Subjective probability of disjunctive hypotheses: local-weight models for decomposition of evidential support

When the probability of a single member of a set of mutually exclusive and exhaustive possibilities is judged, its alternatives are evaluated as a composite "residual" hypothesis. Support theory (Rottenstreich & Tversky, 1997; Tversky & Koehler, 1994) implies that the process of packing alternatives together in the residual reduces the perceived evidential support for the set of alternatives and consequently inflates the judged probability of the focal hypothesis. Previous work has investigated the global weights that determine the extent to which the overall evidential support for the alternatives is discounted by this packing operation (Koehler, Brenner, & Tversky, 1997). In the present investigation, we analyze this issue in greater detail, examining the local weights that measure the specific contribution of each component hypothesis included implicitly in the residual. We describe a procedure for estimating local weights and introduce a set of plausible properties that impose systematic ordinal relationships among local weights. Results from four experiments testing these properties are reported, and a local-weight model is developed that accounts for nearly all of the variance in the probability judgments in these empirical tests. Local weights appear to be sensitive both to the individual component with which they are associated and to the residual hypothesis in which the component resides.     

Effects of conditionality on INDSCAL and ALSCAL weights

The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.     

Determining Predictor Weights in Military Selection: An Application of Dominance Analysis

This article illustrates the use of dominance analysis (DA) to identify the most appropriate set of weights for combining predictor data. We used meta-analytic information on cognitive ability, structured interviews, conscientiousness, and emotional stability to determine relative weights in predicting entry-level job performance and turnover. For job performance, the optimal predictor weights were 51% (cognitive ability), 38% (structured interview), 9% conscientiousness, and 2% (emotional stability). The weights differed considerably for turnover: 48% (structured interview), 27% (emotional stability), 13% (conscientiousness), and 11% (cognitive ability). Using a simulated data set, we showed how the rank-order of applicants can change based on DA and traditional regression analysis. The results suggest that DA has wide applicability for military selection.     

Coefficients alpha and reliabilities of unrotated and rotated components

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.Comments by Henk A.L. Kiers and by anonymous referees are gratefully acknowledged.     

A note on the determination of configuration and weights for a class of individual scaling models

Consider the typical problem in individual scaling, namely finding a common configuration and weights for each individual from the given interpoint distances or scalar products. Within the STRAIN framework it is shown that the problem of determining weights for a given configuration can be posed as a standard quadratic programming problem. A set of necessary conditions for an optimal configuration to satisfy are given. A closed form expression for the configuration is obtained for the one dimensional case and an approach is given for the two dimensional case.     

The simultaneous prediction of any number of criteria by the use of a unique set of weights

When several criteria are available, it is ordinarily necessary (1) to select one of them asthe criterion, (2) to use several and thus arrive at several different sets of weights, or (3) to combine them into a single measure. A formula is derived for the determination of a unique set of weights. The use of these weights will produce the highest possible average coefficient of correlation between the various criteria and two (or more) weighted independent variables. If desired, the criteria may be assigned any predetermined weights. The weights then derived for the independent variables are such that the weighted average of the correlation coefficients between the various criteria and the independent variable composite will be a maximum. In the use of these formulas, no assumptions are necessary regarding the interrelationships existing among the criteria and it is not necessary to compute the intercorrelations among the criteria. A numerical example is included.     

An Alternative Interpretation of the Linearly Weighted Kappa Coefficients for Ordinal Data

When two (or more) observers are independently categorizing a set of observations, Cohen’s kappa has become the most notable measure of interobserver agreement. When the categories are ordinal, a weighted form of kappa becomes desirable. The two most popular weighting schemes are the quadratic weights and linear weights. Quadratic weights have been justified by the fact that the corresponding weighted kappa is asymptotically equivalent to an intraclass correlation coefficient. This paper deals with linear weights and shows that the corresponding weighted kappa is equivalent to the unweighted kappa when cumulative probabilities are substituted for probabilities. A numerical example is provided.     

The role of general ability in prediction

Several data sets were analyzed to compare the prediction possible from a uniform general factor score with that produced by a separately tailored set of regression weights when those weights are applied to a new cross-validation sample. Double cross-validation designs were used. When regression weights were derived from large groups, they provided an increase of 10–15% in the prediction over that possible from a uniform general factor measure. However, with smaller samples, of the size typical of industrial personnel research, the uniform general factor score was clearly superior.     

