Mixture separation for mixed-mode data

One possible approach to cluster analysis is the mixture maximum likelihood method, in which the data to be clustered are assumed to come from a finite mixture of populations. The method has been well developed, and much used, for the case of multivariate normal populations. Practical applications, however, often involve mixtures of categorical and continuous variables. Everitt (1988) and Everitt and Merette (1990) recently extended the normal model to deal with such data by incorporating the use of thresholds for the categorical variables. The computations involved in this model are so extensive, however, that it is only feasible for data containing very few categorical variables. In the present paper we consider an alternative model, known as the homogeneous Conditional Gaussian model in graphical modelling and as the location model in discriminant analysis. We extend this model to the finite mixture situation, obtain maximum likelihood estimates for the population parameters, and show that computation is feasible for an arbitrary number of variables. Some data sets are clustered by this method, and a small simulation study demonstrates characteristics of its performance.     

Grouped penalization estimation of the osteoporosis data in the traditional Chinese medicine

Both continuous and categorical covariates are common in traditional Chinese medicine (TCM) research, especially in the clinical syndrome identification and in the risk prediction research. For groups of dummy variables which are generated by the same categorical covariate, it is important to penalize them group-wise rather than individually. In this paper, we discuss the group lasso method for a risk prediction analysis in TCM osteoporosis research. It is the first time to apply such a group-wise variable selection method in this field. It may lead to new insights of using the grouped penalization method to select appropriate covariates in the TCM research. The introduced methodology can select categorical and continuous variables, and estimate their parameters simultaneously. In our application of the osteoporosis data, four covariates (including both categorical and continuous covariates) are selected out of 52 covariates. The accuracy of the prediction model is excellent. Compared with the prediction model with different covariates, the group lasso risk prediction model can significantly decrease the error rate and help TCM doctors to identify patients with a high risk of osteoporosis in clinical practice. Simulation results show that the application of the group lasso method is reasonable for the categorical covariates selection model in this TCM osteoporosis research.     

Non-parametric smoothing of the location model in mixed variable discrimination

The location model is a familiar basis for discriminant analysis of mixtures of categorical and continuous variables. Its usual implementation involves second-order smoothing, using multivariate regression for the continuous variables and log-linear models for the categorical variables. In spite of the smoothing, these procedures still require many parameters to be estimated and this in turn restricts the categorical variables to a small number if implementation is to be feasible. In this paper we propose non-parametric smoothing procedures for both parts of the model. The number of parameters to be estimated is dramatically reduced and the range of applicability thereby greatly increased. The methods are illustrated on several data sets, and the performances are compared with a range of other popular discrimination techniques. The proposed method compares very favourably with all its competitors.     

Mixture of latent trait analyzers for model-based clustering of categorical data

Model-based clustering methods for continuous data are well established and commonly used in a wide range of applications. However, model-based clustering methods for categorical data are less standard. Latent class analysis is a commonly used method for model-based clustering of binary data and/or categorical data, but due to an assumed local independence structure there may not be a correspondence between the estimated latent classes and groups in the population of interest. The mixture of latent trait analyzers model extends latent class analysis by assuming a model for the categorical response variables that depends on both a categorical latent class and a continuous latent trait variable; the discrete latent class accommodates group structure and the continuous latent trait accommodates dependence within these groups. Fitting the mixture of latent trait analyzers model is potentially difficult because the likelihood function involves an integral that cannot be evaluated analytically. We develop a variational approach for fitting the mixture of latent trait models and this provides an efficient model fitting strategy. The mixture of latent trait analyzers model is demonstrated on the analysis of data from the National Long Term Care Survey (NLTCS) and voting in the U.S. Congress. The model is shown to yield intuitive clustering results and it gives a much better fit than either latent class analysis or latent trait analysis alone.     

Analytical closed-form solution for binary logit regression by categorical predictors

In contrast to the common belief that the logit model has no analytical presentation, it is possible to find such a solution in the case of categorical predictors. This paper shows that a binary logistic regression by categorical explanatory variables can be constructed in a closed-form solution. No special software and no iterative procedures of nonlinear estimation are needed to obtain a model with all its parameters and characteristics, including coefficients of regression, their standard errors and t-statistics, as well as the residual and null deviances. The derivation is performed for logistic models with one binary or categorical predictor, and several binary or categorical predictors. The analytical formulae can be used for arithmetical calculation of all the parameters of the logit regression. The explicit expressions for the characteristics of logit regression are convenient for the analysis and interpretation of the results of logistic modeling.     

On the bias of factor score determinacy coefficients based on different estimation methods of the exploratory factor model

The factor score determinacy coefficient represents the common variance of the factor score predictor with the corresponding factor. The aim of the present simulation study was to compare the bias of determinacy coefficients based on different estimation methods of the exploratory factor model. Overall, determinacy coefficients computed from parameters based on maximum likelihood estimation, unweighted least squares estimation, and principal axis factoring were more precise than determinacy coefficients based on generalized least squares estimation and alpha factoring.     

