The Reduced Monotonic Regression Method

Abstract

Medical researchers often desire to categorize patients into monotonic response groups based on the relationship between continuous variables. Isotonic regression fits consist of level sets of increasing value, for which the estimated response is constant. However, the number of level sets obtained is often large, preventing simple description. This article introduces two new nonparametric methods called reduced isotonic regression and reduced monotonic regression, the latter being a two-sided extension of the former for use when the direction of the trend is unknown. Using a backward elimination algorithm, the new procedures reduce the number of level sets by combining those whose values do not differ greatly. For the statistical relations examined here, the reduced monotonic method averaged at most 30% of the number of level sets obtained for isotonic regression. The method is illustrated with an example that examines the relationship between risk factors for survival among children with leukemia. In simulation studies, the reduced monotonic method fits the data as closely as alternative methods that combine isotonicity and smoothing, while improving greatly on isotonic regression. The method is also related to changepoint models of normally distributed sequences.

Key Words:

• Changepoint analysis
• Isotonic regression
• Monotonicity
• Nonparametric regression
• Nonparametric regression