Zahid, Faisal Maqbool
(2011):
Ordinal Ridge Regression with Categorical Predictors.
Department of Statistics: Technical Reports, No.101

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Abstract
In multicategory response models categories are often ordered. In case of ordinal response models, the usual likelihood approach becomes unstable with illconditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraints on the order of intercept values are not met during the iterative fitting procedure. Proportional odds models are most commonly used for ordinal responses. In this paper penalized likelihood with quadratic penalty is used to address these issues with a special focus on proportional odds models. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized likelihood function penalizes the parameter estimates or differences between the parameters estimates according to the type of predictors. Mean squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application.