Let's begin by taking the \(3^k\) designs and we will describe partitioning where you take one replicate of the design and put it into blocks. We will then take that structure and look at \(3^{k-p}\) factorials. These designs are not used for screening as the \(2^k\) designs were; rather with three levels we begin to think about response surface models. Also, \(3^{k}\) designs become very large as k gets large With just four factors a complete factorial is already 81 observations, i.e. \(N = 3^4\). In general, we won't consider these designs for very large k, but we will point out some very interesting connections that these designs reveal.

Reiterating what was said in the introduction, consider the two-factor design \(3^2\) with factors A and B, each at 3 levels. We denote the levels 0, 1, and 2. The \(A \times B\) interaction, with 4 degrees of freedom, can be split into two orthogonal components. One way to define the components is that AB component will be defined as a linear combination as follows:

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

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

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

In the table above for the \(AB\) and the \(AB^2\) components we have 3 0's, 3 1's and 3 2's, so this modular arithmetic gives us a balanced set of treatments for each component. Note that we could also find the \(A^{2}B\) and \(A^{2}B^{2}\) components but when you do the computation you discover that \(AB^{2}=A^{2}B\) and \(AB=A^{2}B^{2}\).

We will use this to construct the design as shown below.

We will take one replicate of this design and partition it into 3 blocks. Before we do, let’s consider the analysis of variance table for this single replicate of the design.

We have partitioned the \(A \times B\) interaction into \(AB\) and \(AB^2\), the two components of the interaction, each with 2 degrees of freedom. So, by using modular arithmetic, we have partitioned the 4 degrees of freedom into two sets, and these are orthogonal to each other. If you create two dummy variables for each of these factors, \(A\), \(B\), \(AB\) and \(AB^{2}\) you would see that each of these sets of dummy variables are orthogonal to the other.

These pseudo components can also be manipulated using a symbolic notation. This is included here for completeness, but it is not something you need to know to use or understand confounding. Consider the interaction between \(AB\) and \(AB^{2}\). Thus \(AB \times AB^2\) which gives us \(A^2 B^3\) which using modular (3) arithmetic gives us \(A^2 B^0 = A^2 = (A^2)^2 = A\). Therefore, the interaction between these two terms gives us the main effect. If we wanted to look at a term such as \(A^2 B\) or \(A^2 B^2\), we would reduce it by squaring it which would give us: \((A^2 B)^2=AB^2\) and likewise \((A^2 B^2)^2 = AB\). We never include a component that has an exponent on the first letter because by squaring it we obtain an equivalent component. This is just a way of partitioning the treatment combinations and these labels are just an arbitrary identification of them.

Let's now look at the one replicate where we will confound the levels of the AB component with our blocks. We will label these 0, 1, and 2 and we will put our treatment pairs in blocks from the following table.

Now we assign the treatment combinations to the blocks, where the pairs represent the levels of factors A and B.

This is how we get these three blocks confounded with the levels of the \(L_{AB}\) component of interaction.

Now, let's assume that we have four reps of this experiment - all the same - with AB confounding with blocks using the \(L_{AB}\). (each replicate is assigned to 3 blocks with AB confounded with blocks). We have defined one rep by confounding the AB component, and then we will do the same with 3 more reps.

Let's take a look at the AOV resulting from this experiment:

Note that Rep as an overall block has 3 df. Within reps we have variation among the 3 blocks, which are the AB levels - this has 2 df. Then we have Rep by blk or Rep by AB which has 6 df. This is the inter-block part of the analysis. These 11 degrees of freedom represents the variation among the 12 blocks (3*4).

Next we consider the intra-block part: A with 2 df, B with 2 df and the \(A \times B\) or \(AB^{2}\) component that also has 2 df. Finally we have error, which we can get by subtraction, (36 observations = 35 total df, 35 - 17 = 18 df). Another way to think about the Error is the interaction between the treatments and reps which is \(6 \times 3 = 18\), which is the same logic as in a randomized block design, where the SSE is (a-1)(b-1). A possible confusion here is using the terminology of blocks at two levels, the reps are at an overall level, and then within each rep we have the smaller blocks which are confounded with the AB component.

We now examine another experiment, this time confounding the AB² factor. We can construct another design using this component as our generator to confound with blocks.

Using the AB² then gives us the following treatment pairs (A,B) assigned to 3 blocks:

This partitions all nine of the treatment combinations into the three blocks.

We now consider a combination of these experiments, in which we have 2 reps confounding AB and 2 reps confounding \(AB^{2}\). We again will have 4 reps but our AOV will look a little different:

There are only two reps with AB confounded, so \(Rep \times AB = (2-1) * (3-1) = 2 df\) . The same is true for the \(AB^2\) component. This gives us the same 11 df among the 12 blocks. In the intra-block section, we can estimate A and B, so they will have 2 df. \(A \times B\) will have 4 df now, and if we look at what this is in terms of the \(AB\) and the \(AB^2\) component each accounts for 2 df. Then we have Error with 16 df and the total stays the same. The 16 df comes from the unconfounded effects - \(\left( A \colon 2 \times 3 = 6 \text{ and } B \colon 2 \times 3 = 6 \right) \) - that's 12 of these df, plus each of the \(AB\) and the \(AB^{2}\) components which are confounded in two reps, and unconfounded in the other two reps - \( \left(2 \times \left(2-1 \right) = 2 \text { for AB and } 2 \times \left( 2-1 \right) = 2 \text{ for } AB^{2}\right)\) - which accounts for the remaining 4 of the total 16 df for error.

