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In a study of the Krunnit Islands archipelago, researchers presented results of extensive bird surveys taken over four decades. They visited each island several times, cataloguing species. If a species was found on a specific island in 1949, it was considered to be at risk of extinction for the next survey of the island in 1959. If it was not found in 1959, it was counted as an “extinction”, even though it might reappear later. This data frame contains data on island size, number of species at risk to become extinct and number of extinctions.

1 |

A data frame with 18 observations on the following 4 variables.

- Island
Name of Island

- Area
Area of Island

- AtRisk
Number of species at risk

- Extinct
Number of extinctions

Scientists agree that preserving certain habitats in their natural states is necessary to slow the accelerating rate of species extinctions. But they are divided on how to construct such reserves. Given a finite amount of available land, is it better to have many small reserves or a few large one? Central to the debate on this question are observational studies of what has happened in island archipelagos, where nearly the same fauna tries to survive on islands of different sizes.

Ramsey, F.L. and Schafer, D.W. (2013). *The Statistical Sleuth: A
Course in Methods of Data Analysis (3rd ed)*, Cengage Learning.

Väisänen, R.A. and Järvinen, O. (1977). Dynamics of
Protected Bird Communities in a Finnish Archipelago, *Journal of
Animal Ecology* **46**: 891–908.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | ```
str(case2101)
attach(case2101)
## EXPLORATION AND MODEL BUILDING
proportionExtinct <- Extinct/AtRisk
oddsExtinct <- proportionExtinct/(1 - proportionExtinct)
logitExtinct <- log(oddsExtinct) # Logit = Log Odds
logArea <- log(Area)
plot(logitExtinct ~ logArea)
binResponse <- cbind(Extinct,AtRisk-Extinct)
myGlm1 <- glm(binResponse ~ logArea, family=binomial)
summary(myGlm1)
logArea2 <- logArea^2
myGlm2 <- update(myGlm1, ~ . + logArea2)
summary(myGlm2) # p-value for quadratic term: 0.77
## INFERENCE AND INTERPRETATION
myGlm3 <- update(myGlm1, ~ . - logArea)
anova(myGlm3, myGlm1) # Drop in deviance statistic = 33.277 on 1 d.f.
1 - pchisq(33.277,1) # p-value = 7.992234e-09
beta <- myGlm1$coef
1 - 2^beta[2] # 0.1861153
1 - 2^confint(myGlm1,2) #0.2462041 0.1247743
# Interpretation: Associated with each doubling of island area is a 19%
# reduction in the odds of extinction (95% confidence interval: 12% to 25%
# reduction).
## GRAPHICAL DISPLAY FOR PRESENTATION
plot(oddsExtinct ~ Area, log="xy", ylab="Observed Odds of Extinction; log scale",
xlab=expression(paste("Island Area (km"^"2","); log scale")),
main="Extinctions of Bird Species in the Krunnit Island Archipelago",
pch=21, lwd=2, bg="green", cex=2) # Plot odds of extinction vs island area
dummyArea <- seq(min(Area),max(Area),length=50)
lp <- beta[1] + beta[2]*log(dummyArea)
odds <- exp(lp)
lines(odds ~ dummyArea,lwd=2)
plot(proportionExtinct ~ Area, log="xy",
ylab="Proportions of 1949 Species not Found in 1959",
xlab=expression(paste("Island Area (km"^"2","); log scale")),
main="Proportions of 1949 Bird Species Extinct in 1959 on 18 Krunnit Archipelago Islands",
pch=21, lwd=2, bg="green", cex=2) # Plot probability of extinction vs area
dummyArea <- seq(min(Area),max(Area),length=50)
lp <- beta[1] + beta[2]*log(dummyArea)
myProbability <- exp(lp)/(1 + exp(lp))
lines(myProbability ~ dummyArea,lwd=2,col="blue")
legend(.08,.055,legend="Estimated Probability of Extinction",lty=1,lwd=2,col="blue")
detach(case2101)
``` |

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