The multivariate linear model with multivariatet and intra-class covariance structure

The prediction distribution of future responses from a multivariate linear model with error having a multivariatet-distribution and intra-class covariance structure has been derived. The distribution depends on ρ, the intra-class correlation coefficient. For unknown ρ, the marginal likelihood function of ρ has been obtained and the prediction distribution has been approximated by the estimate of ρ. As an application, a β-expectation tolerance region for the model has been constructed.     

Structural inference of the parameters of the heteroscedastic simultaneous equation model

The structural approach of inference for the parameters of a simultaneous equation model with heteroscedastic error variance is investigated in this paper. The joint and the marginal structural distributions for the coefficients of the exogenous variables and the scale parameters of the error variables, and the marginal likelihood function of the coefficients of the endogenous variables have been derived. The estimates are directly obtainable from the structural distribution and the marginal likelihood function of the parameters. The marginal distribution of a subset of coefficients of exogenous variables provides the basis for making inference for a particular subset of parameter of interest.     

Combined estimators for the correlation coefficient

Three combined estimators for the bivariate normal correlation parameter are considered. The data consist of k independent sample correlation coefficients and it is assumed that the underlying correlation parameters are all equal to ρ. Based upon the joint density function of the sample correlations a combined estimator of ρ is obtained as an approximation to the maximum likelihood solution. Two linearly combined estimators are also considered. One of them is based on Fisher's z-transformation of the sample correlations and the other on an unbiased estimator of ρ. The comparison of these three estimators indicates that the combined (approximate) MLE has a slightly smaller estimated mean squared error relative to the other two combined methods of estimation, but it does so at the expense of a relatively larger bias.     

Longitudinal conditional models with intermittent missingness: SAS code and applications

This work provides a set of macros performed with SAS (Statistical Analysis System) for Windows, which can be used to fit conditional models under intermittent missingness in longitudinal data. A formalized transition model, including random effects for individuals and measurement error, is presented. Model fitting is based on the missing completely at random or missing at random assumptions, and the separability condition. The problem translates to maximization of the marginal observed data density only, which for Gaussian data is again Gaussian, meaning that the likelihood can be expressed in terms of the mean and covariance matrix of the observed data vector. A simulation study is presented and misspecification issues are considered. A practical application is also given, where conditional models are fitted to the data from a clinical trial that assessed the effect of a Cuban medicine on a disease of the respiratory system.     

Limited information bayesian analysis of a structural coefficient in a simultaneous equations system

An analytical expression is obtained for the marginal posterior density for a structural coefficient in a simultaneous equations system based on a limited information Bayesian analysis. A con- ditional posterior density is obtained given reduced form para- meters. This conditional posterior density is in univariate student t form. Numerical examples suggest that the conditional density hasa tighter distribution around the posterior mean than the unconditional density when the correlation between the endo- genous variables and the structural error term is high.     

On asymptotic normality of likelihood and conditional analysis

The likelihood function from a large sample is commonly assumed to be approximately a normal density function. The literature supports, under mild conditions, an approximate normal shape about the maximum; but typically a stronger result is needed: that the normalized likelihood itself is approximately a normal density. In a transformation-parameter context, we consider the likelihood normalized relative to right-invariant measure, and in the location case under moderate conditions show that the standardized version converges almost surely to the standard normal. Also in a transformation-parameter context, we show that almost sure convergence of the normalized and standardized likelihood to a standard normal implies that the standardized distribution for conditional inference converges almost surely to a corresponding standard normal. This latter result is of immediate use for a range of estimating, testing, and confidence procedures on a conditional-inference basis.     

Model Selection for Linear Mixed Models Using Predictive Criteria

Predictive criteria, including the adjusted squared multiple correlation coefficient, the adjusted concordance correlation coefficient, and the predictive error sum of squares, are available for model selection in the linear mixed model. These criteria all involve some sort of comparison of observed values and predicted values, adjusted for the complexity of the model. The predicted values can be conditional on the random effects or marginal, i.e., based on averages over the random effects. These criteria have not been investigated for model selection success.

We used simulations to investigate selection success rates for several versions of these predictive criteria as well as several versions of Akaike's information criterion and the Bayesian information criterion, and the pseudo F-test. The simulations involved the simple scenario of selection of a fixed parameter when the covariance structure is known.

