Stat 511
Homework 12 Solution
Spring 2011
Due: 5pm, Tuesday Apr 26
1. If
y
∼
Poisson
(
λ
), the pmf of
y
is
f
(
y
) =
λ
y
e
−
λ
y
!
Equation (1) on slide 2 of section 26 gives the exponentionalscale family form of a distribution.
Write the pmf of
y
in exponentialscale form and identify
θ
,
η
(
θ
),
T
(
y
),
a
(
φ
), and
b
(
θ
).
f
(
y
) =
λ
y
e
−
λ
y
!
=
exp
(
y
log
λ

λ

log
y
!)
Namely,we have
θ
=
λ, η
(
θ
) = log(
θ
)
, T
(
y
) =
y, a
(
φ
) = 1
, b
(
θ
) =
θ
2. The data in birth2.txt are from a sociological study in Baltimore. The investigators believed
that there is an association between loss of a child during pregnancy and the behaviour in
school of a subsequent liveborn child.
The prospective research design (looking forward)
would be to enroll some mothers who have lost a child and some who haven’t, then look at
school behaviour (problem or not). This is very inefficient because the proportion of school
problems is low (thankfully!).
Instead, the investigators used a retrospective design, which is common in epidemiology and
efficient at examining associations with rare events. The investigators identified 255 problem
children and 110 control (nonproblem) children.
The mothers were then asked about the
birth order (2nd child, 3rd child, ...)
and whether they had lost the previous child.
Birth
order is 2 (2nd child), 3.5 (3rd or 4th child) and 5 (5th or higher child).
The response is
whether the previous child was lost or not. (The loss rates seem high to me, but that’s what
they were recorded as). The investigators want to know whether the odds of losing a child
are associated with birth order and/or whether or not the subsequent child was a problem in
school.
(a) Fit an appropriate generalized linear model treating problem as a factor and birth order
as a continuous linear covariate. Do not worry about overdispersion for now. Report the
odds ratio for being a problem child and the odds ratio for a 1 child increase in birth
order. Does being a problem child increase or decrease the probability that your mom
lost the previous child?
d < read.table(’D:/STAT511/data/birth2.txt’,as.is=T,header=T)
> d$problem <as.factor(d$problem)
> d$f < cbind(d$loss,d$noloss)
> d.glm < glm(f~problem+birth,data=d,family=binomial)
> summary(d.glm)
Estimate Std. Error z value Pr(>z)
1
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Stat 511
Homework 12 Solution
Spring 2011
(Intercept) 2.63005
0.39459 6.665 2.64e11 ***
problemY
0.36724
0.24394
1.505
0.132
birth
0.48683
0.09764 4.986 6.17e07 ***
The odds ratio for being a problem child is exp(0.36724)=1.4437 and for a 1 child increase
in birth order is exp(0.48683)=1.6272. Being a problem child increases the probability
that the Mom lost the previous child by about 44%.
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 Spring '08
 Staff
 Statistics, Birth order

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