Testing log-linear models with inequality constraints: a comparison of asymptotic, bootstrap, and posterior predictive p-values

An important aspect of applied research is the assessment of the goodness-of-fit of an estimated statistical model. In the analysis of contingency tables, this usually involves determining the discrepancy between observed and estimated frequencies using the likelihood-ratio statistic. In models with inequality constraints, however, the asymptotic distribution of this statistic depends on the unknown model parameters and, as a result, there no longer exists an unique p -value. Bootstrap p -values obtained by replacing the unknown parameters by their maximum likelihood estimates may also be inaccurate, especially if many of the imposed inequality constraints are violated in the available sample. We describe the various problems associated with the use of asymptotic and bootstrap p -values and propose the use of Bayesian posterior predictive checks as a better alternative for assessing the fit of log-linear models with inequality constraints.     

Bayesian model selection using encompassing priors

This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes.     

On new variance approximations for linear models with inequality constraints

In this paper, we examine the estimation of linear models subject to inequality constraints with a special focus on new variance approximations for the estimated parameters. For models with one inequality restriction, the proposed variance formulas are exact. The variance approximations proposed in this paper can be used in regression analysis, Kalman filtering, and balancing national accounts, when inequality constraints are to be incorporated in the estimation procedure.     

Jackknife estimators of a relative risk in 2×2 and 2×2×K contingency tables

Several jackknife estimators of a relative risk in a single 2×2 contingency table and of a common relative risk in a 2×2× K contingency table are presented. The estimators are based on the maximum likelihood estimator in a single table and on an estimator proposed by Tarone (1981) for stratified samples, respectively. For the stratified case, a sampling scheme is assumed where the number of observations within each table tends to infinity but the number of tables remains fixed. The asymptotic properties of the above estimators are derived. Especially, we present two general results which under certain regularity conditions yield consistency and asymptotic normality of every jackknife estimator of a bunch of functions of binomial probabilities.     

Computations for constrained linear models

The article presents an algorithm for linear regression computations subject to linear parametric equality constraints, linear parametric inequality constraints, or a mixture of the two. No rank conditions are imposed on the regression specification or the constraint specification. The algorithm requires a full Moore-Penrose g-inverse which entails extra computational effort relative to other orthonormalization type algorithms. In exchange, auxiliary statistical information is generated: feasibility of a set of constraints may be checked, estimability of a linear parametric function may be checked, and bias and variance may be decomposed by source.     

Macro‐integration with inequality constraints: an application to the integration of transport and trade statistics

Macro‐integration is the process of combining data from several sources at an aggregate level. We review a Bayesian approach to macro‐integration with special emphasis on the inclusion of inequality constraints. In particular, an approximate method of dealing with inequality constraints within the linear macro‐integration framework is proposed. This method is based on a normal approximation to the truncated multivariate normal distribution. The framework is then applied to the integration of international trade statistics and transport statistics. By combining these data sources, transit flows can be derived as differences between specific transport and trade flows. Two methods of imposing the inequality restrictions that transit flows must be non‐negative are compared. Moreover, the figures are improved by imposing the equality constraints that aggregates of incoming and outgoing transit flows must be equal.     

Time varying VARs with inequality restrictions

In many applications involving time-varying parameter VARs, it is desirable to restrict the VAR coefficients at each point in time to be non-explosive. This is an example of a problem where inequality restrictions are imposed on states in a state space model. In this paper, we describe how existing MCMC algorithms for imposing such inequality restrictions can work poorly (or not at all) and suggest alternative algorithms which exhibit better performance. Furthermore, we show that previous algorithms involve an approximation relating to a key prior integrating constant. Our algorithms are exact, not involving this approximation. In an application involving a commonly used U.S. data set, we present evidence that the algorithms proposed in this paper work well.     

应急物流的系统分析与构建

根据应急物流系统的要素和设计原则,分析了系统的约束条件,提出了构建应急物流系统的目标、系统的结构、层次与保障机制。     

A new look at static spatial price equilibrium models

In this paper we propose a new view and more general formulations of static single commodity spatial equilibrium models, that lead to simpler and more efficient algorithms than those previously employed for solving models of this type. The proposed formulations incorporate general transportation networks and may be extended to allow for multiple commodities. Solution algorithms are suggested for the problem with multiple commodities, when there exists an equivalent optimization problem. We show that the multiple commodity problem may be cast in the form of a variational inequality, when there does not exist an equivalent optimization problem and propose algorithms to solve this version of the problem as well.     

An algebraic generalisation of some variants of simple correspondence analysis

For an analysis of the association between two categorical variables that are cross-classified to form a contingency table, graphical procedures have been central to this analysis. In particular, correspondence analysis has grown to be a popular method for obtaining such a summary and there is a great variety of different approaches that one may consider to perform. In this paper, we shall introduce a simple algebraic generalisation of some of the more common approaches to obtaining a graphical summary of association, where these approaches are akin to the correspondence analysis of a two-way contingency table. Specific cases of the generalised procedure include the classical and non-symmetrical correspondence plots and the symmetrical and isometric biplots.     

A nonlinear approach to a class of combinatorial optimization problems

A special class of combinatorial optimization problems is considered. We develop a compact nonconvex quadratic model for these problems that incorporates all inequality constraints in the objective function, and discuss two approximation algorithms for solving this model. One is inspired by Karmarkar's potential reduction algorithm for solving combinatorial optimization problems; the other is a variant of the reduced gradient method. The paper concludes with computational experiences with both real-life and randomly generated instances of the frequency assignment problem. Large problems are satisfactorily solved in reasonable computation times.     

