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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.44 no.1 Bogotá Jan./June 2021  Epub Feb 25, 2021

https://doi.org/10.15446/rce.v44n1.83987 

Original articles of research

Comparison of Correction Factors and Sample Size Required to Test the Equality of the Smallest Eigenvalues in Principal Component Analysis

Comparación de los factores de corrección y tamaños de muestra requeridos para probar la igualdad de los valores propios más pequeños en el análisis de componentes principales

Eduard Gañan-Cardenas1  a 

Juan Carlos Correa-Morales2  b 

1Departamento de Calidad y Producción, Facultad de Ciencias Económicas y Administrativas, Instituto Tecnológico Metropolitano, Medellín, Colombia

2Escuela de Estadística, Facultad de Ciencias, Universidad Nacional de Colombia, Sede Medellín, Colombia


Abstract

In the inferential process of Principal Component Analysis (PCA), one of the main challenges for researchers is establishing the correct number of components to represent the sample. For that purpose, heuristic and statistical strategies have been proposed. One statistical approach consists in testing the hypothesis of the equality of the smallest eigenvalues in the covariance or correlation matrix using a Likelihood-Ratio Test (LRT) that follows a x2 limit distribution. Different correction factors have been proposed to improve the approximation of the sampling distribution of the statistic. We use simulation to study the significance level and power of the test under the use of these different factors and analyze the sample size required for an adequate approximation. The results indicate that for covariance matrix, the factor proposed by Bartlett offers the best balance between the objectives of low probability of Type I Error and high Power. If the correlation matrix is used, the factors W * B and 2 D are the most recommended. Empirically, we can observe that most factors require sample sizes 10 or 20 times the number of variables if covariance or correlation matrices, respectively, are implemented.

Key words: Chi-square distribution; Likelihood ratio test; Power comparisons; Principal components analysis; Sphericity test

Resumen

Dentro del proceso inferencial del Análisis de Componentes Principales (PCA) uno de los interrogantes principales de los investigadores es sobre el número correcto de componentes para representar la muestra. Para este fin se han propuesto estrategias heurísticas y estadísticas. Un enfoque estadístico consiste en probar la hipótesis sobre la igualdad de los valores propios más pequeños de la matriz de covarianza o correlación a través de una prueba de razón de verosimilitud (LRT) que sigue una distribución límite x2. Diferentes factores de corrección han sido propuestos para mejorar la aproximación de la distribución muestral del estadístico. En este trabajo utilizamos simulación para estudiar el nivel de significancia y la potencia de la prueba bajo el uso de estos diferentes factores, así como una revisión del tamaño de muestra requerido para una adecuada aproximación. Los resultados para la matriz de covarianza indican que el factor propuesto por Bartlett ofrece el mejor equilibrio entre los objetivos de baja probabilidad de Error Tipo I y alta potencia. En caso de la matriz de correlación, los factores W * B y 2 D son los más recomendados. Empíricamente se observa que la mayoría de los factores requieren tamaños de muestra 10 y 20 veces mayores al número de variables en caso de la matriz de covarianza o de correlación respectivamente.

Palabras clave: Análisis de componentes principales; Comparación de potencias; Distribución Chi-cuadrado; Prueba de esfericidad; Prueba de razón de verosimilitud

Full text available only in PDF format

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a M.Sc. E-mail: eduardganan@itm.edu.co

b Ph.D. E-mail: jccorrea@unal.edu.co

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