modelling bivariate ordinal responses smoothly with examples from ophthalmology and genetics a non parametric implementation of the bivariate dale model bdm is presented as an extension of the generalized additive model gam of hastie and tibshirani the original bdm is an example of a bivariate generalized linear model in this paper smoothing is introduced on the marginal as well as on the association level our non parametric procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in the linear bdm hence it also provides a kind of goodness of fit test for a bivariate generalized linear model cubic smoothing spline functions for the covariates are estimated by maximizing a penalized version of the log likelihood the method is applied to two studies the first study is the classical wisconsin epidemiologic study of diabetic retinopathy the second study is a twin study where the association between the elements of twin pairs is of primary interest the results show that smoothing on the association level can give a significant improvement to the model fit

multiple loci mapping via model free variable selection summary despite recent flourish of proposals o nv ariable selection genome wide multiple loci mapping remains to be challenging the majority of existing variable selection methods impose a model and often the homoscedastic linear model prior to selection however the true association between the phenotypical trait and the genetic markers is rarely known ap riori and the presence of epistatic interactions makes the association more complex than al inear relation model free variable selection o ers a useful alternative in th is context but the fact that the number of markers p often far exceeds the number of experimental units n renders all the existing model free solutions that require n p inapplicable in this article we examine a number of model free variable selection methods for small n large p regressions in the context of genome wide multiple loci mapping we propose and advocate a multivariate group wise adaptive penalization solution which requires no model prespecification and thus works for complex trait marker association and handles one variable at a time so that works for n p e ctiveness of the new method is demonstrated through bo th intensive simulations and a comprehensive real data analysis across 6100 gene expression traits

an application of maximum likelihood and generalized estimating equations to the analysis of ordinal data from a longitudinal study with cases missing at random data are analysed from a longitudinal psychiatric study in which there are no dropouts that do not occur completely at random a marginal proportional odds model is fitted that relates the response severity of side effects to various covariates two methods of estimation are used generalized estimating equations gee and maximum likelihood ml both the complete set of data and the data from only those subjects completing the study are analysed for the completers only data the gee and ml analyses produce very similar results these results differ considerably from those obtained from the analyses of the full data set there are also marked differences between the results obtained from the gee and ml analysis of the full data set the occurrence of such differences is consistent with the presence of a non completely random dropout process and it can be concluded in this example that both the analyses of the completers only and the gee analysis of the full data set produce misleading conclusions about the relationships between the response and covariates

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

smoothing spline anova for multivariate bernoulli observations with application to ophthalmology data we combine a smoothing spline analysis of variance ss anova model and a log linear model to build a partly flexible model for multivariate bernoulli data the joint distribution conditioning on the predictor variables is estimated the log odds ratio is used to measure the association between outcome variables a numerical scheme based on the block one step successive over relaxation sor newton ralphson algorithm is proposed to obtain an approximate solution for the variational problem we extend the generalized approximate cross validation gacv and the randomized gacv for choosing smoothing parameters to the case of multivariate bernoulli responses the randomized version is fast and stable to compute and is used to adaptively select smoothing parameters in each block onestep sor iteration approximate bayesian confidence intervals are obtained for the flexible estimates of the conditional logit functions simulation studies are conducted to check the performance of the proposed method using the com

estimation for a partial linear single index model in this paper we study the estimation for a partial linear single index model a two stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index as well as the parameters in the linear component of the model asymptotic normality is established for both parametric components for the index a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance the estimator of the nonparametric link function achieves optimal convergence rates and the structural error variance is obtained in addition the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters a simulation study is performed and an application to a real dataset is illustrated the extension to multiple indices is briefly sketched

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

estimating central subspaces via inverse third moments summary modern graphical tools have enhanced our ability to learn many things from data directly in recent years dimension reduction has proven to be an effective tool for generating low dimensional summary plots without appreciable loss of information some well known inverse regression methods for dimension reduction such as sliced inverse regression li 1991 and sliced average variance estimation cook weisberg 1991 have been developed to estimate summary plots for regression and discriminant analysis in this paper we suggest a new method that makes use of inverse third moments this method can find structure beyond that found by sliced inverse regression and sliced average variance estimation particularly regression mixtures illustrative examples are presented

local versus global association for bivariate ordered responses summary local and global parameters modelling association are examined for contingency tables with both margins ordered responses local association parameters are crossratios of 2 x 2 subtables of adjacent cell probabilities global association parameters are cross ratios of quadrant probabilities the interpretation and various properties of these measures of association are compared and illustrated

marginal methods for correlated binary data with misclassified responses misclassification is a longstanding concern in medical research although there has been much research concerning error prone covariates relatively little work has been directed to problems with response variables subject to error in this paper we focus on misclassification in clustered or longitudinal outcomes we propose marginal analysis methods to handle binary responses which are subject to misclassification the proposed methods have several appealing features including simultaneous inference for both marginal mean and association parameters and they can handle misclassified responses for a number of practical scenarios such as the case with a validation subsample or replicates furthermore the proposed methods are robust to model misspecification in a sense that no full distributional assumptions are required numerical studies demonstrate satisfactory performance of the proposed methods under a variety of settings copyright 2011 oxford university press

