Notes and Communications “A DECISION THEORY EXAMPLE IN FOOTBALL”: A COMMENT*

Janssen and Daniel analyzed the choice between a one- or a two-point conversion for a particular game situation in college football. Their decision criteria was maximum expected utility based on a von Neumann-Morgenstern utility function defined over the games outcomes. An alternative approach based on a stochastic dominance criterion is presented that does not rely on knowledge of the relative importance of tying vs. winning; rather, it relies on a notion of consistency in the sequential problem.     

Note on “many-sided shadow prices”

It has been previously shown, with the aid of a simple example, that linear programmes may have two different non-negative marginal values at the optimum if the solution is degenerate. It is demonstrated in this note that other marginal values exist which may take negative values.     

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis [Decision Sciences (2001), 32(4), 557–573]. These authors present an integer‐programming model for the single‐item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer‐programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question.     

A Note on “An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem”

In a recent paper, McKnew, Saydam, and Coleman [3] presented a novel zero-one integer programming formulation of the multilevel dynamic, deterministic lot-sizing problem in assembly systems. They stated that “the relaxed linear programming solution to this formulation will always be integer’ [3, p. 280] since the constraint matrix is totally unimodular. In this note, we point out that the constraint matrix is not totally unimodular and therefore the authors’claim that a linear relaxation of the zero-one integer formulation always yields an integer solution is not true.     

TOWARD UNDERSTANDING THE CONTENTS OF THE “BLACK BOX” FOR PREDICTING COMPLEX DECISION-MAKING OUTCOMES

The purpose of this research is to show the usefulness of three relatively simple nonlinear classification techniques for policy-capturing research where linear models have typically been used. This study uses 480 cases to assess the decision-making process used by 24 experienced national bank examiners in classifying commercial loans as acceptable or questionable. The results from multiple discriminant analysis (a linear technique) are compared to those of chi-squared automatic interaction detector analysis (a search technique), log-linear analysis, and logit analysis. Results show that while the four techniques are equally accurate in predicting loan classification, chi-squared automatic interaction detector analysis (CHAID) and log-linear analysis enable the researcher to analyze the decision-making structure and examine the “human” variable within the decision-making process. Consequently, if the sole purpose of research is to predict the decision maker's decisions, then any one of the four techniques turns out to be equally useful. If, however, the purpose is to analyze the decision-making process as well as to predict decisions, then CHAID or log-linear techniques are more useful than linear model techniques.     

ON USING LINEAR PROGRAMMING IN DISCRIMINANT PROBLEMS

This commentary evaluates the usefulness of the Freed and Glover [6] linear programming approach to the discriminant problem, relates linear programming to other parametric and nonparametric approaches, and evaluates the linear programming approach.     

FOUR APPROACHES TO THE CLASSIFICATION PROBLEM IN DISCRIMINANT ANALYSIS: AN EXPERIMENTAL STUDY*

Four discriminant models were compared in a simulation study: Fisher's linear discriminant function [14], Smith's quadratic discriminant function [34], the logistic discriminant model, and a model based on linear programming [17]. The study was conducted to estimate expected rates of misclassification for these four procedures when observations were sampled from a variety of normal and nonnormal distributions. In contrast to previous research, data were taken from four types of Kurtotic population distributions. The results indicate the four discriminant procedures are robust toward data from many types of distributions. The misclassification rates for both the logistic discriminant model and the formulation based on linear programming consistently decreased as the kurtosis in the data increased. The decreases, however, were of small magnitude. None of these procedures yielded statistically significant lower rates of misclassification under nonnormality. The quadratic discriminant function produced significantly lower error rates when the variances across groups were heterogeneous.     

SPECIFICATION AND ESTIMATION OF LOGICALLY CONSISTENT LINEAR MODELS

The implications of constrained dependent and independent variables for model parameters are examined. In the context of linear model systems, it is shown that polyhedral constraints on the dependent variables will hold over the domain of the independent variables when a set of polyhedral constraints is satisfied by the model parameters. This result may be used in parameter estimation, in which case all predicted values of the dependent variables are consistent with constraints on the actual values. Also, the implicit constraints that define the set of parameters for many commonly used linear stochastic models with an error term yield values of the dependent variables consistent with the explicit constraints. Models possessing these properties are termed “logically consistent”.     

Applications and Implementation

We propose an alternative solution to the discriminant problem, one that requires little more than a minimum familiarity with linear programming. The approach shows promise for eliminating the complexities of conventional statistical approaches without sacrificing the essential power of existing methods.     

Note on Second-Order Polynomial Regression Models

Polynomial regression models have applications in the social sciences and in business research. Unfortunately, such models have a high degree of multicollinearity that creates problems with the statistical assessment of the model. In fact, the collinearity may be so severe that it could lead to an incorrect conclusion that some of the terms in the model are not statistically significant and should therefore be omitted from the model. This note provides a simple transformation to achieve orthogonality in polynomial models between the linear and quadratic terms, thereby eliminating the collinearity problem. It also shows that the same procedure does not achieve orthogonality for higher-order terms. An example data set is analyzed to show the benefits of such a procedure.     

