Factorial experiment

Remarks

In a $2^k$ experiment with one run per combination and responses $y_t$, the main effect of factor $i$ is the mean of $y_t$ over $t_i = +1$ minus the mean over $t_i = -1$. The interaction of factors $i$ and $j$ is half of the difference between the effect of factor $i$ at $t_j = +1$ and at $t_j = -1$. Equivalently, it is the mean of $y_t$ over $t_i t_j = +1$ minus the mean over $t_i t_j = -1$.

Example: a $2^2$ experiment with factors $A$ and $B$ and responses $y_{--} = 10$, $y_{+-} = 14$, $y_{-+} = 12$, $y_{++} = 20$, where the first sign is the level of $A$. The main effect of $A$ is $(14 + 20)/2 - (10 + 12)/2 = 6$, and that of $B$ is $(12 + 20)/2 - (10 + 14)/2 = 4$. The interaction is $\big( (20 - 12) - (14 - 10) \big)/2 = 2$, which also equals $(10 + 20)/2 - (14 + 12)/2$.

Non-example: the runs $(-,-)$, $(+,-)$, $(-,+)$ change one factor at a time from a baseline. The combination $(+,+)$ is missing, so this is not a full factorial experiment, and the interaction of $A$ and $B$ cannot be estimated.

The data of a factorial experiment are usually analysed by analysis of variance, with one sum of squares for each main effect and each interaction.