A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.We would like to acknowledge the helpful comments of Ruth Silver. We also wish to thank the referees for helping to clarify the paper. This work was partly carried out with research funds provided by the Economic and Social Research Council (U.K.).     

Infantile handling and the extinction of responses acquired under food reward

A comparison was made of the runway behaviour of rats which had been handled from Days 1-21 and their non-handled litter mates. The training began on Day 70 after the animals were habituated to a restricted food schedule for 10 days. The subjects were given six trials each day in the runway and were rewarded with a 0.045 g. Noyes pellet. After 10 days of rewarded training trials, subjects were given 6 extinction trials a day for 10 days. Results showed that handled rats ran faster than non-handled rats during acquisition and during the first 3 days of extinction. The extinction data suggested that the relationship between emotionality and effects of frustrative non-reward should be re-evaluated.     

Three common factor models for groups of variables

For the case of two or more groups of variables (batteries), generalizations on three common factor models, the Equal Residual Variances model, the Image model of Jöreskog, and the Canonical Factor Analysis model, are described and illustrated with empirical examples.The author is indebted to Dr. E. J. Burr for his valuable criticisms.     

A note on the derivation of the general latent class model

A general formulation of the latent structure principle is suggested, from which it is possible to derive Lazarsfeld's accounting equations in their most general form. The basic equations of Gibson's latent profile model can thence be derived in a single step.     

A comparison of four methods of constructing factor scores

Four least-squares methods for constructing factor scores have been described in the literature. The formal properties of these scores are developed, and they are compared in terms of four generally desirable properties of constructed factor scores. In particular, it is shown that two of the methods yield scores that are conditionally unbiassed, and univocal in the sense of Guilford and Michael, though not orthogonal, while one of the other methods yields orthogonal scores.It is shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis.The general problem of choosing between the methods is discussed, on the basis of the theoretical relations obtained.EDITOR'S NOTE: The reader will quickly discover that this article develops several of the generalizations given in the second part of the preceding article, On Factors and Factor Scores. Independent development of the same generalizations is, of course, not a new phenomenon. Because the Presidential Address automatically is accepted for publication and given space in the December issue, it was decided that the only fair thing to do was to print this article in the same issue.     

Numerical methods for polynomial models in nonlinear factor analysis

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.This work was carried out while the author was a Visiting Research Fellow at the Educational Testing Service, Princeton, N. J. It made use of computer facilities at Princeton University that are supported in part by National Science Foundation Grant NSF-Gp579.     

The effect of additional variables on factor indeterminacy in models with a single common factor

Determinate solutions for the indeterminate common factor ofp variables satisfying the single common factor model are not unique. Therefore an infinite sequence of additional variables that conform jointly with the originalp variables to the original single common factor model does not determine a unique solution for the indeterminate factor of thep variables (although the solution is unique for the factor of the infinite sequence). Other infinite sequences may be found to determine different solutions for the factor of the originalp variables. The paper discusses a number of theorems about the effects of additional variables on factor indeterminacy in a model with a single common factor and draws conclusions from them for factor theory in general.