Dummy encoding for the multinomial response variable

Question

I am reading about multinomial response models from the book Multivariate Statistical Modelling Based on Generalized Linear Models by Fahrmeir and Tutz. I am trying to understand the following paragraph from the book (Chapter 3.1, p. 70):

For the categorical responses considered in this chapter, the basic distribution is the multinomial distribution. Let the response variable $Y$ have $k$ possible values, which for simplicity are labeled $1, ... , k$. Sometimes consideration of $Y\in \{1, 2, ..., k\}$ hides the fact that we actually have a multivariate response variable. This becomes obvious by considering the response vector of the dummy variables $y' = (\tilde{y_1}, \tilde{y_2}... ,\tilde{y_q}), q = k - 1$, with components $$ \begin{equation} \tag{3.1.1} \tilde{y_r} = \begin{cases} 1 & \text{if $Y = r$, $r = 1, ..., q$} \newline 0 & \text{otherwise.} \end{cases} \end{equation} $$ Then we have $$ Y = r \iff y = (0, ..., 1, ..., 0). $$ The probabilities are simply connected by $$P(Y = r) = P(y_r =1).$$

I am confused about the use of $r$ in the above set of equations. Is the $r$ in the equation $Y = r$ different than the $r$ in the equation 3.1.1? Because, if we consider $Y$ with $k = 4$, then for $Y = 4$, $r > q$, but considering equation 3.1.1, $r \in \{1, 2, ..., q\}$ and thus, $ r \le q$.

Answer 1

There are $k$ categories of the original response $Y$. When performing the dummy encoding, we omit one of the categories because it is redundant (similar to dummy coding of categorical predictors). In this case, the authors chose to omit the $k$th (last) category when specifying their dummy encoding. Now, there are $k-1$ dummy variables, representing the categories $1$ to $k-1$. The authors simply use $q$ to represent $k-1$.

$r$ is used to index the first $k-1$ categories of $Y$, each of which receive a dummy variable. For $k=4$, we have \begin{align} \tilde{y}_1 &= I(Y=1) \\ \tilde{y}_2 &= I(Y=2) \\ \tilde{y}_3 &= I(Y=3) \end{align} where $I(.)$ is the indicator function, equal to 1 if the condition is met and 0 otherwise. Nothing happens for $Y=4$; because there is no dummy variable for $Y=4$, it doesn't need to be notated. When $Y=4$, $\tilde{y}_1=\tilde{y}_2=\tilde{y}_3=0$.

$r$ is never equal to $k$, because it only indexes the dummy variables used to represent $Y$, and there is no dummy variable for category $k$.