Bayesian semiparametric models for nonignorable missing mechanisms in generalized linear models

Semiparametric models provide a more flexible form for modeling the relationship between the response and the explanatory variables. On the other hand in the literature of modeling for the missing variables, canonical form of the probability of the variable being missing (p) is modeled taking a fully parametric approach. Here we consider a regression spline based semiparametric approach to model the missingness mechanism of nonignorably missing covariates. In this model the relationship between the suitable canonical form of p (e.g. probit p) and the missing covariate is modeled through several splines. A Bayesian procedure is developed to efficiently estimate the parameters. A computationally advantageous prior construction is proposed for the parameters of the semiparametric part. A WinBUGS code is constructed to apply Gibbs sampling to obtain the posterior distributions. We show through an extensive Monte Carlo simulation experiment that response model coefficent estimators maintain better (when the true missingness mechanism is nonlinear) or equivalent (when the true missingness mechanism is linear) bias and efficiency properties with the use of proposed semiparametric missingness model compared to the conventional model.     

Optimal designs for stated choice experiments that incorporate ties

In 1970 Davidson generalised the Bradley–Terry model to allow respondents to say that the two options presented in a choice task were equally attractive. In this paper we extend this idea to the MNL model with m options in each choice set and we show that the optimal designs for the MNL model are also optimal in this setting.     

Testing generalized linear and semiparametric models against smooth alternatives

We propose goodness-of-fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both continous and factorial covariates. As a smooth extension of a parametric or semiparametric model we use generalized varying-coefficient models as proposed by Hastie and Tibshirani. A likelihood ratio statistic is used for testing. Asymptotic expansions allow us to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping of the test statistic. The test is shown to have √ n -power, but in contrast with parametric tests it is powerful against smooth alternatives in general.     

Optimal designs for estimating a choice hierarchy by a general nested multinomial logit model

Abstract

In choice experiments the process of decision-making can be more complex than the proposed by the Multinomial Logit Model (MNL). In these scenarios, models such as the Nested Multinomial Logit Model (NMNL) are often employed to model a more complex decision-making. Understanding the decision-making process is important in some fields such as marketing. Achieving a precise estimation of the models is crucial to the understanding of this process. To do this, optimal experimental designs are required. To construct an optimal design, information matrix is key. A previous research by others has developed the expression for the information matrix of the two-level NMNL model with two nests: Alternatives nest (J alternatives) and No-Choice nest (1 alternative). In this paper, we developed the likelihood function for a two-stage NMNL model for M nests and we present the expression for the information matrix for 2 nests with any amount of alternatives in them. We also show alternative D-optimal designs for No-Choice scenarios with similar relative efficiency but with less complex alternatives which can help to obtain more reliable answers and one application of these designs.     

Semiparametric inference for the recurrent events process by means of a single-index model

In this paper, we introduce new parametric and semiparametric regression techniques for a recurrent event process subject to random right censoring. We develop models for the cumulative mean function and provide asymptotically normal estimators. Our semiparametric model which relies on a single-index assumption can be seen as a dimension reduction technique that, contrary to a fully nonparametric approach, is not stroke by the curse of dimensionality when the number of covariates is high. We discuss data-driven techniques to choose the parameters involved in the estimation procedures and provide a simulation study to support our theoretical results.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

Regression splines in the quasi-likelihood analysis of recurrent event data

In many longitudinal studies of recurrent events there is an interest in assessing how recurrences vary over time and across treatments or strata in the population. Usual analyses of such data assume a parametric form for the distribution of the recurrences over time. Here, we consider a semiparametric model for the analysis of such longitudinal studies where data are collected as panel counts. The model is a non-homogeneous Poisson process with a multiplicative intensity incorporating covariates through a proportionality assumption. Heterogeneity is accounted for in the model through subject-specific random effects. The key feature of the model is the use of regression splines to model the distribution of recurrences over time. This provides a flexible and robust method of relaxing parametric assumptions. In addition, quasi-likelihood methods are proposed for estimation, requiring only first and second moment assumptions to obtain consistent estimates. Simulations demonstrate that the method produces estimators of the rate with low bias and whose standardized distributions are well approximated by the normal. The usefulness of this approach, especially as an exploratory tool, is illustrated by analyzing a study designed to assess the effectiveness of a pheromone treatment in disturbing the mating habits of the Cherry Bark Tortrix moth.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

On minimum Hellinger distance estimation

Efficiency and robustness are two fundamental concepts in parametric estimation problems. It was long thought that there was an inherent contradiction between the aims of achieving robustness and efficiency; that is, a robust estimator could not be efficient and vice versa. It is now known that the minimum Hellinger distance approached introduced by Beran [R. Beran, Annals of Statistics 1977;5:445–463] is one way of reconciling the conflicting concepts of efficiency and robustness. For parametric models, it has been shown that minimum Hellinger estimators achieve efficiency at the model density and simultaneously have excellent robustness properties. In this article, we examine the application of this approach in two semiparametric models. In particular, we consider a two‐component mixture model and a two‐sample semiparametric model. In each case, we investigate minimum Hellinger distance estimators of finite‐dimensional Euclidean parameters of particular interest and study their basic asymptotic properties. Small sample properties of the proposed estimators are examined using a Monte Carlo study. The results can be extended to semiparametric models of general form as well. The Canadian Journal of Statistics 37: 514–533; 2009 © 2009 Statistical Society of Canada     

NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.     

