Note on Selection from a Multivariate Normal Population

Abstract

The problem of statistical “selection” is concerned with the alteration induced in a frequency distribution in several variables by an alteration of the parameters in a subsection of the distribution. It may be illustrated by a simple trivariate case, as follows:From a population characterised by variables x, y, z, correlated and normally distributed, with means 0, 0, 0, variances and product variances r12σ1σ2, r13σ1σ3, r23σ2σ3, a sub-population is extracted by selection in x alone, in such a way that after selection x is still normally distributed, but with mean h and variance s2 . It is required to determine the new values, in the selected population, of the means and variances of y and z, and of the product variances.

A Meta-Analysis of the Validity of the Pharmacy College Admission Test (PCAT) and Grade Predictors of Pharmacy Student Performance

“…Ideally, future research would examine the effect of multivariate restriction of range on PCAT score validities. 42,43 Although much of the variability across studies could simply be attributed to sampling error, some of the remaining variability is likely due to differential restriction of range. Ironically, those schools that are more selective and make more use of PCAT scores are more likely to obtain smaller validity estimates.…”

Replication of Zeidner, Johnson, and Colleagues' Method for Estimating Army Aptitudes Area (AA) Composites

“…After correcting for criterion unreliability, ASVAB-SQT validities were corrected for multivariate range restriction by MOS using formulas originally developed by Aitken (1934) and Lawley (1943), and described by Gulliksen (1950), and Bimbaum, Paulson, and Andrews (1950). Large-scale simulations demonstrate that, when applied appropriately, the AitkenLawley corrections consistently produce estimates that closely approximate validities for the relevant reference population (Sackett & Yang, 2000).…”

The Performance of a Higher Order Invariance Idiographic Filter in Confirmatory Factor Analysis of Cross Sectional and Longitudinal Data

“…In the field of psychology, notions of invariance grew out of selection theory from Pearson (1903); Aitken (1935) and Lawley (1944). The Pearson-Aitken selection formula (Pearson, 1903;Aitken, 1935), later generalized to multiple factors (Thurstone, 1931(Thurstone, , 1947Thomson and Lederunn, 1939;Ahmavaara, 1954;Meredith, 1964a), was concerned with how selection affects model parameters and established that selection can operate in such a way that factor loadings can be transformed to be identical, or invariant.…”

“…The Pearson-Aitken selection formula (Pearson, 1903;Aitken, 1935), later generalized to multiple factors (Thurstone, 1931(Thurstone, , 1947Thomson and Lederunn, 1939;Ahmavaara, 1954;Meredith, 1964a), was concerned with how selection affects model parameters and established that selection can operate in such a way that factor loadings can be transformed to be identical, or invariant. Confirmatory factor analysis (CFA Jöreskog, 1969;Jöreskog, 1970) soon emerged as a superior method of investigating invariance between samples and groups.…”

“…The concept of invariance emerged out of selection theory in factor analysis in the early 20th century (Pearson, 1903;Aitken, 1935;Thurstone, 1931Thurstone, , 1947Thomson and Lederunn, 1939;Ahmavaara, 1954;Meredith, 1964a), and continues to be an important aspect of factor analysis to this day. In multiple group analysis, whether comparing across time or varying cultures or groups based on some other variable, measurement invariance has been considered necessary in order to establish factorial invariance (Cheung and Rensvold, 1999;Schmitt and Kuljanin, 2008;Vandenberg and Lance, 2000;Widaman and Reise, 1997).…”