Parameters Of Log-linear Model Research Articles

On the correspondence of deviances and maximum-likelihood and interval estimates from log-linear to logistic regression modelling.

Consider a set of categorical variables where at least one, denoted by Y, is binary. The log-linear model that describes the contingency table counts implies a logistic regression model, with outcome Y. Extending results from Christensen (1997, Log-linear models and logistic regression, 2nd edn. New York, NY, Springer), we prove that the maximum-likelihood estimates (MLE) of the logistic regression parameters equals the MLE for the corresponding log-linear model parameters, also considering the case where contingency table factors are not present in the corresponding logistic regression and some of the contingency table cells are collapsed together. We prove that, asymptotically, standard errors are also equal. These results demonstrate the extent to which inferences from the log-linear framework translate to inferences within the logistic regression framework, on the magnitude of main effects and interactions. Finally, we prove that the deviance of the log-linear model is equal to the deviance of the corresponding logistic regression, provided that no cell observations are collapsed together when one or more factors in become obsolete. We illustrate the derived results with the analysis of a real dataset.

Estimation of multidrug resistance variability in the National Antimicrobial Monitoring System

Multidrug resistance is a serious problem raising the specter of infections for which there is no treatment. One of the most important tools in combating multidrug resistance is large scale monitoring programs, because they track resistance over large geographic areas and time scales. This large scope, however, can also introduce variability into the data. The primary monitoring program in the United States is the National Antimicrobial Resistance Monitoring System (NARMS). This study examines the variability of a previously identified resistance pattern in Escherichia coli among ampicillin, gentamicin, sulfisoxazole, and tetracycline using samples isolated from chicken during the years 2004 to 2006 and 2008 to 2012. 2007 is excluded because sulfisozaxole resistance was not measured at slaughter that year. To assess variability in this resistance pattern susceptibility/resistance contingency tables were constructed for each of the 15 combinations of the 4 drugs for each of the years. For each table, variability across the years was assessed at the full table multinomial level as a measure of general variability of the resistance pattern and at the level of the highest order interaction term in a log-linear model of the table as a measure of variability in that particular component of the resistance pattern. A power analysis using the traditional asymptotic normal approximation and one using a Dirichlet-multinomial simulation were carried out to determine the effect of variation on ability to detect nonzero highest order loglinear model terms and the validity of the normal approximation in carrying out such tests. All tables exhibit overdispersion at the multinomial level and in their highest order model parameters. The normal approximation performs well for large sample sizes, low levels of dispersion, and small log-linear model parameters. The approximation breaks down as dispersion or the log linear model parameter grows or sample size shrinks. Taken together these analyses indicate that the level of variability in the NARMS dataset makes it difficult to detect multidrug resistance patterns at the current level of sample collection. In order to better control this dispersion NARMS could collect more variables on each of the samples.

Optimal Gaussian Approximations to the Posterior for Log-Linear Models with Diaconis–Ylvisaker Priors

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis–Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. Here we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis–Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback–Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even for modest sample sizes. We also propose a method for model selection using the approximation. The proposed approximation provides a computationally scalable approach to regularized estimation and approximate Bayesian inference for log-linear models.

A new approach to fitness improvement in multi-correlated binary response model

The main purpose of this manuscript is to improve the concordance of multi-correlated binary response problems using heuristic and meta-heuristic methods. For this purpose and modelling the problem, the log-linear model parameters should be estimated. In this research, an iterative nonlinear heuristic method has been proposed for simultaneous parameter estimation of multi-binary response problem. The fitness analysis of the mentioned method is evaluated by comparing the results with their counterparts in the case of independency between responses by three numerical examples in various sizes. Results illustrate the fact that the proposed heuristic method performs well in comparison with the separate estimation of the coefficients with respect to number of concordances criterion. Then, a meta-heuristic approach called simulated annealing (SA) is applied to develop the model's fitness in which the best set of controllable variables is determined in order to maximise the number of concordances.

Measuring and Estimating Treatment Effect on Count Outcome in Randomized Trial and Observational Studies

When estimating treatment effect on count outcome of given population, one uses different models in different studies, resulting in non-comparable measures of treatment effect. Here we show that the marginal rate differences in these studies are comparable measures of treatment effect. We estimate the marginal rate differences by log-linear models and show that their finite-sample maximum-likelihood estimates are unbiased and highly robust with respect to effects of dispersing covariates on outcome. We get approximate finite-sample distributions of these estimates by using the asymptotic normal distribution of estimates of the log-linear model parameters. This method can be easily applied to practice.

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.

