Editing Analysis of variance (section)

===Connection to linear regression===
Below we make clear the connection between multi-way ANOVA and linear regression.

Linearly re-order the data so that <math>k</math>-th observation is associated with a response <math>y_k</math> and factors <math>Z_{k,b}</math> where <math>b \in \{1,2,\ldots,B\}</math> denotes the different factors and <math>B</math> is the total number of factors. In one-way ANOVA <math>B=1</math> and in two-way ANOVA <math>B = 2</math>. Furthermore, we assume the <math>b</math>-th factor has <math>I_b</math> levels, namely <math>\{1,2,\ldots,I_b\}</math>. Now, we can [[one-hot]] encode the factors into the <math display="inline"> \sum_{b=1}^B I_b</math> dimensional vector <math>v_k</math>.

The one-hot encoding function <math>g_b : \{1,2,\ldots,I_b\} \mapsto \{0,1\}^{I_b}</math> is defined such that the <math>i</math>-th entry of <math>g_b(Z_{k,b})</math> is
<math display="block">g_b(Z_{k,b})_i = \begin{cases}
1 & \text{if } i=Z_{k,b} \\
0 & \text{otherwise}
\end{cases}</math>
The vector <math>v_k</math> is the concatenation of all of the above vectors for all <math>b</math>. Thus, <math>v_k = [g_1(Z_{k,1}), g_2(Z_{k,2}), \ldots, g_B(Z_{k,B})]</math>. In order to obtain a fully general <math>B</math>-way interaction ANOVA we must also concatenate every additional interaction term in the vector <math>v_k</math> and then add an intercept term. Let that vector be <math>X_k</math>.

With this notation in place, we now have the exact connection with linear regression. We simply regress response <math>y_k</math> against the vector <math>X_k</math>. However, there is a concern about [[identifiability]]. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use ''F''-statistics or other methods to determine the relevance of the individual factors.

====Example====
We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels.

Define <math>a_i = 1</math> if <math>Z_{k,1}=i</math> and <math>b_i = 1</math> if <math>Z_{k,2} = i</math>, i.e. <math>a</math> is the one-hot encoding of the first factor and <math>b</math> is the one-hot encoding of the second factor.

With that,
<math display="block">
X_k = [a_1, a_2, b_1, b_2, b_3 ,a_1 \times b_1, a_1 \times b_2, a_1 \times b_3, a_2 \times b_1, a_2 \times b_2, a_2 \times b_3, 1]
</math>
where the last term is an intercept term. For a more concrete example suppose that
<math display="block">\begin{align}
Z_{k,1} & = 2 \\
Z_{k,2} & = 1
\end{align}</math>
Then, <math display="block">X_k = [0,1,1,0,0,0,0,0,1,0,0,1]</math>