Factorial Design Model Matrix

For a $2^3$ design, the model matrix typically contains the columns

$$ 1,\ A,\ B,\ C,\ AB,\ AC,\ BC,\ ABC, $$

where the factor columns are coded as $\pm 1$ and interaction columns are formed by multiplying the corresponding factor columns.

This matrix can be used to compute factorial effects. For example, in a $2^3$ design,

$$ \text{effect}(A) = \frac{1}{4}\sum_{r=1}^{8} x_{A,r} y_r. $$

The same pattern applies to $B$, $C$, $AB$, $AC$, $BC$, and $ABC$.

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Related

• STA305
• 2^3 Factorial Design
• Main Effect
• Two-Factor Interaction
• Three-Factor Interaction
• 2^k Factorial Linear Model