Multinomial Regression

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# Multinomial Regression
### Yue Jiang
### STA 210 / Duke University / Spring 2024

---

### Alligator diets

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### Alligator diets

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### Multinomial data

In this case, our outcome is categorical, but *not* ordered - as well, we have
more than two categories, so we can't simply use a straightforward logistic
regression.

.question[
Suppose you wanted to predict the preferred diet of an alligator based on its
sex and its length. How might you do this?
]

---

### Just a bunch of logistic regressions

Our outcome `\(Y\)` has `\(J\)` total categories. We might intuitively choose one of 
these categories to be the referent category, and then compare each of the other
categories against it in a pairwise comparison with logistic regressions.

Suppose `\(j\)` is the reference category. Then we will fit each of the following
models for `\(j = 2, \cdots, J\)`:

`\begin{align*}
\log\left(\frac{P(Y_i = j)}{P(Y_i = 1)}\right) = \beta_{0;j} + \beta_{1;j}x_{i1} + \cdots + \beta_{p;j}x_{ip}
\end{align*}`

.question[
Take a look at this model - what does each term represent? Why do we only fit
`\(j - 1\)` separate models?
]

---

### Fitting a multinomial regression model

```r
library(nnet)
m1 <- multinom(food ~ sex + length, data = gators)
```

```
## # weights:  12 (6 variable)
## initial  value 64.818125 
## iter  10 value 48.651915
## final  value 48.651292 
## converged
```

---

### Fitting a multinomial regression model

```r
summary(m1)
```

```
## Call:
## multinom(formula = food ~ sex + length, data = gators)
## 
## Coefficients:
##              (Intercept)       sexM     length
## invertibrate    4.070252  0.2313371 -2.4218022
## other          -1.543849 -0.7175895  0.2512553
## 
## Std. Errors:
##              (Intercept)      sexM    length
## invertibrate    1.476859 0.6746705 0.8283374
## other           1.313245 0.8485217 0.5486893
## 
## Residual Deviance: 97.30258 
## AIC: 109.3026
```

---

### Fitting a multinomial regression model

```r
exp(coef(m1))
```

```
##              (Intercept)     sexM     length
## invertibrate  58.5717389 1.260284 0.08876151
## other          0.2135574 0.487927 1.28563828
```

---

### Fitting a multinomial regression model

```r
head(round(fitted(m1), 3))
```

```
##    fish invertibrate other
## 1 0.238        0.692 0.069
## 2 0.262        0.660 0.078
## 3 0.232        0.735 0.033
## 4 0.240        0.725 0.035
## 5 0.271        0.649 0.081
## 6 0.305        0.602 0.093
```

---

### An important assumption

One important assumption in the multinomial model is that the probability of
being in category A or B shouldn't depend on whether category C is included or
not as a potential option.

.question[
What does this mean in our alligator diet example?
]