Fungible Weights in Multiple Regression

Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungiblebecause they yield identical SSE (sum of squared errors) and R ² values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression. The author wishes to thank Drs. Robyn Dawes, William Grove, Markus Keel, Leslie Yonce, Joe Rausch, the editor, and three anonymous reviewers for helpful comments on earlier versions of this article.     

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.     

On the reliability of a weighted composite

A general formula for the reliability of a weighted composite has been derived by which that reliability can be estimated from a knowledge of the weights whatever their source, reliabilities, dispersions, and intercorrelations of the components. The Spearman-Brown formula has been shown to be a special case of the general statement. The effect of the internal consistency or intercorrelation of the components has been investigated and the conditions defining the set of weights yielding maximum reliability shown to be that the weight of a component is proportional to the sum of its intercorrelations with the remaining components and inversely proportional to its error variance.     

The Central Role of General Ability in Prediction

Three data sets are analyzed that permit double cross-validation of a test battery against criterion variables in a number of educational programs or jobs. The validity of the first general factor score is compared with that obtained from the set of cross-validated regression weights, and is found to account, respectively, for approximately 85, 90 and 120 percent as much criterion variance as the cross-validated regression weights. Small further contributions appear to be made by a mechanical/technical and by a psychomotor factor. However, for a wide range of criterion variables the major role in validity appears to be played by a common general factor.     

Is the weber fraction a function of physical or perceived input?

It is well known that a given physical input (e.g. intensity of light or sound, length or weight of an object) does not always give rise to the same sensation. For example, arrow heads affect the perceived length of lines (Muller-Lyer illusion) and size affects the apparent weight of an object (size-weight illusion). It is generally assumed that the differential threshold is a simple function of the physical intensity of the stimulus. We may however ask whether the differential threshold is affected by illusions.

To try to answer this question we estimate differential thresholds under a pair of conditions in which the relevant input is the same but appears different. Using the size-weight illusion, we have found that the differential threshold for weight is greater for a set of small weights than for a set of larger and apparently lighter weights.     

Density versus feature weights as predictors of visual identifications: comment on Appelman and Mayzner

Appleman and Mayzner's application of the distance-density model to confusion data is compared with Keren and Baggen's application of the feature-matching model. In both applications, the distinctive features of two stimuli are predictors of the number of confusion errors. However, the models differ in that the feature-matching model assigns weights to the features and assumes that the shared feature weights also affect the probability of confusion. In contrast, the distance-density model assumes that the number of confusions between two stimuli is affected by the number of stimuli in the entire stimulus set that are similar to the two stimuli (density). The two models are compared in the context of a set of digit identification data.     

Methodology to select solutions for multiobjective optimization problems: Weighted stress function method

The weighted stress function method is proposed here as a new way of identifying the best solution from a set of nondominated solutions according to the decision maker's preferences, expressed in terms of weights. The method was tested using several benchmark problems from the literature, and the results obtained were compared with those of other methods, namely, the reference point evolutionary multiobjective optimization (EMO), the weighted Tchebycheff metric, and a goal programming method. The weighted stress function method can be seen to exhibit a more direct correspondence between the weights set by the decision maker and the final solutions obtained than the other methods tested.     

The discriminant function and its use in psychology

R. A. Fisher's method of determining a set of weights for the linear combination of the scores in a battery of tests is explained and illustrated. The relationship of this method to combination by multiple regression methods is shown.     

Simplified regression patterns for classification

A transformation of a set of partial regression equations to a potentially simple form is shown to be permissible for personnel classification purposes when all individuals are assigned to jobs. Simplification is achieved by maximizing the number of near zero partial regression weights.     

A parameter analysis method for the weight‐set to satisfy preference orders of alternatives in additive multi‐criteria value models

In the context of the additive multi‐criteria value model, this paper investigates how the set of criteria weights (weight‐set hereafter) can be determined according to the preference orders of alternatives given by the decision maker. A construction method is proposed for the weight‐set for different intervals of β, where β is a differential amount of value between the preference information on two alternatives. The results of this paper are important for sensitivity analysis in multi‐criteria decision making (MCDM) problems and multi‐criteria group decision analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

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 .