5.
Process Improvement
5.5. Advanced topics


John's designs require only 3/4 of the number of runs a full 2^{n} factorial would require 
Threequarter (¾) designs are twolevel factorial designs that
require only threequarters of the number of runs of the 'original'
design. For example, instead of making all of the sixteen runs required
for a 2^{4} fractional factorial design, we need only run 12 of
them. Such designs were invented by Professor Peter John of the
University of Texas, and are sometimes called 'John's ¾
designs.'
Threequarter fractional factorial designs can be used to save on resources in two different contexts. In one scenario, we may wish to perform additional runs after having completed a fractional factorial, so as to dealias certain specific interaction patterns. Second , we may wish to use a ¾ design to begin with and thus save on 25% of the run requirement of a regular design. 

Semifolding Example  
Four experimental factors  We have four experimental factors to investigate, namely X_{1}, X_{2}, X_{3}, and X_{4}, and we have designed and run a 2^{41} fractional factorial design. Such a design has eight runs, or rows, if we don't count center point runs (or replications).  
Resolution IV design  The 2^{41} design is of resolution IV, which means that main effects are confounded with, at worst, threefactor interactions, and twofactor interactions are confounded with other two factor interactions.  
Design matrix 
The design matrix, in standard order, is shown in Table 5.8 along with
all the twofactor interaction columns. Note that the column for
X_{4} is constructed by multiplying columns for
X_{1}, X_{2}, and X_{3}
together (i.e., 4=123).


Confounding of twofactor interactions  Note also that 12=34, 13=24, and 14=23. These follow from the generating relationship 4=123 and tells us that we cannot estimate any twofactor interaction that is free of some other twofactor alias.  
Estimating twofactor interactions free of confounding 
Suppose that we became interested in estimating some or all of the
twofactor interactions that involved factor X_{1}; that
is, we want to estimate one or more of the interactions 12, 13, and 14
free of twofactor confounding.
One way of doing this is to run the 'other half' of the design: an additional eight rows formed from the relationship 4 = 123. Putting these two 'halves' together, the original one and the new one, we'd obtain a 2^{4} design in sixteen runs. Eight of these runs would already have been run, so all we'd need to do is run the remaining half. 

Alternative method requiring fewer runs  There is a way, however, to obtain what we want while adding only four more runs. These runs are selected in the following manner: take the four rows of Table 5.8 that have '1' in the 'X_{1}' column and switch the '' sign under X_{1} to '+' to obtain the fourrow table of Table 5.9. This is called a foldover on X_{1}, choosing the subset of runs with X_{}1 = 1. Note that this choice of 4 runs is not unique, and that if the initial design suggested that X_{1} = 1 were a desirable level, we would have chosen to experiment at the other four treatment combinations that were omitted from the initial design.  
Table of the additional design points 


Table with new design points added to the original design points 
Add this new block of rows to the bottom of Table 5.8 to obtain a
design in twelve rows. We show this in Table 5.10 and also add in the
twofactor interactions as well for illustration (not needed when we
do the runs).


Design is resolution V  Examine the twofactor interaction columns and convince yourself that no two are alike. This means that no twofactor interaction involving X_{1} is aliased with any other twofactor interaction. Thus, the design is resolution V, which is not always the case when constructing these types of ¾ foldover designs.  
Estimating X1 twofactor interactions  What we now have is a design with 12 runs, with which we can estimate all the twofactor interactions involving X_{1} free of aliasing with any other twofactor interaction. It is called a ¾ design because it has ¾ the number of rows of the next regular factorial design (a 2^{4}).  
Standard errors of effect estimates  If one fits a model with an intercept, a block effect, the four main effects and the six twofactor interactions, then each coefficient has a standard error of σ/8^{1/2}, instead of σ/12^{1/2}, because the design is not orthogonal and each estimate is correlated with two other estimates. Note that no degrees of freedom exists for estimating σ. Instead, one should plot the 10 effect estimates using a normal (or halfnormal) effects plot to judge which effects to declare significant.  
Further information  For more details on ¾ fractions obtained by adding a followup design that is half the size of the original design, see Mee and Peralta (2000).  
Next we consider an example in which a ¾ fraction arises when the (¾) 2^{kp} design is planned from the start because it is an efficient design that allows estimation of a sufficient number of effects.  
A 48Run 3/4 Design Example  
Estimate all main effects and twofactor interactions for 8 factors  Suppose we wish to run an experiment for k=8 factors, with which we want to estimate all main effects and twofactor interactions. We could use the \( 2_{V}^{82} \) design described in the summary table of fractional factorial designs, but this would require a 64run experiment to estimate the 1 + 8 + 28 = 37 desired coefficients. In this context, and especially for larger resolution V designs, ¾ of the design points will generally suffice.  
Construction of the 48run design 
The 48 rundesign is constructed as follows: start by creating the full
\( 2_{V}^{82} \)
design using the generators 7 = 1234 and 8 = 1256. The defining relation
is I = 12347 = 12568 = 345678 (see the summary table
details for this design).
Next, arrange these 64 treatment combinations into four blocks of size 16, blocking on the interactions 135 and 246 (i.e., block 1 has 135 = 246 = 1 runs, block 2 has 135 = 1, 246 = +1, block 3 has 135 = +1, 246 = 1 and block 4 has 135 = 246 = +1). If we exclude the first block in which 135 = 246 = 1, we have the desired ¾ design reproduced below (the reader can verify that these are the runs described in the summary table, excluding the runs numbered 1, 6, 11, 16, 18, 21, 28, 31, 35, 40, 41,46, 52, 55, 58 and 61). 

Table containing the design matrix 


Good precision for coefficient estimates  This design provides 11 degrees of freedom for error and also provides good precision for coefficient estimates (some of the coefficients have a standard error of \( \sigma/\sqrt{32} \) and some have a standard error of \( \sigma/\sqrt{42.55} \).  
Further information  More about John's ¾ designs can be found in John (1971) or Diamond (1989). 