Computational Statistics & Data Analysis 50 (2006) 446 – 462 www.elsevier.com/locate/csda Local minima in categorical multiple regression Anita J. van der Kooij a , ∗ , Jacqueline J. Meulman a , Willem J. Heiser b a Department of Educational Sciences, DataTheory Group, Leiden University, P.O. Box 9555, 2300 RB Leiden, The Netherlands b Department of Psychology, Psychometrics Group, Leiden University, The Netherlands Received 6 November 2003; received in revised form 27 August 2004; accepted 30 August 2004 Available online 18 September 2004 Abstract CATREG is a program for categorical multiple regression, applying optimal scaling methodology to quantify categorical variables, including the response variable, simultaneously optimizing the multiple regression coefficient. The scaling levels that can be applied are nominal, nonmonotonic spline, ordinal, monotonic spline or numerical. When ordinal or monotonic spline scaling levels are applied, local minima can occur. With ordinal or monotonic spline scaling levels, the transformations are required to be monotonically increasing, but this can also be achieved by reflecting a monotonic decreasing transformation. A monotonic transformation is obtained by restricting a nonmonotonic transformation, but the direction of the monotonic restriction (increasing or decreasing) is undefined, and it will be shown that this is the cause of local minima. Several strategies to obtain the global minimum for the ordinal scaling level will be presented. Also, results of a simulation study to assess the performance of these strategies are given. The simulation study is also used to identify data conditions under which local minima are more likely to occur and are more likely to be severe. It was found that local minima more often occur with low to moderately low R 2 values, with higher number of categories and with higher multicollinearity. © 2004 Elsevier B.V.All rights reserved. Keywords: Alternating least squares; Backfitting; Categorical data; CATREG; Local minima; Linearization; Monotonic; Multiple regression; Nonlinear regression; Nonmonotonic; Optimal scaling; Quantification; Multiple systematic starts; Transformation ∗ Corresponding author. Tel.: +31-71-527-3827; fax: +31-71-527-3865. E-mail address: kooij@fsw.leidenuniv.nl (A.J. van der Kooij). 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.08.009