The clustering of mixed-mode data: a comparison of possible approaches

Various methods for clustering mixed-mode data are compared. It is found that a method based on a finite mixture model in which the observed categorical variables are generated from underlying continuous variables out-performs more conventional methods when applied to artificially generated data. This method also performs best when applied to Fisher's iris data in which two of the variables are categorized by applying thresholds.     

Simulation of close-to-reality population data for household surveys with application to EU-SILC

Statistical simulation in survey statistics is usually based on repeatedly drawing samples from population data. Furthermore, population data may be used in courses on survey statistics to explain issues regarding, e.g., sampling designs. Since the availability of real population data is in general very limited, it is necessary to generate synthetic data for such applications. The simulated data need to be as realistic as possible, while at the same time ensuring data confidentiality. This paper proposes a method for generating close-to-reality population data for complex household surveys. The procedure consists of four steps for setting up the household structure, simulating categorical variables, simulating continuous variables and splitting continuous variables into different components. It is not required to perform all four steps so that the framework is applicable to a broad class of surveys. In addition, the proposed method is evaluated in an application to the European Union Statistics on Income and Living Conditions (EU-SILC).     

Bayesian latent variable models for clustered mixed outcomes

A general framework is proposed for modelling clustered mixed outcomes. A mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables, and an arbitrary function relates the underlying variables to be observed outcomes. The model accommodates multilevel data structures, general covariate effects and distinct link functions and error distributions for each underlying variable. Within the framework proposed, novel models are developed for clustered multiple binary, unordered categorical and joint discrete and continuous outcomes. A Markov chain Monte Carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables. Because of the flexibility of the modelling framework and estimation procedure, extensions to ordered categorical outcomes and more complex data structures are straightforward. The methods are illustrated by using data from a reproductive toxicity study.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

Indirect estimation of (latent) linear models with ordinal regressors A Monte Carlo study and some empirical illustrations

Summary This paper investigates the effects of ordinal regressors in linear regression models and in limited dependent variable models. Each ordered categorical variable is interpreted as a rough measurement of an underlying continuous variable as it is often done in microeconometrics for the dependent variable. It is shown that using ordinal indicators only leads to correct answers in a few special cases. In most situations, the usual estimators are biased. In order to estimate the parameters of the model consistently, the indirect estimation procedure suggested by Gourieroux et al. (1993) is applied. To demonstrate this method, first a simulation study is performed and then in a second step, two real data sets are used. In the latter case, continuous regressors are transformed into categorical variables to study the behavior of the estimation procedure. The method is extended to the case of limited dependent variable models. In general, the indirect estimators lead to adequate results. Received: March 27, 2000; revised version: March 6, 2001     

A distance based regression model for prediction with mixed data

A multiple regression method based on distance analysis and metric scaling is proposed and studied. This method allow us to predict a continuous response variable from several explanatory variables, is compatible with the general linear model and is found to be useful when the predictor variables are both continuous and categorical. Real data examples are given to illustrate the results obtained.     

Sample size calculation for a proportional hazards mixture cure model with nonbinary covariates

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced medical technology, some patients can be cured of certain chronic diseases, and the proportional hazards mixture cure model has been developed to handle survival data with potential cure information. Given the needs of survival trials with potential cure proportions, a corresponding sample size formula based on the log-rank test statistic for binary covariates has been proposed by Wang et al. [25]. However, a sample size formula based on continuous variables has not been developed. Herein, we presented sample size and power calculations for the mixture cure model with continuous variables based on the log-rank method and further modified it by Ewell's method. The proposed approaches were evaluated using simulation studies for synthetic data from exponential and Weibull distributions. A program for calculating necessary sample size for continuous covariates in a mixture cure model was implemented in R.     

Mixtures of general location model with factor analyzer covariance structure for clustering mixed type data

Cluster analysis is one of the most widely used method in statistical analyses, in which homogeneous subgroups are identified in a heterogeneous population. Due to the existence of the continuous and discrete mixed data in many applications, so far, some ordinary clustering methods such as, hierarchical methods, k-means and model-based methods have been extended for analysis of mixed data. However, in the available model-based clustering methods, by increasing the number of continuous variables, the number of parameters increases and identifying as well as fitting an appropriate model may be difficult. In this paper, to reduce the number of the parameters, for the model-based clustering mixed data of continuous (normal) and nominal data, a set of parsimonious models is introduced. Models in this set are extended, using the general location model approach, for modeling distribution of mixed variables and applying factor analyzer structure for covariance matrices. The ECM algorithm is used for estimating the parameters of these models. In order to show the performance of the proposed models for clustering, results from some simulation studies and analyzing two real data sets are presented.     