Data augmentation in multi-way contingency tables with fixed marginal totals

We describe and illustrate approaches to data augmentation in multi-way contingency tables for which partial information, in the form of subsets of marginal totals, is available. In such problems, interest lies in questions of inference about the parameters of models underlying the table together with imputation for the individual cell entries. We discuss questions of structure related to the implications for inference on cell counts arising from assumptions about log-linear model forms, and a class of simple and useful prior distributions on the parameters of log-linear models. We then discuss “local move” and “global move” Metropolis–Hastings simulation methods for exploring the posterior distributions for parameters and cell counts, focusing particularly on higher-dimensional problems. As a by-product, we note potential uses of the “global move” approach for inference about numbers of tables consistent with a prescribed subset of marginal counts. Illustration and comparison of MCMC approaches is given, and we conclude with discussion of areas for further developments and current open issues.

Outlying Cell Identification Method Using Interaction Estimates of Log-linear Models

This work is proposed an alternative identification method of outlying cell which is one of important issues in categorical data analysis. One finds that there is a strong relationship between the location of an outlying cell and the corresponding parameter estimates of the well-fitted log-linear model. Among parameters of log-linear model, an outlying cell is affected by interaction terms rather than main effect terms. Hence one could identify an outlying cell by investigating of parameter estimates in an appropriate log-linear model.

Effect Analysis in Loglinear Model Approach to Path Analysis of Categorical Variables

This paper considers a method for measuring causal effects of factors in recursive causal systems described by hierarchical loglinear models. The total, direct and indirect effects of factors are defined by loglinear model parameters. A numerical example is also given to illustrate the present discussion.

Models for Categorical Data with Nonignorable Nonresponse

Abstract When categorical outcomes are subject to nonignorable nonresponse, log-linear models may be used to adjust for the nonresponse. The models are fitted to the data in an augmented frequency table in which one index corresponds to whether or not the subject is a respondent. The likelihood function is maximized over pseudo-observed cell frequencies with respect to this log-linear model using an EM algorithm. Each E step of the EM algorithm determines the pseudo-observed cell frequencies, and the M step yields the maximum likelihood estimators (MLE's) of these pseudo-observed cell frequencies. This approach may produce boundary estimates for the expected cell frequencies of the nonrespondents. In these cases the estimators of the log-linear model parameters are not uniquely determined and may be unstable. Following the approach of Clogg et al., we propose a Bayesian method that uses smoothing constants to adjust the pseudo-observed cell frequencies so that the solution is not on the boundary. The role...

Nonlinear discrete relative population dynamics of the U.S. regions

A new discrete relative-dynamics algorithm is tested using population interactions of U.S. regions. A 133-year time period is used (1850–1983) to calibrate the log-linear model's parameters using standard linear regression tests. We also report certain forecasts under different spatial disaggregation schemes of the U.S. regions, and under two different time-period steps (iterations) to the year 2050. All forecasts imply competitive exclusion for the northern regions; under certain areal disaggregations and time (iteration) steps the southern or western region also experiences conditions leading to drastic declines in their share of the U.S. population. We also report the results of certain random fluctuations on the model's parameters and their implications regarding the dynamics of the U.S. regional population structure. The outcomes seem to be robust under these fluctuations.

Exact and stochastic nonlinear constrained estimation in the conditional poisson log-linear model

A minimum chi-square estimator of the conditional Poisson log-linear model parameters subject to exact and stochastic nonlinear constraints is defined. The Wald test is used to evaluate the exact and stochastic constraints.

Estimating the Size of Closed Populations Using Inverse Multiple-Recapture Sampling

A log-linear model for estimating the size of a closed population is defined for inverse multiple-recapture sampling with dependent samples. Efficient estimators of the log-linear model parameters and the population size are obtained by the method of minimum chi-square. A chi-square test of the general linear hypothesis regarding the log-linear model parameters is defined.

Maximum Likelihood Estimation and Model Selection in Contingency Tables with Missing Data

Abstract In many studies the values of one or more variables are missing for subsets of the original sample. This article focuses on the problem of obtaining maximum likelihood estimates (MLE) for the parameters of log-linear models under this type of incomplete data. The appropriate systems of equations are presented and the expectation-maximization (EM) algorithm (Dempster, Laird, and Rubin 1977) is suggested as one of the possible methods for solving them. The algorithm has certain advantages but other alternatives may be computationally more effective. Tests of fit for log-linear models in the presence of incomplete data are considered. The data from the Protective Services Project for Older Persons (Blenkner, Bloom, and Nielsen 1971; Blenkner, Bloom, and Weber 1974) are used to illustrate the procedures discussed in the article.

Log-Linear Techniques and the Regression Analysis of Dummy Dependent Variables

Although previous comparisons of log-linear techniques with the regression analysis of dummy dependent variables have focused on the statistical superiority of the log-linear techniques, this paper presents three advantages of dummy dependent-variable regressions. First, dummy dependent-variable regression is able to accommodate both categorical and continuous independent variables. Second, the calculation of indirect effects and reciprocal effects is possible only with dummy dependent-variable regression. Third, the slopes yielded by dummy dependent-variable regression possess the properties of fundamental parameters; the effect parameters of log-linear models do not. In particular, controlling for a variable that is irrelevant to the causal system under investigation can yield a partial relationship whose expected value differs from that of the zero-order relationship when using log-linear techniques.