In addition, these standard models do not allow for nonlinear covariate effects

In addition, these standard models do not allow for nonlinear covariate effects. a sparse CR kernel machine (KM) regression method for ordinal outcomes where we use the KM framework to incorporate nonlinearity and impose sparsity on the overall differences between the Sucralose covariate effects of continuation ratios to control for overfitting. In addition, we provide data driven rule to select an optimal kernel to maximize the prediction accuracy. Simulation results show that our proposed procedures perform well under both linear and nonlinear settings, especially when the true underlying model is in-between pCR and fCR models. We apply our procedures to develop a prediction model for levels of anti-CCP among rheumatoid arthritis patients and demonstrate the advantage of our method over other commonly used methods. with a 1 predictor vector x, one may employ regression models relating x to and classify future subjects into different categories based on their predicted = | x). Naive analysis strategies, such as dichotomizing into a binary fitting and variable multinomial regression models, are not efficient as they do not take into account the ordinal property of the outcome. Commonly used traditional methods for modeling ordinal response data include the cumulative proportional odds model, the forward and backward continuation ratio (CR) models and the corresponding proportional odds version of the CR (pCR) model (Ananth and Kleinbaum, 1997). The forward Sucralose full CR (fCR) model assumes that is assumed to take ordered categories, {1, , and but not Sucralose all and Rabbit Polyclonal to OR8K3 thus it is possible to improve the estimation by leveraging the sparsity on independent and identically distributed random vectors, to denote Fubinis norm for matrices. From here onward, for notational ease, we suppress from the kernel function with respect to the eigensystem of has eigenvalues = 1, , with = 1, , such that 0 for any . The basis functions, = 1, , , span the RKHS . All for all is smooth Hence, leading to bounded = 1, , = 1, , ? 1: = [ 1 vector of unknown weights to be estimated as model parameters. This representation reduces (6) to an explicit optimization problem in the dual form: + 1)(? 1) parameters to be estimated, when the sample size is not small especially. On the other hand, if the eigenvalues of decay quickly, then we may reduce the complexity by approximating by a truncated kernel such that can be bounded by is the kernel matrix constructed from kernel is typically fairly small and we can effectively approximate by a finite dimensional space . Although Sucralose = (= diag{ 0 are the eigenvalues of and {u1, , uconverge to the eigenvalues and the projection error can be bounded by and sufficiently fast decay rate for {and applying a variable transformation is the for some close to 1. Let denote the estimator from the maximization of (8). For a future subject with x Then, the probability = ? = = 1= 1within a range of values. For any given and obtained from (10), in (and the resulting classification will outperform the corresponding estimators and classifications derived from the fCRKM model based on and the reduced pCRKM model when the underlying model has but not all. When = can be approximated well with a finite dimensional space with a fixed 1 if and the average size of prediction sets ( ) to be defined below. The OME puts equal weights to any error as long as = 11 = 1 in to fit our proposed procedures with several candidate kernels and obtain the corresponding estimate to calculate their predicted probabilities (= 1 would then be used for prediction in the validation set. In regards to the choice of = 10 as previously suggested in Breiman and Spector (1992). 3. Numerical Studies 3.1 Simulation Study We conducted extensive simulations to evaluate the finite sample performance of our proposed methods and compared with three existing methods: the one-against-one SVM method (Hsu and Lin, 2002), the 1)with continuous covariates under the CRKM model in (3). The 20and = 1 1 were set to be between 0 and 0.4 and the intercept parameters = 1 Sucralose 1 were selected such that there were approximately the same number.