Assignment 3

Statistics 9055
Assignment 3
Due March 10, 2013
1. For an I × J × K contingency table the full (saturated) log-linear model is given by
ln( ijk )  ln( n ijk )    i   j   k  ( )ij  ( )ik  (  ) jk  ( )ijk
with the appropriate restrictions on the parameters.
a. Show that testing the hypothesis
H 0 :  ijk   i  jk
is equivalent to testing the hypothesis
H 0 : ( )ij  ( )ik  ( )ijk  0 .
b. Interpret in words the meaning of H 0 :  ijk   i  jk .
2. In the table below a sample of 800 boys are classified according to their socioeconomic
status (S), whether the boy belongs to the Boy scouts (B), and the boys’ juvenile delinquency
(D) status:
Socioeconomic
status
Low
Medium
High
Boy scout
Yes
No
Yes
No
Yes
No
Delinquent
Yes
No
11
43
42
169
14
104
20
132
8
196
2
59
Is the boy’s delinquent status independent of his socioeconomic status and/or his boy scout
status?
3. The data in crab.txt are taken from a study of nesting horseshoe crabs. Each female
horseshoe crab in the study had a male crab attached to her in her nest. The study
investigated factors that affect whether the female crab had any other males, called satellites,
residing near her. Explanatory variables that are thought to affect this included the female
crab’s colour (C), spine condition (S), weight (Wt), and carapace width (W). The response
outcome for each female crab is her number of satellites (Sa). The data were coded as
follows:
C: 1 – light medium, 2 – medium, 3 – dark medium, 4 – dark
S: 1 – both good, 2 – one worn or broken, 3 – both worn or broken
Wt: carapace weight in kilograms
W: carapace width in centimeters
Analyze these data using Poisson regression.
4. The Gamma distribution is given by the density function
f ( y) 
y  1 exp(  y /  )
,
 ( )
where y > 0,  > 0 and  > 0.
a. When  is considered to be a dispersion parameter (  ), show that the Gamma
distribution belongs to the exponential family of distributions.
b. Find the mean and variance of a Gamma random variable using the method through the
exponential family.
c. What link function g would you suggest to incorporate covariates through g (  )  x T β
where   E ( y ) ? Justify your answer.

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