The Statistical Sleuth Chapter 22, Problem 18 Galapagos Islands. Reanalyze the d

Question

The Statistical Sleuth

Chapter 22, Problem 18

Galapagos Islands. Reanalyze the data in Exercise 12.20 with number of native species as the response, but using log-linear regression. (a) Fit the model with log area, log elevation, log of distance from nearest island, and log area of nearest island as explanatory variables; and then check for extra- Poisson variation. (b) Use backward elimination to eliminate insignificant explanatory variables. (c) Describe the effects of the remaining explanatory variables.

Data set in R:

library(Sleuth3)

ex1220

(Please also include R code)

Explanation / Answer

There are no pointers on the previous excercise , however we shall do this based on the given information in the question

Please see the complete R snippet , we shall use the glm function to fit the log linear model using the poisson family

library(Sleuth3)
data("ex1220")
ex1220

## FIT THE model
mod0 <- glm(Total ~ Area+ Elev +DistNear +AreaNear,
data = ex1220, family = poisson)
summary(mod0)

## check the deviance
pchisq(deviance(mod0), df = df.residual(mod0), lower.tail = F)


## run the backward elemination to check for the insignifcant variables
step(mod0,direction="backward",trace=TRUE)

############################

The results are

> summary(mod0)

Call:
glm(formula = Total ~ Area + Elev + DistNear + AreaNear, family = poisson,
data = ex1220)

Deviance Residuals:
Min 1Q Median 3Q Max
-11.1537 -4.3318 -0.7932 2.3551 9.9384

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.998e+00 4.973e-02 60.285 <2e-16 ***
Area -5.837e-04 2.574e-05 -22.676 <2e-16 ***
Elev 3.606e-03 8.588e-05 41.992 <2e-16 ***
DistNear -3.328e-03 1.445e-03 -2.303 0.0213 *
AreaNear -7.596e-04 2.786e-05 -27.271 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 3510.73 on 29 degrees of freedom
Residual deviance: 812.97 on 25 degrees of freedom
AIC: 983.8

Number of Fisher Scoring iterations: 5

We see that all the variables are significant as the p value is less than 0.05 , hence we conclude that the explanatory variables used for the model are statistically signifcant

The stepwise regression stops after the 1st step as it finds all variables to be signifcant and thus cant remove any explanaoty variable

> step(mod0,direction="backward",trace=TRUE)
Start: AIC=983.8
Total ~ Area + Elev + DistNear + AreaNear

Df Deviance AIC
<none> 812.97 983.8
- DistNear 1 818.37 987.2
- Area 1 1324.29 1493.1
- AreaNear 1 1801.10 1969.9
- Elev 1 2587.27 2756.1

Call: glm(formula = Total ~ Area + Elev + DistNear + AreaNear, family = poisson,
data = ex1220)

Coefficients:
(Intercept) Area Elev DistNear AreaNear
2.9982177 -0.0005837 0.0036061 -0.0033281 -0.0007597

Degrees of Freedom: 29 Total (i.e. Null); 25 Residual
Null Deviance:   3511
Residual Deviance: 813    AIC: 983.8

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