1. Chapter 17.5 Poisson ANCOVA

2. 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 1 st,2 nd, and 3 rd units ?

3. 1. Construct Model – Graphical

4. 1. Construct Model – Formal

5. 2. Execute analysis & 3. Evaluate model glm1 <- glm(deaths~year*corps, family = poisson(link=log), data=horsekick)

6. 2. Execute analysis & 3. Evaluate model glm1 <- glm(deaths~year*corps, family = poisson(link=log), data=horsekick)

7. 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

8. Side note: Over-dispersion

9. > deviance(glm2)/df.residual(glm2) [1] 4.632645

10. 4.State population and whether sample is representative. 5.Decide on mode of inference. Is hypothesis testing appropriate? 6.State H A / H o pair, tolerance for Type I error Statistic: Non-Pearsonian Chisquare (G-statistic) Distribution: Chisquare

11. 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

12. 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

13. 8.Assess table in view of evaluation of residuals. – Residuals acceptable 9.Assess table in view of evaluation of residuals. – Reject H A : 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. – β year was not significant – report mean deaths/unit-yr (56 deaths / 20 years) / 4 units = 0.7 deaths/unit-year

14. 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)