Using compositional and Dirichlet models for market share regression

When the aim is to model market shares, the marketing literature proposes some regression models which can be qualified as attraction models. They are generally derived from an aggregated version of the multinomial logit model. But aggregated multinomial logit models (MNL) and the so-called generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross effect parameters. In this paper, we consider alternative models: the Dirichlet model (DIR) and the compositional model (CODA). DIR allows to introduce brand-specific parameters and CODA allows additionally to consider cross effect parameters. We show that these two models can be written in a similar fashion, called attraction form, as the MNL and the GMCI models. As market share models are usually interpreted in terms of elasticities, we also use this notion to interpret the DIR and CODA models. We compare the properties of the models in order to explain why CODA and DIR models can outperform traditional market share models. An application to the automobile market is presented where we model brands market shares as a function of media investments, controlling for the brands price and scrapping incentive. We compare the quality of the models using measures adapted to shares.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

Asymptotics of bivariate penalised splines

We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), ‘Asymptotic Properties of Penalised Spline Estimators’, Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), ‘Optimal Global Rates of Convergence for Nonparametric Regression’, The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

Pseudomartingale estimating equations for modulated renewal process models

Summary.  We adapt martingale estimating equations based on gap time information to a general intensity model for a single realization of a modulated renewal process. The consistency and asymptotic normality of the estimators is proved under ergodicity conditions. Previous work has considered either parametric likelihood analysis or semiparametric multiplicative models using partial likelihood. The framework is generally applicable to semiparametric and parametric models, including additive and multiplicative specifications, and periodic models. It facilitates a semiparametric extension of a popular parametric earthquake model. Simulations and empirical analyses of Taiwanese earthquake sequences illustrate the methodology's practical utility.     

Estimating the error distribution function in semiparametric additive regression models

We consider semiparametric additive regression models with a linear parametric part and a nonparametric part, both involving multivariate covariates. For the nonparametric part we assume two models. In the first, the regression function is unspecified and smooth; in the second, the regression function is additive with smooth components. Depending on the model, the regression curve is estimated by suitable least squares methods. The resulting residual-based empirical distribution function is shown to differ from the error-based empirical distribution function by an additive expression, up to a uniformly negligible remainder term. This result implies a functional central limit theorem for the residual-based empirical distribution function. It is used to test for normal errors.     

Semiparametric lack-of-fit tests in an additive hazard regression model

In the semiparametric additive hazard regression model of McKeague and Sasieni (Biometrika 81: 501–514), the hazard contributions of some covariates are allowed to change over time, without parametric restrictions (Aalen model), while the contributions of other covariates are assumed to be constant. In this paper, we develop tests that help to decide which of the covariate contributions indeed change over time. The remaining covariates may be modelled with constant hazard coefficients, thus reducing the number of curves that have to be estimated nonparametrically. Several bootstrap tests are proposed. The behavior of the tests is investigated in a simulation study. In a practical example, the tests consistently identify covariates with constant and with changing hazard contributions.     

A lack-of-fit test for parametric zero-inflated Poisson models

Count data often contain many zeros. In parametric regression analysis of zero-inflated count data, the effect of a covariate of interest is typically modelled via a linear predictor. This approach imposes a restrictive, and potentially questionable, functional form on the relation between the independent and dependent variables. To address the noted restrictions, a flexible parametric procedure is employed to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. The semiparametric zero-inflated Poisson regression model is fitted by maximizing the likelihood function through an expectation–maximization algorithm. The smooth estimate of the functional form of the covariate effect can enhance modelling flexibility. Within this modelling framework, a log-likelihood ratio test is used to assess the adequacy of the covariate function. Simulation results show that the proposed test has excellent power in detecting the lack of fit of a linear predictor. A real-life data set is used to illustrate the practicality of the methodology.     

Semiparametric Time-Varying Coefficients Regression Model for Longitudinal Data

Abstract.  In this paper, we consider a semiparametric time-varying coefficients regression model where the influences of some covariates vary non-parametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The weighted least squares type estimators for the unknown parameters of the parametric coefficient functions as well as the estimators for the non-parametric coefficient functions are developed. We show that the kernel smoothing that avoids modelling of the sampling times is asymptotically more efficient than a single nearest neighbour smoothing that depends on the estimation of the sampling model. The asymptotic optimal bandwidth is also derived. A hypothesis testing procedure is proposed to test whether some covariate effects follow certain parametric forms. Simulation studies are conducted to compare the finite sample performances of the kernel neighbourhood smoothing and the single nearest neighbour smoothing and to check the empirical sizes and powers of the proposed testing procedures. An application to a data set from an AIDS clinical trial study is provided for illustration.