Designs that require fewer than 2k runs

In the remainder of this chapter, we describe experimental designs for k factors, each of which has 2 levels, but which use fewer than the 2k runs required for one replicate of a complete factorial design.

Design principles

We first describe some principles that the designs should follow.

Same number of runs for the high and low levels of all factors
This ensures that the mean response is estimated with the same accuracy at the high and low factor level.
For any pair of factors, the four possible combinations of factor levels should be used in the same number of runs.
If this occurs, the factors are said to be orthogonal.

Orthogonality is important because it ensures that the estimate of the main effect of any factor does not depend on whether or not the other factors affect the response.

If the factors are orthogonal, the estimates of their main effects are independent.


Non-orthogonal design

We illustrate the requirement of orthogonality by showing an experiment for two factors with 10 runs that is not orthogonal. In this experiment, changing factor B from low to high increases the response by 5, but factor A has no effect on the response. To simplify, we assume that there is no experimental error. The table below shows the 10 response values:

  Factor B
    low     high  
Factor A low
55
55
10
high 5
10 10
10 10

The main effect of each factor is usually defined to be the difference between the mean response at its low and high level. In this experiment,

main effect for B  =  (mean response at high B) - (mean response at low B)  =  5

main effect for A  =  (mean response at high A) - (mean response at low A)  =  3

Factor A is incorrectly estimated to increase the response by 3 when changed from low to high level — it really has no effect.

Orthogonal design

In orthogonal designs, this cannot happen. The effect of factor B cannot cause factor A to appear to affect the response. In the experimental data below, increasing factor A from low to high is estimated to decrease the response by 4.

  Factor B
    low     high  
Factor A low
5 7
6 8
8 7
5 6
high
4 2
1 3
3 4
1 2

The following data are the same except that the effect of factor B has been increased by 10. Since this equally affects the low and high levels of factor A, the estimated effect of changing A from low to high remains the same — a decrease of 4.

  Factor B
    low     high  
Factor A low
5 7
6 8
18 17
15 16
high
4 2
1 3
13 14
11 12

In later sections, we will describe how to design screening experiments for k factors with 2 levels each. We now present the results from one such experiment.

Hardness of car paint

The following table shows the screening design that was used to estimate the effects on hardness of adding 15 different compounds (A - O) to a car paint. The experiment used 16 different combinations of the additives — the +1 and -1 values in each of the 16 rows of the table indicate whether the additive was present or absent.

The response, Y, was the hardness of the paint on one painted panel, as measured by a scratch test.

  Additive Hardness
Run A B C D E F G H I J K L M N O Y
1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 53.3
2 +1 +1 +1 -1 +1 +1 -1 +1 -1 -1 +1 -1 -1 -1 -1 46.6
3 +1 +1 -1 +1 +1 -1 +1 -1 +1 -1 -1 +1 -1 -1 -1 53.8
4 +1 +1 -1 -1 +1 -1 -1 -1 -1 +1 -1 -1 +1 +1 +1 44.6
5 +1 -1 +1 +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 -1 -1 44.8
6 +1 -1 +1 -1 -1 +1 -1 -1 +1 -1 -1 +1 -1 +1 +1 58.9
7 +1 -1 -1 +1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 56.5
8 +1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 -1 -1 60.5
9 -1 +1 +1 +1 -1 -1 -1 +1 +1 +1 -1 -1 -1 +1 -1 48.2
10 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1 -1 +1 +1 -1 +1 47.4
11 -1 +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 -1 +1 54.8
12 -1 +1 -1 -1 -1 +1 +1 -1 -1 +1 +1 +1 -1 +1 -1 45.9
13 -1 -1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 -1 +1 56.3
14 -1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 50.2
15 -1 -1 -1 +1 +1 +1 -1 +1 -1 -1 -1 +1 +1 +1 -1 62.3
16 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 +1 44.7

This experiment uses far fewer experimental units (runs) than a complete factorial experiment. However all factors are orthogonal so the main effects of all factors can still be independently estimated. These are shown in the table below:

FactorEffect
A      1.15  
B -4.95  
C -2.17  
D 3.90  
E -0.65  
F -0.77  
G -4.45  
H 1.27  
I 2.50  
J -4.02  
K 2.42  
L 6.00  
M 0.87  
N 1.37  
O 0.52  

The negative effects correspond to additives that are estimated to decrease the hardness of the paint, so we would only consider ones with positive effects for further study. Of these, molecules L, D, I and K have the highest estimated effects.