Page 21 - Contributed Paper Session (CPS) - Volume 6

CPS1468 Takeshi Kurosawa et al.

1
̂
̂
̅
̂
̂
(, ) = ∑( − )( − ) = Cov(, ).
− 1 − 1
=1
We consider an another estimator, assuming that is a vector of random
variables with distribution characterized by the vector of parameters . By
Definition 2.1, integrating with respective to the distribution of the covariates,
we can derive an explicit form of (, ) for Poisson GLMs. Substituting
pp
estimates of , , into the explicit form of , we get the estimator
pp
̂ ̂
(̂, |).
pp
The notation makes explicit the dependence of the measure of predictive
̂
power on parameters .

4. Application to Horseshoe Crab Data
This section applies pp to the horseshoe crab data provided in Agresti
(2002). Briefly, the dataset consists of 173 female crabs, with the response
variable being the number of male crabs satelliting with each female crab Sa.
There are also four explanatory variables: 1) weight of a female crab (Wt); 2)
the carapace width a female crab (W); 3) the body color of a female crab (C);
4) the spine condition of a female crab (S). Both weight Wt and carapace width
W are continuous variables, and a test of normality applied to both predictors
suggested no strong evidence that either deviated substantially from a normal
distribution (Takahashi and Kurosawa, 2016). Body color is a factor variable
with levels C = 1: light medium, 2: medium, 3: dark medium, and 4: dark. We
converted body colour into a binary predictor C2, such that C2 = 1 if C = 4 and
C2 = 0 otherwise. Analogously, the spine condition is a factor with levels S = 1:
both good, 2: one worn, 3: both worn. We also converted this to a binary
categorical variable S2 such that S2 = 1 if S = 1 and S2 = 0 if otherwise.
Assuming a Poisson distribution for the count response Sa, we fitted 15
candidate models involving different subsets of the four covariates included
̂ ̂
as main effects, and Table 1: Values of (̂, |) for 15 candidate Poisson
pp
regression models fitted to the horseshoe crab dataset. There were four
explanatory variables: 1) weight of a female crab (continuous variable; Wt); 2)
the carapace width a female crab (continuous variable; W); 3) the body color
of a female crab (binary variable; C2); 4) the spine condition of a female crab

(binary variable; S2).

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