Using a general analysis of variance algorithm for covariance designs

Procedures are described which enable researchers to implement balanced covariance designs of from one to four independent variables. Use is made of three subroutines from IBM’s Scientific Subroutine Package which implement a general decomposition algorithm for balanced designs. FORTRAN instructions, illustrating the main calling program, are given.     

Analysis of covariance structures

A general method is presented for estimating variance components when the experimental design has one random way of classification and a possibly unbalanced fixed classification. The procedure operates on a sample covariance matrix in which the fixed classes play the role of variables and the random classes correspond to observations. Cases are considered which assume (i) homogeneous and (ii) nonhomogeneous error variance, and (iii) arbitrary scale factors in the measurements and homogeneous error variance. The results include maximum-likelihood estimations of the variance components and scale factors, likelihood-ratio tests of the goodness-of-fit of the model assumed for the design, and large-sample variances and covariances of the estimates. Applications to mental test data are presented. In these applications the subjects constitute the random dimension of the design, and a classification of the mental tests according to objective features of format or content constitute the fixed dimensions.Preparation of this paper has been supported in part by NSF Grant GB-939 and U. S. P. H. Grant GM-1286-01. Computer time was donated by the Computation Center, University of Chicago.Now at the University of Chicago.Now at the University of Georgia.     

Comparison of variance and covariance patterns in parallel and serial theories of timing.

Parallel and serial timing processes are analyzed for their account of the dynamics of intertrial responding in the peak procedure. A strictly serial model, such as the behavioral theory of timing (Killeen & Fetterman, 1988), does not fit the dynamic correlation pattern in the location and duration of the middle high-rate responding portion of peak trials. In contrast, the parallel scalar expectancy theory model, with a sample for memory and threshold, does fit this pattern. A modification of the serial model is presented that also accommodates the within-trial covariance pattern. The modification, which is formally equivalent to a model for human tapping (Wing & Kristofferson, 1973), entails the addition of concurrent processes operating in parallel with serial timing.     

A criterion for significant common factor variance

Up to the present only empirical methods have been available for determining the number of factors to be extracted from a matrix of correlations. The problem has been confused by the implicit attitude that a matrix of intercorrelations between psychological variables has a rank which is determinable. A table of residuals always contains error variance and common factor variance. The extraction of successive factors increases the proportion of error variance remaining to common factor variance remaining, and a point is reached where the extraction of more dimensions would contain so much error variance that the common factor variance would be overshadowed. The critical value for this point is determined by probability theory and does not take into account the size of the residuals. Interpretation of the criterion is discussed.     

A note on the factor analysis of partial covariance matrices

A relationship is given between the joint common factor structure of two sets of variables, and the factor structure of the partial covariance matrix of one of the sets with the other partialled out.     

Sensitivity analysis in factor analysis: Difference between using covariance and correlation matrices

Influence curves of some parameters under various methods of factor analysis have been given in the literature. These influence curves depend on the influence curves for either the covariance or the correlation matrix used in the analysis. The differences between the influence curves based on the covariance and the correlation matrices are derived in this paper. Simple formulas for the differences of the influence curves, based on the two matrices, for the unique variance matrix, factor loadings and some other parameter are obtained under scale-invariant estimation methods, though the influence curves themselves are in complex forms.The authors are most grateful to the referees, the Associate Editor, the Editor and Raymond Lam for helpful suggestions for improving the clarity of the paper.     

Factorial design and covariance in the study of individual educational development

This is the report of the application of the principles of factorial design to an investigation of individual educational development. The specific type of factorial design formulated was a 2 × 3 × 3 × 3 arrangement, that is, the effect of sex, grade location, scholastic standing, and individual order, singly and in all possible combinations was studied in relation to educational development as measured by theIowa Tests of Educational Development. An application of the covariance method was introduced which resulted in increased precision of this type of experimental design by significantly reducing experimental error. The two concomitant measures used to increase the sensitiveness of the experiment were initial status of individual development and mental age. Without these statistical controls all main effects and two first-order interactions would have been accepted as significant. With their use only sex (doubtful), scholastic standing, and individual order demonstrated significant effects. The chief beauty of the analysis of variance and covariance as an integral part of a self-contained experiment is demonstrated in the complete single analysis of the data. The statistical utilization of the experimental results has also been developed for purposes of estimation and prediction. The mathematical statistician is being continuously required to develop and analyze experimental designs of increasing complexity since the introduction of the analysis of variance and covariance. The mathematical formulation and solution of the problem of this investigation is carried out. The methods illustrated and explained in this study, and modifications and extensions of them are capable of very wide application. The general principles can be used to various degrees and in a number of ways.For the research grant to finance this study, grateful acknowledgment is given to the Graduate School, the University of Minnesota.     

Analysis of variance for correlated observations

Two different linear models are presented for the four-dimensional classification system in which correlations exist between certain pairs of observations. Except for the assumption of correlated observations, classical assumptions associated with classification systems are made. The models considered are modifications of those which underlie the split-plot design and the split-split-plot design. In the first model the correlations between observations of the levels of one dimension are all set equal to. In the second model the observations of the levels of one dimension are assumed correlated to degree ₁, whereas the observations of a second dimension are correlated to degree ₂. Analyses for the two models and tests of hypotheses for various parameters are indicated.     

Analysis of variance for repeated measures designs with word materials as a nested random or fixed factor

This article is about analysis of data obtained in repeated measures designs in psycholinguistics and related disciplines with items (words) nested within treatment (= type of words). Statistics tested in a series of computer simulations are: F1, F2, F1 & F2, F', min F', plus two decision procedures, the one suggested by Forster and Dickinson (1976) and one suggested by the authors of this article. The most common test statistic, F1 & F2, turns out to be wrong, but all alternative statistics suggested in the literature have problems too. The two decision procedures perform much better, especially the new one, because it systematically takes into account the subject by treatment interaction and the degree of word variability.     

Selecting a linear mixed model for longitudinal data: repeated measures analysis of variance, covariance pattern model, and growth curve approaches

With increasing popularity, growth curve modeling is more and more often considered as the 1st choice for analyzing longitudinal data. Although the growth curve approach is often a good choice, other modeling strategies may more directly answer questions of interest. It is common to see researchers fit growth curve models without considering alterative modeling strategies. In this article we compare 3 approaches for analyzing longitudinal data: repeated measures analysis of variance, covariance pattern models, and growth curve models. As all are members of the general linear mixed model family, they represent somewhat different assumptions about the way individuals change. These assumptions result in different patterns of covariation among the residuals around the fixed effects. In this article, we first indicate the kinds of data that are appropriately modeled by each and use real data examples to demonstrate possible problems associated with the blanket selection of the growth curve model. We then present a simulation that indicates the utility of Akaike information criterion and Bayesian information criterion in the selection of a proper residual covariance structure. The results cast doubt on the popular practice of automatically using growth curve modeling for longitudinal data without comparing the fit of different models. Finally, we provide some practical advice for assessing mean changes in the presence of correlated data.