Multiway Classification Models
Definition
The two-sample model (or the two-sample problem) has input space $\set{1, 2}$ and regression function $\phi : \set{1, 2} \to \R ^2$ where $\phi (1) = \transpose{(1, 0)}$ and $\phi (2) = \transpose{(0, 1)}$. In this case, we model two populations (corresponding to 1 and 2) with different means but a common variance. The regression range is the set $\set{(1, 0), (0, 1)}$.
Generalizing from two to $a$ populations gives the one-way classification model. In that case the input space is $\set{1, \dots , a}$ and the regression function is $\phi : \set{1, \dots , a} \to \R ^a$ defined by $\phi (i) = e_i$ where $e_i$ is the standard unit vector in $\R ^a$.1 The regression range is $\set{e_1, \dots , e_a}$. In this case we say that the factor population takes levels $1, \dots , a$.
If there are more than one factors, then we have a multiway classification model. For example, the two-way classification model with no interaction has input domain $X = \set{1, \dots , a} \times \set{1, \dots , b}$ and the regression function $\phi : X \to \R ^{a + b}$ is defined by $\phi (i, j) = \transpose{(e_i, e_j)}$.
- Future editions will define the standard unit vector. ↩︎