Correction
Factor for the Deviance Distribution of Proportion Data.

José Eduardo CORRENTE[1]

Ana Paula Gomes da Silva GIMENES2

§ ABSTRACT: The analysis of proportion
data generally presents certain difficulties since the distribution underlying
such data can be considered to be binomial, that is, it does not follow some basic
assumptions for the fit of a mathematical model. A few transformations are
suggested; however, good results are not always obtained.In the approach given
by generalized linear models, the statistic that measures the quality of the
model’s fit for such data is called deviance. It so happens that the deviance
distribution is generally unknown. However, it can be approximated by a χ2 distribution for data with a binomial distribution. Nevertheless, that
is not good for small sample sizes. In order to improve this approximation,
some data correction factors are suggested, but the obtained results are still
unsatisfactory. Therefore, this work aims at proposing a new correction factor
for data following a binomial distribution so as to obtain an improvement in
the deviance distribution for any sample size. To that end, a constant is added
to the response variable and, through the expected deviance value, such
constant is calculated so as to reduce the error made in the approximation.
Simulations of the binomial distribution and deviance calculation are made and QQ-plots are
used to compare with distribution χ2.

§ KEYWORDS: Binomial distribution,
deviance, Taylor series, generalized linear models.

[1] 1Departamento
de Ciˆencias Exatas, Escola Superior de Agricultura Luiz de Queiróz da
Universidade de São Paulo, CEP: 13418-900, Piracicaba-SP, Brasil. E-mail:
jecorren@carpa.ciagri.usp.br

2Departamento de Ciˆencias Exatas, Escola
Superior de Agricultura Luiz de Queiróz da Universidade de São Paulo, CEP:
13418-900, Piracicaba-SP, Brasil. E-mail: anapaula@linksat.com.br