Several variance–covariance structures were used. For compound symmetry structures, higher success rates for the predictive criteria were obtained when marginal rather than conditional predicted values were used. Information criteria had higher success rates when a certain term (normally left out in SAS MIXED computations) was included in the criteria. Various penalty functions were used in the information criteria, but these had little effect on success rates. The pseudo F-test performed as expected. For the autoregressive with random effects structure, the results were the same except that success rates were higher for the conditional version of the predictive error sum of squares.

Characteristics of the data, such as the covariance structure, parameter values, and sample size, greatly impacted performance of various model selection criteria. No one criterion was consistently better than the others.     

Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions

In latent variable models parameter estimation can be implemented by using the joint or the marginal likelihood, based on independence or conditional independence assumptions. The same dilemma occurs within the Bayesian framework with respect to the estimation of the Bayesian marginal (or integrated) likelihood, which is the main tool for model comparison and averaging. In most cases, the Bayesian marginal likelihood is a high dimensional integral that cannot be computed analytically and a plethora of methods based on Monte Carlo integration (MCI) are used for its estimation. In this work, it is shown that the joint MCI approach makes subtle use of the properties of the adopted model, leading to increased error and bias in finite settings. The sources and the components of the error associated with estimators under the two approaches are identified here and provided in exact forms. Additionally, the effect of the sample covariation on the Monte Carlo estimators is examined. In particular, even under independence assumptions the sample covariance will be close to (but not exactly) zero which surprisingly has a severe effect on the estimated values and their variability. To address this problem, an index of the sample’s divergence from independence is introduced as a multivariate extension of covariance. The implications addressed here are important in the majority of practical problems appearing in Bayesian inference of multi-parameter models with analogous structures.     

Analysis of the linear correlation coefficient using pseudo-likelihoods

Summary The problem of the inferential analysis of the linear correlation coefficient of normal bivariate populations is tackled, both from the likelihood and Bayesian viewpoints. In particular it is shown how, using pseudo-likelihood (marginal likelihood function and profile likelihood), hypotheses such asH ₀:ϱ=ϱ₀ andH ₀:ϱ_x=ϱ_y can be verified without prohibitive computation effort. The results of marginal and profile likelihood are compared and it is shown that these two methods are virtually equivalent even for small sample sizes. Furthermore, in suitable conditions, the posterior distribution of the coefficient ϱ can be readily obtained, using the exact form or different approximate formulations of the marginal or profile likelihood. Lastly some possible prior distributions of ϱ are illustrated and some explanatory examples are presented.     

Regression analysis for discrete event history or failure time data

The paper deals with discrete-time regression models to analyze multistate—multiepisode models for event history data or failure time data collected in follow-up studies, retrospective studies, or longitudinal panels. The models are applicable if the events are not dated exactly but only a time interval is recorded. The models include individual specific parameters to account for unobserved heterogeneity. The explantory variables may be time-varying and random with distributions depending on the observed history of the process. Different estimation procedures are considered: Estimation of structural as well as individual specific parameters by maximization of a joint likelihood function, estimation of the structural parameters by maximization of a conditional likelihood function conditioning on a set of sufficient statistics for the individual specific parameters, and estimation of the structural parameters by maximization of a marginal likelihood function assuming that the individual specific parameters follow a distribution. The advantages and limitations of the different approaches are discussed.     

Likelihood-Based Inference for the Shape Parameter of a Two Parameter Weibull Distribution

Based on a Type 2 censored sample, we use the likelihood-based approach to draw likelihood inference on the shape parameter gamma of a two-parameter Weibull distribution. In particular, we derive the profile, conditional and marginal likelihoods of gamma. Numerical results along with some concluding remarks regarding the use of likelihood-based methods for inference are provided.     

The unconditional distributions of the OLS,TSLS and LIML estimators in a simple structural equations model

The exact distributions of the standard estimators of the structural coe?cients in a linear structural equations model conditional on the exogenous variables have been shown to have some unexpected and quirky features. Since the argument for conditioning on exogenous (ancillary) variables has been weakened over the past 20 years by the discovery of an “ancillarity paradox,” it is natural to wonder whether such finite sample properties are in fact due to conditioning on the exogenous variables. This article studies the exact distributions of the ordinary least squares (OLS), two-stage least squares (TSLS), and limited information maximum likelihood (LIML) estimators of the structural coe?cients in a linear structural equation without conditioning on the exogenous variables.     

