¾ fractional factorial designs can be used to save on resources in two different contexts. In the first place, we may wish to do more runs after having completed a fractional factorial, so as to de-alias certain specific interaction patterns; in the second place, we may wish to do a ¾ design to begin with and so save on 25% of the run requirement of a regular design. 

Semifolding Example

We have four experimental factors to investigate, namely X1, X2, X3, and X4, and we have designed and run a 2^4-1 fractional factorial design. Such a design has eight runs, or rows, if we don’t count center point runs (or replications). 

The 2^4-1 design is of resolution IV which means that main effects are confounded with, at worst, three-factor interactions, but that two factor interactions are confounded with other two factor interactions. 

The design layout matrix, in standard order, is shown in Table 5.8 along with all the two-factor interaction columns. Note that the column for X4 is constructed by multiplying columns for X1, X2, and X3 together (i.e. 4=123). 

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 two-factor interaction that is free of some other two-factor alias. 

Suppose that we became interested in estimating some or all of the two-factor interactions that involved factor X1; that is, we want to estimate one or more of  the relationships 12, 13, and 14 free of two-factor 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 get a 2⁴ 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. 

There is a way to get what we want while only adding four more runs. These runs are selected in the following manner: take the  four rows of Table 5.8 that have ‘-1’ in the ‘X1’ column, and switch the ‘-‘ sign under X1 to ‘+’ to get the four-row table of Table 5.9. This is called a foldover on X1, choosing the subset of runs with X1 = -1. Note that this choice of 4 runs is not unique, and that if the initial design suggested that X1 = -1 was a desirable level, we would have chosen to experiment at the other four treatment combinations that were omitted from the initial design. 

Add this new block of rows to the bottom of Table 5.8 to give a design in twelve rows. We show this done in Table 5.10, and also add in the two-factor interactions as well for illustration (not needed when we do the runs). 

Examine the two-factor interaction columns and convince yourself that no two are alike. This means that no two-factor interaction involving X1 is aliased with any other two-factor interaction, which means that the design we now have is at least of resolution V when considering factor X1. In fact, it is of resolution V for all factors—which is not always the case when constructing these types of  ¾ foldover designs. 

What we now possess is a design that has 12 runs, and with which we can estimate all the two-factor interaction of X1 free of aliasing with any other two-factor interaction. It is called a ¾ design because it has ¾ the number of rows than the next regular factorial design (a 2⁴). 

If one fits a model with an intercept, a block effect, the four main effects and the six two factor interactions then each coefficient has a standard error of s/8^1/2 - instead of s/12^1/2 - because the design is not orthogonal and each estimates is correlated with two other estimates. Note that no degrees of freedom exist for estimating s. Instead, one should plot the 10 effect estimates using a normal (or half-normal) effects plot to judge which effects to keep. 

For more details on ¾ fractions obtained by adding a follow-up design half the size of the original design, see Mee and Peralta (2000).

Next we consider an example where a ¾ fraction arises when the (¾) 2^k-p design is planned from the start because it is an efficient design that allows estimation of a sufficient number of effects.

A 48 Run 3/4 Design Example

Suppose we wish to run an experiment for k=8 factors, for which we want to estimate all main effects and two factor interactions. We could use the 2_V^8-2 desgn described in the summary table of fractional factorial designs, but this would require a 64 run experiment to estimate the 1 + 8 + 28 = 37 desired coefficients. In this context, and for larger resolution V designs, ¾ of the resolution design will generally suffice.

The 48 run design is constructed as follows: start by creating the full 2_V^8-2
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 combination 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 where 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).
 
 

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 s/32^.5 and some have a standard error of s/42.55^.5).

More about John’s ¾ designs can be found in John (1971) or  Diamond (1989).