Trajectory Modeling of Longitudinal Binary Data: Application of the EM Algorithm for Mixture Models

A developmental trajectory describes the course of behavior over time. Identifying multiple trajectories within an overall developmental process permits a focus on subgroups of particular interest. We introduce a framework for identifying trajectories by using the Expectation-Maximization (EM) algorithm to fit semiparametric mixtures of logistic distributions to longitudinal binary data. For performance comparison, we consider full maximization algorithms (PROC TRAJ in SAS), standard EM, and two other EM-based algorithms for speeding up convergence. Simulation shows that EM methods produce more accurate parameter estimates. The EM methodology is illustrated with a longitudinal dataset involving adolescents smoking behaviors.     

Flexible Covariance Structures for Categorical Dependent Variables Through Finite Mixtures of Generalized Extreme Value Models

A new class of finite mixture discrete choice models, denoted FinMix (fīn m?ks), is introduced. These arise from the combination of a finite number of core Generalized Extreme Value (GEV) models to achieve more flexible functional forms, particularly in terms of error covariance structures. Example members of the class include combinations of (1) Multinomial Logit (MNL) models with differing scales, (2) multinomial logit with nested MNL models, (3) tree extreme value models with differing preference trees, and so on. Compatibility of FinMix models with utility maximization is easily determined, which permits empirical investigation of the suitability of specific model forms for economic evaluation exercises.     

Macro-Economic Factors in Credit Risk Calculations: Including Time-Varying Covariates in Mixture Cure Models

The prediction of the time of default in a credit risk setting via survival analysis needs to take a high censoring rate into account. This rate is because default does not occur for the majority of debtors. Mixture cure models allow the part of the loan population that is unsusceptible to default to be modeled, distinct from time of default for the susceptible population. In this article, we extend the mixture cure model to include time-varying covariates. We illustrate the method via simulations and by incorporating macro-economic factors as predictors for an actual bank dataset.     

The REBMIX Algorithm for the Multivariate Finite Mixture Estimation

The article extends the REBMIX to multivariate data. Random variables may follow normal, lognormal, or Weibull parametric families and should be independent within components. The initial weights and component parameters are not required. Preprocessing of observations folows the histogram, Parzen window, or k-nearest neighbor approach. The number of components, weights, and component parameters are gained iteratively by using information measures of the distance, such as the total of positive relative deviations and the information criterion. The number of classes or the number of the nearest neighbors can be optimized, as well. The REBMIX software is available on http://www.fs.uni-lj.si/lavek.     

A Finite Mixture Three-Parameter Weibull Model for the Analysis of Wind Speed Data

This article presents a mixture three-parameter Weibull distribution to model wind speed data. The parameters are estimated by using maximum likelihood (ML) method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved numerically by the interior-point method. By applying this model to four lattice-point wind speed sequences extracted from National Centers for Environmental Prediction (NCEP) reanalysis data, it is observed that the mixture three-parameter Weibull distribution model proposed in this paper provides a better fit than the existing Weibull models for the analysis of wind speed data under study.     

Edgeworth Expansion of the Moment-Based Test for Homogeneity in an NEF-QVF Mixture Model

In this article, we study the moment-based test procedure for a mixture distribution for the Natural exponential family with quadratic variance functions (NEF-QVF) family proposed by Ning et al. (2009b Ning, W., Zhang, S. G. and Yu, C. 2009b. A moment-based test for the homogeneity in mixture natural exponential family with quadratic variance functions. Statistical and Probability Letters, 79: 828–834. [Crossref] , [Google Scholar]) in the small sample size scenario. We derive the approximation for the null distribution of the test statistic by the Edgeworth expansion. The simulations are conducted for a binomial mixture distribution, which includes the situation corresponding to the detection of the linkage in the genetic analysis, with different sample sizes and family sizes at various significance levels. The simulation results show that our test performs reasonably well. We also apply the proposed method to the real clinical data to verify the significant difference between two drug treatments. The critical values associated with a binomial mixture distribution are also provided.     

