The iterative proportional fitting algorithm for adjusted agreement in a non‐inferiority diagnostic clinical trial

In this paper, we describe a method of comparing agreement between two diagnostic contingency tables after adjustment to more clinically relevant marginal distributions using the iterative proportional fitting algorithm. When the categories of a contingency table represent mild, moderate, and severe outcomes, the majority of patients often are in the mild category. Because it is often of more interest to evaluate agreement when patients are uniformly distributed among categories, we present the primary results of two clinical trials with adjustment to this structure. We also describe the relationship between the sponsor's pre‐specified agreement measure for the observed contingency table and kappa for the adjusted table; and by either criterion, we then show that the agreement of the new diagnostic tool with the standard diagnostic tool is comparably non‐inferior to the agreement of the standard diagnostic tool with itself. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation of contingency tables in complex survey sampling using probabilistic expert systems

In this paper we explore the possibility to use a particular class of models, known as probabilistic expert systems, to define two classes of estimators of a contingency table in case of stratified sampling designs. The two classes are characterized by the different role of the sampling design: in the first, the sampling design is treated as an additional variable; in the second, it is used only for estimation purposes by means of the survey weights. The bias/variance trade off of these estimators is analyzed and the consequences of model misspecification are illustrated. Furthermore, it is shown that the Horvitz–Thompson estimator belongs to both classes of estimators. It comes out that the Horvitz–Thompson estimator is almost always inefficient but robust. Monte Carlo simulations illustrate the efficiency of the proposed estimators.     

Influence diagnostics for stratified ordinal contingency tables

Influence diagnostics are investigated in this study. In particular, an approach based on the generalized linear mixed model setting is presented for formulating ordered categorical counts in stratified contingency tables. Deletion diagnostics and their first-order approximations are developed for assessing the stratum-specific influence on parameter estimates in the models. To illustrate the proposed model diagnostic technique, the method is applied to analyze two sets of data: a clinical trial and a survey study. The two examples demonstrate that the presence of influential strata may substantially change the results in ordinal contingency table analysis.     

Model determination for categorical data with factor level merging

Summary.  We deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors. We assume that such relationships can be adequately described by the cond`itional independence structure that is imposed by an undirected graphical model. If the contingency table is large, a desirable simplified interpretation can be achieved by combining some categories, or levels, of the factors. We introduce conditions under which such an operation does not alter the Markov properties of the graph. Implementation of these conditions leads to Bayesian model uncertainty procedures based on reversible jump Markov chain Monte Carlo methods. The methodology is illustrated on a 2×3×4 and up to a 4×5×5×2×2 contingency table.     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

The Box–Cox Transformation and Non‐Iterative Estimation Methods for Ordinal Log‐Linear Models

Recently Beh and Farver investigated and evaluated three non‐iterative procedures for estimating the linear‐by‐linear parameter of an ordinal log‐linear model. The study demonstrated that these non‐iterative techniques provide estimates that are, for most types of contingency tables, statistically indistinguishable from estimates from Newton's unidimensional algorithm. Here we show how two of these techniques are related using the Box–Cox transformation. We also show that by using this transformation, accurate non‐iterative estimates are achievable even when a contingency table contains sampling zeros.     

Expert knowledge elicitation using item response theory

In an expert knowledge elicitation exercise, experts face a carefully constructed list of questions that they answer according to their knowledge. The elicitation process concludes when a probability distribution is found that adequately captures the experts' beliefs in the light of those answers. In many situations, it is very difficult to create a set of questions that will efficiently capture the experts' knowledge, since experts might not be able to make precise probabilistic statements about the parameter of interest. We present an approach for capturing expert knowledge based on item response theory, in which a set of binary response questions is proposed to the expert, trying to capture responses directly related to the quantity of interest. As a result, the posterior distribution of the parameter of interest will represent the elicited prior distribution that does not assume any particular parametric form. The method is illustrated by a simulated example and by an application involving the elicitation of rain prophets' predictions for the rainy season in the north-east of Brazil.     

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.     

Detection of outlying proportions

In this paper we introduce a new method for detecting outliers in a set of proportions. It is based on the construction of a suitable two-way contingency table and on the application of an algorithm for the detection of outlying cells in such table. We exploit the special structure of the relevant contingency table to increase the efficiency of the method. The main properties of our algorithm, together with a guide for the choice of the parameters, are investigated through simulations, and in simple cases some theoretical justifications are provided. Several examples on synthetic data and an example based on pseudo-real data from biological experiments demonstrate the good performances of our algorithm.     

Relationships Between Full and Layer Models with Applications to Level Merging

Analysis of a large dimensional contingency table is quite involved. Models corresponding to layers of a contingency table are easier to analyze than the full model. Relationships between the interaction parameters of the full log-linear model and that of its corresponding layer models are obtained. These relationships are not only useful to reduce the analysis but also useful to interpret various hierarchical models. We obtain these relationships for layers of one variable, and extend the results for the case when layers of more than one variable are considered. We also establish, under conditional independence, relationships between the interaction parameters of the full model and that of the corresponding marginal models. We discuss the concept of merging of factor levels based on these interaction parameters. Finally, we use the relationships between layer models and full model to obtain conditions for level merging based on layer interaction parameters. Several examples are discussed to illustrate the results.     

Quasi local odds symmetry model for square contingency table with ordinal categories

This paper proposes a new model for square contingency tables. The proposed model tests the equality of local odds ratios between the one side of the main diagonal and corresponding other side and it represents the non-symmetric structure of the square contingency table. The proposed model is compared with twenty-five models introduced for analysing the square contingency tables for both symmetric and non-symmetric structures. The results show that the proposed model provides best fit performance than other existing models for square contingency tables.     