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

generalized estimating equations for correlated binary data using the odds ratio as a measure of association summary moment methods for analyzing repeated binary responses have been proposed by liang zeger 1986 and extended by prentice 1988 in their generalized estimating equations both liang zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses because the odds ratio has many desirable properties and some investigators may find the odds ratio is easier to interpret we discuss modelling the association between binary responses at pairs of times with the odds ratio we then modify the estimating equations of prentice to estimate the odds ratios in simulations the parameter estimates for the logistic regression model for the marginal probabilities appear slightly more efficient when using the odds ratio parameterization

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

determining the dimensionality in sliced inverse regression abstract a general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term sliced inverse regression is a method for determining these linear combinations in this article we address the problem of determining how many linear combinations are involved procedures based on conditional means and conditional covariance matrices as well as a procedure combining the two approaches are considered in each case we develop a test that has an asymptotic chi squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

approximating conditional distribution functions using dimension reduction motivated by applications to prediction and forecasting we sug gest methods for approximating the conditional distribution function of a random variable y given a dependent random d vector x the idea is to estimate not the distribution of y x but that of y t x where the unit vectoris selected so that the approximation is opti mal under a least squares criterion we show thatmay be estimated root n consistently furthermore estimation of the conditional distri bution function of y givent x has the same first order asymptotic properties that it would enjoy ifwere known the proposed method is illustrated using both simulated and real data examples showing its effectiveness for both independent datasets and data from time series numerical work corroborates the theoretical result thatcan be estimated particularly accurately

multivariate logistic models summary when data composed of several categorical responses together with categorical or continuous predictors are observed the multivariate logistic transform introduced by mccullagh and nelder can be used to define a class of regression models that is in many applications particularly suitable for relating the joint distribution of the responses to predictors in this paper we give a general definition of this class of models and study their properties a computational scheme for performing maximum likelihood estimation for data sets of moderate size is described and a system of model formulae that succinctly define particular models is introduced applications of these models to longitudinal problems are illustrated by numerical examples

parametric robust inferences for correlated ordinal data the aim of this article is to provide asymptotically valid likelihood inferences about regression parameters for correlated ordinal response variables the legitimacy of this novel approach requires no knowledge of the underlying joint distributions so long as their second moments exist the efficacy of the proposed parametric approach is demonstrated via simulations and the analyses of two real data sets

an appraisal of methods for the analysis of longitudinal categorical data with mar drop outs a number of methods for analysing longitudinal ordinal categorical data with missing at random drop outs are considered two are maximum likelihood methods maxlik which employ marginal global odds ratios to model associations the remainder use weighted or unweighted generalized estimating equations gee two of the gee use cholesky decomposed standardized residuals to model the association structure while another three extend methods developed for longitudinal binary data in which the association structures are modelled using either gaussian estimation multivariate normal estimating equations or conditional residuals simulated data sets were used to discover differences among the methods in terms of biases variances and convergence rates when the association structure is misspecified the methods were also applied to a real medical data set two of the gee methods referred to as cond and ml norm in this paper and by their originators were found to have relatively good convergence rates and mean squared errors for all sample sizes 80 120 300 considered and one more referred to as mgee in this paper and by its originators worked fairly well for all but the smallest sample size 80

an application of isotonic longitudinal marginal regression to monitoring the healing process this paper discusses marginal regression for repeated ordinal measurements that are isotonic over time such data are often observed in longitudinal studies on healing processes where due to recovery the status of patients only improves or stays the same we show how this prior information can be used to construct appropriate and parsimoniously parametrized marginal models as a second aspect we also incorporate nonparametric fitting of covariate effects via a penalized quasi likelihood or gee approach we illustrate our methods by an application to injuries from sporting activities

