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Statistical Inference Involving Binomial and Negative Binomial Parameters

Published online by Cambridge University Press:  10 January 2013

Miguel A. García-Pérez  and
Vicente Núñez-Antón
Miguel A. García-Pérez*
Affiliation:
Universidad Complutense (Spain)
Vicente Núñez-Antón
Affiliation:
Universidad del País Vasco (Spain)
*
Corresponding Author's Address: Miguel A. García-Pérez, Departamento de Metodología, Facultad de Psicología, Universidad Complutense, Campus de Somosaguas, 28223 Madrid, Spain. Phone: +34 913 943 061; Fax: +34 913 943 189; E-mail: miguel@psi.ucm.es
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Abstract

Statistical inference about two binomial parameters implies that they are both estimated by binomial sampling. There are occasions in which one aims at testing the equality of two binomial parameters before and after the occurrence of the first success along a sequence of Bernoulli trials. In these cases, the binomial parameter before the first success is estimated by negative binomial sampling whereas that after the first success is estimated by binomial sampling, and both estimates are related. This paper derives statistical tools to test two hypotheses, namely, that both binomial parameters equal some specified value and that both parameters are equal though unknown. Simulation studies are used to show that in small samples both tests are accurate in keeping the nominal Type-I error rates, and also to determine sample size requirements to detect large, medium, and small effects with adequate power. Additional simulations also show that the tests are sufficiently robust to certain violations of their assumptions.

El contraste de hipótesis acerca de dos proporciones supone que cada una de ellas se ha estimado mediante muestreo binomial, pero hay ocasiones en que interesa evaluar la hipótesis de que la probabilidad de éxito a medida que se repite una determinada tarea varía una vez que se ha obtenido el primer éxito. En estos casos, la probabilidad de éxito antes de que ocurra el primer éxito se estima mediante muestreo binomial negativo, en tanto que la probabilidad de éxito después del primer éxito se estima mediante muestreo binomial, y ambas estimaciones están relacionadas. En este trabajo se presentan procedimientos para contrastar dos hipótesis aplicables a esta situación. Una es la de que las dos probabilidades son iguales y tienen un determinado valor; la otra es más general y sólo expresa que las dos probabilidades son iguales. El comportamiento de estos dos contrastes en muestras finitas se analiza mediante simulaciones cuyos resultados muestran que en ambos casos se preserva adecuadamente la tasa nominal de error de tipo I. También se ha determinado mediante simulación los tamaños muestrales necesarios para detectar efectos grandes, medianos o pequeños con potencia suficiente. Finalmente, otro grupo de simulaciones muestra que ambos contrastes son suficientemente robustos ante violaciones de sus supuestos.

Keywords

statistical testsbinomial parameterspowereffect sizeMonte Carlo simulationscontraste de hipótesisproporcionespotenciatamaño del efectosimulación
Type
Research Article
Information
The Spanish Journal of Psychology , Volume 12 , Issue 1 , May 2009 , pp. 288 - 307
Copyright
Copyright © Cambridge University Press 2009

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