Week 3

This week we learned about poisson regression. With poisson regression, there are 4 assumptions. 1) The response variable is a count per unit of time 2) All observations are independent 3) E(X) = Var(X) 4) Log of the mean (lambda) is a linear function of x

library(faraway)
data(gala)
attach(gala)

This example uses data of the number of tortoise species. Since it is a count, it makes sense to use poisson regression since we can’t have a negative count.

mod <- glm(Species ~ Area + Elevation + Nearest+Scruz+Adjacent, family = poisson, data = gala)
summary(mod)

## 
## Call:
## glm(formula = Species ~ Area + Elevation + Nearest + Scruz + 
##     Adjacent, family = poisson, data = gala)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -8.2752  -4.4966  -0.9443   1.9168  10.1849  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  3.155e+00  5.175e-02  60.963  < 2e-16 ***
## Area        -5.799e-04  2.627e-05 -22.074  < 2e-16 ***
## Elevation    3.541e-03  8.741e-05  40.507  < 2e-16 ***
## Nearest      8.826e-03  1.821e-03   4.846 1.26e-06 ***
## Scruz       -5.709e-03  6.256e-04  -9.126  < 2e-16 ***
## Adjacent    -6.630e-04  2.933e-05 -22.608  < 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:  716.85  on 24  degrees of freedom
## AIC: 889.68
## 
## Number of Fisher Scoring iterations: 5

pchisq(716.85, 24, lower.tail = FALSE)

## [1] 7.058684e-136

halfnorm(residuals(mod))

dp <- sum(residuals(mod, type = "pearson")^2/mod$df.residual)
summary(mod, dispersion = dp)

## 
## Call:
## glm(formula = Species ~ Area + Elevation + Nearest + Scruz + 
##     Adjacent, family = poisson, data = gala)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -8.2752  -4.4966  -0.9443   1.9168  10.1849  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  3.1548079  0.2915897  10.819  < 2e-16 ***
## Area        -0.0005799  0.0001480  -3.918 8.95e-05 ***
## Elevation    0.0035406  0.0004925   7.189 6.53e-13 ***
## Nearest      0.0088256  0.0102621   0.860    0.390    
## Scruz       -0.0057094  0.0035251  -1.620    0.105    
## Adjacent    -0.0006630  0.0001653  -4.012 6.01e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 31.74914)
## 
##     Null deviance: 3510.73  on 29  degrees of freedom
## Residual deviance:  716.85  on 24  degrees of freedom
## AIC: 889.68
## 
## Number of Fisher Scoring iterations: 5