Appendix. The main definitions of experimental design

About gap between statisticians and experimenters (scientists and engineers)

This is reality: two areas of knowledge which must not only coexist but closely collaborate. Typical statement of engineers “ I would like to study the effect of different variables on the process to improve quality of product or to find new effective regime of process”. Sometimes they say more indefinitely: “I would like to improve anything”.

Montgomery gives two formulations of problem from engineer (a) and statisticians (b):

(a) I’d like to show that catalyst Z14 works better than catalyst D12 if the technician adjusts the electrode voltage just right.

(b) To quantify the efficiency difference between catalysts Z14 and D12 for electrode voltage 7, 8, and 9 in the ABC conversion process and asses statistical significance (compare to 95%) and practical significance (> 3%), perhaps economically justifying one catalyst over the other.

He presents the last (b) as more correct one.

Definition 1. The set of runs x_u = { x_u0, x_u1,..., x_ui,..., x_un }, u = 1,..., N, i = 0, 1,..., n (for N runs and n factors) or

X = or X =

is called by the experimental design.

Definition 2. The point x° = N^-1 is the centre of design.

Definition 3. The design is called the central one if the points of design’s centre equals zero: x° = 0. Example. Let X is given by matrix

X =

or graphically

This design is central one since x°1 = (+1 -1 +1 -1) / 4 = 0, x°2 = (+1 +1 -1 -1) / 4 = 0.

Definition 4. Any design Z with points Z_u can be transform to the central design X by the change:

X = Z - Z°, Z° = N^-1 .

Consequence from definition 4. If we choose both the high value and low value of factor , it is always possible to calculate the interval of factor’s variation as follows:

, i = 1, 2,..., N, and zero level of factor:

or .

Then, the values

are called the standardised or standard levels of factor x_i. It is clear the standard levels are equal +1, -1, 0.

Definition 5. The area of possible natural values of factors is called by the area of experiment or operability region x_i^* Î M_x^* (see a figure). Usually, the experimenter explores only the limited region of interest for standardised value of factors x_i Î M_x which is entirely contained within the operability region, that is M_x Ì M_x^* . If M_x is restricted by boundaries -1 x_i +1, the designed experimental area is called to be the hypercube.

Definition 6. The matrix M = X*X (X* is transposed matrix) by the size (n + 1) ´ (n + 1) is called to be information matrix. The matrix M^-1 = (X*X)^-1 is the dispersion matrix or matrix of variance.

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