A Graphical Tool for Interpreting Regression Coefficients of Trinomial Logit Models

Multinomial logit (also termed multi-logit) models permit the analysis of the statistical relation between a categorical response variable and a set of explicative variables (called covariates or regressors). Although multinomial logit is widely used in both the social and economic sciences, the interpretation of regression coefficients may be tricky, as the effect of covariates on the probability distribution of the response variable is nonconstant and difficult to quantify. The ternary plots illustrated in this article aim at facilitating the interpretation of regression coefficients and permit the effect of covariates (either singularly or jointly considered) on the probability distribution of the dependent variable to be quantified. Ternary plots can be drawn both for ordered and for unordered categorical dependent variables, when the number of possible outcomes equals three (trinomial response variable); these plots allow not only to represent the covariate effects over the whole parameter space of the dependent variable but also to compare the covariate effects of any given individual profile. The method is illustrated and discussed through analysis of a dataset concerning the transition of master’s graduates of the University of Trento (Italy) from university to employment.     

An Alternative Estimation Method for the Semiparametric Accelerated Failure Time Mixture Cure Model

We propose an alternative estimation method for the semiparametric accelerated failure time mixture cure model by incorporating the profile likelihood into the M-step of the EM algorithm. The proposed method performs as well as the existing methods when the censoring is light and better than the existing methods when the censoring is moderate from the simulation studies. Regarding to the computational time, the proposed method runs faster than the existing methods.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Mixture Distributions Based Methods of Calibration for the Empirical Log-Likelihood Ratio

Empirical likelihood ratio confidence regions based on the chi-square calibration suffer from an undercoverage problem in that their actual coverage levels tend to be lower than the nominal levels. The finite sample distribution of the empirical log-likelihood ratio is recognized to have a mixture structure with a continuous component on [0, + ∞) and a point mass at + ∞. The undercoverage problem of the Chi-square calibration is partly due to its use of the continuous Chi-square distribution to approximate the mixture distribution of the empirical log-likelihood ratio. In this article, we propose two new methods of calibration which will take advantage of the mixture structure; we construct two new mixture distributions by using the F and chi-square distributions and use these to approximate the mixture distributions of the empirical log-likelihood ratio. The new methods of calibration are asymptotically equivalent to the chi-square calibration. But the new methods, in particular the F mixture based method, can be substantially more accurate than the chi-square calibration for small and moderately large sample sizes. The new methods are also as easy to use as the chi-square calibration.     

Mixture model on the variance for the differential analysis of gene expression data

Summary.  In microarray experiments, accurate estimation of the gene variance is a key step in the identification of differentially expressed genes. Variance models go from the too stringent homoscedastic assumption to the overparameterized model assuming a specific variance for each gene. Between these two extremes there is some room for intermediate models. We propose a method that identifies clusters of genes with equal variance. We use a mixture model on the gene variance distribution. A test statistic for ranking and detecting differentially expressed genes is proposed. The method is illustrated with publicly available complementary deoxyribonucleic acid microarray experiments, an unpublished data set and further simulation studies.     

A Bayesian Approach to the Estimation of Expected Cell Counts by Using Log-Linear Models

ABSTRACT

In this article, Bayesian estimation of the expected cell counts for log-linear models is considered. The prior specified for log-linear parameters is used to determine a prior for expected cell counts, by means of the family and parameters of prior distributions. This approach is more cost-effective than working directly with cell counts because converting prior information into a prior distribution on the log-linear parameters is easier than that of on the expected cell counts. While proceeding from the prior on log-linear parameters to the prior of the expected cell counts, we faced with a singularity problem of variance matrix of the prior distribution, and added a new precision parameter to solve the problem. A numerical example is also given to illustrate the usage of the new parameter.     

Estimation of principal points for a multivariate binary distribution using a log-linear model

This article proposes a method for estimating principal points for a multivariate binary distribution, assuming a log-linear model for the distribution. Through numerical simulation studies, the proposed parametric estimation method using a log-linear model is compared with a nonparametric estimation method.     

