Chapter 17.5Poisson 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 0.4343corps 1.27787 3 0.7344year: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 0.4343corps 1.27787 3 0.7344year: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 0.9630corps 3 1.14678 75 94.617 0.7658year: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 onlyglm1 <- 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)