Bayesian estimation of log odds ratios from R × C and 2 × 2 × K contingency tables

In this paper, Bayesian estimation of log odds ratios over R × C and 2 × 2 × K contingency tables is considered, which is practically reasonable in the presence of prior information. Likelihood functions for log odds ratios are derived for each table structure. A prior specification strategy is proposed. Posterior inferences are drawn using Gibbs sampling and Metropolis–Hastings algorithm. Two numerical examples are given to illustrate the matters argued.     

Exact inference in the inequality constrained normal linear regression model

Inference in the inequality constrained normal linear regression model is approached as a problem in Bayesian inference, using a prior that is the product of a conventional uninformative distribution and an indicator function representing the inequality constraints. The posterior distribution is calculated using Monte Carlo numerical integration, which leads directly to the evaluation of expected values of functions of interest. This approach is compared with others that have been proposed. Three empirical examples illustrate the utility of the proposed methods using an inexpensive 32-bit microcomputer.     

Conditional independence and log linear models for multi-dimensional contingency tables

Interference about conditional independence in relation to log linear models are discussed for contingency tables. The parameters and likelihood ratios for a log linear model with a dependent variable are shown to be identical to those for a multivariate model. An approximaate method of calculating log likelihood ratios, even when all dimensions of the table have more than two levels (no binary variables) is derived. The implications for sociological “causal” models are discussed.     

Estimation in Surveys Using Conditional Inclusion Probabilities: Simple Random Sampling

In survey sampling, auxiliary information on the population is often available. The aim of this paper is to develop a method which allows one to take into account such auxiliary information at the estimation stage by means of conditional bias adjustment. The basic idea is to attempt to construct a conditionally unbiased estimator. Four estimators that have a small conditional bias with respect to a statistic are proposed. It is shown that many of the estimators used in the literature in the case of simple random sampling can be obtained by using this estimation principle. The problem of simple random sampling with replacement, poststratification, and adjustment of a 2 x 2 dimensional contingency table to marginal totals are discussed in the conditional framework. Finally it is shown that the regression estimator can be viewed as an approximation of an application of the conditional principle.     

审计师行业专长与或有事项信息披露——基于客户重要性的调节效应研究

以企业年报中表内列报的预计负债金额与表外附注中披露的或有事项频数为考察对象，研究审计师行业专长对企业或有事项信息披露的影响以及客户重要性对上述影响的调节效应。在控制自选择等内生性影响后发现，审计师行业专长越高，被其审计的公司表内列报的预计负债金额越大、表外附注中披露的或有事项频数越高；同时，客户越重要，审计师行业专长对企业或有事项信息披露的促进作用越强。按客户重要性分组的检验结果表明，审计师行业专长对表外或有事项信息披露的促进作用仅存在于中型和大型客户中。     

The causal analysis of contingency tables: A return to first principles

Given the undoubtedly major advances in the analysis of contingency tables which have been achieved over the past ten years or so (see for example, Fienberg, 1980; Upton, 1978; Everitt, 1977; Haberman, 1978, 1979), it might seem rather unnecessary to want to return to first princples again. However, the need arises precisely because of these advances; for progress in the specifically causal analysis of contingency tables has not matched the other advances at all. Whilst Fienberg devoted a chapter to causal analysis, he made it clear that he views “the assignment of numerical values [to the arrows in a path diagram] as problematic, and [he] would limit [the analysis] to an indication of sign for causal relationships, in a fashion similar to that described by Blalock (1964)” (Fienberg, 1980, pp. 91–92). Considering how far quantitative-variable causal analysis has developed since Blalock (1964), it becomes clear that the causal analysis of qualitative data is still at a rather primitive stage. Indeed, Haberman (1978, 1979), in his two-volume survey of the analysis of qualitative data, does not mention it at all. The problem, I believe, is that log-linear and logit methodology are not particularly suited to the logic of causality in contingency tables. In order to derive a suitable method, it is necessary to uncover the logic underlying causality when applied to qualitative variables. A few others have taken seriously the idea that a direct analysis of the form of a contingency table can lead to fruitful results (see, especially, for example, Boudon, 1967), but their work has been overshadowed by the statistically more profound advances made in log-linear methods. This article is an attempt to provide a statistically rigorous analysis based on the direct interpretation of causality embodied in a contingency table.     

Empirical Bayes analysis of log-linear models for a generalized finite stationary Markov chain

This article presents the empirical Bayes method for estimation of the transition probabilities of a generalized finite stationary Markov chain whose ith state is a multi-way contingency table. We use a log-linear model to describe the relationship between factors in each state. The prior knowledge about the main effects and interactions will be described by a conjugate prior. Following the Bayesian paradigm, the Bayes and empirical Bayes estimators relative to various loss functions are obtained. These procedures are illustrated by a real example. Finally, asymptotic normality of the empirical Bayes estimators are established.     

Nonparametric estimation and inference on conditional quantile processes

This paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process. Two estimators based on local linear regressions are proposed. The first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity. The bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity. For inference, the paper first establishes a uniform Bahadur representation and then shows that the two estimators converge weakly to the same limiting Gaussian process. As an empirical illustration, the paper considers a dataset from Project STAR and delivers two new findings.     

Breakdown concepts for contingency tables

Loglinear Poisson models are commonly used to analyse contingency tables. So far, robustness of parameter estimators as well as outlier detection have rarely been treated in this context. We start with finite-sample breakdown points. We yield that the breakdown point of mean value estimators determines a lower bound for a masking breakdown point of a class of one-step outlier identification rules. Within a more refined breakdown approach, which takes account of the structure of the contingency table, a stochastic breakdown function is defined. It returns the probability that a given proportion of outliers is randomly placed at such a pattern, where breakdown is possible. Finally, the introduced breakdown concepts are applied to characterise the maximum likelihood estimator and a median-polish estimator.