Revisiting ISO 14000 Diffusion: A New “Look” at the Drivers of Certification

This replication study of the drivers of ISO 14000 certifications extends the work of Corbett and Kirsch (2001) and provides a different and simpler predictive model of the factors contributing to the growth of ISO 14000 certifications. The main finding is that at national level ISO 14000 certification densities can be explained by two factors: the installed base of ISO 9000 certificates and the number of environmental treaties signed and ratified. The first factor, considering the common elements of the two standards, points to conditions of infrastructural convenience, while the second highlights the importance of political‐economic considerations. The study uses a new set of tools (graphical displays, distribution‐free computer intensive methods) that are better suited for exploratory research when discontinuities and existence of subgroups in the data set may make findings from a linear regression suspect.     

Necessary and Sufficient Conditions of Unacceptable Solutions in LP Discriminant Analysis*

Characterization of unacceptable solutions for linear programming (LP) discriminant models have been discussed in the literature and the results presented so far are not satisfactory. This paper establishes necessary and sufficient conditions of unacceptable solutions for a number of LP models, addresses the practical implications of these conditions, and discusses the relationship between unacceptable solutions and multiple solutions.     

EVALUATING ALTERNATIVE LINEAR PROGRAMMING MODELS TO SOLVE THE TWO-GROUP DISCRIMINANT PROBLEM

The two-group discriminant problem has applications in many areas, for example, differentiating between good credit risks and poor ones, between promising new firms and those likely to fail, or between patients with strong prospects for recovery and those highly at risk. To expand our tools for dealing with such problems, we propose a class of nonpara-metric discriminant procedures based on linear programming (LP). Although these procedures have attracted considerable attention recently, only a limited number of computational studies have examined the relative merits of alternative formulations. In this paper we provide a detailed study of three contrasting formulations for the two-group problem. The experimental design provides a variety of test conditions involving both normal and nonnormal populations. Our results establish the LP model which seeks to minimize the sum of deviations beyond the two-group boundary as a promising alternative to more conventional linear discriminant techniques.     

Theoretical Distributions of Optima for Univariate Discrimination of Random Data*

Optimal linear discriminant models maximize percentage accuracy for dichotomous classifications, but are rarely used because a theoretical framework that allows one to make valid statements about the statistical significance of the outcomes of such analyses does not exist. This paper describes an analytic solution for the theoretical distribution of optimal values for univariate optimal linear discriminant analysis, under the assumption that the data are random and continuous. We also present the theoretical distribution for sample sizes up to N= 30. The discovery of a statistical framework for evaluating the performance of optimal discriminant models should greatly increase their use by scientists in all disciplines.     

Solving the Classification Problem in Discriminant Analysis Via Linear and Nonlinear Programming Methods*

This paper demonstrates the feasibility of applying nonlinear programming methods to solve the classification problem in discriminant analysis. The application represents a useful extension of previously proposed linear programming-based solutions for discriminant analysis. The analysis of data obtained by conducting a Monte Carlo simulation experiment shows that these new procedures are promising. Future research that should promote application of the proposed methods for solving classification problems in a business decision-making environment is discussed.     

A Comparison of Linear Programming and Parametric Approaches to the Two-Group Discriminant Problem*

Recent simulation-based studies of linear programming models for discriminant analysis have used the Fisher linear discriminant function as the benchmark for parametric methods. This article reports experimental evidence which suggests that, while some linear programming models may match or even exceed the Fisher approach in classification accuracy, none of the fifteen models tested is as accurate on normally distributed data as the Smith quadratic discriminant function. At the minimum, further testing is warranted with an emphasis on data sets that arise from significantly non-Gaussian populations.     

Unacceptable Solutions and the Hybrid Discriminant Model

A new family of linear programming (LP) discriminant models, the hybrid discriminant model, was recently presented with claims that it avoids all the problems of its predecessor LP models. However, this note shows that it does not avoid unacceptable solutions.     

Characterization of Unacceptable Solutions in LP Discriminant Analysis*

Many linear programming models have been proposed for performing discriminant analysis. Partial characterizations for unacceptable solutions have been presented and new models proposed to circumvent these problems. In this paper those conditions leading to unacceptable solutions for all two-group models are characterized.     

EFFECT-SIZE ESTIMATION: A COMMENTARY ON WOLF AND BASSLER

Wolf and Bassler [10] have presented a general linear model approach to the estimation of a variance-accounted-for measure of effect magnitude. This commentary questions the usefulness of such a measure, citing its inconsistency with contrast estimates, problems with its interpretation in nonorthogonal designs, and its susceptibility to extraneous influence unrelated to observed differences among group means. In addition, it questions whether the term “importance” should be applied to such a measure.     

A Multivariate Methodology for Simultaneously Capturing and Clustering Judgment Policies

This paper presents a new linear model methodology for clustering judges with homogeneous decision policies and differentiating dimensions which distinguish judgment policies. This linear policy capturing model based on canonical correlation analysis is compared to the standard model based on regression analysis and hierarchical agglomerative clustering. Potential advantages of the new methodology include simultaneous instead of sequential consideration of information in the dependent and independent variable sets, decreased interpretational difficulty in the presence of multicollinearity and/or suppressor/moderator variables, and a more clearly defined solution structure allowing assessment of a judge's relationship to all of the derived, ideal policy types. An application to capturing policies of information systems recruiters responsible for hiring entry-level personnel is used to compare and contrast the two techniques.