Determining the Effect of a Factor
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Determining the Effect of a Factor
Basic 2 level, 2 variable Design (22 Design)
22 Experimental Design with a Measured ResponseExperiment (Yates Order) / Factor A / Factor B / Response
1 (1) / -1 / -1 / 6
2 (a) / 1 / -1 / 3
3 (b) / -1 / 1 / 7
4 (ab) / 1 / 1 / 10
Computing the Average Effect of Factor A
The Effect of a factor is defined as the change in the response produced by a change in the level of that factor.
The AVERAGE effect of factor A at its LOW level is (-6 -7)/2 = - 6.5
The AVERAGE effect of factor A at its HIGH level is (+3 + 10)/2 = + 6.5
The Average effect of A FOR THE ENTIRE EXPERIMENTAL EFFORT is the sum of the average of the low and the average high level settings
= [(3 + 10)2] – [(6 + 7)/2] = ((3 + 10) – (6 + 7))/2 = 0
Thus the average effect of factor A = 6.5 – 6.5 = 0
Computing the Average Effect of Factor B
In a similar fashion – assigning the signs of the column associated with Factor B to the responses we have the following
The AVERAGE effect of factor B at its LOW level is (-6 -3)/2 = - 4.5
The AVERAGE effect of factor B at its HIGH level is (+7 + 10)/2 = + 8.5
The Average effect of B FOR THE ENTIRE EXPERIMENTAL EFFORT is the sum of the average of the low and the average high level settings
Thus the average effect of factor B = 8.5 – 4.5 = 4
The expression of the calculation in Terms of Yates Order
Effect of Factor A = ((a + ab) – (1 + b))/2 which gives ((3 + 10) – (6 + 7))/2 = 0
Effect of Factor B = ((b + ab) – (1 + a))/2 which gives ((7 + 10) – (6 + 3))/2 = 4
Item of Interest – note well – Yates order is just a short hand way of describing the makeup of a particular experiment. If the letter is present then that corresponds to the factor setting at a high level (+1) and if it is absent then that corresponds to the factor setting at a low level (-1). Thus for the design above, the first experiment has both factor A and factor B at a low level corresponding to the Yates designation of (1). The second experiment has factor A at a high level and factor B at a low level thus the Yates designation of (a) etc.
How Can the Average Effect of A be Distinguished from the Effect of B using the same set of Experiments?
The Average effort of A = Average of the sum of the values of the response when A is at the high level minus the average of the sum of the values of the response when A is at the low level = [ (a +ab)/2] – [(1 + b)/2] = ((a +ab) – (1 + b))/2.
Experiment (a) has factor B at its low level
Experiment (1) has factor B at its low level
Experiment (b) has factor B at its high level
Experiment (ab) has factor B at its high level
THEREFORE the AVERAGE EFFECT of FACTOR B when computing THE AVERAGE EFFECT OF A IS:
((Blow + Bhigh) -(Blow + Bhigh))/2 = 0
The Extrapolation
If we have a three variable, 2 level design (23) the factors will be A, B, and C.
The 8 experiments in the design will be (remember the Yates designation): (1), a, b, ab, c, ac, bc, and abc.
Therefore the average effect of Factor A is ((a + ab + ac+ abc) – (1 + b+ c +bc))/4
Note the division by 4 since we are now taking averages of 4 experiments with the factor settings at either the high or the low level.