Contingency Tables

Se hace  una sucinta presentación del desarrollo de las tablas de contingencia,  así como  los elementos básicos de las tablas de contingencia discutiendo algunos detalles que, al no considerarlos invalidan, las generalizaciones que se hacen usando los métodos que comúnmente se introducen en los cursos de estadística.  Se presentan los elementos básicos de las  medidas de asociación y se analizan algunos coeficientes que caracterizan las propiedades de las relaciones que aparecen entre las variables.

KEYWORDS: Tablas de contingencia, medidas de asociación, pruebas

El presente artículo presenta la metodología para
analizar las expectativas y percepciones del cliente en tres
tipos de empresas de servicio distintas utilizando el cuestionario SERVQUAL propuesto por Parasuraman, la metodología hace hincapié en el uso de pruebas no paramétricas, concretamente tablas de contingencia y pruebas de Chi-cuadrada para verificar su dependencia o independencia y, de esta manera, determinar el futuro plan de mejora de cada dimensión en particular. Se muestra además un caso que no cumple con la condición de uso de la prueba Chi-cuadrada y su posterior tratamiento.

'CAinterprTools' is a R package for visual aid to Correspondence Analysis interpretation. The package (which is described in Alberti G, CAinterprTools: An R package to help interpreting Correspondence Analysis’ results, SoftwareX, Volumes 1–2, September 2015, Pages 26-31) has been improved several times since 2015.
An updated description of the facilities it provides can be found in this technical report.

Categorical data are often stratified into two dimensional I × J tables in order to test the independence of attributes. Agresti (1999) has discussed testing methods with partitioning property of chi square to extract components that describe certain aspects of the overall association in a table. In this work an attempt has been made to study the pattern in which sub-tables exhibit a sign of reverse association when compared to a significant association of attributes with regard to the original I × J table. Simulation studies and subsequent results of vote counting method indicate that 2 × 2 tables have the reversal component association when compared to higher order sub-tables; interestingly, in more than 90% cases. The computationally extensive and exhaustive procedure provide a better tool to understand the association between the categories that focus on the strongest differences among all comparisons, and could be practically and other aspects in experimental studies such as cl...

This paper introduces and investigates the notion of a hyper Markov law, which is a probability distribution over the set of probability measures on a multivariate space that (i) is concentrated on the set of Markov probabilities over some decomposable graph, and (ii) satisfies ...

The relevance of a fund manager's educational and experience profile to the size of investment portfolio return has been the subject of recurrent research in the last decade. While previous research considered an external reference point of view analysing industry ...

RESUMEN
Se realizó un estudio comparativo entre pruebas de permutación y asintóticas, aplicadas a tablas de contingencia de dimensión 2×2, utilizando como medida de comparación la diferencia entre el p-valor exacto y su equivalente asintótico. Se analizaron cinco ejemplos publicados en la literatura científica internacional con el objeto de mostrar bajo cuáles condiciones ambos enfoques presentan un comportamiento diferente o un comportamiento equivalente para las pruebas de independencia de ChI-2, razón de verosimilitud y prueba exacta de Fisher. Los resultados mostraron que el comportamiento de las metodologías exacta y asintótica depende del tamaño de muestra, del balanceo y la dispersión de la tabla de contingencia y la prueba aplicada. Para los casos estudiados, se encontró que los p-valores exactos y asintóticos presentaron diferencias notables para tamaños de muestras pequeños, sobre todo en tablas de contingencia desbalanceadas y dispersas, y tendieron a mostrar convergencia en la medida que el tamaño de muestra era mayor.
PALABRAS CLAVE: p-valor, tablas de contingencia balanceadas, tablas de contingencia desbalanceadas, tablas de contingencia dispersas, pruebas de independencia.
ABSTRACT
A comparative study between exact and asymptotic tests applied to 2×2 dimension contingency tables using the difference between the exact p-value and the asymptotic counterpart as a comparison measure was conducted. Five examples published in the international scientific literature were analyzed in order to show under what conditions the two approaches present a different or equivalent behavior for independence tests Chi-2, likelihood ratio, and Fisher exact test. The results showed that the behavior of exact and asymptotic methods depends on the sample size, the balance and the dispersion of the contingency table and the independency test applied. For the studied cases it was found that exact p-values differ from the asymptotic one for small sample sizes, especially in unbalanced and sparse contingency tables, and tended to show convergence as the sample size was increased.
KEYWORDS: p-value, balanced contingency tables, unbalanced contingency tables, sparse contingency tables, independence tests.

