Free Knot Splines with RJMCMC in Survival Data Analysis

The B-spline representation is a common tool to improve the fitting of smooth nonlinear functions, it offers a fitting as a piecewise polynomial. The regions that define the pieces are separated by a sequence of knots. The main difficulty in this type of modeling is the choice of the number and the locations of these knots. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm provides a solution to simultaneously select these two parameters by considering the knots as free parameters. This algorithm belongs to the MCMC techniques that allow simulations from target distributions on spaces of varying dimension. The aim of the present investigation is to use this algorithm in the framework of the analysis of survival time, for the Cox model in particular. In fact, the relation between the hazard ratio function and the covariates being assumed to be log-linear, this assumption is too restrictive. Thus, we propose to use the RJMCMC algorithm to model the log hazard ratio function by a B-spline representation with an unknown number of knots at unknown locations. This method is illustrated with two real data sets: the Stanford heart transplant data and lung cancer survival data. Another application of the RJMCMC is selecting the significant covariates, and a simulation study is performed.     

Predictive performance of missing data methods for logistic regression,classification trees and neural networks

Although the effect of missing data on regression estimates has received considerable attention, their effect on predictive performance has been neglected. We studied the performance of three missing data strategies—omission of records with missing values, replacement with a mean and imputation based on regression—on the predictive performance of logistic regression (LR), classification tree (CT) and neural network (NN) models in the presence of data missing completely at random (MCAR). Models were constructed using datasets of size 500 simulated from a joint distribution of binary and continuous predictors including nonlinearities, collinearity and interactions between variables. Though omission produced models that fit better on the data from which the models were developed, imputation was superior on average to omission for all models when evaluating the receiver operating characteristic (ROC) curve area, mean squared error (MSE), pooled variance across outcome categories and calibration X ² on an independently generated test set. However, in about one-third of simulations, omission performed better. Performance was also more variable with omission including quite a few instances of extremely poor performance. Replacement and imputation generally produced similar results except with neural networks for which replacement, the strategy typically used in neural network algorithms, was inferior to imputation. Missing data affected simpler models much less than they did more complex models such as generalized additive models that focus on local structure For moderate sized datasets, logistic regressions that use simple nonlinear structures such as quadratic terms and piecewise linear splines appear to be at least as robust to randomly missing values as neural networks and classification trees.     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

广义部分线性单指数模型的惩罚样条估计

 本文讨论了指数族广义部分线性单指数模型(Generalized Partially Linear Single Index Models, GPLSIM) 的惩罚样条迭代估计,提出了基于惩罚似然和一组预先取定的单指数参数向量 的初始估计的迭代估计算法。另外本文还通过一组模拟数据的分析对所提出的迭代算法进行了验证。     

Identification of non-linear additive autoregressive models

Summary.  We propose a lag selection method for non-linear additive autoregressive models that is based on spline estimation and the Bayes information criterion. The additive structure of the autoregression function is used to overcome the 'curse of dimensionality', whereas the spline estimators effectively take into account such a structure in estimation. A stepwise procedure is suggested to implement the method proposed. A comprehensive Monte Carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local-polynomial-based methods. Consistency of the lag selection method based on the Bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing, which provides the first theoretical result of this kind for spline smoothing of weakly dependent data.     

Splines as a Useful and Convenient Statistical Tool

The framework for a unified statistical theory of spline regression assuming fixed knots using the truncated polynomial or “+” function representation is presented. In particular, a partial ordering of some spline models is introduced to clarify their relationship and to indicate the hypotheses that can be tested by using either standard multiple regression procedures or a little-used conditional test developed by Hotelling (1940). The construction of spline models with polynomial pieces of different degrees is illustrated. A numerical example from a chemical experiment is given by using the GLM procedure of the statistical software package SAS (Barr et al. 1976).     

Performance Measures for Selection of Metamodels to be Used in Simulation Optimization

This paper points out the need for performance measures in the context of simulation optimization and suggests six such measures. Two of the measures are indications of absolute performance, whereas the other four are useful in assessing the relative performance of various candidate metamodels. The measures assess performance on three fronts: accuracy of placing optima in the correct location, fit to the response, and fit to the character of the surface (expressed in terms of the number of optima). Examples are given providing evidence of the measures' utility—one in a limited scenario deciding which of two competing metamodels to use as simulation optimization response surfaces vary, and the other in a scenario of a researcher developing a new, sequential optimization search procedure.     

Generalized monotonic functional mixed models with application to modelling normal tissue complications

Summary.  Normal tissue complications are a common side effect of radiation therapy. They are the consequence of the dose of radiation that is received by the normal tissue surrounding the site of the tumour. Within a specified organ each voxel receives a certain dose of radiation, leading to a distribution of doses over the organ. It is often not known what aspect of the dose distribution drives the presence and severity of the complications. A summary measure of the dose distribution can be obtained by integrating a weighting function of dose ( w ( d )) over the density of dose. For biological reasons the weight function should be monotonic. We propose a generalized monotonic functional mixed model to study the dose effect on a clinical outcome by estimating this weight function non-parametrically by using splines and subject to the monotonicity constraint, while allowing for overdispersion and correlation of multiple obervations within the same subject. We illustrate our method with data from a head and neck cancer study in which the irradiation of the parotid gland results in loss of saliva flow.     

Consistency of Posterior Distributions for Heteroscedastic Nonparametric Regression Models

In this article, we consider Bayesian inferences for the heteroscedastic nonparametric regression models, when both the mean function and variance function are unknown. We demonstrated consistency of posterior distributions for this model using priors induced by B-splines expansion, treating both random and deterministic covariates in a uniform manner.