diagnostics for heteroscedasticity in regression summary for the usual regression model without replication we provide a diagnostic test for heteroscedasticity based on the score statistic a graphical procedure to complement the score test is also presented some key u ords influence linear model residual score test diagnostic methods in linear regression are used to examine the appropriateness of assumptions underlying the modelling process and to locate unusual characteristics of the data that may influence conclusions the recent literature on diagnostics is dominated by studies of methods for the detection of influential observations cook weisberg 1982 provide a review diagnostics for the relevance of specific assumptions however have not received the same degree of attention even though these may be of equal importance our purpose here is to provide appropriate diagnostic techniques to aid in an assessment of the validity of the usual assumption of homoscedasticity when little or no replication is present available methods for studying this assumption include both graphical and nongraphical procedures the usual graphical procedure consists of plotting the ordinary least squares residuals against fitted values or an explanatory variable a megaphone shaped pattern is taken as evidence that the variance depends on the quantity plotted on tlle abscissa weisberg 1980 chapter 6 in 3 we suggest several ways in which this

global cross ratio models for bivariate discrete ordered responses summary a family of statistical models is presented for bivariate discrete response to a regressor when both components of the response have ordered categories association between components is expressed in terms of global cross ratios cross product ratios of quadrant probabilities for each double dichotomy of the response table of probabilities into quadrants pearson and heron 1913 biometrika 9 159 315 these models are extensions to the work of plackett 1965 journal of the american statistical association 60 516 522 and mantel and brown 1973 biometrics 29 649 665 the marginal cumulative probabilities may satisfy linear logistic or other generalized linear models mccullagh 1980 journal of the royal statistical society series b 42 109 142 an analysis of patients postoperative pain level and medication frequency illustrates these methods

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

a proportional odds model with subject specific effects for repeated ordered categorical responses summary suppose subjects make repeated responses on the same ordered categorical scale we propose a generalization of the rasch model that expresses the cumulative logit of the response distribution using subject parameters and a proportional odds structure for item effects parameters in the model describe subject specific rather than populationaveraged effects consistent estimation of the effects requires eliminating the subject parameters we accomplish this by simultaneous fitting of rasch models conditional on sufficient statistics for those parameters for the possible binary collapsings of the response the fitting process uses an improved newton raphson algorithm for fitting generalized loglinear models by maximum likelihood estimation subject to constraints for the case of two items we give simple expressions for an effect estimate and its standard error and suggest a test of marginal homogeneity for ordinal matched pairs

marginal modelling of multivariate categorical data this paper describes likelihood methods of analysis for multivariate categorical data the joint distribution is specified in terms of marginal mean functions and pairwise and higher order association measures for the association the emphasis is on global odds ratios the method allows flexible formulation of a broad class of designs such as repeated measurements longitudinal studies interrater agreement and cross over trials the proposed model can be used for parameter estimation and hypothesis testing simple fitting algorithms are proposed the method is illustrated using a data example

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

sliced regression for dimension reduction a new dimension reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed compared with the traditional inverse regression methods e g sliced inverse regression sir the new method is free of the linearity condition and has much better estimation accuracy compared with the direct estimation methods e g mave the new method is much more robust against extreme values and can capture the entire central subspace cs exhaustively to determine the cs dimension a consistent cross validation criterion is developed extensive numerical studies including a real example confirm our theoretical findings

assessing the number of linear components in a general regression problem abstract this article presents a method for testing the dimension of a general regression model without assuming any specific model for the distribution of the regressors examples of application and comparisons with existing methods are also studied

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

simultaneously modeling joint and marginal distributions of multivariate categorical responses abstract we discuss model fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses the models are members of a broad class of generalized logit and loglinear models we fit them by improving a maximum likelihood algorithm that uses lagrange s method of undetermined multipliers and a newton raphson iterative scheme we also discuss goodness of fit tests and adjusted residuals and give asymptotic distributions of model parameter estimators for this class of models inferences are equivalent for poisson and multinomial sampling assumptions simultaneous models for joint and marginal distributions may be useful in a variety of applications including studies dealing with longitudinal data multiple indicators in opinion research cross over designs social mobility and inter rater agreement the models are illustrated for one such application using data from a recent general social survey regarding opinions about various types of government s

comparing estimation approaches for the illness death model under left truncation and right censoring left truncated data arise when lifetimes are observed only if they are larger than independent truncation times for example in a cross sectional sampling only individuals who live long enough to be present on the sampling day are observed there are several ways to perform statistical inference under this setting one can do the following i use an unconditional approach ii condition on the value of the truncation variable or iii condition on all the history up to the time of truncation the latter two approaches are equivalent when analyzing univariate survival outcomes but differ under the multi state framework in this paper we consider the illness death model and compare between the three estimation approaches in a parametric regression framework we show that approach ii is more efficient than the standard approach iii although it requires more computational effort approach i is the most efficient approach but it requires knowledge on the distribution of the truncation variable and hence is less robust the methods are compared using a theoretical example and simulations and are applied to intensive care units data collected in a cross sectional design where the illness state corresponds to a bloodstream infection

estimating the structural dimension of regressions via parametric inverse regression a new estimation method for the dimension of a regression at the outset of an analysis is proposed a linear subspace spanned by projections of the regressor vector x which contains part or all of the modelling information for the regression of a vector y on x and its dimension are estimated via the means of parametric inverse regression smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model no restrictions are placed on the distribution of the regressors the estimate of the dimension of the regression is based on optimal estimation procedures a simulation study shows the method to be more powerful than sliced inverse regression in some situations

