Locating the Extrema of Fungible Regression Weights

In a multiple regression analysis with three or more predictors, every set of alternate weights belongs to an infinite class of “fungible weights” (Waller, Psychometrica, in press) that yields identical SSE (sum of squared errors) and R ² values. When the R ² using the alternate weights is a fixed value, fungible weights (a _i ) that yield the maximum or minimum cosine with an OLS weight vector (b) are called “fungible extrema.” We describe two methods for locating fungible extrema and we report R code (R Development Core Team, 2007) for one of the methods. We then describe a new approach for populating a class of fungible weights that is derived from the geometry of alternate regression weights. Finally, we illustrate how fungible weights can be profitably used to gauge parameter sensitivity in linear models by locating the fungible extrema of a regression model of executive compensation (Horton & Guerard, Commun. Stat. Simul. Comput. 14:441–448, 1985).     

Fungible Correlation Matrices: A Method for Generating Nonsingular,Singular, and Improper Correlation Matrices for Monte Carlo Research

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.     

Investigating the Performance of Alternate Regression Weights by Studying All Possible Criteria in Regression Models with a Fixed Set of Predictors

We describe methods for assessing all possible criteria (i.e., dependent variables) and subsets of criteria for regression models with a fixed set of predictors, x (where x is an n×1 vector of independent variables). Our methods build upon the geometry of regression coefficients (hereafter called regression weights) in n-dimensional space. For a full-rank predictor correlation matrix, R _xx, of order n, and for regression models with constant R ² (coefficient of determination), the OLS weight vectors for all possible criteria terminate on the surface of an n-dimensional ellipsoid. The population performance of alternate regression weights—such as equal weights, correlation weights, or rounded weights—can be modeled as a function of the Cartesian coordinates of the ellipsoid. These geometrical notions can be easily extended to assess the sampling performance of alternate regression weights in models with either fixed or random predictors and for models with any value of R ². To illustrate these ideas, we describe algorithms and R (R Development Core Team, 2009) code for: (1) generating points that are uniformly distributed on the surface of an n-dimensional ellipsoid, (2) populating the set of regression (weight) vectors that define an elliptical arc in ℝ ⁿ , and (3) populating the set of regression vectors that have constant cosine with a target vector in ℝ ⁿ . Each algorithm is illustrated with real data. The examples demonstrate the usefulness of studying all possible criteria when evaluating alternate regression weights in regression models with a fixed set of predictors.     

Use of the Jackknife Statistic to Evaluate Result Replicability

Result replicability has often been confused with significance testing because of a misinformed view that statistical significance evaluates result importance and result replicability. Several alternatives, of which the jackknife statistic is one, provide researchers with the tools for estimating result replicability. In the present study, an available data set from A. L. Edwards (1985, p. 57) is used to illustrate the use of the jackknife statistic to assess the replicability of multiple regression results in a concrete fashion. The jackknifed coefficients are computed to assess the stability of beta weights and the multiple R ² value. Confidence intervals and t statistics are also calculated to facilitate the interpretation of these jackknifed coefficients.     

Flexible Approaches to Computing Mediated Effects in Generalized Linear Models: Generalized Estimating Equations and Bootstrapping

In behavioral research, interest is often in examining the degree to which the effect of an independent variable X on an outcome Y is mediated by an intermediary or mediator variable M. This article illustrates how generalized estimating equations (GEE) modeling can be used to estimate the indirect or mediated effect, defined as the amount by which the regression coefficient of X on Y changes after adjusting for M. Advantages of this method are: (a) it applies to the class of generalized linear models, including linear, logistic, and Poisson regression as special cases; (b) it allows multiple independent variables and mediators in the same model; and (c) asymptotically valid standard errors and confidence intervals are obtained using standard software. This methodology is compared with the bootstrap, another general methodology that can be applied to the same broad class of models, and is evaluated using simulation in both linear and logistic regression scenarios. The methods are utilized to examine the degree to which the effect of low birthweight status on internalizing symptoms at age 20 is mediated through IQ at age 8.     

Correlation Weights in Multiple Regression

A general theory on the use of correlation weights in linear prediction has yet to be proposed. In this paper we take initial steps in developing such a theory by describing the conditions under which correlation weights perform well in population regression models. Using OLS weights as a comparison, we define cases in which the two weighting systems yield maximally correlated composites and when they yield minimally similar weights. We then derive the least squares weights (for any set of predictors) that yield the largest drop in R ² (the coefficient of determination) when switching to correlation weights. Our findings suggest that two characteristics of a model/data combination are especially important in determining the effectiveness of correlation weights: (1) the condition number of the predictor correlation matrix, R _xx , and (2) the orientation of the correlation weights to the latent vectors of R _xx .     

Selecting K of N alternatives

In some decision contexts (e.g. graduate admissions) the task to select a subset of k distinct alternatives from among the N > k > 1 options available. For moderate values of N and k, the number of possible subsets of size k becomes quite large. Assuming alternatives are evaluated using consistent, though not explicit weights, selecting k, rather than 1, empirically excludes a substantial proportion (more than 80%) of the subsets; unfortunately to the unaided decision maker it is not obvious which subsets should be excluded. The tendency of subjects to select excludable subsets can be called an ‘infeasibility’ bias. The experimental evidence surprisingly shows no ‘infeasibility’ bias, as subjects overwhelmingly chose non-excluded selections.     

