9.3 - Mixed Factorials

We have been talking about 2-level designs and 3-level designs. 2-level designs for screening factors and 3-level designs analogous to the 2-level designs, but the beginning of our discussion of response surface designs.

Since a 2-level design only has two levels of each factor, we can only detect linear effects. We have been mostly thinking about quantitative factors but especially when screening two level designs the factors can be presence / absence, or two types and you can still throw it into that framework and decide whether that's an important factor. If we go to three level designs we are almost always thinking about quantitative factors. But, again, it doesn't always have to be, it could be three types of something. However, in the general application we are talking about quantitative factors.

If we take a 2-level design that has center points.

Then, if you project into the A axis or the B axis, you have three distinct values, -1, 0, and +1.

In the main effect sense, a two level design with center points gives you three levels. This was our starting point towards moving to a three level design. Three-level designs require a whole lot more observations. With just two factors, i.e., \(k = 2\), you have \(3^k = 9\) observations, but as soon as we get to \(k = 4\), now you already have \(3^4 = 81\) observations, and with \(k = 5\) becomes out of reach - \(3^5 = 243\) observations. These designs grow very fast so obviously we are going to look for more efficient designs.

Mixed Level Designs Section

When we think of next level designs we think of factors with 4 or 5 levels, or designs with combinations of 2, 3, 4, or 5 levels of factors. In an Analysis of Variance course, which most of you have probably taken, it didn't distinguish between these factors. Instead, you looked at general machinery for factors with any numbers of level. What is new here is thinking about writing efficient designs. Let's say you have a \(2^3\times 3^2\) - this would be a mixed level design with \(8\times 9 = 72\) observations in a single replicate. So this is growing pretty rapidly! As this gets even bigger we could trim the size of this by looking at fractions for instance, \(2^{3-1}\), a fractional factorial of the first part. And, as these numbers of observations get larger you could look at crossing fractions of factorial designs.

A Note about Factors with 4 levels - \(2^2\) Section

This design is \(2^2\), so in some sense there is nothing new here. By using the machinery of the \(2^k\) designs you can always take a factor with four levels and call it the four combinations of \(2^2\).

A Note about Factors with 5 levels Section

Design with factors with 5 levels... Think quantitative - if it is quantitative then you have five levels, and we should then be thinking about fitting a polynomial regression function.


X 1 X 2 = star points -1 -1 +1 +1

This leads us to a whole new class of designs that we will look at next - Response Surface Designs.

What we have plotted here is a \(2^2\) design, which are the four corners of a \(2^2\). We have center points. And then to achieve what we will refer to as a central composite design we will add what are called star points (axial points). These are points that are outside the range of -1 and 1 in each dimension. If you think in terms of projecting, we now have 5 levels of each of these 2 factors obtained in some automatic way. Instead of having 25 points which is what a \(5\times 5\) requires, we only have 9 points. It is a more efficient design but still in a projection we have five levels in each direction. What we want is enough points to estimate a response surface but at the same time keep the design as simple and with as few observations as possible.

The primary reason that we looked at the \(3^k\) designs is to understand the confounding that occurs. When we have quantitative variables we will generally not use a 3 level designs. We use this more for understanding of what is going on. In some sense 3 level designs are not as practical as CCD designs. We will next consider response surface designs to address to goals of fitting a response surface model.