Lesson 9: 3-level and Mixed-level Factorials and Fractional Factorials

Overview Section

These designs are a generalization of the \(2^k\) designs. We will continue to talk about coded variables so we can describe designs in general terms, but in this case we will be assuming in the \(3^k\) designs that the factors are all quantitative. With \(2^k\) designs we weren't as strict about this because we could have either qualitative or quantitative factors. Most \(3^k\) designs are only useful where the factors are quantitative. With \(3^k\) designs we are moving from screening factors to analyzing them to understand what their actual response function looks like.

With 2 level designs, we had just two levels of each factor. This is fine for fitting a linear, straight line relationship. With three level of each factor we now have points at the middle so we will are able to fit curved response functions, i.e. quadratic response functions. In two dimensions with a square design space, using a \(2^k\) design we simply had corner points, which defined a square that looked like this:

b ab a (1) (0,0) +1 +1 -1 -1

In three dimensions the design region becomes a cube and with four or more factors it is a hypercube which we can't draw.

We can label the design points, similar to what we did before – see the columns on the left. However for these design we prefer the other way of coding, using {0,1,2} which is a generalization of the {0,1} coding that we used in the \(2^k\) designs. This is shown in the columns on the right in the table below:

A B   A B
- -   0 0
0 -   1 0
+ -   2 0
- 0   0 1
0 0   1 1
+ 0   2 1
- +   0 2
0 +   1 2
+ +   2 2

For either method of coding, the treatment combinations represent the actual values of \(X_1\) and \(X_2\), where there is some high level, a middle level and some low level of each factor. Visually our region of experimentation or region of interest is highlighted in the figure below when \(k = 2\):

low low high high

If we look at the analysis of variance for a \(k = 2\) experiment with n replicates, where we have three levels of both factors we would have the following:

AOV df
A 2
B 2
A x B 4
Error 9(n-1)
Total 9n-1

Important idea used for confounding and taking fractions

How we consider three level designs will parallel what we did in two level designs, therefore we may confound the experiment in incomplete blocks or simply utilize a fraction of the design. In two-level designs, the interactions each have 1 d.f. and consist only of +/- components, so it is simple to see how to do the confounding. Things are more complicated in 3 level designs, since a p-way interaction has \(2^p\) d.f. If we want to confound a main effect (2 d.f.) with a 2-way interaction (4 d.f.) we need to partition the interaction into 2 orthogonal pieces with 2 d.f. each. Then we will confound the main effect with one of the 2 pieces. There will be 2 choices. Similarly, if we want to confound a main effect with a 3-way interaction, we need to break the interaction into 4 pieces with 2 d.f. each. Each piece of the interaction is represented by a psuedo-factor with 3 levels. The method given using the Latin squares is quite simple . There is some clever modulus arithmetic in this section, but the details are not important. The important idea is that just as with the \(2^k\)designs, we can purposefully confound to achieve designs that are efficient either because they do not use the entire set of \(3^k\)runs or because they can be run in blocks which do not disturb our ability to estimate the effects of most interest.

Following the text, for the A*B interaction, we define the pseudo factors, which are called the AB component and the \(AB^2\) component. These components could be called pseudo-interaction effects. The two components will be defined as a linear combination as follows, where \(X_1\) is the level of factor A and \(X_2\) is the level of factor B using the {0,1,2} coding system. Let the \(AB\) component be defined as

\(L_{AB}=X_{1}+X_{2}\ (mod3)\)

and the \(AB^2\) component will be defined as:

\(L_{AB^2}=X_{1}+2X_{2}\ (mod3)\)

Using these definitions we can create the pseudo-interaction components. Below you see that the AB levels are defined by \(L_{AB}\) and the \(AB^2\) levels are defined by \(L_{AB^2}\).

\(A\) \(B\)   \(AB\) \(AB^2\)
0 0   0 0
1 0   1 1
2 0   2 2
0 1   1 2
1 1   2 0
2 1   0 1
0 2   2 1
1 2   0 2
2 2   1 0

This table has entries {0, 1, 2} which allow us to confound a main effect or either component of the interaction A*B. Each of these main effects or pseudo interaction components have three levels and therefore 2 degrees of freedom.

This section will also discuss partitioning the interaction SS's into 1 d.f. sums of squares associated with a polynomial, however, this is just polynomial regression. This method does not seem to be readily applicable to creating interpretable confounding patterns.


Upon successful completion of this lesson, you should be able to:

  • Application of \(3^k\) factorial designs, the interaction components and relative degrees of freedom
  • How to perform blocking of \(3^k\) designs in \(3^p\) number of blocks and how to choose the effect(s) which should be confounded with blocks
  • Concept of “Partial Confounding” in replicated blocked designs and its advantages
  • How to generate reasonable \(3^{k-p}\) fractional factorial designs and understand the alias structure
  • The fact that Latin square and Graeco-Latin square designs are special cases of \(3^k\) fractional  factorial design
  • Mixed level factorial designs and their applications