In building a proportional odds model, like other model building problems, the decision of which covariates to include in the final model has always been an important task for investigators. A successful variable selection can result in better risk assessment and model interpretation. For proportional odds model, variable selection is a more challenging task not only because of its nature of censored data, but also because of the unavailability of its partial likelihood. In this dissertation, we investigate the variable selection problem for proportional odds model. The proportional odds model fit by maximizing the marginal likelihood is proposed subject to the elastic net penalty. We also impose different weights on different coefficients so that important variables are most retained in the proposed model while the unimportant ones are most likely to be eliminated. This method combines the strength of the adaptively weighted lasso shrinkage and the quadratic regularization. It ensures the optimal large sample performance and handles collinearity simultaneously. We extend this method to ordinal regression with cumulative logit. We develop the computational algorithm for the proposed method and compare its performance with lasso, elastic net and adaptive lasso methods in simulation studies as well as in applications to real datasets. Results show that the proposed method works better than the existing ones.
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