possible in a well defined (least squares) sense. Potential applications to learning data, electrophysiological responses, and growth data are indicated.Portions of this research were supported by the National Research Council of Canada, Grant A8615 to the second author. We thank Jagdeth Sheth for supplying his raw data.     

Text comprehension in aphasia

The semantic and pragmatic strategies in the comprehension of spoken texts are investigated in four subgroups of aphasic patients and in normal and brain-damaged controls. Short texts of similar linguistic structure were read to the subjects, who were required to choose the picture, from a multiple choice set of five, that was appropriate to the story. Besides a picture showing the main event of the story, one picture depicted the literal sense of a metaphorical comment, and the others misrepresented semantic functions expressed in the text. With respect to these types of responses, both aphasics and controls exhibited the same pattern of reaction. It is concluded that the redundancy of texts makes up for the difficulties aphasics have in comprehending isolated words and sentences.     

The sources of developmental differences in children's incidental processing during discrimination trials

Crane and Ross reported that second graders learned more than sixth graders about attributes made relevant after solution of a discrimination task. Here two experiments are reported that enlighten the sources of this developmental difference. Both make use of an experimental technique whereby children verbalize their hypotheses during solution of a discrimination problem. The results indicate that ten-year-olds do not learn about incidental attributes that they tested while irrelevant in the pre-solution period, but that five-year-olds and seven-year-olds do. Children of all three ages process incidental information about attributes that they did not sample pre-solution. With some qualification, the incidentally processed information is retained throughout a five-minute delay interval. The results bear on developmental trends in the distribution of attention and on theoretical accounts of incidental learning in discrimination tasks.     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Comments on “the measurement of factorial indeterminacy”

Guttman's index of indeterminacy (2² – 1) measures the potential amount of uncertainty in picking the right alternative interpretation for a factor. When alternative solutions for a factor are equally likely to be correct, then the squared multiple correlation ² for predicting the factor from the observed variables is the average correlation _AB between independently selected alternative solutionsA andB, while var ( _AB )=(1 – ²)²/s, wheres is the dimensionality of the space in which unpredicted components of alternative solutions are to be found. When alternative solutions for the factor are not equally likely to be chosen, ² is the lower bound for E( _AB ); however, E( _AB ) need not be a modal value in the distribution of _AB . Guttman's index and E( _AB ) measure different aspects of the same indeterminacy problem.     

A look,by simulation,at the validity of some asymptotic distribution results for rotated loadings

In the last few years, a number of asymptotic results for the distribution of unrotated and rotated factor loadings have been given. This paper investigates the validity of some of these results based on simulation techniques. In particular, it looks at principal component extraction and quartimax rotation on a problem with 13 variables. The indication is that the asymptotic results are quite good.     

Effect of sample presentation time on long-delay matching in the pigeon

The effect of sample stimulus presentation time on long-delay matching in highly practiced pigeons was investigated. The birds were found capable of above chance matching performance at a delay of 60 sec provided the sample stimulus was presented for 4 sec or longer. Matching accuracy increased as a negatively accelerated function of sample stimulus presentation time and decreased as a negatively accelerated function of time since the termination of the sample. The rate of forgetting was found to be independent of sample stimulus presentation time. The data were inconsistent with a temporal discrimination interpretation of the effect of presentation time on delayed matching. The data were interpreted as supporting a simple trace strength and decay model of pigeon delayed matching.     

The sex by race interaction in cognitive measures

Data consisting of 79 cognitive measures from Project TALENT for 100,000 white and black high school students have been analyzed for possible race by sex interactions. Control variables included geographical area, grade in high school, and socio-economic class. Race by sex interactions are largest when socioeconomic class is controlled and are highly related to the size of the main effect of sex as well. White boys and girls differ more than black boys and girls whether the overall sex difference favors males or females. Sex by grade and, to a lesser extent, sex by area interactions were also moderately large and consistent with the size of the main effect of sex. Since there are no consistent relationships with other main effects, and since the measures producing sex differences also tend to produce the interactions, it is concluded that sex differences and the interactions with sex share the same causes. Differences between these results and those of Jensen are also discussed.