Mixture Representation of the Maximum Entropy Density Through Archimedean Copulas

In this article, we obtain a mixture representation of the maximum entropy density introduced by Rodrigues (2004 Rodrigues , J. ( 2004 ). An entropy model for dependent variables . Commun. Statist. Theor. Meth. 4 ( 33 ): 979 – 990 . [Google Scholar]) via Laplace approximation. This representation suggests, as in Sklar (1959 Sklar , A. ( 1959 ). Fonctions de répartition à ndimensions et marges . Publications de l 'Université de Paris 8 : 229 – 231 . [Google Scholar]), a dependence structure through Archimedean copulas independently of the specified marginal distributions. This result can be used as a natural Bayesian and non Bayesian procedure to estimate the dependence function and the marginal, separately.     

Testing the Homogeneity of Two Survival Functions Against a Mixture Alternative Based on Censored Data

When the survival distribution in a treatment group is a mixture of two distributions of the same family, traditional parametric methods that ignore the existence of mixture components or the nonparametric methods may not be very powerful. We develop a modified likelihood ratio test (MLRT) for testing homogeneity in a two sample problem with censored data and compare the actual type I error and power of the MLRT with that nonparametric log-rank test and parametric test through Monte-Carlo simulations. The proposed test is also applied to analyze data from a clinical trial on early breast cancer.     

On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis

In this article, we propose a new generalized multivariate log-gamma distribution. We consider the usage of the proposed multivariate distribution as the prior distribution in the Bayesian analysis. The generalized multivariate log-gamma distribution allows for the inclusion of prior knowledge on correlations between model parameters when likelihood is not in the form of a normal distribution. Use of the proposed distribution in the Bayesian analysis of log-linear models is also discussed.     

On the Characterization of a Mixture of Generalized Power Function Distributions by Conditional Expectation of Order Statistics

In the present article, we give some theorems to characterize the mixture of two generalized power function distributions based on conditional expectation of order statistics.     

Statistical Modeling for the Evaluation of Consumers’ Preferences

In 2009 a survey was performed in Veneto, a region in the north-east of Italy, to study the demand for wine and specifically for Passito, a typical Italian wine. The main goal of the study consisted in analyzing how the preferences and consumption habits of Passito vary depending on consumers’ characteristics. Specifically two kinds of statistical methods were applied: Covariate Uniform Binomial (CUB) model, a statistical approach for ordinal data to study the feeling toward Passito and the uncertainty of the respondents; classical logistic regression analysis, to describe how the attitude toward passito can be modeled as function of consumers’ covariates. Gender and residence were the most important covariates, useful in defining segments of consumers with significant differences in terms of Passito's preferences and consumption behavior. The logistic regression analysis allowed to complete the statistical analysis based on CUB models validating the results of the CUB model and estimating a model useful to predict the attitude toward the considered product for specific sub-groups of consumers.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

A Test for Multivariate Analysis of Variance in High Dimension

In this article, we consider the problem of testing a general multivariate linear hypothesis in a multivariate linear model when the N × p observation matrix is normally distributed with unknown covariance matrix, and N ≤ p. This includes the case of testing the equality of several mean vectors. A test is proposed which is a generalized version of the two-sample test proposed by Srivastava and Du (2008 Srivastava , M. S. , Du , M. ( 2008 ). A test for the mean vector with fewer observations than the dimension . J. Multivariate Anal. 99 : 386 – 402 .[Crossref], [Web of Science ®] , [Google Scholar]). The asymptotic null and nonnull distributions are obtained. The performance of this test is compared, theoretically as well as numerically, with the corresponding generalized version of the two-sample Dempster (1958 Dempster , A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29:995–1010.[Crossref] , [Google Scholar]) test, or more appropriately Bai and Saranadasa (1996 Bai , Z. , Saranadasa , H. ( 1996 ). Effect of high dimension: an example of a two sample problem . Statistica Sinica 6 : 311 – 329 .[Web of Science ®] , [Google Scholar]) test who gave its asymptotic version.