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.     

The mixture index of fit and minimax regression

A measure of the fit of a statistical model can be obtained by estimating the relative size of the largest fraction of the population where a distribution belonging to the model may be valid. This is the mixture index of fit that was suggested for models for contingency tables by Rudas, Clogg, Lindsay (1994) and it is extended here for models involving continuous observations. In particular, the approach is applied to regression models with normal and uniform error structures. Best fit, as measured by the mixture index of fit, is obtained with minimax estimation of the regression parameters. Therefore, whenever minimax estimation is used for these problems, the mixture index of fit provides a natural approach for measuring model fit and for variable selection. Received: September 1997     

Stochastic monotonicity in intergenerational mobility tables

The aim of this paper is to test for stochastic monotonicity in intergenerational socio‐economic mobility tables. In other words, we question whether having a parent from a high socio‐economic status is never worse than having one with a lower status. Using existing inferential procedures for testing unconditional stochastic monotonicity, we first test a set of 149 intergenerational mobility tables in 35 different countries and find that monotonicity cannot be rejected in hardly any table. In addition, we propose new testing procedures for testing conditional stochastic monotonicity and investigate whether monotonicity still holds after conditioning on a number of covariates such as education, cognitive and non‐cognitive skills. Based on the NCDS cohort data from the UK, our results provide evidence that monotonicity holds, even conditionally. Moreover, we do not find large differences in our results when comparing social class and wage class mobility. Copyright © 2010 John Wiley & Sons, Ltd.     

On fitting the conditional difference asymmetry models to square contingency tables with nominal categories

In this paper, we implement the conditional difference asymmetry model (CDAS) for square tables with nominal categories proposed by Tomizawa et al. (J. Appl. Stat. 31(3): 271–277, 2004) with the use of the non-standard log-linear model formulation approach. The implementation is carried out by refitting the model in the 3 ×  3 table in (Tomizawa et al. J. Appl. Stat. 31(3): 271–277, 2004). We extend this approach to a larger 4 ×  4 table of religious affiliation. We further calculated the measure of asymmetry along with its asymptotic standard error and confidence bounds. The procedure is implemted with SAS PROC GENMOD but can also be implemented in SPSS by following the discussion in (Lawal, J. Appl. Stat. 31(3): 279–303, 2004; Lawal, Qual. Quant. 38(3): 259–289, 2004).     

The Structure of the Log Odds-Ratios in Non-Independence and Symmetry Diagonal Models for Square Contingency Tables

This paper examines the structure of the interaction term when expressed as the log ofthe odds-ratios ij in the log-linear model formulation of non-independence and symmetry diagonalmodels applied to an I x I contingency table having ordinal classificatory variables. The class ofprincipal diagonal, diagonal band, full diagonal and diagonal-parameters symmetry models areexamined in the study. The general structure of Φ_ij is examined for each class of diagonal models.The 5 x 5 British Social Mobility data (Glass, 1954) will be employed as an example in this paper.The structure of the log odds-ratios are formulated in terms of the parameter estimates obtained from the application of SAS PROC GENMOD. Goodman in several of his papers on the subject has also derived some of these log odds ratios but our focus here is the structure of these odds, when viewed from the non-standard log-linear perspective (Lawal, 2001, 2002).     

Assessing the Italian tax courts system by weighted three-way log-ratio analysis

Deciding on what should be the most suitable reforms of a national judicial system, whether this means reorganising a courts' structure or offering incetives to judges, requires more empirical evidence. The efficiency of the public judicial system is important to any functioning democracy as it affects several aspects of citizens’ life. As a consequence, new statistical methodologies are required for further valuable insights into the issues that affect the effectiveness of a judicial system.The current study attempts to fill this gap by introducing a new statistical method to investigate the heterogeneity among Italian judicial courts. Here we propose a generalisation of log-ratio analysis for the analysis of the association between the variables of a three-way contingency table. Our focus of this investigation will be on the tax court system of the Italian judicial system.Log-ratio analysis presents many advantages, two of which are the computation and the visual representation of the odds ratios on which the analysis is based. For three-way tables, a conditional odds ratio reflects the likelihood that an event (say tax disputes) will take place in a particular geographical area (say North Italy) instead of an another (say South Italy) given a specific condition (type of justice court).Interestingly, the three-way log-ratio analysis presented in this paper allows one to visually describe these conditional odds-ratios in terms of point distances in a biplot between the types of justice court located in different geographical areas of Italy.     

