Sample Correlation Matrices Research Articles

Note: Inter-Rater Reliability of Scree Test and Mean Square Ratio Test of Number of Factors

The inter-rater reliability of Cattell's scree and Linn's mean square ratio test of the number of factors was studied. Sample correlation matrices were generated from a population correlation matrix by means of standard Monte Carlo procedures such that there were 100 samples based on each of 3 sample sizes. Each matrix was factored and the scree test and the mean square ratio test were each applied by five raters. For both tests, the inter-rater reliabilities were very low. These results suggest that inexperienced factor analysts should be wary of these tests of the number of factors.

A Note on Some Sampling Characteristics of the Measure of Sampling Adequacy (MSA).

The distributional characteristics of the measure of sampling adequacy (MSA) were investigated in sample correlation matrices generated from multivariate normal populations with covariance matrix equal to the identity. Systematic variation of sample size and number of variables resulted in minimal fluctuation of the overall MSA from .50. Implications for the use of the index were discussed.

A Computer Program to Generate Sample Correlation and Covariance Matrices

Given a population-covariance matrix, this program can generate any number of sample-covariance (or correlation) matrices, based on any sample sizes. If no population matrix is input, the program generates random correlation matrices.

A NOTE ON HORN'S TEST FOR THE NUMBER OF FACTORS IN FACTOR ANALYSIS

To examine the statistical sampling distributions of Horn's test for the number of factors, 100 sample correlation matrices at each of three sample sizes were generated from a structured population correlation matrix and an unstructured (identity) population correlation matrix of the same order. Each of the 600 sample matrices was submitted to principal components, SMC, image, and Harris rescaled farings. Two variations of the Horn test (which differed in the number of sets of unstructured Eigen-values averaged for the comparison) were applied to each factoring of each structured sample matrix. It was found that the 2 variations were not significantly different but that the type of factoring did have a significant effect. It was also found that the test tended to fail to give an estimate of the number of factors in many samples following the SMC and image factorings.

Analysis of Growth and Dose Response Curves

SUMMARY A method of analysis of 'growth curves' that utilizes the technique of analysis of covariance is developed and illustrated. The method yields results identical to those obtained by weighting inversely by the sample covariance matrix, but has the additional feature of allowing flexibility in weighting by choosing subsets of covariates that have special properties. Thus, exploitation of Rao's observation that selected subsets of the covariates may in some cases yield 'better' estimates than the complete set is easily implemented. Examples are included to illustrate the calculations and how to determine when it will be advantageous to use all the vectors in the error space as covariates (weight inversely by the sample covariance matrix) or to use only a subset, and how to choose the 'best' subset from examination of sample covariance or correlation matrices.

The influence of communality, factor strength, and loading size on the sampling characteristics of factor loadings.

A Monte Carlo approach is employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings. A number of sample correlation matrices were generated from a specified population, factored, and transformed to a least-squares fit to the population values. Influences of factor strength, communality and loading size are discussed in relation to the statistics summarizing the results of the above procedures. Influences producing biased estimators of the population values are also discussed.

On Testing Correlation Matrices

Procedures and illustrative examples are given to test a null hypothesis specifying the population correlation matrix and the homogeneity of several independent sample correlation matrices.

Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix

A method for generating sample and population score matrices and sample correlation matrices from a given population correlation matrix is developed. An example giving the desired matrices for a population Guttman simplex correlation matrix is presented.