In this section, the following kinds of *factorial designs* will be described:

Hint

All available designs can be accessed after a simple import statement:

```
>>> from pyDOE import *
```

This kind of design offers full flexibility as to the number of discrete levels for each factor in the design. Its usage is simple:

```
>>> fullfact(levels)
```

where `levels` is an array of integers, like:

```
>>> fullfact([2, 3])
array([[ 0., 0.],
[ 1., 0.],
[ 0., 1.],
[ 1., 1.],
[ 0., 2.],
[ 1., 2.]])
```

As can be seen in the output, the design matrix has as many columns as items in the input array.

This function is a convenience wrapper to `fullfact` that forces all the
factors to have two levels each, you simple tell it how many factors to
create a design for:

```
>>> ff2n(3)
array([[-1., -1., -1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., -1.],
[-1., -1., 1.],
[ 1., -1., 1.],
[-1., 1., 1.],
[ 1., 1., 1.]])
```

This function requires a little more knowledge of how the *confounding*
will be allowed (this means that some factor effects get muddled with
other interaction effects, so it’s harder to distinguish between them).

Let’s assume that we just can’t afford (for whatever reason) the number
of runs in a *full-factorial* design. We can systematically decide on a
*fraction of the full-factorial* by allowing some of the factor *main
effects* to be confounded with other factor *interaction effects*. This
is done by defining an *alias* structure that defines, symbolically,
these interactions. These alias structures are written like “C = AB” or
“I = ABC”, or “AB = CD”, etc. These define how one column is related to
the others.

For example, the alias “C = AB” or “I = ABC” indicate that there are
three factors (A, B, and C) and that the main effect of factor
C is confounded with the interaction effect of the product AB, and by
extension, A is confounded with BC and B is confounded with AC. A full-
factorial design with these three factors results in a design matrix with
8 runs, but we will assume that we can only afford 4 of those runs. To
create this *fractional* design, we need a matrix with three columns, one
for A, B, and C, only now where the levels in the C column is created by
the product of the A and B columns.

The input to `fracfact` is a generator string of symbolic characters
(lowercase or uppercase, but not both) separated by spaces, like:

```
>>> gen = 'a b ab'
```

This design would result in a 3-column matrix, where the third column is
implicitly defined as `"c = ab"`. This means that the factor in the third
column is confounded with the interaction of the factors in the first two
columns. The design ends up looking like this:

```
>>> fracfact('a b ab')
array([[-1., -1., 1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., 1.]])
```

Fractional factorial designs are usually specified using the notation
2^(k-p), where k is the number of columns and p is the number
of effects that are confounded. In terms of *resolution* level, higher is
“better”. The above design would be considered a 2^(3-1)
fractional factorial design, a 1/2-fraction design, or a *Resolution III*
design (since the smallest alias “I=ABC” has three terms on the right-hand
side). Another common design is a Resolution III, 2^(7-4)
fractional factorial and would be created using the following string
generator:

```
>>> fracfact('a b ab c ac bc abc')
array([[-1., -1., 1., -1., 1., 1., -1.],
[ 1., -1., -1., -1., -1., 1., 1.],
[-1., 1., -1., -1., 1., -1., 1.],
[ 1., 1., 1., -1., -1., -1., -1.],
[-1., -1., 1., 1., -1., -1., 1.],
[ 1., -1., -1., 1., 1., -1., -1.],
[-1., 1., -1., 1., -1., 1., -1.],
[ 1., 1., 1., 1., 1., 1., 1.]])
```

More sophisticated generator strings can be created using the “+” and “-” operators. The “-” operator swaps the levels of that column like this:

```
>>> fracfact('a b -ab')
array([[-1., -1., -1.],
[ 1., -1., 1.],
[-1., 1., 1.],
[ 1., 1., -1.]])
```

In order to reduce confounding, we can utilize the `fold` function:

```
>>> m = fracfact('a b ab')
>>> fold(m)
array([[-1., -1., 1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., 1.],
[ 1., 1., -1.],
[-1., 1., 1.],
[ 1., -1., 1.],
[-1., -1., -1.]])
```

Applying the fold to all columns in the design breaks the alias chains
between every *main factor and two-factor interactions*. This means that
we can then estimate *all the main effects clear of any two-factor
interactions*. Typically, when all columns are folded, this “upgrades”
the resolution of the design.

By default, `fold` applies the level swapping to all columns, but we can
fold specific columns (first column = 0), if desired, by supplying an array
to the keyword `columns`:

```
>>> fold(m, columns=[2])
array([[-1., -1., 1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., 1.],
[-1., -1., -1.],
[ 1., -1., 1.],
[-1., 1., 1.],
[ 1., 1., -1.]])
```

Note

Care should be taken to decide the appropriate alias structure for your design and the effects that folding has on it.

Another way to generate fractional-factorial designs is through the use
of **Plackett-Burman** designs. These designs are unique in that the
number of trial conditions (rows) expands by multiples of four (e.g. 4,
8, 12, etc.). The max number of columns allowed before a design increases
the number of rows is always one less than the next higher multiple of four.

For example, I can use up to 3 factors in a design with 4 rows:

```
>>> pbdesign(3)
array([[-1., -1., 1.],
[ 1., -1., -1.],
[-1., 1., -1.],
[ 1., 1., 1.]])
```

But if I want to do 4 factors, the design needs to increase the number of rows up to the next multiple of four (8 in this case):

```
>>> pbdesign(4)
array([[-1., -1., 1., -1.],
[ 1., -1., -1., -1.],
[-1., 1., -1., -1.],
[ 1., 1., 1., -1.],
[-1., -1., 1., 1.],
[ 1., -1., -1., 1.],
[-1., 1., -1., 1.],
[ 1., 1., 1., 1.]])
```

Thus, an 8-run Plackett-Burman design can handle up to (8 - 1) = 7 factors.

As a side note, It just so happens that the Plackett-Burman and 2^(7-4) fractional factorial design are identical:

```
>>> np.all(pbdesign(7)==fracfact('a b ab c ac bc abc'))
True
```

If the user needs more information about appropriate designs, please consult the following articles on Wikipedia:

There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general.

Any questions, comments, bug-fixes, etc. can be forwarded to the author.