A Multidimensional Model to Facilitate Within Person Comparison of Attributes.

In psychological research and practice, a person's scores on two different traits or abilities are often compared. Such within-person comparisons require that measurements have equal units (EU) and/or equal origins: an assumption rarely validated. We describe a multidimensional SEM/IRT model from the literature and, using principles of conjoint measurement, show that its expected response variables satisfy the axioms of additive conjoint measurement for measurement on a common scale. In an application to Quality of Life data, the EU analysis is used as a pre-processing step to derive a simple structure Quality of Life model with three dimensions expressed in equal units. The results are used to address questions that can only be addressed by scores expressed in equal units. When the EU model fits the data, scores in the corresponding simple structure model will have added validity in that they can address questions that cannot otherwise be addressed. Limitations and the need for further research are discussed.

Linear equality constraints: Reformulations of criterion related profile analysis with extensions to moderated regression for multiple groups.

Criterion-related profile analysis (CPA) is a least squares linear regression technique for identifying a criterion-related pattern (CRP) among predictor variables and for quantifying the variance accounted for by the pattern. A CRP is a pattern, described by a vector of contrast coefficients, such that predictor profiles with higher similarity to the pattern have higher expected criterion scores. A review of applications shows that researchers have extended the analysis to meta-analyses, logit regression, canonical regression, and structural equation modeling. It also reveals a need for better methods of comparing CRPs across populations. While the original method for identifying the CRP tends to underestimate the variance accounted for by pattern only, both the pattern identified by the original method and the pattern identified by the new method proposed here have useful and complementary interpretations. Imposing linear equality constraints on regression coefficients yields a more accurate method of estimating the variance accounted for by pattern only, and this constrained approach leads to moderated regression models for investigating whether the CRP is the same in two or more populations. Finally, we show how the elements in Cronbach and Gleser's (1953) classic profile decomposition are related to the linear regression model and the CPA model. Academic ability tests as predictors of college GPA are used to illustrate the analyses. Implications of the profile pattern models for psychological theory and applied decision-making are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Regression with reduced rank predictor matrices: A model of trade-offs.

A regression model of predictor trade-offs is described. Each regression parameter equals the expected change in Y obtained by trading 1 point from one predictor to a second predictor. The model applies to predictor variables that sum to a constant T for all observations; for example, proportions summing to T = 1.0 or percentages summing to T = 100 for each observation. If predictor variables sum to a constant T for all observations and if a least squares solution exists, the predicted values for the criterion variable Y will be uniquely determined, but there will be an infinite set of linear regression weights and the familiar interpretation of regression weights does not apply. However, the regression weights are determined up to an additive constant and thus differences in regression weights ßv-ßv∗ are uniquely determined, readily estimable, and interpretable. ßv-ßv∗ is the expected increase in Y given a transfer of 1 point from variable v∗ to variable v. The model is applied to multiple-choice test items that have four response categories, one correct and three incorrect. Results indicate that the expected outcome depends, not just on the student's number of correct answers, but also on how the student's incorrect responses are distributed over the three incorrect response types. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Addressing Quantitative and Qualitative Hypotheses Using Regression Models with Equality Restrictions and Predictors Measured in Common Units.

In regression, some or all of the predictors may be measured in common units: e.g. X1 = carbohydrate calories, X2 = protein calories, X3 = fat calories. Such predictors can occur in disciplines as diverse as business, economics, education, medicine, nutrition, psychology, sport science, etc. Predictors in common units can lead to unique quantitative and qualitative hypotheses that can be addressed by imposing equality restrictions on the regression weights (e.g. b1=b2=b3). A simple device, total score substitution, is available for constraining regression coefficients to be equal in a variety of regression applications. Applications to linear, moderated linear, and polynomial models are described, but extensions to generalized linear models and multilevel linear models are also possible. Total score substitution in linear and moderated regression is illustrated using high school coursework and mathematics achievement data. Data, code (R, SPSS, SAS), and output are publicly available.

Identifying criterion-related patterns of predictor scores using multiple regression.

Along with examples involving vocational interests and mathematics achievement, the authors describe a multiple regression based, pattern recognition procedure that can be used to identify a pattern of predictor scores associated with high scores on a criterion variable. This pattern is called the criterion pattern. After the criterion pattern has been identified, a second regression procedure can be used to estimate the proportion of variation attributable to the criterion pattern. Cross-validation can then be used to estimate the variation attributable to a criterion pattern derived from regression weights estimated in another sample. Finally, issues of criterion pattern invariance and interpretation are discussed.