RESUMO
We obtain expressions for the asymptotic distributions of the Rényi and Tsallis of order q entropies and Fisher information when computed on the maximum likelihood estimator of probabilities from multinomial random samples. We verify that these asymptotic models, two of which (Tsallis and Fisher) are normal, describe well a variety of simulated data. In addition, we obtain test statistics for comparing (possibly different types of) entropies from two samples without requiring the same number of categories. Finally, we apply these tests to social survey data and verify that the results are consistent but more general than those obtained with a χ2 test.
RESUMO
Teorias sobre fenômenos psicológicos frequentemente fazem referência a processos que não são diretamente observáveis (processos latentes). Tradicionalmente, no entanto, a investigação desses fenômenos é feita de forma indireta aos processos latentes. O objetivo deste artigo é introduzir os conceitos fundamentais de modelagem multinomial. Aqui mostramos como modelos de processos latentes são derivados de modelos puramente descritivos através da redução do espaço de parâmetros motivada por uma ou mais teorias psicológicas. Os resultados são os modelos multinomiais que fornecem medidas simples de processos psicológicos (probabilidades) e que podem ser quantitativamente testados com dados reais. O uso de modelagem multinomial permite a análise direta dos efeitos de variáveis independentes nos próprios processos latentes que controlam o desempenho em uma ou mais tarefas experimentais, assim, facilitando o teste de predições e explicações teóricas sobre fenômenos psicológicos.
Theories about psychological phenomena often refer to unobservable processes (latent processes). Traditionally, however, the psychological investigation of these phenomena is done indirectly to the latent processes themselves. The objective of this article is to introduce fundamental concepts about multinomial modeling. Here we show that latent processes models are derived from purely descriptive models by reducing the parameter space according to one or more psychological theories. The result is multinomial models that deliver simple measures of psychological processes (probabilities) and that can be tested quantitatively with real data. The use of multinomial modeling allows direct analysis of the effects of independent variables on the latent processes that control performance on one or more experimental tasks, thus making it easier to test theoretical predictions and explanations about psychological phenomena.
Teorías sobre fenómenos psicológicos a menudo se refieren a procesos que no son directamente observables (procesos latentes). Sin embargo, la investigación de estos fenómenos se realiza tradicionalmente de manera indirecta con respecto a los procesos latentes. El propósito de este artículo es presentar los conceptos fundamentales del modelado multinomial. Aquí mostramos cómo los modelos de procesos latentes se derivan de modelos puramente descriptivos al reducir el espacio de parámetros motivado por una o más teorías psicológicas. El resultado son modelos multinomiales que proporcionan medidas simples de procesos psicológicos (probabilidades) y que pueden probarse cuantitativamente con datos reales. El uso de modelos multinomiales permite el análisis directo de los efectos de variables independientes en los procesos latentes que controlan el rendimiento en una o más tareas experimentales, lo que facilita la prueba de predicciones y explicaciones teóricas sobre fenómenos psicológicos.
RESUMO
Transition models are an important framework that can be used to model longitudinal categorical data. They are particularly useful when the primary interest is in prediction. The available methods for this class of models are suitable for the cases in which responses are recorded individually over time. However, in many areas, it is common for categorical data to be recorded as groups, that is, different categories with a number of individuals in each. As motivation we consider a study in insect movement and another in pig behaviou. The first study was developed to understand the movement patterns of female adults of Diaphorina citri, a pest of citrus plantations. The second study investigated how hogs behaved under the influence of environmental enrichment. In both studies, the number of individuals in different response categories was observed over time. We propose a new framework for considering the time dependence in the linear predictor of a generalized logit transition model using a quantitative response, corresponding to the number of individuals in each category. We use maximum likelihood estimation and present the results of the fitted models under stationarity and non-stationarity assumptions, and use recently proposed tests to assess non-stationarity. We evaluated the performance of the proposed model using simulation studies under different scenarios, and concluded that our modeling framework represents a flexible alternative to analyze grouped longitudinal categorical data.
RESUMO
The multinomial logistic regression model (MLRM) can be interpreted as a natural extension of the binomial model with logit link function to situations where the response variable can have three or more possible outcomes. In addition, when the categories of the response variable are nominal, the MLRM can be expressed in terms of two or more logistic models and analyzed in both frequentist and Bayesian approaches. However, few discussions about post modeling in categorical data models are found in the literature, and they mainly use Bayesian inference. The objective of this work is to present classic and Bayesian diagnostic measures for categorical data models. These measures are applied to a dataset (status) of patients undergoing kidney transplantation.
