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Interactions are represented in the model matrix by cross-products. The elements in X 12 are the products
of X 1 and X 2 (for example, (−1)(−1) = 1, (1)(−1) = −1, (−1)(1) = −1, (1)(1) = 1, etc.). Similarly, X 13 is
X 1 times X 3 . X 23 is X 2 times X 3 . Likewise, X 123 is found by multiplying the elements of X 1 , X 2 , and X 3
(or the equivalent, X 12 times X 3 , or X 13 times X 2 ). The order of the X vectors in the model matrix is not
important, but the order shown (a column of +1’s, the factors, the two-factor interactions, followed by
higher-order interactions) is a standard and convenient form.
From the eight response measurements y 1 , y 2 ,…, y 8 , we can form eight statistically independent
quantities by multiplying the y vector by each of the X vectors. The reason these eight quantities are
1
statistically independent derives from the fact that the X vectors are orthogonal. The independence of
the estimated effects is a consequence of the orthogonal arrangement of the experimental design.
This multiplication is done by applying the signs of the X vector to the responses in the y vector and
then adding the signed y’s. For example, y multiplied by X 0 gives the sum of the responses: X 0 ⋅ y =
y 1 + y 2 + … + y 8 . Dividing the quantity X 0 ⋅ y by 8 gives the average response of the whole experiment.
Multiplying the y vector by an X i vector yields the sum of the four differences between the four y’s at
the +1 levels and the four y’s at the −1 levels. The effect is estimated by the average of the four differences;
that is, the effect of factor X i is X i ⋅ y/4.
The eight effects and interactions that can be calculated from a full eight-run factorial design are:
y 7 +
y 5 +
y 6 +
y 2 +
y 3 +
y 4 +
y 1 +
y 8
Average X 0 y⋅ = ----------------------------------------------------------------------------------
8
Main effect of factor 1 X 1 y⋅ = – y 1 + y 2 – y 3 + y 4 – y 5 + y 6 – y 7 + y 8
-------------------------------------------------------------------------------------
4
y 6 +
y 4 +
y 2 +
= -------------------------------------- – y 1 + y 3 + y 5 + y 7
y 8
--------------------------------------
4 4
y 7 +
y 4 +
y 3 +
y 8
Main effect of factor 2 X 2 y⋅ = -------------------------------------- – y 1 + y 2 + y 5 + y 6
--------------------------------------
4 4
y 7 +
y 6 +
y 5 +
y 8
Main effect of factor 3 X 3 y⋅ = -------------------------------------- – y 1 + y 2 + y 3 + y 4
--------------------------------------
4 4
y 4 +
y 5 +
y 1 +
y 8
Interaction of factors 1 and 2 X 12 y⋅ = -------------------------------------- – y 2 + y 3 + y 6 + y 7
--------------------------------------
4 4
y 6 +
y 3 +
y 1 +
y 8
Interaction factors 1 and 3 X 13 y⋅ = -------------------------------------- – y 2 + y 4 + y 5 + y 7
--------------------------------------
4 4
y 2 +
y 7 +
y 1 +
y 8
Interaction of factors 2 and 3 X 23 y⋅ = -------------------------------------- – y 3 + y 4 + y 5 + y 6
--------------------------------------
4 4
y 5 +
y 3 +
y 2 +
y 8
Interaction of factors 1, 2, and 3 X 123 y⋅ = -------------------------------------- – y 1 + y 4 + y 6 + y 7
--------------------------------------
4 4
2
If the variance of the individual measurements is σ , the variance of the mean is:
1
1
--- [
()
--- 8σ =
Var y() =  2 Var y 1 + Var y 2 + … + ()] =  2 2 σ 2
()
------
8  Var y 8 8  8
The variance of each main effect and interaction is:
1
1
--- [
()
()
Var effect) =  2 Var y 1 + Var y 2 + … + ()] =  2 2 σ 2
(
--- 8σ =
------
4  Var y 8 4  2
Orthogonal means that the product of any two-column vectors is zero. For example, X 3 ⋅ X 123 = (−1)(−1) + … + (+1)(+1) =
1
1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 = 0.
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