Comparing the squared multiple correlation coefficients of non‐nested models: An examination of confidence intervals and hypothesis testing

The performance of the asymptotic method for comparing the squared multiple correlations of non‐nested models was investigated. Specifically, the increase in a given regression model's R² when one predictor is added was compared to the increase in the same model's R² when another predictor is added. This comparison can be used to determine predictor importance and is the basis for procedures such as Dominance Analysis. Results indicate that the asymptotic procedure provides the expected coverage rates for sample sizes of 200 or more, but in many cases much higher sample sizes are required to achieve adequate power. Guidelines and computations are provided for the determination of adequate sample sizes for hypothesis testing.     

Statistical guidelines for studies of the theory of reasoned action and the theory of planned behaviour

Abstract

We identify potential problems in the statistical analysis of social cognition model data, with special emphasis on the theories of reasoned action (TRA) and planned behaviour (TPB). Some statistical guidelines are presented for empirical studies of the TRA and the TPB based upon multiple linear regression and structural equation modelling (SEM). If the model is tested using multiple regression, the assumptions of this technique must be considered and variables transformed if necessary. Adjusted R² (not R²) should be used as a measure of explained variance and semipartial correlations are useful in assessing each component's unique contribution to explained variance. R² is not an indicator of model adequacy and residuals should be examined. Expectancy-value variables that are the product of expectancy and value measures represent the interaction term in a multiple regression and should not be used. SEM approaches make explicit the assumptions of unidimensionality of constructs in the TRA/TPB, assumptions that might usefully be challenged by competing models with multidimensional constructs. Finally, statistical power and sample size should be considered for both approaches. Inattention to any of these aspects of analysis threatens the validity of TRA/TPB research.     

Analysis of Interactions and Nonlinear Effects with Missing Data: A Factored Regression Modeling Approach Using Maximum Likelihood Estimation

Abstract

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.     

Predicted differences and differences between predictions

WhenK tests are given toN individuals, and for each individual there are two criterion measures, then (1) the multiple regression weight to be applied to the standard score for each test to predict the criterion-difference score equals the difference of the weights for predicting each criterion separately; (2) the difference between the predicted scores equals the predicted difference (each test being assigned the appropriate multiple regression weight); (3) the square of the multiple correlation between predicted and actual criterion-difference scores equals the sum of squares of the multiple correlations of the battery with each criterion less the product of these correlations and the correlation between predicted scores all divided by twice the quantity one minus the criterion intercorrelation; and (4) the variance of errors of estimating the criterion-difference score equals the sum of the variances of errors of estimating each criterion score minus twice the criterion intercorrelation, plus twice the correlation between predicted scores multiplied by the product of the square root of one minus the variance of errors of estimating one criterion and the corresponding square root for the second criterion.The author wishes to express his appreciation for the suggestions and guidance given by Dr. Harold Gulliksen in the preparation of this article. He also wishes to acknowledge the helpful comments of Dr. Paul Horst and Dr. Ledyard Tucker on certain phases of the development.     

Multiple regression analysis of twin and sibling data

DeFries and Fulker (1985) employed a multiple regression analysis of twin data in which a cotwin's score is predicted from the proband's score, the coefficient of relationship (R = 1.0 for monozygotic twins and 0.5 for dizygotic twins), and their interaction to yield direct estimates of heritability (h²) and the proportion of variance due to common or shared environmental influences (c²). The purpose of the report is to extend this model to allow for the analysis of data from more than two familial relationships simultaneously. Data from identical twins, fraternal twins, and siblings pairs, in which one member of each pair was selected based on low reading performance, were analyzed. To analyze the data simultaneously, additional coefficients were added to the regression equation in order to differentiate shared environmental influences of twins and siblings. When the model was applied to the combined cognitive data, significant estimates for heritability were obtained in three of the six tests analyzed. The results also indicated a lack of justification for the separate twin and sibling environmental parameters, and a discussion of more parsimonious models is included.     

Dominance Measuring Method Performance under Incomplete Information about Weights

In multi‐attribute utility theory, it is often not easy to elicit precise values for the scaling weights representing the relative importance of criteria. A very widespread approach is to gather incomplete information. A recent approach for dealing with such situations is to use information about each alternative's intensity of dominance, known as dominance measuring methods. Different dominance measuring methods have been proposed, and simulation studies have been carried out to compare these methods with each other and with other approaches but only when ordinal information about weights is available. In this paper, we use Monte Carlo simulation techniques to analyse the performance of and adapt such methods to deal with weight intervals, weights fitting independent normal probability distributions or weights represented by fuzzy numbers. Moreover, dominance measuring method performance is also compared with a widely used methodology dealing with incomplete information on weights, the stochastic multicriteria acceptability analysis (SMAA). SMAA is based on exploring the weight space to describe the evaluations that would make each alternative the preferred one. Copyright © 2012 John Wiley & Sons, Ltd.     