Decomposition of the main effects and interaction term by using orthogonal polynomials in multiple non symmetrical correspondence analysis

The multiple non symmetric correspondence analysis (MNSCA) is a useful technique for analyzing a two-way contingency table. In more complex cases, the predictor variables are more than one. In this paper, the MNSCA, along with the decomposition of the Gray–Williams Tau index, in main effects and interaction term, is used to analyze a contingency table with two predictor categorical variables and an ordinal response variable. The Multiple-Tau index is a measure of association that contains both main effects and interaction term. The main effects represent the change in the response variables due to the change in the level/categories of the predictor variables, considering the effects of their addition, while the interaction effect represents the combined effect of predictor categorical variables on the ordinal response variable. Moreover, for ordinal scale variables, we propose a further decomposition in order to check the existence of power components by using Emerson's orthogonal polynomials.     

A Multistage Chi-square Test for Measurement of the Degree of Association in Two-way Table

The classical chi-square test for independence cannot convey additional information and the degree of the association of two factors in two-way contingency table. Besides, measures of association by contingency coefficient need to be used with care, because the association measures depend on the number of rows r and the number of columns c. This article proposes a multistage chi-square test to measure the degree of the association between the two factors in two-way contingency table. We also give simulation and real examples to assess the performance of the proposed method. The results show that our proposed method can effectively investigate the degree of association of two factors in two-way contingency table.     

Measure of departure from symmetry for the analysis of collapsed square contingency tables with ordered categories

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper considers the symmetry model for collapsed square contingency tables and proposes a measure to represent the degree of departure from symmetry. The proposed measure is defined as the arithmetic mean of submeasures each of which represents the degree of departure from symmetry for each collapsed 3×3 table. Each submeasure also represents the mean of power-divergence or diversity index for each collapsed table. Examples are given.     

Bayesian disclosure risk assessment: predicting small frequencies in contingency tables

Summary.  We propose an approach for assessing the risk of individual identification in the release of categorical data. This requires the accurate calculation of predictive probabilities for those cells in a contingency table which have small sample frequencies, making the problem somewhat different from usual contingency table estimation, where interest is generally focused on regions of high probability. Our approach is Bayesian and provides posterior predictive probabilities of identification risk. By incorporating model uncertainty in our analysis, we can provide more realistic estimates of disclosure risk for individual cell counts than are provided by methods which ignore the multivariate structure of the data set.     

A knowledge acquisition inductive system driven by empirical interpretation of derived results

This paper describes the design philosophy of and current issues concerning a knowledge acquisition system namedkaiser. This system is an intelligent workbench for construction of knowledge bases for classification tasks by domain experts themselves. It first learns classification knowledge inductively from the examples given by a human expert, then analyzes the result and process based on abstract domain knowledge which is also given by the expert. Based on this analysis, it asks sophisticated questions for acquiring new knowledge. The queries stimulate the human expert and help him to revise the learned results, control the learning process and prepare new examples and domain knowledge. Viewed from an AI aspect, it aims at integrating similarity-based inductive learning and explanation-based deductive reasoning by guiding inductive inference with theoretical and/or heuristic knowledge about the domain. This interactive induce-evaluate-ask cycle produces a rational interview which promotes incremental acquisition of domain knowledge as well as efficient induction of operational and reasonable knowledge proved by the domain knowledge.     

Uncertain evidence and artificial analysis

A belief function measure of uncertainty, associated with a network based knowledge structure, effectively defines an artificial analyst (e.g., an expert system) capable of making uncertain judgements. In practice, a belief function is typically constructed by combining independent components developed on local areas of the network from inputs such as accepted causal theories, probabilistic judgements by experts, or empirical sample data. Representation of the network in a special form called a ‘tree of cliques’ leads to a locally controlled algorithm that propagates locally defined beliefs through the tree, and fuses the resulting beliefs at the nodes, in such a way as to simultaneously compute marginal beliefs over all nodes of the tree. The paper develops a simple hypothetical example from the field of reliability to illustrate these ideas.     

Empirical bayes estimation in contingency tables

In the estimation of cell probabilities from a two–way contingency table, suppose that a priori the classification variables are believed independent. New empirical Bayes and Bayes estimators are proposed which shrink the observed proportions towards classical estimates under the model of independence. The estimators, based on a Dirichlet mixture class of priors, compare favorably to an estimator of Laird (1978) that is based on a normal prior on terms of a log–linear model. The methods are generalized to three–way tables.     

A NOTE ON MULTIVARIATE LOGISTIC MODELS FOR CONTINGENCY TABLES

The log-linear model is a tool widely accepted for modelling discrete data given in a contingency table. Although its parameters reflect the interaction structure in the joint distribution of all variables, it does not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters, recently introduced by Glonek & McCullagh (1995), which have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. Glonek & McCullagh give the link between the cell probabilities and the multivariate logistic parameters, in an algebraic fashion. The present paper focuses on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff-Nielsen, 1978) is used; this also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. The paper emphasises the analysis of independence hypotheses in margins of a contingency table.     

Expressing imprecision in probabilistic knowledge

Summary The representation of imprecise knowledge in probabilistic expert systems is investigated. It is argued that Bayesian statistics offers a multitude of models and methods that may be employed to express probabilistic knowledge by first and second order probability distributions. Distributions offer a much richer repertoire to express uncertainty that point probabilities. Special attention is payed to Dirichletian weights of evidence in discrete systems and to predictons and inverse predictions in linear regression. Approximate methods for non-conjugate distributions and for incomplete data are discussed. An important property of imprecision in complex systems is that it explodes rapidly when the conclusions are derived from long chains of premises. Thus, imprecision is a pruning criterion is a pruning criterion in complex knowledge systems. This research was supported by the Fonds zur F?rderung der wissenschftlichen Forschung, Vienna.