RESUMO
This paper describes two new methods for comparing two independent, discrete distributions, when the sample space is small, using an extension of the Storer-Kim method for comparing independent binomials. These methods are relevant, for example, when comparing groups based on a Likert scale, which was the motivation for the paper. In essence, the goal is to test the hypothesis that the cell probabilities associated with two independent multinomial distributions are equal. Both a global test and a multiple comparison procedure are proposed. The small-sample properties of both methods are compared to four other techniques via simulations: Cliff's generalization of the Wilcoxon-Mann-Whitney test that effectively deals with heteroscedasticity and tied values, Yuen's test based on trimmed means, Welch's test and Student's t test. For the simulations, data were generated from beta-binomial distributions. Both symmetric and skewed distributions were used. The sample space consisted of the integers 0(1)4 or 0(1)10. For the global test that is proposed, when testing at the 0.05 level, simulation estimates of the actual Type I error probability ranged between 0.043 and 0.059. For the new multiple comparison procedure, the estimated family wise error rate ranged between 0.031 and 0.054 for the sample space 0(1)4. But for 0(1)10, the estimates dropped as low as 0.016 in some situations. Given the goal of comparing means, Student's t is well known to have practical problems when distributions differ. Similar problems are found here among the situations considered. No single method dominates in terms of power, as would be expected, because different methods are sensitive to different features of the distributions being compared. But in general, one of the new methods tends to have relatively good power based on both simulations and experience with data from actual studies. If, however, there is explicit interest in comparing means, rather than comparing the cell probabilities, Welch's test was found to perform well. The new methods are illustrated using data from the Well-Elderly Study where the goal is to compare groups in terms of depression and the strategies used for dealing with stress.
En este artículo se describen dos nuevos métodos para comparar dos distribuciones discretas independientes, cuando el espacio muestral es pequeño, usando una extensión del método Storer-Kim para comparar binomios independientes. Estos métodos son relevantes, por ejemplo, cuando se comparan grupos basados en una escala Likert, la cual motivó la escritura del artículo. En esencia, el objetivo es evaluar la hipótesis de que las probabilidades de células asociadas con dos distribuciones multinominales independientes son iguales. Se propone una prueba global y un procedimiento de comparación múltiple. Las propiedades de las muestras pequeñas de ambos métodos fueron comparadas con otras cuatro técnicas a través de simulaciones: generalización de Cliff de la prueba de Wilcoxon-Mann-Whitney que trata eficazmente con heteroscedasticidad y valores vinculados, la prueba de Yuen basada en medias truncadas, la prueba de Welch y la prueba t de Student. Para las simulaciones, los datos se generaron a partir de distribuciones beta-binomiales. Se utilizaron distribuciones tanto simétricas como asimétricas. El espacio muestral consistió en los enteros 0(1)4 o 0(1)10. Para la prueba global que se propone, cuando se evaluó al nivel de 0.05, la simulación estimó la probabilidad del error tipo I osciló entre 0.043 y 0.059. Para el nuevo procedimiento de comparación múltiple, la tasa de error estimada oscilaba entre 0.031 y 0.054 para el espacio de la muestra 0(1)4. Pero para 0(1)10, las estimaciones fueron tan bajas como 0.016 en algunas situaciones. Teniendo en cuenta el objetivo de la comparación de medias, la prueba t de Student es bien conocida por tener problemas prácticos cuando distribuciones difieren. Problemas similares se encuentraron entre las situaciones consideradas. No existe un único método que domina en términos de poder, como sería de esperar, debido a que los diferentes métodos son sensibles a las diferentes características de las distribuciones que son comparadas. Pero en general, uno de los nuevos métodos tiende a tener relativamente buen poder basado tanto en simulaciones y la experiencia con los datos de estudios reales. Si, sin embargo, existe un interés explícito en comparar medias, en lugar de comparar las probabilidades de celda, la prueba de Welch se encuentra que tiene un buen desempeño. Los nuevos métodos se ilustran usando datos del estudio Well-Elderly donde el objetivo es comparar los grupos en cuanto a la depresión y las estrategias utilizadas para hacer frente al estrés.