Chapter 17.5
Poisson ANCOVA
Classic Poisson Example
• Number of deaths by horse kick, for each of 16 corps in
the Prussian army, from 1875 to 1894
• The risk of death did not change over time in Guard
Corps.
• Is there a similar lack of trend in the 1st,2nd, and 3rd
units ?
1. Construct Model – Graphical
1. Construct Model
– Formal
2. Execute analysis & 3. Evaluate model
glm1 <- glm(deaths~year*corps, family =
poisson(link=log), data=horsekick)
2. Execute analysis & 3. Evaluate model
glm1 <- glm(deaths~year*corps, family =
poisson(link=log), data=horsekick)
2. Execute analysis & 3. Evaluate model
glm1 <- glm(deaths~year*corps, family =
poisson(link=log), data=horsekick)
deviance(glm1)/df.residual(glm1)
[1] 1.134671
• Dispersion parameter assumed to be 1
• As a general rule, dispersion parameters approaching 2 (or
0.5) indicate possible violations of this assumption
Side note: Over-dispersion
Side note: Over-dispersion
> deviance(glm2)/df.residual(glm2)
[1] 4.632645
4. State population and whether sample is
representative.
5. Decide on mode of inference. Is hypothesis
testing appropriate?
6. State HA / Ho pair, tolerance for Type I error
Statistic: Non-Pearsonian Chisquare (G-statistic)
Distribution: Chisquare
7. ANODEV. Calculate change in fit
(ΔG) due to explanatory variables.
> library(car)
> Anova(glm1, type=3)
Analysis of Deviance Table (Type III tests)
Response: deaths
LR Chisq Df Pr(>Chisq)
year
0.61137 1
0.4343
corps
1.27787 3
0.7344
year:corps 1.27073 3
0.7361
7. ANODEV. Calculate change in fit
(ΔG) due to explanatory variables.
> Anova(glm1, type=3)
…
LR Chisq Df Pr(>Chisq)
year
0.61137 1
0.4343
corps
1.27787 3
0.7344
year:corps 1.27073 3
0.7361
> anova(glm1, test="LR")
…
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev Pr(>Chi)
NULL
79
95.766
year
1 0.00215
78
95.764
0.9630
corps
3 1.14678
75
94.617
0.7658
year:corps 3 1.27073
72
93.347
0.7361
8. Assess table in view of evaluation of
residuals.
–
Residuals acceptable
9. Assess table in view of evaluation of
residuals.
–
Reject HA: The four corps show the same lack of trend in
deaths by horsekick over two decades (ΔG=1.27,
p=0.736)
10.Analysis of parameters of biological interest.
–
βyear was not significant – report mean deaths/unit-yr
• (56 deaths / 20 years) / 4 units = 0.7 deaths/unit-year
library(pscl)
library(Hmisc)
library(car)
corp.id <- c("G","I","II","III")
horsekick <- subset(prussian, corp %in% corp.id)
names(horsekick) <- c("deaths","year","corps")
glm0 <- glm(deaths ~ 1, family = poisson(link = log), data = horsekick) # intercept only
glm1 <- glm(deaths ~ year*corps, family = poisson(link = log), data = horsekick)
plot(fitted(glm1),residuals(glm1),pch=16, xlab="Fitted values", ylab="Residuals")
plot(residuals(glm1), Lag(residuals(glm1)), xlab="Residuals", ylab="Lagged residuals", pch=16)
sum(residuals(glm1, type="pearson")^2)/df.residual(glm1)
deviance(glm1)/df.residual(glm1)
plot(horsekick$year,horsekick$deaths, pch=16, axes=F, xlab="Year", col=horsekick$corps,
ylab="Deaths")
axis(1, at=75:94, labels=1875:1894)
axis(2, at=0:4)
box()
Anova(glm1, type=3, test.statistic="LR")
anova(glm1, test="LR")
species <- read.delim("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/species.txt")
plot(Species~Biomass, data=species, pch=16)
lm1 <- lm(Species~Biomass, data=species)
plot(fitted(lm1),residuals(lm1), pch=16, xlab="Fitted values", ylab="Residuals", main="GLM")
glm2<-glm(Species~Biomass, data=species, family=poisson)
plot(fitted(glm2),residuals(glm2), pch=16, xlab="Fitted values", ylab="Residuals",
main="GzLM")
deviance(glm2)/df.residual(glm2)