Covariate Selection for the Semiparametric Additive Risk Model

Abstract.  This paper considers covariate selection for the additive hazards model. This model is particularly simple to study theoretically and its practical implementation has several major advantages to the similar methodology for the proportional hazards model. One complication compared with the proportional model is, however, that there is no simple likelihood to work with. We here study a least squares criterion with desirable properties and show how this criterion can be interpreted as a prediction error. Given this criterion, we define ridge and Lasso estimators as well as an adaptive Lasso and study their large sample properties for the situation where the number of covariates p is smaller than the number of observations. We also show that the adaptive Lasso has the oracle property. In many practical situations, it is more relevant to tackle the situation with large p compared with the number of observations. We do this by studying the properties of the so-called Dantzig selector in the setting of the additive risk model. Specifically, we establish a bound on how close the solution is to a true sparse signal in the case where the number of covariates is large. In a simulation study, we also compare the Dantzig and adaptive Lasso for a moderate to small number of covariates. The methods are applied to a breast cancer data set with gene expression recordings and to the primary biliary cirrhosis clinical data.     

Sparse additive models

Summary.  We present a new class of methods for high dimensional non-parametric regression and classification called sparse additive models. Our methods combine ideas from sparse linear modelling and additive non-parametric regression. We derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size. Sparse additive models are essentially a functional version of the grouped lasso of Yuan and Lin. They are also closely related to the COSSO model of Lin and Zhang but decouple smoothing and sparsity, enabling the use of arbitrary non-parametric smoothers. We give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data, showing that they can be effective in fitting sparse non-parametric models in high dimensional data.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Missing item imputation for quality-of-life instruments with application to asthma quality-of-life questionnaires

There has been increasing use of quality-of-life (QoL) instruments in drug development. Missing item values often occur in QoL data. A common approach to solve this problem is to impute the missing values before scoring. Several imputation procedures, such as imputing with the most correlated item and imputing with a row/column model or an item response model, have been proposed. We examine these procedures using data from two clinical trials, in which the original asthma quality-of-life questionnaire (AQLQ) and the miniAQLQ were used. We propose two modifications to existing procedures: truncating the imputed values to eliminate outliers and using the proportional odds model as the item response model for imputation. We also propose a novel imputation method based on a semi-parametric beta regression so that the imputed value is always in the correct range and illustrate how this approach can easily be implemented in commonly used statistical software. To compare these approaches, we deleted 5% of item values in the data according to three different missingness mechanisms, imputed them using these approaches and compared the imputed values with the true values. Our comparison showed that the row/column-model-based imputation with truncation generally performed better, whereas our new approach had better performance under a number scenarios.     

Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.     

Quasi-minimax estimation in the general linear regression model

This paper is mainly concerned with minimax estimation in the general linear regression model y=Xβ+ε$y=X\beta +\epsilon$ under ellipsoidal restrictions on the parameter space and quadratic loss function. We confine ourselves to estimators that are linear in the response vector y  . The minimax estimators of the regression coefficient β$\beta$ are derived under homogeneous condition and heterogeneous condition, respectively. Furthermore, these obtained estimators are the ridge-type estimators and mean dispersion error (MDE) superior to the best linear unbiased estimator b=(X^′W^-1X)^-1X^′W^-1y$b=(X^{\prime }{W}^{-1}X{)}^{-1}{X}^{\prime }{W}^{-1}y$ under some conditions.     

Nonparametric estimation of waiting time distributions in a Markov model based on current status data

We construct nonparametric estimators of state waiting time distribution functions in a Markov multistate model using current status data. This is a particularly difficult problem since neither the entry nor the exit times of a given state are directly observed. These estimators are obtained, using the Markov property, from estimators of counting processes of state entry and exit times, as well as, the size of “at risk” sets of state entry and transitions out of that state. Consistency of our estimators is established. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on current status data compare well with those based on complete data. We also illustrate our method using a pubertal development data set obtained from the NHANES III [1997. NHANES III Reference Manuals and Reports (CD-ROM). Analytic and Reporting Guidelines: The Third National Health and Nutrition Examination Survey (1988–94). National Center for Health Statistics, Centers for Disease Control and Prevention, Hyattsville, MD] study.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

Analysis of microtubule dynamics using growth curve models

Microtubules are part of the structural network within a cell's cytoplasm, providing structural support as well as taking part in many of the cellular processes. A large body of data provide evidence that dynamics of microtubules in a cell is reponsible for the performance of many critical cellular functions such as cell division. In this article, we study the effect of four different isoforms of a protein tau on microtubule dynamics using growth curve models. The results show that a linear growth curve model is sufficient to explain the data. Moreover, we find that a mutated version of a 3-repeat tau protein has a similar effect as a 4-repeat tau protein on microtubule dynamics. The latter findings conform with the biological understanding of the effect of the protein tau on microtubule dynamics.