Page 18 - Contributed Paper Session (CPS) - Volume 6

CPS1468 Takeshi Kurosawa et al.

Estimators of goodness-of-fit measures for a
Poisson regression model
2
3
3
1
Takeshi Kurosawa , Kousuke Shinmura , Francis K. C. Hui , A. H. Welsh ,
Nobuoki Eshima
4
1 Department of Applied Mathematics Science, Faculty of Science, Tokyo University of Science,
Tokyo, Japan
2 Department of Applied Mathematics Science, Graduate School of Science, Tokyo University of
Science, Tokyo, Japan
3 Mathematical Sciences Institute, The Australian National University, Acton ACT, 2601,
Australia
4 Center for Educational Outreach and Admissions, Kyoto University, Kyoto, Japan

Abstract
In this study, we discuss a measure of predictive power which is one of
pp
the goodness-of-fit measures for generalized linear models (GLMs) proposed
by Eshima and Tabata (2007). This measure expresses average amount of
decreasing uncertainness of a response variable by a vector of regressors.
We apply it to a Poisson regression model with a random vector of the
regressors. Moreover, we propose an estimator of pp and compare it with
other estimators.

Keyword
coefficient of determination; entropy; generalized linear models; measure of
predictive power; correlation coefficient

1. Introduction
In regression analysis, it is often desirable to numerically summarize the
overall fitted model through a goodness-of-fit measure. Perhaps the most well
known of these is the Akaike Information Criterion (AIC). Being a relative
measure, the actual values of AIC do not have a clear interpretation, and
instead it is differences in AIC values between candidate models which drive
its usage. This is in contrast to the multiple correlation coefficient R in the
linear model, whose value can be interpreted explicitly as a ratio between the
̂
conditional variance of the fitted values based on the candidate model, and
the overall variance of . This article focuses on one particular goodness-of-
fit measure, based on the covariance between the response and the canonical
parameter in a exponential family of distributions.
Approaching the problem of goodness-of-fit measures for GLMs, Zheng and
Agresti (2000) proposed the regression correlation coefficient (RCC) which is
the correlation between the response variable and the its conditional

7 | I S I W S C 2 0 1 9