A penalized estimation for the Cox model with ordinal multinomial covariates

Models dealing with ordinal multinomial(OM) variables have drawn much attention in regression analysis. Studies on the ordinal response variable have been solidly established and widely applied. However, few studies have investigated generalized linear models with OM covariates, especially in high-dimensional situations. For this problem, the detection of pseudo categories for the OM covariates and the selection of other important covariates need to be concerned simultaneously. This paper proposes an $L_1$-norm penalized estimation procedure to detect pseudo categories of OM covariates in high-dimensional( $p_n\le n,p_n=O\left(n^{\sigma }\right),0<\sigma \le 1)$) Cox models. The estimation approach is based on the combination of a transformation method of dummy variables and the penalized partial likelihood. Theoretical properties, such as consistency and oracle property of the proposed estimators, are rigorously established under some regularity conditions. The performance of the proposed method is illustrated by analyses on both simulated data and real data.