Cohen’s Linearly Weighted Kappa is a Weighted Average of 2×2 Kappas

An agreement table with n∈ℕ_≥3 ordered categories can be collapsed into n−1 distinct 2×2 tables by combining adjacent categories. Vanbelle and Albert (Stat. Methodol. 6:157–163, 2009c) showed that the components of Cohen’s weighted kappa with linear weights can be obtained from these n−1 collapsed 2×2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2×2 tables, where the weights are the denominators of the 2×2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.     

A distance-based variety of nonlinear multivariate data analysis, including weights for objects and variables

In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects in the analysis. In this paper two methods are presented for including weights in distance-based nonlinear multivariate data analysis. In the first method, weights are assigned to the objects while the second method is concerned with differential weighting of groups of variables. When each analysis variable defines a group the latter method becomes a variable weighting method. For objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combined and have several important applications. For example, they make it possible to perform efficient analyses of large data sets, to use the distance-based variety of nonlinear multivariate data analysis as an addition to loglinear analysis of multiway contingency tables, and to do stability studies of the solutions by applying the bootstrap on the objects or the variables in the analysis. These and other applications are discussed, and an efficient algorithm is proposed to minimize the corresponding loss function.This study is funded by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the PIONEER project Subject Oriented Multivariate Analysis to the third author.     

Estimating the weights for multiple attributes in a composite criterion using pairwise judgments

This paper presents a new methodology for estimating the weights or saliences of subcriteria (attributes) in a composite criterion measure. The inputs to the estimation procedure consist of (i) a set of stimuli or objects with each stimulus defined by its subcriteria profile (set of attribute values) and (ii) the set of paired comparison dominance (e.g., preference) judgments on the stimuli made by a single judge (expert) in terms of the global criterion. A criterion of fit is developed and its optimization via linear programming is illustrated with an example. The procedure is generalized to estimate a common set of weights when the pairwise judgments on the stimuli are made by more than one judge. The procedure is computationally efficient and has been applied in developing a composite criterion of managerial success yielding high concurrent validity.This methodology can also be used to perform ordinal multiple regression—i.e., multiple regression with an ordinally scaled dependent variable and a set of intervally scaled predictor variables. The approach is further extended to internal analysis (unfolding) using the vector model of preference and to the additive model of conjoint measurement.     

Media experiences and associations with mental health among the bereaved of the MH17‐disaster: A latent profile analysis

Research has shown that the amount of media exposure is associated with post‐event mental health problems. Whether bereaved individuals have negative experiences with media reports and whether they are associated with post‐event mental health is unclear. This study evaluated these experiences and associations following the MH 17‐disaster. How media reports were experienced (nine topics, modified MAS ), depression symptoms (QIDS ‐SR ), functional problems (WSAS ) and event‐related coping‐self‐efficacy (CSE ) were assessed about one year post‐disaster (May‐August 2015) among Dutch bereaved (N  = 152). A substantial minority reported negative experiences such as reports made me angry (30%) and made me sad (48%). Latent profile analysis with symptoms, problems and coping self‐efficacy as indicators, identified four classes of post‐disaster mental health: a Well‐functioning^{(class 1)}, 35.1%; a Mild‐problems^{(class 2)}, 30.4%; a Sub‐clinical^{(class 3)}, 27.0%; and a Clinical^{(class 4)}, 7.4%. Differences in symptoms, problems and coping self‐efficacy levels between classes were large according to Cohen's d s. Multivariate logistic regression (MLR ) showed that the Clinical^{(class 4)} compared to the Well‐functioning^{(class 1)}, more often that felt that reports strongly “embarrassed me,” “made me feel sad,” “filled me with fear” and “served as a magnifying glass.” Future research should assess opportunities and effects of limiting media consumption.     

A Dollar Is a Dollar Is a Dollar,or Is It? Insights From Children's Reasoning About “Dirty Money”

Money can take many forms—a coin or a bill, a payment for an automobile or a prize for an award, a piece from the 1989 series or the 2019 series, and so on—but despite this, money is designed to represent an amount and only that. Thus, a dollar is a dollar, in the sense that money is fungible. But when adults ordinarily think about money, they think about it in terms of its source, and in particular, its moral source (e.g., dirty money). Here we investigate the development of the belief that money carries traces of its moral history. We study children ages 5–6 and 8–9, who are sensitive to both object history and morality, and thus possess the component pieces needed to think that a dollar may not be like any other. Across three principal studies (and three additional studies in Appendix S1 ; N = 327; 219 five- and six-year-olds; 108 eight- and nine-year-olds), we find that children are less likely to want money with negative moral history, a pattern that was stronger and more consistent among 8- and 9-year-olds than 5- and 6-year-olds. These findings highlight pressing directions for future research that could help shed light on the mechanisms that contribute to the belief that money carries traces of its moral history.     

To what extent can communalities reduce rank?

The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrixR and not at, say,R ^–1. What may be best forR may be worst forR ^–1, and conversely, yetR ^–1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to whichR can be reduced, when modified by communalities, is indeed false.This research was facilitated by a noncommitted grant-in-aid to the writer from the Ford Foundation.