Collapsibility of regression coefficients and its extensions

Collapsibility with respect to a measure of association implies that the measure of association can be obtained from the marginal model. We first discuss model collapsibility and collapsibility with respect to regression coefficients for linear regression models. For parallel regression models, we give simple and different proofs of some of the known results and obtain also certain new results. For random coefficient regression models, we define (average) A$A$-collapsibility and obtain conditions under which it holds. We consider Poisson regression and logistic regression models also, and derive conditions for collapsibility and A$A$-collapsibility, respectively. These results generalize some of the results available in the literature. Some suitable examples are also discussed.     

A model of uniform association plus two-diagonalsparameter and its application to occupational mobility table data

For the analysis of square contingency tables with ordered categories, Agresti (1988) introduced a model having the structure of uniform association plus a main-diagonal parameter. This paper extends that model. The extended model has the structure of uniform association plus two-diagonals-parameter, and it is a special case of the quasi-uniform association model introduced by Goodman (1979). The Danish occupational mobility table data are analyzed using the models introduced here.     

Stochastic volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

Designs for cropping systems research

The present article establishes equivalence between extended group divisible (EGD) designs and designs for crop sequence experiments. This equivalence has encouraged the agricultural experimenters to use EGD designs for their experimentation. Some real life applications of EGD designs have been given. It has also been shown that several existing association schemes are special cases of EGD association scheme. Some methods of construction of EGD designs are also given. A catalogue of EGD designs obtainable through methods of construction along with efficiency factors of various factorial effects is also presented. In some crop sequence experiments that are conducted to develop suitable integrated nutrient supply system of a crop sequence, the treatments do not comprise of a complete factorial structure. The experimenter is interested in estimating the residual and direct effect of the treatments along with their cumulative effects. For such experimental settings block designs with two sets of treatments applied in succession are the appropriate designs. The correspondence established between row–column designs and block designs for two stage experiments by Parsad et al. [2003. Structurally incomplete row–column designs. Comm. Statist. Theory Methods 32(1), 239–261] has been exploited in obtaining designs for such experimental situations. Some open problems related to designing of crop sequence experiments are also given.     

Robust prediction and extrapolation designs for censored data

In this paper we present the construction of robust designs for a possibly misspecified generalized linear regression model when the data are censored. The minimax designs and unbiased designs are found for maximum likelihood estimation in the context of both prediction and extrapolation problems. This paper extends preceding work of robust designs for complete data by incorporating censoring and maximum likelihood estimation. It also broadens former work of robust designs for censored data from others by considering both nonlinearity and much more arbitrary uncertainty in the fitted regression response and by dropping all restrictions on the structure of the regressors. Solutions are derived by a nonsmooth optimization technique analytically and given in full generality. A typical example in accelerated life testing is also demonstrated. We also investigate implementation schemes which are utilized to approximate a robust design having a density. Some exact designs are obtained using an optimal implementation scheme.     

Some sufficient conditions for establishing (M,S)-optimality

Two sufficient conditions are given for an incomplete block design to be (M,S- optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.     

Analysis of Double Single Index Models

Motivated from problems in canonical correlation analysis, reduced rank regression and sufficient dimension reduction, we introduce a double dimension reduction model where a single index of the multivariate response is linked to the multivariate covariate through a single index of these covariates, hence the name double single index model. Because nonlinear association between two sets of multivariate variables can be arbitrarily complex and even intractable in general, we aim at seeking a principal one‐dimensional association structure where a response index is fully characterized by a single predictor index. The functional relation between the two single‐indices is left unspecified, allowing flexible exploration of any potential nonlinear association. We argue that such double single index association is meaningful and easy to interpret, and the rest of the multi‐dimensional dependence structure can be treated as nuisance in model estimation. We investigate the estimation and inference of both indices and the regression function, and derive the asymptotic properties of our procedure. We illustrate the numerical performance in finite samples and demonstrate the usefulness of the modelling and estimation procedure in a multi‐covariate multi‐response problem concerning concrete.     

