Owl Data
Explore Owl Data
(28.5,29.3]
(27.7,28.5]
(27,27.7]
(26.2,27]
(25.5,26.2]
(24.7,25.5]
(24,24.7]
(23.2,24]
(22.5,23.2]
(21.7,22.5]
M Sat
F Sat
M Dep
Biologists observer 27 nests of barn owls.
Response: Number of Calls
Predictors:
Size of brood (1 to 7)
Sex of parent( 245 female, 354 male)
Food ”Treatment” (satiated vs deprived)
Arrival Time (21.7 to 29.2 hours)
F Dep
0
2
4
6
8
0
2
4
6
8
8
CallsPrChick
6
4
2
0
22
24
26
28
ArrvTime
Possible interaction between treatment and gender.
Stat 506
& Hill, Chapter 15
Calls per chick seems to have
peaksGelman
at 2230
and 2530. I’m using a
Random
Effects?
cubic spline
of degree 5 to model the curve.
fit1
fit2
fit3
BIC
4895
4750
4733
logLik
-2416
-2337
-2312
deviance
4831
4673
4624
Chisq
157.7
49.2
Chi Df
Pr(>Chisq)
2
5
Evidence that slopes vary from nest to nest. Large interaction.
Stat 506
Gelman & Hill, Chapter 15
0.0000
0.0000
Random Slopes
0.4
Random Intercepts
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−2
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0.2
0.5
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0.0
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Sample Quantiles
AIC
4851
4697
4658
1], main = "Random Intercepts")
1])
2], main = "Random Slopes")
2])
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−0.4
Df
10
12
17
par(mfrow = c(1, 2))
qqnorm(ranef(fit3)$Nest[,
qqline(ranef(fit3)$Nest[,
qqnorm(ranef(fit3)$Nest[,
qqline(ranef(fit3)$Nest[,
0.0
fit1 <- glmer(Ncalls ~ offset(log(BroodSize)) + FoodTrt *
SexParent + bs(cTime, df = 5) + (1 | Nest), data = owls,
family = poisson)
fit2 <- glmer(Ncalls ~ offset(log(BroodSize)) + FoodTrt *
SexParent + bs(cTime, df = 5) + (1 + cTime | Nest),
data = owls, family = poisson)
fit3 <- glmer(Ncalls ~ offset(log(BroodSize)) + FoodTrt *
SexParent + SexParent * bs(cTime, df = 5) + (1 + cTime |
Nest), data = owls, family = poisson)
xtable(anova(fit1, fit2, fit3), digits = c(0, 0, 0, 0, 0,
0, 1, 0, 4))
−0.5
Fit Owl Data
−1.0
Gelman & Hill, Chapter 15
Sample Quantiles
Stat 506
●
●
−1
0
1
2
Theoretical Quantiles
−2
−1
0
1
Theoretical Quantiles
May not be “Gaussian” errors. Are we missing a predictor?
Stat 506
Gelman & Hill, Chapter 15
2
Gelman §15.1 Traffic Stops
Gelman Frisk 2
We don’t have access to the data. The Poisson Mixed Model:
yi ∼ Poisson(ui e Xi β+i ), i ∼ N(0, σ2 )
Their example adds random effects [βp ∼ N(0, σp2 )] for Precints
and (indep of) and extra overdispersion. Race effects, αe are of
interest, and there is an general intercept, µ.
Stat 506
Gelman & Hill, Chapter 15
Gelman Storable Votes §15.2
invlogit[Pr(yi > 1) = Xi β; invlogit[Pr(yi > 2) = Xi β − c2

 1 if zi < c1.5
2 if zi ∈ (c1.5 , c2.5 ) ; zi ∼ logistic(xi , σ 2 )
yi =

3 if zi > c2.5
Use Normal distributions for each of cj1.5 , cj,2.5 , log(σj ) to allow
varying cutoffs and variances for each student. Estimate
σ1.5 , σ2.5 , µlog(σ) , σlog(σ) from the data.
Figure 15.6 looks at optimal versus observed strategies for 2, 3, 6
player games.
Stat 506
Gelman & Hill, Chapter 15
Social Networks §15.3
Multinomial is like 2 logistic regressions:
or
Stat 506
Gelman & Hill, Chapter 15
How many people do you know?
If you know 2 Nicoles out of 358K in the US, we compute size of
your network as:
2
× 280000000 = 1560
358000
Cross check with other names.
But there are “clumps” of data (JayCees), so we need
overdispersed Poisson.
Additional analysis looks at selective memory, positive/negative
experiences and residuals.
Stat 506
Gelman & Hill, Chapter 15