require(Sleuth3)
## Loading required package: Sleuth3
elephants <- case2201

Let’s plot the data

qplot(Age,Matings,data=elephants) +  geom_smooth(method=lm)

qplot(log(Age), Matings,data=elephants) +  geom_smooth(method=lm)

It looks like the residuals increase with age. Let’s visualize the residuals directly.

linear_fit <- lm(Matings~Age, data=elephants)
plot(linear_fit, 1)

That’s what we would expect from Poisson-distributed counts – the variance of \(Poisson(\lambda)\) is \(\lambda\).

fit <- glm(Matings ~ Age, family= "poisson", data= elephants)
summary(fit)
## 
## Call:
## glm(formula = Matings ~ Age, family = "poisson", data = elephants)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.80798  -0.86137  -0.08629   0.60087   2.17777  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.58201    0.54462  -2.905  0.00368 ** 
## Age          0.06869    0.01375   4.997 5.81e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 75.372  on 40  degrees of freedom
## Residual deviance: 51.012  on 39  degrees of freedom
## AIC: 156.46
## 
## Number of Fisher Scoring iterations: 5