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.     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.     

Comparison of intracalss correlation coefficients with the ratio of two independent f-statistics

The ratio of analysis of variance F-statistics is proposed as a test criterion for comparing the intraclass correlation coefficients of two independent groups of classes. Selected percentage points of this ratio’s distribution are tabulated for the special case when there are two observations in each class. Such tables should be useful when investigating heritability from a study of identical and nonidentical twin pairs.     

New critical region tables for Fisher's exact test

The best-known non-asymptotic method for comparing two independent proportions is Fisher's exact text. The usual critical region (CR) tables for this test contain one or more of the following defects:they distinguish between rows and columns; they distinguish between the alternatives H = p1 < p2 and H = p1 > p2; they assume that the error for the two-tailed test is twice that of the one-tailed test; they do not use the optimal version of the test; they do not give both CRs for one and two tails at the same time. All this results in the unnecessary duplication of the space required for the tables, the construction of tables of low-powered methods, or the need to manipulate two different tables (one for the one-tailed test, the other for the two-tailed test). This paper presents CR tables which have been obtained from the most powerful version of Fisher's exact test and which occupy the minimum space possible. The tables, which are valid for one- or two-tailed tests, have levels of significance of 10%, 5% and 1% and values for N (the total size of both samples) of less than or equal to 40. This article shows how to calculate the P value in a specific problem, using the tables as a means of partial checking and as a preliminary step to determining the exact P value.     

Large sample bias of maximum likeihood estimates from a series of sparse ixj tables

For a large series of IxJ tables, each containing two observations, the bias of the maximum likelihood estimates of log linear partial association parameters is shown to be equal to the parameters, regardless of the size of I and J. The partial association considered is that between row and column variables; the three way interactions are assumed to be O. This is a generalization of Andersen's results (1973a, 1973b) for a series of 2x2 tables.     

Testing stochastic orders in tails of contingency tables

Testing for the difference in the strength of bivariate association in two independent contingency tables is an important issue that finds applications in various disciplines. Currently, many of the commonly used tests are based on single-index measures of association. More specifically, one obtains single-index measurements of association from two tables and compares them based on asymptotic theory. Although they are usually easy to understand and use, often much of the information contained in the data is lost with single-index measures. Accordingly, they fail to fully capture the association in the data. To remedy this shortcoming, we introduce a new summary statistic measuring various types of association in a contingency table. Based on this new summary statistic, we propose a likelihood ratio test comparing the strength of association in two independent contingency tables. The proposed test examines the stochastic order between summary statistics. We derive its asymptotic null distribution and demonstrate that the least favorable distributions are chi-bar distributions. We numerically compare the power of the proposed test to that of the tests based on single-index measures. Finally, we provide two examples illustrating the new summary statistics and the related tests.     

Optimal correction for continuity in the chi-squared test in 2×2 tables

The most common asymptotic procedure for analyzing a 2 × 2 table (under the conditioning principle) is the ‰ chi-squared test with correction for continuity (c.f.c). According to the way this is applied, up to the present four methods have been obtained: one for one-tailed tests (Yates') and three for two-tailed tests (those of Mantel, Conover and Haber). In this paper two further methods are defined (one for each case), the 6 resulting methods are grouped in families, their individual behaviour studied and the optimal is selected. The conclusions are established on the assumption that the method studied is applied indiscriminately (without being subjected to validity conditions), and taking a basis of 400,000 tables (with the values of sample size n between 20 and 300 and exact P-values between 1% and 10%) and a criterion of evaluation based on the percentage of times in which the approximate P-value differs from the exact (Fisher's exact test) by an excessive amount. The optimal c.f.c. depends on n, on E (the minimum quantity expected) and on the error α to be used, but the rule of selection is not complicated and the new methods proposed are frequently selected. In the paper we also study what occurs when E ≥ 5, as well as whether the chi-squared by factor (n-1).     

Robust principal component analysis for compositional tables

A data table arranged according to two factors can often be considered a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information consists of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decomposed into independent and interactive parts, and orthonormal coordinates can be assigned to these parts. However, these coordinates usually require some prior knowledge about the data, and they are not easy to handle for exploring the relationships between the given factors. Here we propose a special choice of coordinates with direct relation to centered logratio (clr) coefficients, which are particularly useful for an interpretation in terms of the original cells of the tables. With these coordinates, robust principal component analysis (rPCA) is performed for dimension reduction, allowing to investigate relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply rPCA, which would otherwise suffer from the singularity of clr coefficients.     

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure

For a postulated common odds ratio for several 2 × 2 contingency tables one may, by conditioning on the marginals of the seperate tables, determine the exact expectation and variance of the entry in a particular cell of each table, hence for the total of such cells across all tables. This makes it feasible to determine limiting values, via single-degree-of-freedom, continuity-corrected chi-square tests on the common odds ratio–one determines lower and upper limits corresponding to just barely significant chi-square values. The Mantel-Haenszel approach can be viewed as a special application of this, but directed specifically to the case of unity for the odds ratio, for which the expectation and variance formulas are particularly simple. Computation of exact expectations and variances may be feasible only for 2 × 2 tables of limited size, but asymptotic formulas can be applied in other instances.Illustration is given for a particular set of four 2 × 2 tables in which both exact limits and limits by the proposed method could be applied, the two methods giving reasonably good agreement. Both procedures are directed at the distribution of the total over the designated cells, the proposed method treating that distribution as being asymptotically normal. Especially good agreement of proposed with exact limits could be anticipated in more asymptotic situations (overall, not for individual tables) but in practice this may not be demonstrable as the computation of exact limits is then unfeasible.     

