A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.     

Minimum Risk Invariant Estimators of a Continuous Cumulative Distribution Function

The problem of nonparametric minimum risk invariant estimation has engaged a good deal of attention in the literature and minimum risk invariant estimators (MRIE's) have been constructed for some special statistical models. We present a new and simple method of obtaining the MRIE's of a continuous cumulative distribution function (cdf) under a general invariant loss function. All the MRIE's, which are known from the literature, can be constructed by the method presented in the article, in particular, under the weighted quadratic, LINEX and entropy loss functions. This method enables also to construct the MRIE's in nonparametric statistical models which have not been considered until now. In particular, considering a family of nonparametric precautionary loss functions, a new class of MRIE's of the cdf has been found. We also give some general remarks on obtaining the MRIE's and a review concerning minimaxity and admissibility of MRIE's.     

Non parametric mixture priors based on an exponential random scheme

We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions, the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion ofFeller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, namedFeller prior. Feller priors are related to mixture models with unknown number of components or, more generally, to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance of our procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.     

Generalized additive models for location, scale and shape

Summary.  A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y , as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

General partially linear additive transformation model with right-censored data

We propose a class of general partially linear additive transformation models (GPLATM) with right-censored survival data in this work. The class of models are flexible enough to cover many commonly used parametric and nonparametric survival analysis models as its special cases. Based on the B spline interpolation technique, we estimate the unknown regression parameters and functions by the maximum marginal likelihood estimation method. One important feature of the estimation procedure is that it does not need the baseline and censoring cumulative density distributions. Some numerical studies illustrate that this procedure can work very well for the moderate sample size.     

On the analysis of unbalanced two-level supersaturated designs via generalized linear models

Supersaturated designs (SSDs) are factorial designs in which the number of experimental runs is smaller than the number of parameters to be estimated in the model. While most of the literature on SSDs has focused on balanced designs, the construction and analysis of unbalanced designs has not been developed to a great extent. Recent studies discuss the possible advantages of relaxing the balance requirement in construction or data analysis of SSDs, and that unbalanced designs compare favorably to balanced designs for several optimality criteria and for the way in which the data are analyzed. Moreover, the effect analysis framework of unbalanced SSDs until now is restricted to the central assumption that experimental data come from a linear model. In this article, we consider unbalanced SSDs for data analysis under the assumption of generalized linear models (GLMs), revealing that unbalanced SSDs perform well despite the unbalance property. The examination of Type I and Type II error rates through an extensive simulation study indicates that the proposed method works satisfactorily.     

A pattern-mixture odds ratio model for incomplete categorical data

Most models for incomplete data are formulated within the selection model framework. Pattern-mixture models are increasingly seen as a viable alternative, both from an interpretational as well as from a computational point of view (Little 1993, Hogan and Laird 1997, Ekholm and Skinner 1998). Whereas most applications are either for continuous normally distributed data or for simplified categorical settings such as contingency tables, we show how a multivariate odds ratio model (Molenberghs and Lesaffre 1994, 1998) can be used to fit pattern-mixture models to repeated binary outcomes with continuous covariates. Apart from point estimation, useful methods for interval estimation are presented and data from a clinical study are analyzed to illustrate the methods.     

Depth Measures for Multivariate Functional Data

In this article, we address the problem of mining and analyzing multivariate functional data. That is, data where each observation is a set of possibly correlated functions. Complex data of this kind is more and more common in many research fields, particularly in the biomedical context. In this work, we propose and apply a new concept of depth measure for multivariate functional data. With this new depth measure it is possible to generalize robust statistics, such as the median, to the multivariate functional framework, which in turn allows the application of outlier detection, boxplots construction, and nonparametric tests also in this more general framework. We present an application to Electrocardiographic (ECG) signals.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

The analysis of incontinence episodes and other count data in patients with overactive bladder by Poisson and negative binomial regression

Clinical studies in overactive bladder have traditionally used analysis of covariance or nonparametric methods to analyse the number of incontinence episodes and other count data. It is known that if the underlying distributional assumptions of a particular parametric method do not hold, an alternative parametric method may be more efficient than a nonparametric one, which makes no assumptions regarding the underlying distribution of the data. Therefore, there are advantages in using methods based on the Poisson distribution or extensions of that method, which incorporate specific features that provide a modelling framework for count data. One challenge with count data is overdispersion, but methods are available that can account for this through the introduction of random effect terms in the modelling, and it is this modelling framework that leads to the negative binomial distribution. These models can also provide clinicians with a clearer and more appropriate interpretation of treatment effects in terms of rate ratios. In this paper, the previously used parametric and non‐parametric approaches are contrasted with those based on Poisson regression and various extensions in trials evaluating solifenacin and mirabegron in patients with overactive bladder. In these applications, negative binomial models are seen to fit the data well. Copyright © 2014 John Wiley & Sons, Ltd.     

