Sliced inverse regression for survival data

We apply the univariate sliced inverse regression to survival data. Our approach is different from the other papers on this subject. The right-censored observations are taken into account during the slicing of the survival times by assigning each of them with equal weight to all of the slices with longer survival. We test this method with different distributions for the two main survival data models, the accelerated lifetime model and Cox’s proportional hazards model. In both cases and under different conditions of sparsity, sample size and dimension of parameters, this non-parametric approach finds the data structure and can be viewed as a variable selector.     

Fitting survival data with penalized Poisson regression

Cox’s proportional hazards model is the most common way to analyze survival data. The model can be extended in the presence of collinearity to include a ridge penalty, or in cases where a very large number of coefficients (e.g. with microarray data) has to be estimated. To maximize the penalized likelihood, optimal weights of the ridge penalty have to be obtained. However, there is no definite rule for choosing the penalty weight. One approach suggests maximization of the weights by maximizing the leave-one-out cross validated partial likelihood, however this is time consuming and computationally expensive, especially in large datasets. We suggest modelling survival data through a Poisson model. Using this approach, the log-likelihood of a Poisson model is maximized by standard iterative weighted least squares. We will illustrate this simple approach, which includes smoothing of the hazard function and move on to include a ridge term in the likelihood. We will then maximize the likelihood by considering tools from generalized mixed linear models. We will show that the optimal value of the penalty is found simply by computing the hat matrix of the system of linear equations and dividing its trace by a product of the estimated coefficients.     

Auxiliary covariate in additive hazards regression for survival data

We consider the additive hazards regression analysis by utilising auxiliary covariate information to improve the efficiency of the statistical inference when the primary covariate is ascertained only for a randomly selected subsample. We construct a martingale-based estimating equation for the regression parameter and establish the asymptotic consistency and normality of the resultant estimator. Simulation study shows that our proposed method can improve the efficiency compared with the estimator which discards the auxiliary covariate information. A real example is also analysed as an illustration.     

Nonparametric and semiparametric regression estimation for length-biased survival data

For the past several decades, nonparametric and semiparametric modeling for conventional right-censored survival data has been investigated intensively under a noninformative censoring mechanism. However, these methods may not be applicable for analyzing right-censored survival data that arise from prevalent cohorts when the failure times are subject to length-biased sampling. This review article is intended to provide a summary of some newly developed methods as well as established methods for analyzing length-biased data.     

Parameter estimation in regression for long-term survival rate from censored data

In recent years, regression models have been shown to be useful for predicting the long-term survival probabilities of patients in clinical trials. The importance of a regression model is that once the regression parameters are estimated information about the regressed quantity is immediate. A simple estimator is proposed for the regression parameters in a model for the long-term survival rate. The proposed estimator is seen to arise from an estimating function that has the missing information principle underlying its construction. When the covariate takes values in a finite set, the proposed estimating function is equivalent to an ad hoc estimating function proposed in the literature. However, in general, the two estimating functions lead to different estimators of the regression parameter. For discrete covariates, the asymptotic covariance matrix of the proposed estimator is simple to calculate using standard techniques involving the predictable covariation process of martingale transforms. An ad hoc extension to the case of a one-dimensional continuous covariate is proposed. Simplicity and generalizability are two attractive features of the proposed approach. The last mentioned feature is not enjoyed by the other estimator.     

Sparse Bayesian variable selection in multinomial probit regression model with application to high-dimensional data classification

Here we consider a multinomial probit regression model where the number of variables substantially exceeds the sample size and only a subset of the available variables is associated with the response. Thus selecting a small number of relevant variables for classification has received a great deal of attention. Generally when the number of variables is substantial, sparsity-enforcing priors for the regression coefficients are called for on grounds of predictive generalization and computational ease. In this paper, we propose a sparse Bayesian variable selection method in multinomial probit regression model for multi-class classification. The performance of our proposed method is demonstrated with one simulated data and three well-known gene expression profiling data: breast cancer data, leukemia data, and small round blue-cell tumors. The results show that compared with other methods, our method is able to select the relevant variables and can obtain competitive classification accuracy with a small subset of relevant genes.     

Linear regression analysis of survival data with missing censoring indicators

Linear regression analysis has been studied extensively in a random censorship setting, but typically all of the censoring indicators are assumed to be observed. In this paper, we develop synthetic data methods for estimating regression parameters in a linear model when some censoring indicators are missing. We define estimators based on regression calibration, imputation, and inverse probability weighting techniques, and we prove all three estimators are asymptotically normal. The finite-sample performance of each estimator is evaluated via simulation. We illustrate our methods by assessing the effects of sex and age on the time to non-ambulatory progression for patients in a brain cancer clinical trial.     

Generalised joint regression for count data: a penalty extension for competitive settings

Statistics and Computing - We propose a versatile joint regression framework for count responses. The method is implemented in the R add-on package GJRM and allows for modelling linear and...     

Pruning regression trees for censored survival data: the recpam approach

RECPAM is a methodology, implemented in a computer program of the same name, for the construction of tree-structured models in Biostatistics. In this work we present algorithms for pruning and amalgamating terminal nodes of a tree, within the RECPAM approach. These algorithms construct sequences of nested models and calculate at each step the AIC of the corresponding model and correct significance levels, according to Gabriel's theory of Simultaneous Test Procedures. As an example, the analysis of data from clinical trials involving patients with Small Cell Carcinoma of the Lung is presented.     

