Louis E. Guttman (1916–1987)

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45742/1/11336_2005_Article_BF02294129.pd

Comment on a note on a base-free measure of change

change scores,

Obtaining squared multiple correlations from a correlation matrix which may be singular

A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures. © 1972 Psychometric Society

Estimators of the squared cross-validity coefficient: A Monte Carlo investigation

A monte carlo experiment was used to evaluate four procedures for estimating the population squared cross-validity of a sample least squares regression equation. Four levels of population squared multiple correlation (Rp2) and three levels of number of predictors (n) were factorially crossed to produce 12 population covariance matrices. Random samples at four levels of sample size (N) were drawn from each population. The levels of N, n, and RP2 were carefully selected to ensure relevance of simulation results for much applied research. The least squares regression equation from each sample was applied in its respective population to obtain the actual population squared cross-validity (Rcv2). Estimates of Rcv2 were computed using three formula estimators and the double cross-validation procedure. The results of the experiment demonstrate that two estimators which have previously been advocated in the literature were negatively biased and exhibited poor accuracy. The negative bias for these two estimators increased as Rp2 decreased and as the ratio of N to n decreased. As a consequence, their biases were most evident in small samples where cross-validation is imperative. In contrast, the third estimator was quite accurate and virtually unbiased within the scope of this simulation. This third estimator is recommended for applied settings which are adequately approximated by the correlation model