Screening experiments

Latin square and Graeco-Latin square designs in which the 'rows' and 'columns' of the design refer to levels of two controlled factors are examples of designs called fractional factorial designs. For example, a complete experiment for three r-level factors would require r3 experimental units, but a r×r Latin square design could be used with only 1/r of these experimental units.

Fractional factorial designs like Latin and Graeco-Latin squares are pairwise orthogonal designs for three or more factors, so the order of adding their terms to the analysis of variance table does not affect their sums of squares.

In this section, we concentrate on fractional factorial designs for several factors with two levels.

This section therefore describes experimental designs for k 2-level factors that use fewer than the 2k experimental units required for one replicate of a complete factorial design.

Such experiments are used more often in industry than in agricultural research. They are often used in the first stage of a research project to identify which of many potential factors are the most important ones whose effects will be examined in much more detail in the main part of the project. These initial experiments are called screening experiments.

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.
Pairwise orthogonal
Each of the four combinations of levels of any pair of factors should be used for the same number of experimental units to ensure that the factors are 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.

Orthogonality of the factors ensures that there is only a single analysis of variance table — the order of adding the factors does not affect their sums of squares.


Hardness of biscuits

The following table shows results from a screening experiment that was used to estimate the effects on hardness of biscuits of adding 15 different edible compounds (A - O) to the ingredients. 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 one biscuit, as measured by an impact 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 the design is pairwise balanced — for each pair of columns, there are four rows in which both values are +1, four in which both are -1, and similarly with the two mixed levels.

The above design is an extreme one for 15 factors with 16 experimental units, so the main effects of the factors cannot be tested — there are no degrees of freedom left for the residual sum of squares and we therefore cannot measure unexplained variation.

However the orthogonality of the factors in the design means that we can estimate the effects of all factors — each of the following parameter estimates is half the difference between the average hardness with and without the additive.

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 biscuits, so we would only consider ones with positive effects for further study. Of these, compounds L, D, I and K have the highest estimated effects.