Union–intersection principle and constrained statistical inference

Most statistical models arising in real life applications as well as in interdisciplinary research are complex in their designs, sampling plans, and associated probability laws, which in turn are often constrained by inequality, order, functional, shape or other restraints. Optimality of conventional likelihood ratio based statistical inference may not be tenable here, although the use of restricted or quasi-likelihood has spurred in such environments. S.N. Roy's ingenious union–intersection principle provides an alternative avenue, often having some computational advantages, increased scope of adaptability, and flexibility beyond conventional likelihood paradigms. This scenario is appraised here with some illustrative examples, and with some interesting problems of inference on stochastic ordering (dominance) in parametric as well as beyond parametric setups.     

Generalized Least Squares Estimation in Contingency Tables Analysis: Asymptotic Properties and Applications

Different sorts of bilinear models (models with bilinear interaction terms) are currently used when analyzing contingency tables: association models, correlation models... All these can be included in a general family of bilinear models: power models. In this framework, Maximum Likelihood (ML) estimation is not always possible, as explained in an introductory example. Thus, Generalized Least Squares (GLS) estimation is sometimes needed in order to estimate parameters. A subclass of power models is then considered in this paper: separable reduced-rank (SRR) models. They allow an optimal choice of weights for GLS estimation and simplifications in asymptotic studies concerning GLS estimators. Power 2 models belong to the subclass of SRR models and the asymptotic properties of GLS estimators are established. Similar results are also established for association models which are not SRR models. However, these results are more difficult to prove. Finally, 2 examples are considered to illustrate our results.     

The effect of area of residence over the life course on subsequent mortality

Summary.  Life course epidemiology concentrates on the contribution that social or physical exposures have across the life course on adult health. It is known that the area of residence can affect health, but little is known about the effect of the area of residence across the life course. We examine the contribution that area of residence in 1960, 1970, 1980 and 1990 made on subsequent mortality for 49736 male inhabitants of Oslo in 1990. We compare the performance of multiple-membership and cross-classified multilevel models on these data with a correlated cross-classified model that was developed for this.     

Stopping times of one-sample rank order sequential probability ratio tests

Weed, Bradley and Grovindarajulu (1974) propose one-sample probability ratio tests based on Lehmann alternatives. They also study the finite sure termination of the stopping times. Motivated by Stein's proof of (1946) of the termination of a sequential probability ratio test (SPRT) in the case of independent and identically distributed (i.i.d.) random variables and the work of Sethuraman (1970) for the two- sample rank order SPRT, we obtain a very mild condition (namely, that a certain random variable U(Z) is not identically zero) for the finite sure termination of the existence of the moment generating function (m.g.f.) for the stopping time of one- sample rank order SPRT's.     

Model‐based clustering of longitudinal data

A new family of mixture models for the model‐based clustering of longitudinal data is introduced. The covariance structures of eight members of this new family of models are given and the associated maximum likelihood estimates for the parameters are derived via expectation–maximization (EM) algorithms. The Bayesian information criterion is used for model selection and a convergence criterion based on the Aitken acceleration is used to determine the convergence of these EM algorithms. This new family of models is applied to yeast sporulation time course data, where the models give good clustering performance. Further constraints are then imposed on the decomposition to allow a deeper investigation of the correlation structure of the yeast data. These constraints greatly extend this new family of models, with the addition of many parsimonious models. The Canadian Journal of Statistics 38:153–168; 2010 © 2010 Statistical Society of Canada     

EDF tests of goodness of fit for transform-both-sides models

Very often in regression analysis, a particular functional form connecting known covariates and unknown parameters is either suggested by previous work or demanded by theoretical considerations so that the deterministic part of the responses has a known form. However, the underlying error structure is often less well understood. In this case, the transform-both-sides (TBS) models are appropriate. In this paper we generalize the usual TBS models and develop tests to assess goodness of fit when fitting TBS or GTBS models. Parameter estimation is discussed, and tests based on the Cramér-von Mises statistic and the Anderson-Darling statistic are presented with a table suitable for finite-sample applications.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

Conditional variance estimation via nonparametric generalized additive models

Generalized additive models provide a way of circumventing curse of dimension in a wide range of nonparametric regression problem. In this paper, we present a multiplicative model for conditional variance functions where one can apply a generalized additive regression method. This approach extends Fan and Yao (1998) to multivariate cases with a multiplicative structure. In this approach, we use squared residuals instead of using log-transformed squared residuals. This idea gives a smaller variance than Yu (2017) when the variance of squared error is smaller than the variance of log-transformed squared error. We provide estimators based on quasi-likelihood and an iterative algorithm based on smooth backfitting for generalized additive models. We also provide some asymptotic properties of estimators and the convergence of proposed algorithm. A numerical study shows the empirical evidence of the theory.     