Review of Non-Independence,Asymmetry, Skew-Symmetry and Point-Symmetry Models in the Analysis of Social Mobility Data

A non-standard log-linear approach is employed to fit the class of non-independence,asymmetric, skew-symmetric and inclined point-symmetry models to intergenerationalmobility table. While these models are not new, our goal in this paper is to present theimplementation of the models that are discussed here using SAS PROC GENMOD. This approach is used to fit these class of models to the 6 × 6 Brazilian social mobility data which has received extensive consideration in the literature, as well as to the 9 × 9 mobility table of men in England and Wales. These are further extended to two 5 × 5 mobility tables of men in the US aged 20–64. Parsimonious models are sought for each table and expressions for estimated odds-ratios under the appropriate models are provided based on our approach.     

A relative mobility table

In this paper a tool of representation of the relative mobility is presented, that couples to the ordinary mobility table as a means of representation of the absolute mobility. From the statistical point of view it is connected to the tradition of “ransacking” mobility tables through odds ratios (Goodman, 1969), whereas the sociological inspiration came from the work of Goldthorpe (1980) on relative mobility. The first section presents the underlying logic and the simple computations involved in the construction of the “relative mobility table” and shows the main properties of the coefficients (“generalized odds ratios”) here used for the study of the “mobility regime”. The second section discusses some other uses of the ralative mobility table data.     

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.     

GENERALIZED AUTOREGRESSIVE SCORE MODELS WITH APPLICATIONS

We propose a class of observation‐driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time‐varying parameters in a wide class of nonlinear models. The GAS model encompasses other well‐known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time‐varying mean. In addition, our approach can lead to new formulations of observation‐driven models. We illustrate our framework by introducing new model specifications for time‐varying copula functions and for multivariate point processes with time‐varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.     

Alternative conceptual frameworks for the analysis of mobility tables and the log-linear models

This article contrasts two alternative conceptual frameworks for the analysis of mobility tables: the “structure vs. circulation” framework that in the past inspired the construction of “pure” mobility indices, and the “absolute vs. relative mobility” one, more recently proposed by Goldthorpe. Contrary to the beliefs of the past, the former cannot conveniently be expressed by the parameters of the saturated log-linear model. The latter, on the contrary, permits (via the language of odds and odds ratios) the coherent application of log-linear models. Moreover, it does not incur those theoretical difficulties which even the most sophisticated attempts to save the old framework incur (e.g. Sobel et al., 1985). In the third section of the article a contribution to the understanding of the analysis of mobility tables through odds ratios is given and the relation between odds ratios and the interaction parameters of the saturated model is shown.     

Testing conditional symmetry against one-sided alternatives in square contingency tables

For a (k×k) square contingency table with ordered categories, letX(Y) denote the row (column) number. The conditional symmetry model is given byP(X=i, Y=j|X<Y)=P(X=j, Y=i |X>Y), ∀i<j. In this paper, we study the likelihood ratio tests of conditional symmetry in a square contingency table against two particular classes of one-sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A simulation study is performed by comparing the powers of different tests. The theory developed is illustrated by using the famous eye vision data from Stuart (1953).     

Beyond Interaction: Theories

An interaction between two factors in their effects on a dependent variable is here modelled as (first) summation of quantities derived from the two factors individually, and (second) a nonlinear relation between that sum and the dependent variable. There are only a few possible nonlinear functions that need to be considered. The dataset used as an example is a 3 × 3 table showing crossover interaction. The proposed explanation uses a quadratic curve (which has a descending limb and an ascending limb) to achieve reversal of the effect of one factor for some categories of the other factor.     