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

marginal modeling of correlated ordinal data using a multivariate plackett distribution abstract an extension of the bivariate model suggested by dale is proposed for the analysis of dependent ordinal categorical data the so called multivariate dale model is constructed by first generalizing the bivariate plackett distribution to any dimensions because the approach is likelihood based it satisfies properties that are not fulfilled by other popular methods such as the generalized estimating equations approach the proposed method models both the marginal and the association structure in a flexible way the attractiveness of the multivariate dale model is illustrated in three key examples covering areas such as crossover trials longitudinal studies with patients dropping out from the study and discriminant analysis applications the differences and similarities with the generalized estimating approach are highlighted

maximum likelihood methods for log linear models when expected frequencies are subject to linear constraints abstract hypotheses relating to log linear models may be defined in terms of constraints on the frequencies or on the logarithms of the frequencies a two step algorithm is developed to allow fitting models containing constraints of both types this algorithm is shown to yield maximum likelihood estimates of the log linear parameters three examples are presented to illustrate cases in which these models are of interest

the analysis of longitudinal ordinal data with nonrandom drop out a model is proposed for longitudinal ordinal data with nonrandom drop out which combines the multivariate dale model for longitudinal ordinal data with a logistic regression model for drop out since response and drop out are modelled as conditionally independent given complete data the resulting likelihood can be maximised relatively simply using the em algorithm which with acceleration is acceptably fast and with appropriate additions can produce estimates of precision the approach is illustrated with an example such modelling of nonrandom drop out requires caution because the interpretation of the fitted models depends on assumptions that are unexaminable in a fundamental sense and the conclusions cannot be regarded as necessarily robust the main role of such modelling may be as a component of a sensitivity analysis

marginal regression models for the analysis of positive association of ordinal response variables given a set of discrete response variables some of which are ordinal and an arbitrary set of discrete explanatory variables we propose a simple matrix formulation for parameterising the saturated model as in glonek 1996 this is such that within a hierarchical structure marginal logits and log odds ratios of various possible types together with the remaining log linear interactions of high order may be modelled by equality and inequality constraints inequality constraints are particularly relevant for specifying models of positive association efficient algorithms are provided for computing maximum likelihood estimates under such constraints the asymptotic distribution of the likelihood ratio test is derived and an extension of the usual analysis of deviance is outlined which incorporates inequality constraints

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

multivariate analysis of longitudinal ordinal data with mixed effects models with application to clinical outcomes in osteoarthritis our objective was to evaluate the efficacy of robenacoxib in osteoarthritic dogs using four ordinal responses measured repeatedly over time we propose a multivariate probit mixed effects model to describe the joint evolution of endpoints and to evidence the intrinsic correlations between responses that are not due to treatment effect maximum likelihood computation is intractable within reasonable time frames we therefore use a pairwise modeling approach in combination with a stochastic em algorithm multidimensional ordinal responses with longitudinal measurements are a common feature in clinical trials however the standard methods for data analysis use unidimensional models resulting in a loss of information our methodology provides substantially greater insight than these methods for the evaluation of treatment effects and shows a good performance at low computational cost we thus believe that it could be used in routine practice to optimize the evaluation of treatment efficacy

composite multiclass losses we consider loss functions for multiclass prediction problems we show when a multiclass loss can be expressed as a proper composite loss which is the composition of a proper loss and a link function we extend existing results for binary losses to multiclass losses we subsume results on classification calibration by relating it to properness we determine the stationarity condition bregman representation order sensitivity and quasi convexity of multiclass proper losses we then characterise the existence and uniqueness of the composite representation for multiclass losses we show how the composite representation is related to other core properties of a loss mixability admissibility and strong convexity of multiclass losses which we characterise in terms of the hessian of the bayes risk we show that the simple integral representation for binary proper losses can not be extended to multiclass losses but offer concrete guidance regarding how to design different loss functions the conclusion drawn from these results is that the proper composite representation is a natural and convenient tool for the design of multiclass loss functions