The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an ...The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having any degrees of freedom. As the degrees of freedom becomes larger, the proposed model approaches the extended linear diagonals-parameter symmetry model, which may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution. The simulation study based on bivariate t-distribution is given. An example is given.展开更多
Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables wi...Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables with sufficiently large sample size, have an asymptotic chi-square distribution with degrees of freedom (df) equal to the number of free parameters in the saturated model. For example under the hypothesis of independence of the row and column conditioned on the layer in an I × J × K contingency table, the df is K(I –1)(J– 1). These test statistics, in large sized tables, will have less power since they have large degrees of freedom. This paper proposes a product effect model, which combines the advantages of the multiplicative models over the additive, for analysing the interaction between the row and column of the 3-way table conditioned on the layer. The derived statistics is shown to be asymptotically chi-square with a small degree of freedom, K?– 1, for the I × J × K contingency table. The performance of the developed statistic is compared with the Pearson’s chi-square statistic and the likelihood ratio statistic test using an illustrative example. The results show that the product effect test can detect interaction even when some of the main effects are not significant and can perform better than the other competitors having smaller degree of freedom in large sized tables.展开更多
For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The pro...For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.展开更多
For square contingency tables with ordered categories, this article proposes new models, which are the extension of Tomizawa’s [1] diagonal exponent symmetry model. Also it gives the decomposition of proposed model, ...For square contingency tables with ordered categories, this article proposes new models, which are the extension of Tomizawa’s [1] diagonal exponent symmetry model. Also it gives the decomposition of proposed model, and shows the orthogonality of the test statistics for decomposed models. Examples are given and the simulation studies based on the bivariate normal distribution are also given.展开更多
In this paper we examine 5 indexes (the two Yule’s indexes, the chi square, the odds ratio and an elementary index) of a two-by-two table, which estimate the correlation coefficient ρ in a bivariate Bernoulli distri...In this paper we examine 5 indexes (the two Yule’s indexes, the chi square, the odds ratio and an elementary index) of a two-by-two table, which estimate the correlation coefficient ρ in a bivariate Bernoulli distribution. We will find the compact expression of the influence functions, which allow the quantification of the effect of an infinitesimal contamination of the probability of any pair of attributes of the bivariate random variable distributed according to the above-mentioned model. We prove that the only unbiased index is the chi square. In order to determine the indexes, which are less sensitive to contamination, we obtain the expressions of three synthetic measures of the influence function, which are the maximum contamination (gross sensitivity error), the mean square deviation and the variance. These results, even if don’t allow a definitive assessment of the overall optimum properties of the five indexes, as not all of them are unbiased, nevertheless they allow to appreciating the synthetic entity of the effect of the contaminations in the estimation of the parameter ρ of the bivariate Bernoulli distribution.展开更多
In this paper, we propose an information-theoretic-criterion-based modelselection procedure for log-linear model of contingency tables under multinomial sampling, andestablish the strong consistency of the method unde...In this paper, we propose an information-theoretic-criterion-based modelselection procedure for log-linear model of contingency tables under multinomial sampling, andestablish the strong consistency of the method under some mild conditions. An exponential bound ofmiss detection probability is also obtained. The selection procedure is modified so that it can beused in practice. Simulation shows that the modified method is valid. To avoid selecting the penaltycoefficient in the information criteria, an alternative selection procedure is given.展开更多
An iterative method of De Jong,Greig and Madan(1985)for obtaining maximum likelihoodestimates for contingency tables derived from a Mover-Stayer model is considered.It is shown that thismethod works for other sampling...An iterative method of De Jong,Greig and Madan(1985)for obtaining maximum likelihoodestimates for contingency tables derived from a Mover-Stayer model is considered.It is shown that thismethod works for other sampling schemes leading to incomplete contingency tables.Moreover,compar-isons indicate that the method is very fast.展开更多
Various quasi-independence models which have been considered in the literature are examined. Methods of obtaining maximum likelihood estimates for these modelsare compared. A new, very simple method is obtained.
文摘The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having any degrees of freedom. As the degrees of freedom becomes larger, the proposed model approaches the extended linear diagonals-parameter symmetry model, which may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution. The simulation study based on bivariate t-distribution is given. An example is given.
文摘Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables with sufficiently large sample size, have an asymptotic chi-square distribution with degrees of freedom (df) equal to the number of free parameters in the saturated model. For example under the hypothesis of independence of the row and column conditioned on the layer in an I × J × K contingency table, the df is K(I –1)(J– 1). These test statistics, in large sized tables, will have less power since they have large degrees of freedom. This paper proposes a product effect model, which combines the advantages of the multiplicative models over the additive, for analysing the interaction between the row and column of the 3-way table conditioned on the layer. The derived statistics is shown to be asymptotically chi-square with a small degree of freedom, K?– 1, for the I × J × K contingency table. The performance of the developed statistic is compared with the Pearson’s chi-square statistic and the likelihood ratio statistic test using an illustrative example. The results show that the product effect test can detect interaction even when some of the main effects are not significant and can perform better than the other competitors having smaller degree of freedom in large sized tables.
文摘For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.
文摘For square contingency tables with ordered categories, this article proposes new models, which are the extension of Tomizawa’s [1] diagonal exponent symmetry model. Also it gives the decomposition of proposed model, and shows the orthogonality of the test statistics for decomposed models. Examples are given and the simulation studies based on the bivariate normal distribution are also given.
文摘In this paper we examine 5 indexes (the two Yule’s indexes, the chi square, the odds ratio and an elementary index) of a two-by-two table, which estimate the correlation coefficient ρ in a bivariate Bernoulli distribution. We will find the compact expression of the influence functions, which allow the quantification of the effect of an infinitesimal contamination of the probability of any pair of attributes of the bivariate random variable distributed according to the above-mentioned model. We prove that the only unbiased index is the chi square. In order to determine the indexes, which are less sensitive to contamination, we obtain the expressions of three synthetic measures of the influence function, which are the maximum contamination (gross sensitivity error), the mean square deviation and the variance. These results, even if don’t allow a definitive assessment of the overall optimum properties of the five indexes, as not all of them are unbiased, nevertheless they allow to appreciating the synthetic entity of the effect of the contaminations in the estimation of the parameter ρ of the bivariate Bernoulli distribution.
基金This research is partially supported by National Natural Science Foundation of China (10171094),Ph.D. Program Foundation of Ministry of Education of China and Special Foundations of the Chinese Academy of SciencesUSTC.
文摘In this paper, we propose an information-theoretic-criterion-based modelselection procedure for log-linear model of contingency tables under multinomial sampling, andestablish the strong consistency of the method under some mild conditions. An exponential bound ofmiss detection probability is also obtained. The selection procedure is modified so that it can beused in practice. Simulation shows that the modified method is valid. To avoid selecting the penaltycoefficient in the information criteria, an alternative selection procedure is given.
文摘An iterative method of De Jong,Greig and Madan(1985)for obtaining maximum likelihoodestimates for contingency tables derived from a Mover-Stayer model is considered.It is shown that thismethod works for other sampling schemes leading to incomplete contingency tables.Moreover,compar-isons indicate that the method is very fast.
文摘Various quasi-independence models which have been considered in the literature are examined. Methods of obtaining maximum likelihood estimates for these modelsare compared. A new, very simple method is obtained.
