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Researchers are often interested in comparing predictors, a practice commonly done via informal comparisons of standardized regression slopes. However, formal interval-based approaches offer advantages over informal comparison. Specifically, this article examines a delta-method-based confidence interval for the difference between two standardized regression coefficients, building upon previous work on confidence intervals for single coefficients. Using Monte Carlo simulation studies, the proposed approach is evaluated at finite sample sizes with respect to coverage rate, interval width, Type I error rate, and statistical power under a variety of conditions, and is shown to outperform an alternative approach that uses the standard covariance matrix found in regression textbooks. Additional simulations evaluate current software implementations, small sample performance, and multiple comparison procedures for simultaneously testing multiple differences of interest. Guidance on sample size planning for narrow confidence intervals, an R function to conduct the proposed method, and two empirical demonstrations are provided. The goal is to offer researchers a different tool in their toolbox for when comparisons among standardized coefficients are desired, as a supplement to, rather than a replacement for, other potentially useful analyses. [ABSTRACT FROM AUTHOR]

An important goal for psychological science is developing methods to characterize relationships between variables. Customary approaches use structural equation models to connect latent factors to a number of observed measurements, or test causal hypotheses between observed variables. More recently, regularized partial correlation networks have been proposed as an alternative approach for characterizing relationships among variables through off-diagonal elements in the precision matrix. While the graphical Lasso (glasso) has emerged as the default network estimation method, it was optimized in fields outside of psychology with very different needs, such as high dimensional data where the number of variables (p) exceeds the number of observations (n). In this article, we describe the glasso method in the context of the fields where it was developed, and then we demonstrate that the advantages of regularization diminish in settings where psychological networks are often fitted ( p≪n ). We first show that improved properties of the precision matrix, such as eigenvalue estimation, and predictive accuracy with cross-validation are not always appreciable. We then introduce nonregularized methods based on multiple regression and a nonparametric bootstrap strategy, after which we characterize performance with extensive simulations. Our results demonstrate that the nonregularized methods can be used to reduce the false-positive rate, compared to glasso, and they appear to provide consistent performance across sparsity levels, sample composition (p/n), and partial correlation size. We end by reviewing recent findings in the statistics literature that suggest alternative methods often have superior performance than glasso, as well as suggesting areas for future research in psychology. The nonregularized methods have been implemented in the R package GGMnonreg.

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. b 1 = b 2 = b 3 ). 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. [ABSTRACT FROM AUTHOR]

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a multivariate normal distribution, which is also the default in many statistical software packages. This distribution will in general be misspecified if predictors with missing data have nonlinear effects (e.g., x2) or are included in interaction terms (e.g., x·z). In the present article, we introduce a factored regression modeling approach for estimating regression models with missing data that is based on maximum likelihood estimation. In this approach, the model likelihood is factorized into a part that is due to the model of interest and a part that is due to the model for the incomplete predictors. In three simulation studies, we showed that the factored regression modeling approach produced valid estimates of interaction and nonlinear effects in regression models with missing values on categorical or continuous predictor variables under a broad range of conditions. We developed the R package mdmb, which facilitates a user-friendly application of the factored regression modeling approach, and present a real-data example that illustrates the flexibility of the software. [ABSTRACT FROM AUTHOR]

Whenever multiple regression is applied to a multiply imputed data set, several methods for combining significance tests for and the change in across imputed data sets may be used: the combination rules by Rubin, the Fisher z-test for by Harel, and F-tests for the change in by Chaurasia and Harel. For pooling itself, Harel proposed a method based on a Fisher z transformation. In the current article, it is argued that the pooled based on the Fisher z transformation, the Fisher z-test for , and the F-test for the change in have some theoretical flaws. An argument is made for using Rubin's method for pooling significance tests for instead, and alternative procedures for pooling are proposed: simple averaging and a pooled constructed from the pooled significance test by Rubin. Simulations show that the Fisher z-test and Chaurasia and Harel's F-tests generally give inflated type-I error rates, whereas the type-I error rates of Rubin's method are correct. Of the methods for pooling the point estimates of no method clearly performs best, but it is argued that the average of 's across imputed data set is preferred. [ABSTRACT FROM AUTHOR]

In multiple regression researchers often follow up significant tests of the interaction between continuous predictors X and Z with tests of the simple slope of Y on X at different sample-estimated values of the moderator Z (e.g., ±1 SD from the mean of Z). We show analytically that when X and Z are randomly sampled from the population, the variance expression of the simple slope at sample-estimated values of Z differs from the traditional variance expression obtained when the values of X and Z are fixed. A simulation study using randomly sampled predictors compared four approaches: (a) the Aiken and West (1991) test of simple slopes at fixed population values of Z, (b) the Aiken and West test at sample-estimated values of Z, (c) a 95% percentile bootstrap confidence interval approach, and (d) a fully Bayesian approach with diffuse priors. The results showed that approach (b) led to inflated Type 1 error rates and 95% confidence intervals with inadequate coverage rates, whereas other approaches maintained acceptable Type 1 error rates and adequate coverage of confidence intervals. Approach (c) had asymmetric rejection rates at small sample sizes. We used an empirical data set to illustrate these approaches. [ABSTRACT FROM AUTHOR]

For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials. [ABSTRACT FROM PUBLISHER]

Whenever multiple regression is applied to a multiply imputed data set, several methods for combining significance tests for R 2 and the change in R 2 across imputed data sets may be used: the combination rules by Rubin, the Fisher z -test for R 2 by Harel, and F -tests for the change in R 2 by Chaurasia and Harel. For pooling R 2 itself, Harel proposed a method based on a Fisher z transformation. In the current article, it is argued that the pooled R 2 based on the Fisher z transformation, the Fisher z -test for R 2 , and the F -test for the change in R 2 have some theoretical flaws. An argument is made for using Rubin's method for pooling significance tests for R 2 instead, and alternative procedures for pooling R 2 are proposed: simple averaging and a pooled R 2 constructed from the pooled significance test by Rubin. Simulations show that the Fisher z -test and Chaurasia and Harel's F -tests generally give inflated type-I error rates, whereas the type-I error rates of Rubin's method are correct. Of the methods for pooling the point estimates of R 2 no method clearly performs best, but it is argued that the average of R 2 's across imputed data set is preferred.

In multiple regression researchers often follow up significant tests of the interaction between continuous predictors X and Z with tests of the simple slope of Y on X at different sample-estimated values of the moderator Z (e.g., ±1 SD from the mean of Z). We show analytically that when X and Z are randomly sampled from the population, the variance expression of the simple slope at sample-estimated values of Z differs from the traditional variance expression obtained when the values of X and Z are fixed. A simulation study using randomly sampled predictors compared four approaches: (a) the Aiken and West ( 1991 ) test of simple slopes at fixed population values of Z, (b) the Aiken and West test at sample-estimated values of Z, (c) a 95% percentile bootstrap confidence interval approach, and (d) a fully Bayesian approach with diffuse priors. The results showed that approach (b) led to inflated Type 1 error rates and 95% confidence intervals with inadequate coverage rates, whereas other approaches maintained acceptable Type 1 error rates and adequate coverage of confidence intervals. Approach (c) had asymmetric rejection rates at small sample sizes. We used an empirical data set to illustrate these approaches.