Parametric and semiparametric hypotheses in the linear model

The independent additive errors linear model consists of a structure for the mean and a separate structure for the error distribution. The error structure may be parametric or it may be semiparametric. Under alternative values of the mean structure, the best fitting additive errors model has an error distribution which can be represented as the convolution of the actual error distribution and the marginal distribution of a misspecification term. The model misspecification term results from the covariates' distribution. Conditions are developed to distinguish when the semiparametric model yields sharper inference than the parametric model and vice versa. The main conditions concern the actual error distribution and the covariates' distribution. The theoretical results explain a paradoxical finding in semiparametric Bayesian modelling, where the posterior distribution under a semiparametric model is found to be more concentrated than is the posterior distribution under a corresponding parametric model. The paradox is illustrated on a set of allometric data. The Canadian Journal of Statistics 39: 165–180; 2011 ©2011 Statistical Society of Canada     

Test for homogeneity in Hardy–Weinberg normal mixture model

The phenotype of a quantitative trait locus (QTL) is often modeled by a finite mixture of normal distributions. If the QTL effect depends on the number of copies of a specific allele one carries, then the mixture model has three components. In this case, the mixing proportions have a binomial structure according to the Hardy–Weinberg equilibrium. In the search for QTL, a significance test of homogeneity against the Hardy–Weinberg normal mixture model alternative is an important first step. The LOD score method, a likelihood ratio test used in genetics, is a favored choice. However, there is not yet a general theory for the limiting distribution of the likelihood ratio statistic in the presence of unknown variance. This paper derives the limiting distribution of the likelihood ratio statistic, which can be described by the supremum of a quadratic form of a Gaussian process. Further, the result implies that the distribution of the modified likelihood ratio statistic is well approximated by a chi-squared distribution. Simulation results show that the approximation has satisfactory precision for the cases considered. We also give a real-data example.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Univariate case

This paper gives a comparative study of the K-means algorithm and the mixture model (MM) method for clustering normal data. The EM algorithm is used to compute the maximum likelihood estimators (MLEs) of the parameters of the MM model. These parameters include mixing proportions, which may be thought of as the prior probabilities of different clusters; the maximum posterior (Bayes) rule is used for clustering. Hence, asymptotically the MM method approaches the Bayes rule for known parameters, which is optimal in terms of minimizing the expected misclassification rate (EMCR).     

Log-linear models for mutations in the HIV genome

We discuss a general application of categorical data analysis to mutations along the HIV genome. We consider a multidimensional table for several positions at the same time. Due to the complexity of the multidimensional table, we may collapse it by pooling some categories. However, the association between the remaining variables may not be the same as before collapsing. We discuss the collapsibility of tables and the change in the meaning of parameters after collapsing categories. We also address this problem with a log-linear model. We present a parameterization with the consensus output as the reference cell as is appropriate to explain genomic mutations in HIV. We also consider five null hypotheses and some classical methods to address them. We illustrate methods for six positions along the HIV genome, through consideration of all triples of positions.     

The random-effects proportional hazards model with grouped survival data: a comparison between the grouped continuous and continuation ratio versions

Summary.  When analysing grouped time survival data having a hierarchical structure it is often appropriate to assume a random-effects proportional hazards model for the latent continuous time and then to derive the corresponding grouped time model. There are two formally equivalent grouped time versions of the proportional hazards model obtained from different perspec-tives, known as the continuation ratio and the grouped continuous models. However, the two models require distinct estimation procedures and, more importantly, they differ substantially when extended to time-dependent covariates and/or non-proportional effects. The paper discusses these issues in the context of random-effects models, illustrating the main points with an application to a complex data set on job opportunities for a cohort of graduates.