A number of interesting packages are available to perform Correspondence Analysis in R. At the best of my knowledge, they lack some tools to help users to eyeball some critical CA aspects (e.g., contribution of rows/cols categories to the principal axes, quality of the display,correlation of rows/cols categories with dimensions, etc). Besides providing those facilities, this package allows the users to calculate the significance of the CA dimensions by means of the 'Average Rule', the Malinvaud test, and by permutation test. Further, it allows to also calculate the permuted significance of the CA total inertia. The package comes with a dataset (greenacre_data) after Greenacre 2007 (p. 90, exhibit 12.1).

The package allows to plot a number of Correspondence Analysis information such as the contribution of rows and columns categories to the principal axes, the quality of points display on selected dimensions, the correlation of row and column categories to selected dimensions, etc. It also allows to assess which dimension(s) is important for the data structure interpretation by means of the so called 'Average Rule'. Moreover, it implements the Malinvaud test, which test the significance of the table dimensions. The package also offers the facility to plot the permuted distribution of the table total inertia as well as of the inertia accounted for by pairs of selected dimensions. The two latter facilities allows to test the significance of the total inertia and of the dimensions the user is interest in. For more details about the rationale behind the use of the implemented CA interpretation facilities, see the journal article Alberti 2013 cited below. As to the situations in which permutation test can be applied to CA (e.g., to assess the significance of the dimensions), see Greenacre 2007 (198-199).

... 1981), the integral in (2) is a Stieltjes integral, which permits, for example, a discrete, continuous or ... Let the array X[_fl consist of all mathematics items on the November 1991 SAT excluding ... the function 9(V) is required to be a nondecreasing item summary (Rosen-baum, 1984 ...

Conditions are presented for local identifiability of discrete undi-rected graphical models with a binary hidden node. These models can be obtained by extending the latent class model to allow for conditional associations between the observed variables. We establish a necessary and sufficient condition for the model to be locally identified almost everywhere in the parameter space and we provide expressions of the subspace where identifiability breaks down. The condition is based on the topology of the undirected graph and relies on the faithfulness assumption.

One of the oldest problems in statistics is how to analyze a 2 x 2 table and it is still very much with us today. The case of zero fixed marginals, analyzed via the unconditional non-asymptotic method, is perhaps the one that has received least attention, owing to its computational complexity. This paper presents a critical review of existing publications

We discuss a situation in which, once a logit model is fitted to the data in a contingency table, some factor levels are grouped. Generally, researchers reapply a logit model on the pooled data, however, this approach leads to the violation of the original distributional assumption, when the probabilities of success of the random variables of aggregation differ. We suggest an alternative procedure that operates under the unsaturated, multifactorial, binomial, logit model. Based on asymptotic theory and taking advantage of the decrease in the variance when a correct distributional assumption is made, the suggested procedure significantly improves the estimates, reduces the standard errors, produces lower residuals and is less likely to reject the goodness of fit test on the model. We present the necessary theory, results of an extensive simulation designed for this purpose, and the suggested procedure contrasted with the usual approach, through a complete numerical example.

Se discute el efecto que se produce sobre el modelo logit binario con un único factor explicativo cuando el investigador decide agrupar algunos niveles de dicho factor. Con base en la parametrización de referencia y el modelo saturado se sugiere un procedimiento que, aprovechando los cómputos de un primer ajuste logit y corrigiendo el supuesto distribucional sobre la varianza, produce estimaciones más eficientemente y con mayor precisión que las que se producen si solo se decide reiterar un ajuste logit. Una vez colocado el tema en perspectiva, se desarrollan las ecuaciones que sustentan el procedimiento sugerido, apelando a la teoría asintótica. Se ilustra mediante un ejemplo la diferencia entre el procedimiento sugerido y el habitual y, con base en una extensa simulación, se muestran tendencias sólidas a favor del primero, en la medida en que las probabilidades de éxito de la variable respuesta (Y=1), asociadas con las categorías del factor explicativo incluidas en la agrupación, ...

'chi.perm' is an R function which allows to perform the chi-square test of independence on the basis of  permuted tables, whose number is selected by user. For the rationale of this approach, see for instance the nice description provided by Beh E.J., Lombardo R. 2014, Correspondence Analysis: Theory, Practice and New Strategies, Chichester, Wiley, at pages 62-64.