Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two-Dimensional Marginals

This paper considers a connected Markov chain for sampling 3 × 3 ×K contingency tables having fixed two‐dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three‐factor interactions. A Markov chain algorithm is a valuable tool for evaluating P‐values, especially for sparse datasets where large‐sample theory does not work well. To construct a connected Markov chain over high‐dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time‐consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3 × 3 ×K contingency tables. The minimal basis is unique. Some numerical examples illustrate the practicality of our algorithms.     

Marginal inhomogeneity models for square contingency tables with nominal categories

For the analysis of square contingency tables with nominal categories, this paper proposes two kinds of models that indicate the structure of marginal inhomogeneity. One model states that the absolute values of log odds of the row marginal probability to the corresponding column marginal probability for each category i are constant for every i. The other model states that, on the condition that an observation falls in one of the off-diagonal cells in the square table, the absolute values of log odds of the conditional row marginal probability to the corresponding conditional column marginal probability for each category i are constant for every i. These models are used when the marginal homogeneity model does not hold, and the values of parameters in the models are useful for seeing the degree of departure from marginal homogeneity for the data on a nominal scale. Examples are given.     

Fractional Regression Hot Deck Imputation Weight Adjustment

Fractional regression hot deck imputation (FRHDI) imputes multiple values for each instance of a missing dependent variable. The imputed values are equal to the predicted value plus multiple random residuals. Fractional weights enable variance estimation and preserve correlations. In some circumstances with some starting weight values, existing procedures for computing FRHDI weights can produce negative values. We discuss procedures for constructing non-negative adjusted fractional weights for FRHDI and study performance of the algorithm using simulation. The algorithm can be used effectively with FRDHI procedures for handling missing data in the context of a complex sample survey.     

Conditional Difference Asymmetry Model for Square Contingency Tables with Nominal Categories

This paper proposes a model, which is an extension-of-symmetry model, for square contingency tables with the same nominal row and column classifications. The model states that the absolute values of difference between the conditional probability that an observation will fall in cell (i, j) on condition that it falls in cell (i, j) or (j, i) and the conditional probability that it falls in cell (j, i) on the same condition, are constant for every i≠j. The model describes a structure of asymmetry (not symmetry), and it is applied to the data on a nominal scale. An example is given.     

SIMULATIONS AND DERIVED APPROXIMATIONS FOR THE MEANS AND STANDARD DEVIATIONS OF THE CHARACTERISTIC ROOTS OF A WISHART MATRIX

Monte Carlo simulations were done to estimate the means and standard deviations of the characteristic roots of a Wishart matrix which can be used in computing tests of hypotheses concerning multiplicative terms in balanced linear-bilinear (multiplicative) models for an m × n table of data. In this report we extend the previous results (Mandel, 1971; Cornelius, 1980) to r ≤ 199, c ≤ 149 or r ≤ 149, c ≤ 199, where r and c are row and column degrees of freedom, respectively, of the two-way array of residuals (with total degrees of freedom rc) after fitting the linear effects. For 187 combinations of r and c at intervals over this domain, we used 5000 simulations to estimate expectations and standard deviations of the Wishart roots. Using weighted linear regression variable selection techniques, symmetric functions of r and c were obtained for approximating the simulated means and standard deviations. Use of these approximating functions will avoid the need for reference to tables for input to computer programs which require these values for tests of significance of sequentially fitted terms in the analyses of balanced linear-bilinear models.     

Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j (> i ) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.     

对外贸易隐含SO2测算及影响因素的结构分解分析

 本研究根据中国2002、2005、2007年非竞争型投入产出表、OECD的国外投入产出表和GTAP Version 7.0的SO2排放强度等数据,利用多国投入产出模型方法测算了中国对外贸易中隐含SO2,相关的行业结构分布以及隐含污染物在主要贸易伙伴国的贸易流向,并通过结构分解方法分析了三大因素对出口贸易隐污染物的贡献大小。研究结果表明：2002年至2007年间,我国对外贸易创造了巨大贸易顺差,同时也带来了大量隐含SO2顺差;迅速增长的贸易规模是出口贸易隐含污染物增加的主要因素,但是,环境保护和节能减排技术改进导致的SO2排放强度下降抑制出口贸易隐含SO2增加,而投入产出中间结构变化对其影响较小。     

Bayesian clustering for row effects models

We deal with two-way contingency tables having ordered column categories. We use a row effects model wherein each interaction term is assumed to have a multiplicative form involving a row effect parameter and a fixed column score. We propose a methodology to cluster row effects in order to simplify the interaction structure and to enhance the interpretation of the model. Our method uses a product partition model with a suitable specification of the cohesion function, so that we can carry out our analysis on a collection of models of varying dimensions using a straightforward MCMC sampler. The methodology is illustrated with reference to simulated and real data sets.     

Generalized linear spatial models in epidemiology: A case study of zoonotic cutaneous leishmaniasis in Tunisia

Generalized linear spatial models (GLSM) are used here to study spatial characters of zoonotic cutaneous leishmaniasis (ZCL) in Tunisia. The response variable stands for the number of affected by district during the period 2001–2002. The model covariates are: climates (temperature and rainfall), humidity and surrounding vegetation status. As the environmental and weather data are not available for all the studied districts, Kriging based on linear interpolation was used to estimate the missing data. To account for unexplained spatial variation in the model, we include a stationary Gaussian process S with a powered exponential spatial correlation function. Moran coefficient, DIC criterion and residuals variograms are used to show the high goodness-of-fit of the GLSM. When compared with the statistical tools used in the previous ZCL studies, the optimal GLSM found here yields a better assessment of the impact of the risk factors, a better prediction of ZCL evolution and a better comprehension of the disease transmission. The statistical results show the progressive increase in the number of affected in zones with high temperature, low rainfall and high surrounding vegetation index. Relative humidity does not seem to affect the distribution of the disease in Tunisia. The results of the statistical analyses stress the important risk of misleading epidemiological conclusions when non-spatial models are used to analyse spatially structured data.     

