Showing papers on "Ordinal regression published in 1976"

Linear Programming Computational Procedures for Ordinal Regression

Abstract: The ordinal regression problem is an extension to the standard multiple regression problem in terms of assuming only ordinal properties for the dependent variable (rank order of preferred brands in a product class, academic ranks for students in a class, etc.) while retaining the interval scale assumption for independent (or predictor) variables. The linear programming formulation for obtaining the regression weights for ordinal regression, developed in an earlier paper, is outlined and computational improvements and alternatives which utilize the special structure of this linear program are developed and compared for their computational efficiency and storage requirements. A procedure which solves the dual of the original linear programming formulation by the dual simplex method with upper bounded variables, in addition to utilizing the special structure of the constraint matrix from the point of view of storage and computation, performs the best in terms of both computational efficiency and storage requirements. Using this special procedure, problems with 100 observations and 4 independent variables take less than 1/2 minute, on an average, on the IBM 360/67. Results also show that the linear programming solution procedure for ordinal regression is valid — the correlation coefficient between “true” and predicted values for the dependent variable was greater than .9 for most of the problems tested.

Analysis of an exponential equation with ordinal variables

Abstract: This paper is concerned with the analysis of the equation xY yZ, where x, y, z are variables ranging over ordinals, and where both sides of the equation are transfinite in value. The method used for this analysis consists in regarding y as a parameter and x as an independent variable, and determining necessary and sufficient conditions to be placed upon x so that the resulting equation in z has a solution. Extensive use is made of normal form, as well as results in ordinal arithmetic by both Bachmann and Sherman. 0. We are interested in determining the ordinal solutions of the threevariable equation xy = yZ in which each side assumes a transfinite value. Our procedure consists in taking y = a as a parameter, and then finding those ordinals /3 for which /8a > and the equation 3a = a Z has an ordinal solution. Since the function z e a Z is normal for any given a > 1, it is obvious that for any given /3, the equation 3a = a Z has at most one solution. This of course would not be the case if we interchanged the roles of x and z. The paper is divided into four sections. This first section is devoted entirely to the introduction of terminology and the statements of a few known results that will be used extensively throughout the remainder of the paper. In the second section we list our results concerning the equation xy = y2, and the last two sections are devoted to the proofs of these results. Lower case Greek letters always denote ordinals. Whilst we do not preclude these from taking finitevalues,we shall generally use small Latin letters "i", "j", . . ., "s", "t" for finite ordinals (numbers), and such a letter will invariably denote a number. The first transfinite ordinal is denoted by "w", and we include 0 among the limit ordinals. For any ordinal a, we put I(a) = max{fw; o 0, there is a unique number n, a unique decreasing n + 1-sequence (ai),n of ordinals, and a unique n + 1-sequence (Pi)i