## Chapter 6.3.2, Nominal response, Alligator Food Choice

Alligators <- read.table("Alligators.dat", header = T)
Alligators
• y1: Fish, y2: Invertebrate, y3: Reptile, y4: Bird, y5: Other
• lake: 1 = Hancock, 2 = Ocklawaha, 3 = Trafford, 4 = George
• size: 1 = Alligator size > 2.3m, 0 = Alligator size <= 2.3m

This is a dataset with grouped multinomial responses.

We can use a Baseline-category logit model. Using the first category as the baseline, we have

library(VGAM)
## Loading required package: stats4
## Loading required package: splines
fit <- vglm(formula = cbind(y2, y3, y4, y5, y1) ~ size + factor(lake), family = multinomial, data = Alligators)
summary(fit)
##
## Call:
## vglm(formula = cbind(y2, y3, y4, y5, y1) ~ size + factor(lake),
##     family = multinomial, data = Alligators)
##
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)
## (Intercept):1    -3.2074     0.6387  -5.021 5.13e-07 ***
## (Intercept):2    -2.0718     0.7067  -2.931 0.003373 **
## (Intercept):3    -1.3980     0.6085  -2.297 0.021601 *
## (Intercept):4    -1.0781     0.4709  -2.289 0.022061 *
## size:1            1.4582     0.3959   3.683 0.000231 ***
## size:2           -0.3513     0.5800  -0.606 0.544786
## size:3           -0.6307     0.6425  -0.982 0.326296
## size:4            0.3316     0.4482   0.740 0.459506
## factor(lake)2:1   2.5956     0.6597   3.934 8.34e-05 ***
## factor(lake)2:2   1.2161     0.7860   1.547 0.121824
## factor(lake)2:3  -1.3483     1.1635  -1.159 0.246529
## factor(lake)2:4  -0.8205     0.7296  -1.125 0.260713
## factor(lake)3:1   2.7803     0.6712   4.142 3.44e-05 ***
## factor(lake)3:2   1.6925     0.7804   2.169 0.030113 *
## factor(lake)3:3   0.3926     0.7818   0.502 0.615487
## factor(lake)3:4   0.6902     0.5597   1.233 0.217511
## factor(lake)4:1   1.6584     0.6129   2.706 0.006813 **
## factor(lake)4:2  -1.2428     1.1854  -1.048 0.294466
## factor(lake)4:3  -0.6951     0.7813  -0.890 0.373608
## factor(lake)4:4  -0.8262     0.5575  -1.482 0.138378
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Names of linear predictors: log(mu[,1]/mu[,5]), log(mu[,2]/mu[,5]),
## log(mu[,3]/mu[,5]), log(mu[,4]/mu[,5])
##
## Residual deviance: 17.0798 on 12 degrees of freedom
##
## Log-likelihood: -47.5138 on 12 degrees of freedom
##
## Number of Fisher scoring iterations: 5
##
## Warning: Hauck-Donner effect detected in the following estimate(s):
## '(Intercept):1'
##
##
## Reference group is level  5  of the response

We can check whether we need a more complicated model by LRT test of the residual deviance

1 - pchisq(17.0798, df = 12)
## [1] 0.1466201

This indicates that interactions are not needed.

Both the lake and size main effects are needed.

library(VGAM)
fit.nosize <- vglm(formula = cbind(y2, y3, y4, y5, y1) ~ factor(lake), family = multinomial, data = Alligators)
anova(fit.nosize, fit, type = "I")

The fitted probabilities

fitted(fit)
##           y2         y3          y4         y5        y1
## 1 0.09309880 0.04745657 0.070401523 0.25373963 0.5353035
## 2 0.02307168 0.07182461 0.140896287 0.19400964 0.5701978
## 3 0.60189675 0.07722761 0.008817482 0.05387208 0.2581861
## 4 0.24864518 0.19483742 0.029416085 0.06866281 0.4584385
## 5 0.51683852 0.08876722 0.035894709 0.17420051 0.1842990
## 6 0.19296122 0.20239954 0.108225068 0.20066164 0.2957525
## 7 0.41285579 0.01156654 0.029671169 0.09380245 0.4521040
## 8 0.13967784 0.02389871 0.081067366 0.09791362 0.6574425

y1: Fish, y2: Invertebrate, y3: Reptile, y4: Bird, y5: Other