Sequential design for nonparametric inference

The performance of nonparametric function estimates often depends on the choice of design points. Based on the mean integrated squared error criterion, we propose a sequential design procedure that updates the model knowledge and optimal design density sequentially. The methodology is developed under a general framework covering a wide range of nonparametric inference problems, such as conditional mean and variance functions, the conditional distribution function, the conditional quantile function in quantile regression, functional coefficients in varying coefficient models and semiparametric inferences. Based on our empirical studies, nonparametric inference based on the proposed sequential design is more efficient than the uniform design and its performance is close to the true but unknown optimal design. The Canadian Journal of Statistics 40: 362–377; 2012 © 2012 Statistical Society of Canada     

A General Quantile Function Model for Economic and Financial Time Series

This article proposed a general quantile function model that covers both one- and multiple-dimensional models and that takes several existing models in the literature as its special cases. This article also developed a new uniform Bayesian framework for quantile function modelling and illustrated the developed approach through different quantile function models. Many distributions are defined explicitly only via their quanitle functions as the corresponding distribution or density functions do not have an explicit mathematical expression. Such distributions are rarely used in economic and financial modelling in practice. The developed methodology makes it more convenient to use these distributions in analyzing economic and financial data. Empirical applications to economic and financial time series and comparisons with other types of models and methods show that the developed method can be very useful in practice.     

A maximin criterion for the logistic random intercept model with covariates

A maximin criterion is used to find optimal designs for the logistic random intercept model with dichotomous independent variables. The dichotomous independent variables can be subdivided into variables for which the distribution is specified prior to data sampling, called variates, and into variables for which the distribution is not specified prior to data sampling, but is obtained from data sampling, called covariates. The proposed maximin criterion maximizes the smallest possible relative efficiency not only with respect to all possible values of the model parameters, but also with respect to the joint distribution of the covariates. We have shown that, under certain conditions, the maximin design is balanced with respect to the joint distribution of the variates. The proposed method will be used to plan a (stratified) clinical trial where variates and covariates are involved.     

Bayesian Kernel Mixtures for Counts

Although Bayesian nonparametric mixture models for continuous data are well developed, there is a limited literature on related approaches for count data. A common strategy is to use a mixture of Poissons, which unfortunately is quite restrictive in not accounting for distributions having variance less than the mean. Other approaches include mixing multinomials, which requires finite support, and using a Dirichlet process prior with a Poisson base measure, which does not allow smooth deviations from the Poisson. As a broad class of alternative models, we propose to use nonparametric mixtures of rounded continuous kernels. An efficient Gibbs sampler is developed for posterior computation, and a simulation study is performed to assess performance. Focusing on the rounded Gaussian case, we generalize the modeling framework to account for multivariate count data, joint modeling with continuous and categorical variables, and other complications. The methods are illustrated through applications to a developmental toxicity study and marketing data. This article has supplementary material online.     

Nonparametric Estimation of Distribution Functions of Nonstandard Mixtures

ABSTRACT

Nonstandard mixtures are those that result from a mixture of a discrete and a continuous random variable. They arise in practice, for example, in medical studies of exposure. Here, a random variable that models exposure might have a discrete mass point at no exposure, but otherwise may be continuous. In this article we explore estimating the distribution function associated with such a random variable from a nonparametric viewpoint. We assume that the locations of the discrete mass points are known so that we will be able to apply a classical nonparametric smoothing approach to the problem. The proposed estimator is a mixture of an empirical distribution function and a kernel estimate of a distribution function. A simple theoretical argument reveals that existing bandwidth selection algorithms can be applied to the smooth component of this estimator as well. The proposed approach is applied to two example sets of data.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Nonparametric estimation in the illness-death model using prevalent data

We study nonparametric estimation of the illness-death model using left-truncated and right-censored data. The general aim is to estimate the multivariate distribution of a progressive multi-state process. Maximum likelihood estimation under censoring suffers from problems of uniqueness and consistency, so instead we review and extend methods that are based on inverse probability weighting. For univariate left-truncated and right-censored data, nonparametric maximum likelihood estimation can be considerably improved when exploiting knowledge on the truncation distribution. We aim to examine the gain in using such knowledge for inverse probability weighting estimators in the illness-death framework. Additionally, we compare the weights that use truncation variables with the weights that integrate them out, showing, by simulation, that the latter performs more stably and efficiently. We apply the methods to intensive care units data collected in a cross-sectional design, and discuss how the estimators can be easily modified to more general multi-state models.