Algebraic method for independence model of two-way contingency tables

The main purpose of the study is to propose an algebraic method to obtain the set of all independence models of I×J two-way contingency tables with the same row sums and column sums which is called fiber in algebraic statistics. This method involves solving a system of linear algebraic equations that only rely on row sums and column sums of the I×J two-way contingency table. The MATLAB software was used to solve this system. The effectiveness of the purposed method is illustrated by applying to a contingency table of agriculture teachers’ perception of secondary school agriculture

Bayesian estimation of the orthogonal decomposition of a contingency table

In a multinomial sampling, contingency tables can be parametrized by probabilities of each cell. These probabilities constitute the joint probability function of two or more discrete random variables. These probability tables have been previously studied from a compositional point of view. The compositional analysis of probability tables ensures coherence when analysing sub-tables. The main results are: (1) given a probability table, the closest independent probability table is the product of their geometric marginals; (2) the probability table can be orthogonally decomposed into an independent table and an interaction table; (3) the departure of independence can be measured using simplicial deviance, which is the Aitchison square norm of the interaction table. In previous works, the analysis has been performed from a frequentist point of view. This contribution is aimed at providing a Bayesian assessment of the decomposition. The resulting model is a log-linear one, which parameters are the centered log-ratio transformations of the geometric marginals and the interaction table. Using a Dirichlet prior distribution of multinomial probabilities, the posterior distribution of multinomial probabilities is again a Dirichlet distribution. Simulation of this posterior allows to study the distribution of marginal and interaction parameters, checking the independence of the observed contingency table and cell interactions. The results corresponding to a two-way contingency table example are presented.Peer ReviewedPostprint (published version

Tables de contingence à trois dimensions : aspects théoriques, applications et analogie avec l'analyse de la variance à trois critères de classification

For contingency table, concepts of independence are not so easy to understand in statistical practice. So, theoretical aspects of the log-linear model are described and applied to a three levels example; an analogy with analysis of variance is also established.Les notions d'indépendance mises en évidence lors de l'étude d'une table de contingence, à trois dimensions au moins, ne semblent pas être évidentes à comprendre en pratique. C'est pourquoi, des aspects théoriques faisant référence au modèle log-linéaire sont détaillés et illustrés, et une analogie avec l'analyse de la variance à trois critères de classification, modèle fixe est établie

Markov bases and structural zeros

AbstractIn this paper we apply the elimination technique to the computation of Markov bases, paying special attention to contingency tables with structural zeros. An algebraic relationship between the Markov basis for a table with structural zeros and the corresponding complete table is proved. In order to find the relevant Markov basis, it is enough to eliminate the indeterminates associated with the structural zeros from the toric ideal for the complete table. Moreover, we use this result for the computation of Markov bases for some classical log-linear models, such as quasi-independence and quasi-symmetry, and computations in the multi-way setting are presented

Evaluation of the weighted least squares method for the analysis of categorical data.

Hypotheses about the relationship among variables in a multiway contingency table may be tested by analysis of the probability distribution of observed frequencies or transformation of these frequencies. Two model-based approaches for the testing of structural hypotheses are the log-linear model, using iterative maximum-likelihood (ML) estimation procedures and the weighted least squares (WLS) linear model method of Grizzle, Starmer and Koch (GSK), a general noniterative procedure. Both methods asymptotically provide the same estimates and test statistics. This study compared the GSK and log-linear approaches for testing hypotheses in r x c contingency tables. Tables were simulated under various conditions of table, sample, row-, and column-effect sizes. Test statistics for row and column effects, and interaction were calculated using: (i) GSK linear model, untransformed proportion (p); (ii) GSK linear model, logarithm of the proportion (log p); (iii) GSK linear model, log-odds (log p/(1-p)); and (iv) log-linear model. Type I error rates were examined, and the relative power of the procedures was studied. The log-linear model yielded Type I error rates close to the expected values; all GSK models yielded error rates higher than expected, with smallest error rates associated with logarithmic transformations. Sample size and table size had no effect on Type I error rates. All GSK procedures were uniformly more powerful than the log-linear procedure. Differences were most noticeable with medium effect sizes and diminished as sample and effect sizes increased. There were no systematic differences due to table size. Findings from this study are pertinent to applied researchers who wish to test hypotheses other than those of independence with categorical data. Hypothesis testing and interpretation of results are straightforward with a model-based approach and are thus encouraged. The results indicate that GSK methods provide the most powerful tests. Since the GSK method is easily implemented and can be understood by researchers familiar with linear regression analysis, it is recommended that the GSK method be used to analyze categorical data

ANALISIS DATA KATEGORI DENGAN LOG LINIER MENGGUNAKAN PRINSIP HIRARKI (STUDI KASUS JUMLAH KECELAKAAN LALU LINTAS DI KOTA MAKASSAR TAHUN 2011)

Presentation of data commonly used frequency tables, but for categorical data, the table used is the contingency table, it is a table in the form of rows and columns and can be used for two or more variables. As with the variable type of vehicle, age, and education level in the case of traffic accidents of Makassar in 2011. If the case stated in the table is included in the category of three-dimensional contingency table, then the appropriate analysis use is an analysis of the log linear analysis, it is techniques to determine the cause dependency category. The parameters of the log linear models estimated using Maximum Likelihood. Test used is the Goodness of Fit test aims to determine independence between variables with the statistical Chi-Square test or Likelihood Ratio. Furthermore, the selection of the best model using backward Elimination matode which basically uses the principle of hierarchy. In the estimation of log linear models for three-dimensional model has several estimators corresponding to such possible models: estimator for model (X, Y, Z) is = .. .. .. …2, then the other estimators in accordance with their respective models. The produced analysis of log linear analysis of traffic accidents in the city of Makassar in 2011 is a model of interaction between the types of vehicles with Last Education and age of the rider with driver education last driver