Fit Two Level Screening as an Orthogonal Rotation

Mathematically, the Fit Two Level Screening platform takes the n values in the response vector and rotates them into n new values. The rotated values are then mapped by the space of the factors and their interactions.

Contrasts = T’ × Responses

In the above equation, T is an n by n orthonormalized set of values starting with the intercept, main effects of factors, two-way interactions, three-way interactions, and so on, until n values have been obtained. Since the first column of T is an intercept, and all the other columns are orthogonal to it, these other columns are all contrasts, that is, they sum to zero. Since T is orthogonal, it can serve as X in a linear model. It does not need inversion, since T’ is equivalent to T-1 and (T’T)T’. The contrasts are the parameters estimated in a linear model.

If no effect in the model is active other than the intercept, the contrasts are just an orthogonal rotation of random independent variates into different random independent variates. These orthogonally-rotated random variates have the same variance as the original random independent variates. To the extent that some effects are active, the inactive effects still represent the same variation as the error in the model. The hope is that the effects and the design are strong enough to separate the active effects from the random error effects.