Comparison of Methods for Monotone Nonparametric Multiple Regression

Abstract

When multiple regression is applied to data, in certain cases the researcher might expect the mean response function to be monotone (nondecreasing or nonincreasing). Standard parametric regression produces a smooth function that may be monotone, despite violations to the assumption that occur among individual points. Due to the flexibility of nonparametric regression, there is usually a greater chance that the estimated function will not be monotone somewhere. Nonparametric regression may still be preferred if the researcher is not willing to assume that a certain ‘smooth parametric function’ model is appropriate. A technique proposed by Mukerjee and Stern (Mukerjee, H., Stern, S. (1994). Feasible nonparametric estimation of multiargument monotone functions. Journal of the American Statistical Association 89:77–80) that takes averages of maximums and minimums of subsets of the data can be applied to the initial nonparametric regression estimates to achieve a monotone function. In this article, a generalization of this technique will be shown to be MLE in the uniform distribution case. A new method will be presented, which stems from Mukerjee's technique and capitalizes on least squares. Results of Monte Carlo simulations will be presented that compare these methods along with more traditional methods of isotonic regression. Lastly, an application to real data will be given.

Keywords:

• Isotonic regression
• Monotone
• Nonparametric regression
• Uniform distribution

Acknowledgments

The author would like to thank Lee Newman and Annyce Mayer of National Jewish Medical and Research Center for providing the data that was used for the application.