Double symmetry model and its decompositions in square contingency tables having ordered categories

In this paper, we have employed the non-standard log-linear models to fit the double symmetry models and some of its decompositions to square contingency tables having ordered categories. SAS PROC GENMOD was employed to fit these models although we could similarly have used GENLOG in SPSS or GLM in STATA. A SAS macro generates the factor or scalar variables required to fit these models. Two sets of \(4 \times 4\) unaided distance vision data that have been previously analyzed in (Tahata and Tomizawa, Journal of the Japan Statistical Society 36:91–106, 2006) were employed for verification of results. We also extend the approach to the Danish \(5 \times 5\) Mobility data as well as to the \(3 \times 3\) Danish longitudinal study data of subjective health, firstly reported in (Andersen, The Statistical Analysis of Categorical Data, Springer:Berlin, 1994) and analyzed in (Tahata and Tomizawa, Statistical Methods and Applications 19:307–318, 2010). Results obtained agree with those published in previous literature on the subject. The approaches suggest here eliminate any programming that might be required in order to apply these class of models to square contingency tables.     

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.     

Monte Carlo theoretical trials of methods for assessing statistical significance for differences between adjusted odds ratios

Odds are generally defined as the number of successes divided by the number of failures in a given number of trials. An odds ratio is the ratio of one odds divided by another. Odds ratios can be adjusted to reflect associations with the outcome independently of the influence of associations with other variables. These are adjusted odds ratios. There are several well known methods for comparing odds ratios and testing for statistically significant differences between them. Analogous methods for adjusted odds ratios are not well known or well documented. One method for comparing adjusted odds ratios is explained by Hosmer and Lemeshow (Applied logistic regression, 2000). This method is used for the odds ratios for two variables from the same data set. The purpose of this analysis was to apply this method to a different situation: comparing odds ratios for the same variable from two different data sets. Monte Carlo trials were used to assess the performance of the method and these indicated the method performed well.     

Tests with correct size when instruments can be arbitrarily weak

This paper applies classical exponential-family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced-form variance matrix. The results can be divided into the limited-information and full-information categories. In the limited-information model, it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable. In the full-information framework, this paper proposes a family of similar tests for subsets of endogenous variables’ coefficients. For both limited- and full-information models, there exist power upper bounds for unbiased tests. When the model is just-identified, the Anderson–Rubin, score, and (pseudo) conditional likelihood ratio tests are optimal. When the model is over-identified, the (pseudo) conditional likelihood ratio test has power close to the power envelope when identification is strong.     

Tests of Predictive Ability for Vector Autoregressions Used for Conditional Forecasting

Many forecasts are conditional in nature. For example, a number of central banks routinely report forecasts conditional on particular paths of policy instruments. Even though conditional forecasting is common, there has been little work on methods for evaluating conditional forecasts. This paper provides analytical, Monte Carlo and empirical evidence on tests of predictive ability for conditional forecasts from estimated models. In the empirical analysis, we examine conditional forecasts obtained with a VAR in the variables included in the DSGE model of Smets and Wouters (American Economic Review 2007; 97 : 586–606). Throughout the analysis, we focus on tests of bias, efficiency and equal accuracy applied to conditional forecasts from VAR models. Copyright © 2016 John Wiley & Sons, Ltd.     

An evaluation of the performance and suitability of R × C methods for ecological inference with known true values

Ecological inference refers to the study of individuals using aggregate data and it is used in an impressive number of studies; it is well known, however, that the study of individuals using group data suffers from an ecological fallacy problem (Robinson in Am Sociol Rev 15:351–357, 1950). This paper evaluates the accuracy of two recent methods, the Rosen et al. (Stat Neerl 55:134–156, 2001) and the Greiner and Quinn (J R Stat Soc Ser A (Statistics in Society) 172:67–81, 2009) and the long-standing Goodman’s (Am Sociol Rev 18:663–664, 1953; Am J Sociol 64:610–625, 1959) method designed to estimate all cells of R × C tables simultaneously by employing exclusively aggregate data. To conduct these tests we leverage on extensive electoral data for which the true quantities of interest are known. In particular, we focus on examining the extent to which the confidence intervals provided by the three methods contain the true values. The paper also provides important guidelines regarding the appropriate contexts for employing these models.     

A generalization of Cochran's procedure for the combining of r×c contingency tables*

Abstract Cochran [3] derives a test of association when k 2 × 2 contingency tables are combined. We show in this paper how to extend Cochran's test to the combining of k r×c contingency tables using a multiple comparison technique similar to the one presented by Dunn [4]. An example is included.