The asymptotic covariance structure of log-linear model estimated parameters for the multiple recapture census

This paper presents a matrix formulation for log-linear model analysis of the incomplete contingency table which arises from multiple recapture census data. Explicit matrix product expressions are given for the asymptotic covariance structure of the maximum likelihood estimators of both the log-linear model parameter vector and the predicted value vector for the observed and missing cells. These results are illustrated for data pertaining to a population of children possessing a common congenital anomaly.     

Bayesian selection of log-linear models

A general methodology is presented for finding suitable Poisson log-linear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two- and three-way contingency tables, in which the regression coefficients are interpretable in terms of odds ratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of two-way simulated tables and for two three-way tables. These methods appear to be useful in selecting the best log-linear model and in estimating parameters of interest that reflect uncertainty in the true model.     

On bayes-type estimators of linearly or log-linearly constrained multinomial cell probabilities

Two Bayes-type procedures for estimating a multinomial cell probabilities vector, P, in the presence of linear constraints on the parameters are proposed and illustrated by examples from contingency table analysis. Estimation under log-linear constraints is also considered.     

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.     

Linear Models Methods for Some Rank Function Analyses of Ordinal Categorical Data

This paper is concerned with the analysis of ordinal data through linear models for rank function measures.Primary attention is directed at pairwise Mann-Whitney statistics for which dimension reduction is managed by use of a Bradley-Terry log-linear structure.The nature of linear models for such quantities is contrasted with that for mean ranks (or ridits).Aspects of application are illustrated with an example for which results of other methods are also given.     

USE OF A LOG-LINEAR MODEL TO COMPUTE THE EMPIRICAL SURVIVAL CURVE FROM INTERVAL-CENSORED DATA, WITH APPLICATION TO DATA ON TESTS FOR HIV POSITIVITY

We describe how a log-linear model can be used to compute the nonparametric maximum likelihood estimate of the survival curve from interval-censored data. This permits such computation to be performed with the aid of readily available statistical software such as GLIM or SAS. The method is illustrated with reference to data from a cohort of Danish homosexual men, each of whom was tested for HIV positivity on one or more of six possible follow-up times.     

Semiparametric multiparameter regression survival modeling

We consider a log-linear model for survival data, where both the location and scale parameters depend on covariates, and the baseline hazard function is completely unspecified. This model provides the flexibility needed to capture many interesting features of survival data at a relatively low cost in model complexity. Estimation procedures are developed, and asymptotic properties of the resulting estimators are derived using empirical process theory. Finally, a resampling procedure is developed to estimate the limiting variances of the estimators. The finite sample properties of the estimators are investigated by way of a simulation study, and a practical application to lung cancer data is illustrated.     

Equivalence of Certain Chi-Squared Test Statistics

In likelihood analysis of categorized data, it is well known that within a restricted class of log-linear models the likelihood kernels for multinomial and product multinomial sampling distributions are identical. In practical terms the estimation procedure for one is appropriate for the other. There does not appear to be a widespread realization that a similar result holds for a wide class of the Grizzle, Starmer, and Koch (1969) weighted least squares techniques. In this report such a fundamental relationship is explicitly presented and illustrated through two analyses of Bartlett's (1935) data.     

Classical multilevel and Bayesian approaches to population size estimation using multiple lists

One of the major objections to the standard multiple-recapture approach to population estimation is the assumption of homogeneity of individual 'capture' probabilities. Modelling individual capture heterogeneity is complicated by the fact that it shows up as a restricted form of interaction among lists in the contingency table cross-classifying list memberships for all individuals. Traditional log-linear modelling approaches to capture–recapture problems are well suited to modelling interactions among lists but ignore the special dependence structure that individual heterogeneity induces. A random-effects approach, based on the Rasch model from educational testing and introduced in this context by Darroch and co-workers and Agresti, provides one way to introduce the dependence resulting from heterogeneity into the log-linear model; however, previous efforts to combine the Rasch-like heterogeneity terms additively with the usual log-linear interaction terms suggest that a more flexible approach is required. In this paper we consider both classical multilevel approaches and fully Bayesian hierarchical approaches to modelling individual heterogeneity and list interactions. Our framework encompasses both the traditional log-linear approach and various elements from the full Rasch model. We compare these approaches on two examples, the first arising from an epidemiological study of a population of diabetics in Italy, and the second a study intended to assess the 'size' of the World Wide Web. We also explore extensions allowing for interactions between the Rasch and log-linear portions of the models in both the classical and the Bayesian contexts.     

Collapsibility of contingency tables based on conditional models

Strict collapsibility and model collapsibility are two important concepts associated with the dimension reduction of a multidimensional contingency table, without losing the relevant information. In this paper, we obtain some necessary and sufficient conditions for the strict collapsibility of the full model, with respect to an interaction factor or a set of interaction factors, based on the interaction parameters of the conditional/layer log-linear models. For hierarchical log-linear models, we present also necessary and sufficient conditions for the full model to be model collapsible, based on the conditional interaction parameters. We discuss both the cases where one variable or a set of variables is conditioned. The connections between the strict collapsibility and the model collapsibility are also pointed out. Our results are illustrated through suitable examples, including a real life application.     

Optimal designs for accelerated life tests with multiple stress factors and heteroscedasticity

Accelerated life testing (ALT) provides a means of obtaining data on product lifetime and reliability relatively quickly by subjecting products to higher-than-usual levels of stress factors. We present methods for obtaining optimal designs for multiple-factor ALTs with time censoring and heteroscedasticity in order to estimate percentiles of product life at usage conditions. We assume a Weibull life distribution and log-linear life–stress relationships with non constant shape parameter for the ALT stress factors. The primary optimality criterion is the minimization of the asymptotic variance of maximum likelihood estimator of the percentile estimator at usage stress. We also consider a secondary criterion for our design optimization. The design construction methods are illustrated by two practical examples.     

