The crabs data set
The crabs data set is derived from Agresti (2007, Table 3.2, pp.76-77). It gives 4 variables for each of 173 female horseshoe crabs.
- Satellites number of male partners in addition to the female's primary partner
- Width width of the female in centimeters
- Dark a binary factor indicating whether the female has dark coloring (yes or no)
- GoodSpine a binary factor indicating whether the female has good spine condition (yes or no)
library(glm2)
data(crabs)
head(crabs) %>% kable()| Satellites | Width | Dark | GoodSpine | Rep1 | Rep2 |
|---|---|---|---|---|---|
| 8 | 28.3 | no | no | 2 | 2 |
| 0 | 22.5 | yes | no | 4 | 5 |
| 9 | 26.0 | no | yes | 5 | 6 |
| 0 | 24.8 | yes | no | 6 | 6 |
| 4 | 26.0 | yes | no | 6 | 8 |
| 0 | 23.8 | no | no | 8 | 8 |
summary(crabs[,1:4]) %>% kable()| Satellites | Width | Dark | GoodSpine | |
|---|---|---|---|---|
| Min. : 0.000 | Min. :21.0 | no :107 | no :121 | |
| 1st Qu.: 0.000 | 1st Qu.:24.9 | yes: 66 | yes: 52 | |
| Median : 2.000 | Median :26.1 | NA | NA | |
| Mean : 2.919 | Mean :26.3 | NA | NA | |
| 3rd Qu.: 5.000 | 3rd Qu.:27.7 | NA | NA | |
| Max. :15.000 | Max. :33.5 | NA | NA |
Question
Fit a Poisson regression model with the number of Satellites as the outcome and the width of the female as the covariate.
What is the multiplicative change in the expected number of crabs for each additional centimeter of width?
crabs.pois <- glm2( Satellites ~ Width, data=crabs, family="poisson")
summary(crabs.pois)##
## Call:
## glm2(formula = Satellites ~ Width, family = "poisson", data = crabs)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.8526 -1.9884 -0.4933 1.0970 4.9221
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -3.30476 0.54224 -6.095 1.1e-09 ***
## Width 0.16405 0.01997 8.216 < 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: 632.79 on 172 degrees of freedom
## Residual deviance: 567.88 on 171 degrees of freedom
## AIC: 927.18
##
## Number of Fisher Scoring iterations: 6exp(-3.30476)*exp(0.164*25)## [1] 2.214973plot(crabs$Width, crabs$Satellites,
pch=16, col="darkred")
points(crabs$Width, crabs.pois$fitted.values,
col="green", type="p", lwd=3)
Question
What is the expected number of Satellites for a female of width 22cm?
Given a set of parameters \(\{\beta_0,\beta_1, \ldots, \beta_n\}\) and an input vector \(x\) (i.e \(\{x_1,x_2, \ldots x_n\}\)), the mean of the predicted Poisson distribution is given by \[\operatorname{E}(Y|x)=e^{\beta_0+\beta_1x_1 + \ldots + \beta_nx_n}\, \] In the case of one predictor variable, we can say \[\operatorname{E}(Y|x)=e^{\beta_0+\beta_1x}=e^{\beta_0}\times e^{\beta_1x}\,\]
Remark: The expected value does not have to be an integer.
exp(-3.30476)*exp(0.16405*22)## [1] 1.35573