The Coverage Probability of Confidence Intervals in One‐Way Analysis of Covariance after Two F Tests

Volume 3 of Analysis of Messy Data by Milliken & Johnson (2002) provides detailed recommendations about sequential model development for the analysis of covariance. In his review of this volume, Koehler (2002) asks whether users should be concerned about the effect of this sequential model development on the coverage probabilities of confidence intervals for comparing treatments. We present a general methodology for the examination of these coverage probabilities in the context of the two‐stage model selection procedure that uses two F tests and is proposed in Chapter 2 of Milliken & Johnson (2002). We apply this methodology to an illustrative example from this volume and show that these coverage probabilities are typically very far below nominal. Our conclusion is that users should be very concerned about the coverage probabilities of confidence intervals for comparing treatments constructed after this two‐stage model selection procedure.     

Multiple comparisons of exponential location parameters with a control based on doubly censored sample under heteroscedasticity

In this article, a one-sample procedure for multiple comparisons of exponential location parameters with a control under heteroscedasticity is proposed. The observations are obtained by doubly censored samples. A one-sided and two-sided confidence intervals are used to perform such multiple comparisons. Statistical tables of critical values and an example of comparing four drugs in treating leukemia are provided.     

Logratio approach to statistical analysis of 2×2 compositional tables

Compositional tables represent a continuous counterpart to well-known contingency tables. Their cells contain quantitatively expressed relative contributions of a whole, carrying exclusively relative information and are popularly represented in proportions or percentages. The resulting factors, corresponding to rows and columns of the table, can be inspected similarly as with contingency tables, e.g. for their mutual independent behaviour. The nature of compositional tables requires a specific geometrical treatment, represented by the Aitchison geometry on the simplex. The properties of the Aitchison geometry allow a decomposition of the original table into its independent and interactive parts. Moreover, the specific case of 2×2 compositional tables allows the construction of easily interpretable orthonormal coordinates (resulting from the isometric logratio transformation) for the original table and its decompositions. Consequently, for a sample of compositional tables both explorative statistical analysis like graphical inspection of the independent and interactive parts or any statistical inference (odds-ratio-like testing of independence) can be performed. Theoretical advancements of the presented approach are demonstrated using two economic applications.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

Wilson Confidence Intervals for the Two-Sample Log-Odds-Ratio in Stratified 2 × 2 Contingency Tables

Large-sample Wilson-type confidence intervals (CIs) are derived for a parameter of interest in many clinical trials situations: the log-odds-ratio, in a two-sample experiment comparing binomial success proportions, say between cases and controls. The methods cover several scenarios: (i) results embedded in a single 2 × 2 contingency table; (ii) a series of K 2 × 2 tables with common parameter; or (iii) K tables, where the parameter may change across tables under the influence of a covariate. The calculations of the Wilson CI require only simple numerical assistance, and for example are easily carried out using Excel. The main competitor, the exact CI, has two disadvantages: It requires burdensome search algorithms for the multi-table case and results in strong over-coverage associated with long confidence intervals. All the application cases are illustrated through a well-known example. A simulation study then investigates how the Wilson CI performs among several competing methods. The Wilson interval is shortest, except for very large odds ratios, while maintaining coverage similar to Wald-type intervals. An alternative to the Wald CI is the Agresti-Coull CI, calculated from the Wilson and Wald CIs, which has same length as the Wald CI but improved coverage.     

Evaluation of missing data mechanisms in two and three dimensional incomplete tables

The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies.     

Dynamic Bayesian analysis of generalized odds ratios assuming multivariate skew-normal distribution for the error terms in the system equation

In this paper, we develop a methodology for the dynamic Bayesian analysis of generalized odds ratios in contingency tables. It is a standard practice to assume a normal distribution for the random effects in the dynamic system equations. Nevertheless, the normality assumption may be unrealistic in some applications and hence the validity of inferences can be dubious. Therefore, we assume a multivariate skew-normal distribution for the error terms in the system equation at each step. Moreover, we introduce a moving average approach to elicit the hyperparameters. Both simulated data and real data are analyzed to illustrate the application of this methodology.     

All-Union population census: family structure

This is a continuation of a previous series of tables on family structure in the USSR, based on data from the 1979 census. Data are included on the size and nationality of families among the urban, rural, and total populations of each Union Republic.     

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.     

Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman considered the diagonals-parameter symmetry (DPS) model. This paper proposes a measure to represent the degree of departure from the DPS model. The proposed measure is expressed by applying Read and Cressie’s power-divergence or Patil and Taillie’s diversity index. The measure would be useful for comparing the degree of departure from the DPS model in several tables. Examples are given.