Log-linear model analysis of the semi-symmetric intraclass contingency table

This paper addresses the problem of analyzing a three-way contingency table that is upper-triangular, and a priori symmetric within layers. The log-linear model is modified to handle this kind of table, and maximum likelihood estimation is carried out for the modified log-linear model. This leads to an expression of the maximum likelihood estimates exclusively in terms of the observed cell counts. It is skin this analysis is equivalent to an application of the gone log-linear model to an artificially complete table, obtain. by splitting the off-diagonal cells in half within layers. This analysis is used in analyzing the results of a study done to determine the effect of the sex-linked dwarfing gene in male chickens on resistance to E. coli infection; the conclusion differs from that of a previous analysis of the same data (see Norwood and Hinkelmann 1978). It is found, in fact, that the structure of association among the two allele variables and the disease variable is somewhat more complex than previously proposed. A second example is taken from Ishii (1960). Finally, collapsibility conditions for the modified log-linear model, as well as various other sampling plans and limitations to the testing procedure, are discussed.     

Log-linear models for correlated marginal totals of a contingency table

The marginal totals of a contingency table can be rearranged to form a new table. If at least twoof these totals include the same cell of the original table, the new table cannot be treated as anordinary contingency table. An iterative method is proposed to calculate maximum likelihood estimators for the expected cell frequencies of the original table under the assumption that some marginal totals (or more generally, some linear functions) of these expected frequencies satisfy a log-linear model.In some cases, a table of correlated marginal totals is treated as if it was an ordinary contingency table. The effects of ignoring the special structure of the marginal table on thedistributionof the goodness-of-fit test statistics are discussed and illustrated, with special reference to stationary Markov chains.     

Some simple log-linear models for the analysis of ordinal response data

A family of log-linear models are proposed to describe contingency tables in which one variable can be considered as the response to the remaining. The proposed models take into account the ordering nature of the response categories and have structure similar to that employed in polynomial regression. Stochastic ordering of the response distributions under the proposed models is discussed and the model-reduction techniques are developed. The proposed models are applied to two data sets previously analysed in the literature.     

An estimated‐score approach for dealing with missing covariate data in matched case–control studies

Matched case–control designs are commonly used in epidemiological studies for estimating the effect of exposure variables on the risk of a disease by controlling the effect of confounding variables. Due to retrospective nature of the study, information on a covariate could be missing for some subjects. A straightforward application of the conditional logistic likelihood for analyzing matched case–control data with the partially missing covariate may yield inefficient estimators of the parameters. A robust method has been proposed to handle this problem using an estimated conditional score approach when the missingness mechanism does not depend on the disease status. Within the conditional logistic likelihood framework, an empirical procedure is used to estimate the odds of the disease for the subjects with missing covariate values. The asymptotic distribution and the asymptotic variance of the estimator when the matching variables and the completely observed covariates are categorical. The finite sample performance of the proposed estimator is assessed through a simulation study. Finally, the proposed method has been applied to analyze two matched case–control studies. The Canadian Journal of Statistics 38: 680–697; 2010 © 2010 Statistical Society of Canada     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Mutual information and redundancy for categorical data

Most methods for describing the relationship among random variables require specific probability distributions and some assumptions concerning random variables. Mutual information, based on entropy to measure the dependency among random variables, does not need any specific distribution and assumptions. Redundancy, which is an analogous version of mutual information, is also proposed as a method. In this paper, the concepts of redundancy and mutual information are explored as applied to multi-dimensional categorical data. We found that mutual information and redundancy for categorical data can be expressed as a function of the generalized likelihood ratio statistic under several kinds of independent log-linear models. As a consequence, mutual information and redundancy can also be used to analyze contingency tables stochastically. Whereas the generalized likelihood ratio statistic to test the goodness-of-fit of the log-linear models is sensitive to the sample size, the redundancy for categorical data does not depend on sample size but depends on its cell probabilities.     

Testing non-nested log-linear models with pseudo estimator

As the number of random variables for the categorical data increases, the possible number of log-linear models which can be fitted to the data increases rapidly, so that various model selection methods are developed. However, we often found that some models chosen by different selection criteria do not coincide. In this paper, we propose a comparison method to test the final models which are non-nested. The statistic of Cox (1961, 1962) is applied to log-linear models for testing non-nested models, and the Kullback-Leibler measure of closeness (Pesaran 1987) is explored. In log-linear models, pseudo estimators for the expectation and the variance of Cox's statistic are not only derived but also shown to be consistent estimators.     

Latent class based multiple imputation approach for missing categorical data

In this paper we propose a latent class based multiple imputation approach for analyzing missing categorical covariate data in a highly stratified data model. In this approach, we impute the missing data assuming a latent class imputation model and we use likelihood methods to analyze the imputed data. Via extensive simulations, we study its statistical properties and make comparisons with complete case analysis, multiple imputation, saturated log-linear multiple imputation and the Expectation–Maximization approach under seven missing data mechanisms (including missing completely at random, missing at random and not missing at random). These methods are compared with respect to bias, asymptotic standard error, type I error, and 95% coverage probabilities of parameter estimates. Simulations show that, under many missingness scenarios, latent class multiple imputation performs favorably when jointly considering these criteria. A data example from a matched case–control study of the association between multiple myeloma and polymorphisms of the Inter-Leukin 6 genes is considered.     

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.