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Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.23 n.1 San José Jan./Jun. 2016
Artículos
An alternative to classical latent class models selection methods for sparse binary data: an illustration with simulated data
Un método alternativo para la selección de modelos de clases latentes en datos binarios escasos: una ilustración con datos simulados
1Sede de Occidente, Universidad de Costa Rica, San Ramón, Costa Rica. E-Mail: carlo.araya@ucr.ac.cr
Within the context of a latent class model with manifest binary variables, we propose an alternative method that solves the problem of estimating empirical distribution with sparse contingency tables and the chisquare approximation for goodness-of-fit will not be valid. We analyze sparse binary data, where there are many response patterns with very small expected frequencies in several data sets varying in degree of sparseness from 1 to 5 defined d = n/2
p
= n/R is a factor that is mentioned in almost all prior literature as being an important determinant of how well the distribution is represented by the chi-squared.The proposed approach produced results that were valid and reliable under the mentioned problematic data conditions. Results from the proposal presented compare the rates of Type I for traditional goodness-of-fit tests. We also show that with data density d ≤ 5, Pearson's statistic 
should not be used to select latent class models using the Patterns Method, given that this has the probability of Type I error being greater than 5%. By comparing the Patterns Method and the Parametric Bootstrap for data density d = 2, we show that the Patterns Method has more accurate Type I error probabilities since the likelihood ratio, Read-Cressie and Freeman-Tukey statistics afford values of α<0.05. In contrast, the Parametric Bootstrap provides values in these statistics that surpass 5%.
Keywords: sparse data; latent class; goodness-of-fit; binary data
En el contexto de modelos de clases latentes con variables manifiestas binarias, se propone un método alternativo para resolver el problema de la estimación de la distribución empírica con tablas de contingencias escasas, donde la aproximación de los estadísticos de bondad de ajuste por la distribución Chi-Cuadrada no es válida. Se analiza datos binarios escasos, donde muchos patrones de respuesta que tienen frecuencias esperadas pequeñas, en conjuntos de datos con grados de datos escasos de 1 a 5, donde d = n/2
p
= n/R es un factor es mencionado en la literatura como determinante de la bondad de ajuste a la distribución Chi-Cuadrada. La propuesta presenta resultados válidos y confiables en las condiciones de los datos mencionadas. Para los resultados se presenta tasas de error tipo I para las pruebas tradiciones de bondad de ajuste. También se muestra que para niveles de densidad de datos d ≤ 5, el estadístico Pearson 
no es el apropiado para seleccionar modelos de clases latentes utilizando el Método de Patrones, dado que presenta probabilidad de error de tipo I más grandes que 5%. Al comparar el Método de Patrones y el Bootstrap Paramétrico para la densidad d = 2, se muestra que el Método de Patrones tiene probabilidades de error de tipo I menores de 5% en los estadísticos de razón de verosimilitud, Read-Cressie y Freeman-Tukey. En contraste, el Bootstrap Paramétrico produce valores en estos estadísticos que superan un 5%.
Palabras clave: datos escasos; clases latentes; bondad de ajuste; datos binarios
Acknowledgments
The author thanks the Sede de Occidente of the Universidad de Costa Rica for assistance in pursuing his PhD studies, and for a grant to stay at the University of Salamanca (Spain) while this work was under preparation.
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Received: June 25, 2014; Revised: August 28, 2015; Accepted: October 19, 2015
This is an open-access article distributed under the terms of the Creative Commons Attribution License