Isotontic regression for interval censored survival data using an e-m algorithm

This paper develops a nonparametric model of the relationship between survival S and a dichotomous random variable X under the order constraint that P(X=1|S=s) is increasing (or decreasing) with s. The estimation procedure, called isotonic regression, has been studied in some depth for the case of uncensored data, but we give a methodology which is appropriate in the more general context of right, left, and interval censored data. An E-M Algorithm (Dempster et. al., 1977) is used for maximum likelihood estimation.     

A new threshold regression model for survival data with a cure fraction

Due to the fact that certain fraction of the population suffering a particular type of disease get cured because of advanced medical treatment and health care system, we develop a general class of models to incorporate a cure fraction by introducing the latent number N of metastatic-competent tumor cells or infected cells caused by bacteria or viral infection and the latent antibody level R of immune system. Various properties of the proposed models are carefully examined and a Markov chain Monte Carlo sampling algorithm is developed for carrying out Bayesian computation for model fitting and comparison. A real data set from a prostate cancer clinical trial is analyzed in detail to demonstrate the proposed methodology.     

A class of mean residual life regression models with censored survival data

When describing a failure time distribution, the mean residual life is sometimes preferred to the survival or hazard rate. Regression analysis making use of the mean residual life function has recently drawn a great deal of attention. In this paper, a class of mean residual life regression models are proposed for censored data, and estimation procedures and a goodness-of-fit test are developed. Both asymptotic and finite sample properties of the proposed estimators are established, and the proposed methods are applied to a cancer data set from a clinic trial.     

Sparse estimation and inference for censored median regression

Censored median regression has proved useful for analyzing survival data in complicated situations, say, when the variance is heteroscedastic or the data contain outliers. In this paper, we study the sparse estimation for censored median regression models, which is an important problem for high dimensional survival data analysis. In particular, a new procedure is proposed to minimize an inverse-censoring-probability weighted least absolute deviation loss subject to the adaptive LASSO penalty and result in a sparse and robust median estimator. We show that, with a proper choice of the tuning parameter, the procedure can identify the underlying sparse model consistently and has desired large-sample properties including root-n consistency and the asymptotic normality. The procedure also enjoys great advantages in computation, since its entire solution path can be obtained efficiently. Furthermore, we propose a resampling method to estimate the variance of the estimator. The performance of the procedure is illustrated by extensive simulations and two real data applications including one microarray gene expression survival data.     

Maximum likelihood estimation for tied survival data under Cox regression model via EM-algorithm

We consider tied survival data based on Cox proportional regression model. The standard approaches are the Breslow and Efron approximations and various so called exact methods. All these methods lead to biased estimates when the true underlying model is in fact a Cox model. In this paper we review the methods and suggest a new method based on the missing-data principle using EM-algorithm that leads to a score equation that can be solved directly. This score has mean zero. We also show that all the considered methods have the same asymptotic properties and that there is no loss of asymptotic efficiency when the tie sizes are bounded or even converge to infinity at a given rate. A simulation study is conducted to compare the finite sample properties of the methods.     

Sparse group lasso for multiclass functional logistic regression models

Sparsity-inducing penalties are useful tools for variable selection and are also effective for regression problems where the data are functions. We consider the problem of selecting not only variables but also decision boundaries in multiclass logistic regression models for functional data, using sparse regularization. The parameters of the functional logistic regression model are estimated in the framework of the penalized likelihood method with the sparse group lasso-type penalty, and then tuning parameters for the model are selected using the model selection criterion. The effectiveness of the proposed method is investigated through simulation studies and the analysis of a gene expression data set.     

Semiparametric multiparameter regression survival modeling

We consider a log-linear model for survival data, where both the location and scale parameters depend on covariates, and the baseline hazard function is completely unspecified. This model provides the flexibility needed to capture many interesting features of survival data at a relatively low cost in model complexity. Estimation procedures are developed, and asymptotic properties of the resulting estimators are derived using empirical process theory. Finally, a resampling procedure is developed to estimate the limiting variances of the estimators. The finite sample properties of the estimators are investigated by way of a simulation study, and a practical application to lung cancer data is illustrated.     

Sparse alternatives to ridge regression: a random effects approach

In a calibration of near-infrared (NIR) instrument, we regress some chemical compositions of interest as a function of their NIR spectra. In this process, we have two immediate challenges: first, the number of variables exceeds the number of observations and, second, the multicollinearity between variables are extremely high. To deal with the challenges, prediction models that produce sparse solutions have recently been proposed. The term ‘sparse’ means that some model parameters are zero estimated and the other parameters are estimated naturally away from zero. In effect, a variable selection is embedded in the model to potentially achieve a better prediction. Many studies have investigated sparse solutions for latent variable models, such as partial least squares and principal component regression, and for direct regression models such as ridge regression (RR). However, in the latter, it mainly involves an L₁ norm penalty to the objective function such as lasso regression. In this study, we investigate new sparse alternative models for RR within a random effects model framework, where we consider Cauchy and mixture-of-normals distributions on the random effects. The results indicate that the mixture-of-normals model produces a sparse solution with good prediction and better interpretation. We illustrate the methods using NIR spectra datasets from milk and corn specimens.