Robust Linear Regression Analysis


On many occasions we desire to explain changes in a dependent variable (Y) as a response to changes in (a single or multiple) explanatory variables (X) and we hypothesize that the relationship between Y and X is linear. To accomplish this, the method of Ordinary Least Squares (OLS) is almost universally applied. However, if data are infested with outliers, the use of OLS is not the best choice. The outliers in a dataset are the points in a minority that are highly unlikely to belong to the population from which the other points (i.e. inliers), which are in a majority, have been drawn. Alternatively, the outliers exhibit a pattern or characteristics that are alien or non-conformal to those of the inliers. These outliers in a data set pull the measures of central tendency towards themselves and also inflate the measures of dispersion leading to biased and inefficient estimators. The pulled measures of location and inflated measures of dispersion often lead to masking of the outliers. A single prominent outlier can mask other relatively less prominent outliers and thus may cause delusion and evade their detection by a cursory inspection. There are a number of methods available to work in this situation (e.g. Andrews, 1974; Rupert and Carrol, 1980; Rousseeuw and Leroy, 1987; Kashyap and Maiyuran, 1993, etc). It is also possible to obtain the robust regression estimators based on Campbell's robust covariance estimation method Campbell, 1980) with no or some slight modifications. The program (source codes) presented here is based on this method (Mishra, 2008).

References :
  • Andrews, D.F. (1974) "A Robust Method for Multiple Linear Regression", Technometrics, 16: 523-531.
  • Campbell, N. A. (1980) "Robust Procedures in Multivariate Analysis I: Robust Covariance Estimation", Applied Statistics, 29 (3): 231-237.
  • Kashyap, R.L and Maiyuran, S. (1993) "Robust Regression and Outlier Set Estimation using Likelihood Reasoning", Electrical and Computer Engineering ECE Technical Reports, TR-EE 93-8, Purdue University School of Electrical Engineering. http://docs.lib.purdue.edu/ecetr/33/.
  • Rousseeuw, P.J., and Leroy, A.M. (1987) Robust Regression and Outlier Detection, Wiley. New York.
  • Rupert, D. and Carrol, R.J. (1980) "Trimmed Least Squares Estimation in the Linear Model," Journal of American Statistical Association, 75: 828-838.
  • Mishra, S.K. (2008) "A New Method of Robust Linear Regression Analysis: Some Monte Carlo Experiments", Journal of Applied Economic Sciences, III-3(5): 261-269. Download.

 

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