On the use of diagnostic biplots in model screening for genotype by environment tables

Popular rank-2 and rank-3 models for two-way tables have geometrical properties which can be used as diagnostic keys in screening for an appropriate model. Row and column levels of two-way tables are represented by points in two or three dimensional space, whereupon collinearity and coplanarity of row and column points provide diagnostic keys for informal model choice. Coordinates are obtained from a factorization of the two-way table Y in the matrix product UV ^T. The rows of U then contain row-point coordinates and the rows of V column-point coordinates. Illustrations of applications of diagnostic biplots in the literature were restricted to data from chemistry and physics with little or no noise. In plant breeding, two-way tables containing substantial amounts of noise regularly arise in the form of genotype by environment tables. To investigate the usefulness of diagnostic biplots for model screening for genotype by environment tables, data tables were generated from a range of two-way models under the addition of various amounts of noise. Chances for correct diagnosis of the generating model depended on the type of model. Diagnostic biplots on their own do not seem to provide a sufficient means for model selection for genotype by environment tables, but in combination with other methods they certainly can provide extra insight into the structure of the data.     

Compositional Tables Analysis in Coordinates

Compositional tables – a continuous counterpart to the contingency tables – carry relative information about relationships between row and column factors; thus, for their analysis, only ratios between cells of a table are informative. Consequently, the standard Euclidean geometry should be replaced by the Aitchison geometry on the simplex that enables decomposition of the table into its independent and interactive parts. The aim of the paper is to find interpretable coordinate representation for independent and interaction tables (in sense of balances and odds ratios of cells, respectively), where further statistical processing of compositional tables can be performed. Theoretical results are applied to real‐world problems from a health survey and in macroeconomics.     

Correspondence analysis of square asymmetric matrices

The application of correspondence analysis to square asymmetric tables is often unsuccessful because of the strong role played by the diagonal entries of the matrix, obscuring the data off the diagonal. A simple modification of the centring of the matrix, coupled with the corresponding change in row and column masses and row and column metrics, allows the table to be decomposed into symmetric and skew symmetric components, which can then be analysed separately. The symmetric and skew symmetric analyses can be performed by using a simple correspondence analysis program if the data are set up in a special block format. The methodology is demonstrated on a social mobility table from the first democratically elected Parliament in Germany, the Frankfurter Nationalversammlung . The table cross-tabulates the jobs of parliamentarians when first entering the labour market and their jobs in May 1848 when the Parliament started its first session.     

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.     

Diagonal uniform association symmetry modelsfor cumulative probabilities in square tables

For the analysis of square contingency tables with ordered categories, Tomizawa (1991) considered the diagonal uniform association symmetry (DUS) model, which has a multiplicative form for cell probabilities and has the structure of uniform association in the tables constructed using two diagonals that are equidistant from the main diagonal. This paper proposes another DUS model which has a similar multiplicative form for cumulative probabilities. The model indicates that the odds that an observation will fall in row category i or below and column category i+k or above, instead of in column category i or below and row category i+k or above, increase (decrease) exponentially as the cutpoint i increases for a fixed k. Examples are given.     

The effect of interaction and rounding error in two-way ANOVA: example of impact on testing for normality

A key issue in various applications of analysis of variance (ANOVA) is testing for the interaction and the interpretation of resulting ANOVA tables. In this note it is demonstrated that for a two-way ANOVA, whether interactions are incorporated or not may have a dramatic influence when considering the usual statistical tests for normality of residuals. The effect of numerical rounding is also discussed.     

Detection of outliers in simple circular regression models using the mean circular error statistic

The investigation on the identification of outliers in linear regression models can be extended to those for circular regression case. In this paper, we propose a new numerical statistic called mean circular error to identify possible outliers in circular regression models by using a row deletion approach. Through intensive simulation studies, the cut-off points of the statistic are obtained and its power of performance investigated. It is found that the performance improves as the concentration parameter of circular residuals becomes larger or the sample size becomes smaller. As an illustration, the statistic is applied to a wind direction data set.     

Marginal asymmetry model for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, this paper proposes a model which indicates the structure of marginal asymmetry. The model states that the absolute values of logarithm of ratio of the cumulative probability that an observation will fall in row category i or below and column category i+1 or above to the corresponding cumulative probability that the observation falls in column category i or below and row category i+1 or above are constant for every i. We deal with the estimation problem for the model parameter and goodness-of-fit tests. Also we discuss the relationships between the model and a measure which represents the degree of departure from marginal homogeneity. Examples are given.     

A model of uniform association plus two-diagonalsparameter and its application to occupational mobility table data

For the analysis of square contingency tables with ordered categories, Agresti (1988) introduced a model having the structure of uniform association plus a main-diagonal parameter. This paper extends that model. The extended model has the structure of uniform association plus two-diagonals-parameter, and it is a special case of the quasi-uniform association model introduced by Goodman (1979). The Danish occupational mobility table data are analyzed using the models introduced here.     

Influence diagnostics in the Gamma regression model with adjusted deviance residuals

In this study, we develop the adjusted deviance residuals for the gamma regression model (GRM) by following Cordeiro's (2004) method. These adjusted deviance residuals under the GRM are used for influence diagnostics. A comparative analysis has been sorted out between our proposed method of the adjusted deviance residuals and an existing method for influence diagnostics. These results are illustrated by a simulation study and using a real data set. They are presented for different values of dispersion and sample sizes and indicate the significant role of the GRM inferences.