Theory & Methods: Residual Diagnostic Plots for Checking for Model Mis‐specification in Time Series Regression

This paper considers residuals for time series regression. Despite much literature on visual diagnostics for uncorrelated data, there is little on the autocorrelated case. To examine various aspects of the fitted time series regression model, three residuals are considered. The fitted regression model can be checked using orthogonal residuals; the time series error model can be analysed using marginal residuals; and the white noise error component can be tested using conditional residuals. When used together, these residuals allow identification of outliers, model mis‐specification and mean shifts. Due to the sensitivity of conditional residuals to model mis‐specification, it is suggested that the orthogonal and marginal residuals be examined first.     

Maximum likelihood estimation of polychoric correlations in r×s ×t contingency tables

This paper discusses the maximum likelihood estimation of the polychoric correlation coefficient based on observed frequencies of three polytomous ordinal variables. The underlying latent variables are assumed to have a standardized trivariate normal distribution. The thresholds and correlations are estimated simultaneously via the scoring algorithm. Some practical applications of the method are discussed. An example is reported to illustrate the theory and some technical details are presented in the Appendix.     

Estimation of odds ratios under order restrictions

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.     

Prediction Regions for Bivariate Extreme Events

This paper suggests using a mixture of parametric and non‐parametric methods to construct prediction regions in bivariate extreme‐value problems. The non‐parametric part of the technique is used to estimate the dependence function, or copula, and the parametric part is employed to estimate the marginal distributions. A bootstrap calibration argument is suggested for reducing coverage error. This combined approach is compared with a more parametric one, relative to which it has the advantages of being more flexible and simpler to implement. It also enjoys these features relative to predictive likelihood methods. The paper shows how to construct both compact and semi‐infinite bivariate prediction regions, and it treats the problem of predicting the value of one component conditional on the other. The methods are illustrated by application to Australian annual maximum temperature data.     

A multivariate logit model with marginal canonical association

In this work, a generalization of the Goodman Association Model to the case of q, q > 2, categorical variables which is based on the idea of marginal modelling discussed by Gloneck–McCullagh is introduced; the difference between the proposed generalization and two models, previously introduced by Becker and Colombi, is discussed. The Becker generalization is not a marginal model because it does not imply Logit Models for the marginal probabilities, and because it is based on the conditional approach of modelling the association. The Colombi model is only partially a marginal model because it uses simple logit models for the univariate marginal probabilities but is based on the conditional approach of modelling the association. It is also shown that the maximum likelihood estimation of the parameters of the new model is feasible and, to compute the maximum likelihood estimates, an algorithm is proposed, which is a numerically convenient compromise between the constrained optimization approach of Lang and the straightforward use of the Fisher Scoring Algorithm suggested by Glonek–McCullagh.Finally, the proposed model is used to analyze a data set concerning work accidents which occurred to workers at some Italian firms during the years 1994–1996.     

Bivariate inverse Weibull distribution

ABSTRACT

Recently it is observed that the inverse Weibull (IW) distribution can be used quite effectively to analyse lifetime data in one dimension. The main aim of this paper is to define a bivariate inverse Weibull (BIW) distribution so that the marginals have IW distributions. It is observed that the joint probability density function and the joint cumulative distribution function can be expressed in compact forms. Several properties of this distribution such as marginals, conditional distributions and product moments have been discussed. We obtained the maximum likelihood estimates for the unknown parameters of this distribution and their approximate variance– covariance matrix. We perform some simulations to see the performances of the maximum likelihood estimators. One data set has been re-analysed and it is observed that the bivariate IW distribution provides a better fit than the bivariate exponential distribution.     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

Approximate multivariate conditional inference using the adjusted profile likelihood

The author proposes saddlepoint approximation methods that are adapted to multivariate conditional inference in canonical exponential familles. Several approaches to approximating conditional discrete distributions involve dividing an approximation to the full joint mass function, summed over tail regions of interest, by an approximate marginal density. The author first approximates this conditional likelihood by the adjusted profile likelihood, and then applies a multivariate saddlepoint approximation. He also presents formulas to aid in performing simultaneously the profiling and maximizing steps.