Proportional Reduction In Error Measures Research Articles

Harmonic, Geometric and Arithmetic Means Type Measures of Proportional Reduction in Error for Ordered Two-Way Contingency Tables

For two-way contingency tables with ordered categories, Yamamoto et al. (2011) and Yamamoto et al. (2012b) considered proportional reduction in error (PRE) measures, which describe the relative decrease in the probability of making an error in predicting the value of one variable when the value of the other variable is known, as opposed to when it is not known. The present paper proposes a new PRE measure for the symmetric situation in which explanatory and response variables are not defined clearly for two-way contingency tables with ordered categories. The measure proposed includes measures of forms of harmonic, geometric and arithmetic means. An example is also given.

Index for Proportional Reduction in Error in Two-way Contingency Tables with Ordinal Categories

For two-way contingency tables with nominal categories in which the explanatory and response variables are not defined clearly, Goodman and Kruskal (1954) considered a proportional reduction in error (PRE) measure, which describes the relative decrease in the probability of making an error in predicting the value of one variable when the value of the other is known, as opposed to when it is not known. The present paper proposes a new PRE measure for two-way contingency tables with ordered categories in which the explanatory and response variables are not defined clearly. The proposed measure lies between 0 and 1. The proposed measure is useful for comparing the degree of PRE in several tables with ordered categories. Examples are given.

Measures on Proportional Reduction in Error by Arithmetic, Geometric and Harmonic Means for Multi-way Contingency Tables

For multi-way contingency tables with nominal categories, this paper proposes three kinds of proportional reduction in error measures, which describe the relative decrease in the probability of making an error in predicting the value of one variable when the values of the other variables are known, as opposed to when they are not known. The measures have forms of arithmetic, geometric and harmonic means. An example is shown.

On measure of proportional reduction in error for nominal-ordinal contingency tables

For two-way contingency tables with a nominal explanatory and an ordinal response variables, the present paper proposes a proportional reduction in error measure, which describes the relative decrease in the probability of making an error in predicting the value of the response variable when the value of the explanatory variable is known, as opposed to when it is not known. The measure depends on the order of listing the response categories. Examples are shown.

Revisiting Predictions of War Duration

We reexamine the fit of Bennett and Stam's 1996 model of war duration, correcting errors in the reported estimates of prediction accuracy.We discuss how to assess fit in the absence of standard or widely accepted measures of fit in duration models. We introduce a proportional reduction in error measure for duration models and report new estimates of model fit from the Bennett and Stam model.The model does significantly reduce prediction error relative to a naïve model.

An asymptotic partial correlation test for the Goodman-Kruskal lambda.

Goodman and Kruskal introduced a measure of predictive association when predicting the category of a variable A from a category of a variable B. This measure, denoted lambda, is the asymmetric proportional reduction in error measure in predicting an individual's A category that can be eliminated by using knowledge of the B classification. It takes values on the unit interval, with a zero value meaning no predictive gain, while a value of unity indicates a perfect predictive association between A and B. A test of H(0): lambda = 0 versus H(1): lambda > 0 is analogous to a test for the significance of the correlation coefficient. A test of the partial lambda coefficient, which is analogous to a test of the partial correlation coefficient, answers the question of whether knowledge of an additional third (or higher) classification or categorical variable results in a significant increase in predicting the variable A. Suich and Turek developed an exact test for the partial lambda coefficient, but only for the situation where the predicted categorical variable A is dichotomous. The present paper completes the previous work by developing an asymptotic test where the predicted category A is any polytomous variable.

Effect of Choice of Measure on the Size of a Racial Disparity

In criminological studies of racial disparities the effect of race on outcomes is assessed after statistical controls for other variables. There is no universally accepted measure of size of disparity after controls in the most common type of disparity study: the study of binary outcomes. Some such studies use a measure that lends itself to interpretation in terms of the proportional reduction in error (PRE). Most use a non-PRE measure of association. This study investigates the effect of choice of measure on conclusions about the size of a disparity after controls. For illustrative purposes the study analyzes data on the awarding of sentence reductions to drug traffickers. After controls, blacks are found to be substantially less likely than whites to receive a reduction according to a type of non-PRE measure: an odds ratio derived from a logit model. But using the same data and model, PRE measures obtained from an ROC analysis indicate that, after controls, there is hardly any difference between the races in their likelihood of sentence reduction. Results illustrate the study's thesis: that the choice of measure can powerfully affect conclusions. The findings' implications—for policymakers and for researchers who conduct racial disparity studies—are discussed.

PRU Measures of Association for Contingency Table Analysis

This paper introduces new predictive measures of association that may be used for contingency table analysis much as the simple, partial, and multiple correlation coefficients are used for regression analysis. The new measures are based on the combination of (1) the logic of the PRE (proportional reduction in error) measures of association, (2) the use of uncertainty (or maximum-likelihood-ϰ2) as indirect measures of errors in prediction, and (3) analysis strategy of log-linear models for contingency tables.

Interpreting Proportional Reduction in Error Measures as Percentage of Variation Explained

Costner (1965) showed that many of the most commonly used measures of association may be conceptually interpreted as indicating proportional reduction in prediction error. But because it is verbally cumbersome, and conceptually complex for some measures, the proportional reduction in error interpretation has not been widely incorporated in research reports where appropriate. Instead, measures of association continue to be interpreted in abstract terms distinguishing between broad levels of strength. This paper demonstrates that all proportional reduction in error measures of association may be alternately interpreted as indicating the percent of variation explained. Because this interpretation is conceptually meaningful in a manner highly relevant for scientific investigation, more convenient to apply in research reports, and already familiar to most social scientists, it is argued that it be standardly applied to all proportional reduction in error measures.

Information-Theoretic Measures and the Analysis of Social Structures

This paper presents a conceptual and methodological framework for the multivariate, nonparametric analysis of models of social structure. Following McFarland (1969), an information-theoretic measure of the openness of a social structure is developed. A Proportional-Reduction-in-Error (PRE) interpretation is presented for this measure and appropriate statistical testing procedures are indicated. This measure is applied to an analysis of the social structure of university teaching and its performance is compared with that of other PRE measures. Finally, the multivariate application of the measure is outlined.