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Chapter 17.5

Chapter 17.5. Poisson ANCOVA. Classic Poisson Example. N umber 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 ?.

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Chapter 17.5

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  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 1st,2nd, and 3rd 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. Side note: Over-dispersion > deviance(glm2)/df.residual(glm2) [1] 4.632645

  10. State population and whether sample is representative. • Decide on mode of inference. Is hypothesis testing appropriate? • State HA / Ho pair, tolerance for Type I error Statistic: Non-PearsonianChisquare(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 ChisqDfPr(>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 ChisqDfPr(>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. DfResid. DevPr(>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. Assess table in view of evaluation of residuals. • Residuals acceptable • 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) • 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

  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)

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