Identifiable finite mixtures of location models for clustering mixed-mode data

For clustering mixed categorical and continuous data, Lawrence and Krzanowski (1996) proposed a finite mixture model in which component densities conform to the location model. In the graphical models literature the location model is known as the homogeneous Conditional Gaussian model. In this paper it is shown that their model is not identifiable without imposing additional restrictions. Specifically, for g groups and m locations, (g!)m–1 distinct sets of parameter values (not including permutations of the group mixing parameters) produce the same likelihood function. Excessive shrinkage of parameter estimates in a simulation experiment reported by Lawrence and Krzanowski (1996) is shown to be an artifact of the model's non-identifiability. Identifiable finite mixture models can be obtained by imposing restrictions on the conditional means of the continuous variables. These new identified models are assessed in simulation experiments. The conditional mean structure of the continuous variables in the restricted location mixture models is similar to that in the underlying variable mixture models proposed by Everitt (1988), but the restricted location mixture models are more computationally tractable.     

PERFORMANCE OF THE LOCATION LINEAR DISCRIMINANT FUNCTION UNDER ACROSS-LOCATION HETEROSCEDASTICITY

ABSTRACT

Classification of data consisting of both categorical and continuous variables between two groups is often handled by the sample location linear discriminant function confined to each of the locations specified by the observed values of the categorical variables. Homoscedasticity of across-location conditional dispersion matrices of the continuous variables is often assumed. Quite often, interactions between continuous and categorical variables cause across-location heteroscedasticity. In this article, we examine the effect of heterogeneous across-location conditional dispersion matrices on the overall expected and actual error rates associated with the sample location linear discriminant function. Performance of the sample location linear discriminant function is evaluated against the results for the restrictive classifier adjusted for across-location heteroscedasticity. Conclusions based on a Monte Carlo study are reported.     

Analyzing Designed Experiments with Multiple Responses

This paper is an overview of a unified framework for analyzing designed experiments with univariate or multivariate responses. Both categorical and continuous design variables are considered. To handle unbalanced data, we introduce the so-called Type II* sums of squares. This means that the results are independent of the scale chosen for continuous design variables. Furthermore, it does not matter whether two-level variables are coded as categorical or continuous. Overall testing of all responses is done by 50-50 MANOVA, which handles several highly correlated responses. Univariate p-values for each response are adjusted by using rotation testing. To illustrate multivariate effects, mean values and mean predictions are illustrated in a principal component score plot or directly as curves. For the unbalanced cases, we introduce a new variant of adjusted means, which are independent to the coding of two-level variables. The methodology is exemplified by case studies from cheese and fish pudding production.     

Empirical Bayes Small-Area Estimation Using Logistic Regression Models and Summary Statistics

Many of the available methods for estimating small-area parameters are model-based approaches in which auxiliary variables are used to predict the variable of interest. For models that are nonlinear, prediction is not straightforward. MacGibbon and Tomberlin and Farrell, MacGibbon, and Tomberlin have proposed approaches that require microdata for all individuals in a small area. In this article, we develop a method, based on a second-order Taylor-series expansion to obtain model-based predictions, that requires only local-area summary statistics for both continuous and categorical auxiliary variables. The methodology is evaluated using data based on a U.S. Census.     

Semiparametric multinomial logit models for analysing consumer choice behaviour

The multinomial logit model (MNL) is one of the most frequently used statistical models in marketing applications. It allows one to relate an unordered categorical response variable, for example representing the choice of a brand, to a vector of covariates such as the price of the brand or variables characterising the consumer. In its classical form, all covariates enter in strictly parametric, linear form into the utility function of the MNL model. In this paper, we introduce semiparametric extensions, where smooth effects of continuous covariates are modelled by penalised splines. A mixed model representation of these penalised splines is employed to obtain estimates of the corresponding smoothing parameters, leading to a fully automated estimation procedure. To validate semiparametric models against parametric models, we utilise different scoring rules as well as predicted market share and compare parametric and semiparametric approaches for a number of brand choice data sets.     

Extending dual multiple factor analysis to categorical tables

This paper describes a proposal for the extension of the dual multiple factor analysis (DMFA) method developed by Lê and Pagès 15 to the analysis of categorical tables in which the same set of variables is measured on different sets of individuals. The extension of DMFA is based on the transformation of categorical variables into properly weighted indicator variables, in a way analogous to that used in the multiple factor analysis of categorical variables. The DMFA of categorical variables enables visual comparison of the association structures between categories over the sample as a whole and in the various subsamples (sets of individuals). For each category, DMFA allows us to obtain its global (considering all the individuals) and partial (considering each set of individuals) coordinates in a factor space. This visual analysis allows us to compare the set of individuals to identify their similarities and differences. The suitability of the technique is illustrated through two applications: one using simulated data for two groups of individuals with very different association structures and the other using real data from a voting intention survey in which some respondents were interviewed by telephone and others face to face. The results indicate that the two data collection methods, while similar, are not entirely equivalent.