We could determine the Error df simply by subtracting from the Total df, but, if it is helpful to think about randomized block designs where you have blocks and treatments and the error is the interaction between them. Note that here we use the term replicates instead of blocks, so actually we consider replicates as sort of super-blocks. In this case, the error would be the interaction between replicates and unconfounded treatments. This RCBD framework is a foundational structure that we use again and again in experimental design.

This is a good example of the benefit of partial confounding because the interaction of the pseudo factors are confounded in only half of the design, so we can estimate the interaction A*B from the other half. You get overall exactly half the information on the interaction from this partially confounded design.

Now let’s think further outside of the box. What if we confound the main effect A? What would this do to our design? What kind of experimental design would this be?

Now we define or construct our blocks by using levels of A from the table above. A single replicate of the design would look like this.

Then we could replicate this design four times. Let's consider an agricultural application and say that A = irrigation method, B = crop variety, and the Blocks = whole plots of land to which we apply the irrigation type. By confounding a main effect we're going to get a split-plot design in which the analysis will look like this:

In this design, there are four reps (3 df), and the blocks within reps are actually the levels of A which has 2 df, \(Rep \times A\) has 6 df. The interblock part of the analysis here is just a randomized complete block analysis of four reps, three treatments, and their interactions. The intra-block part contains B which has 2 df, and the \(A \times B\) interaction which has 4 df. Therefore this is another way to understand a split-plot design, where you confound one of the main effects.

We again start out with a \(3^3\) design which has 27 treatment combinations and assign them to 3 blocks. What we want to do in this lesson, going beyond the \(3^2\) design, is to describe the AOV for this \(3^3\) design. Then we also want to look at the connection between confounding in blocks and \(3^{k-p}\)fractional factorials. This story will be very similar to what we did in the \(2^{k-p}\)designs previously. There is a direct analogue here that you will see.

From the previous section we had the following design, \(3^3\) treatments in 3 blocks with the \(ABC\) pseudo factor confounded with blocks, i.e.,

The three (color coded) blocks are determined by the levels of the \(ABC\) component of the three-way interaction which is confounded with blocks. If we only had one replicate of this design we would have 26 degrees of freedom. So, let's pretend that this design is Rep 1 and we will add Reps 2, 3, 4, just as we did with the two-factor case. This would result in a total of 12 blocks.

If we did this as our basic design and replicate it three more times our AOV would look like the following:

We would have Reps with \(3\ df\), blocks nested in Reps with \(2\ df\times 4\ Reps = 8\ df\), then we would have all of the unconfounded effects as shown above. The \(A\times B\times C\) would only have \(6\ df\) because one component (\(ABC\)) is confounded with blocks. Error is \(24\times 3=72\ df\) and our total is \((3^3\times 4)-1=107\ df\).

Now, we have written Blocks(Rep) with \(8\ df\) equivalently (in the blue font above) as \(ABC\) with \(2\ df\), and \(Rep\times ABC\) with \(6\ df\), but now we are considering the 0, 1, and 2 as levels of the ABC factor. In this case, \(ABC\) is one component of the interaction and still has meaning in terms of the levels of \(ABC\), just not very interesting since it is part of the three-way interaction. Had we confounded the main effect with blocks, we certainly would have wanted to analyze it, as seen above where the main effect was confounded with blocks. Then it had important meaning and you certainly would want to pull this out and be able to test it.

Now we have a total of \(3\times 4=12\) blocks and the \(11\ df\) among them are the interblock part of the analysis. If we averaged the nine observations in each block and got a single number, we could analyze those 12 numbers and this would be the inter-block part of this analysis.

How do we accomplish this in Minitab? If you have a set of data labeled by rep, blocks, and A, B, and C, then you would have everything you need and you can fit a general linear model:

\(Y = Rep\ Blocks(Rep)\ A\ |\ B\ |\ C\)

This would generate the analysis since \(A\ |\ B\ |\ C\) expands to all main effects and all interactions in GLM of Minitab.

In thinking about how this design should be implemented a good idea would be to followed this first Rep with a second Rep that confounds \(L_{AB^2 C}\), confound \(L_{ABC^2}\) in Rep three, and finally confound \(L_{AB^2 C^2}\) in fourth Rep. Now we could estimate all four components of the three-way interactions because in three of the Reps they would be unconfounded. There is no information available in the way we had approached it previously. There is lots of information available using this partial confounding strategy of the three-way interactions.

The whole point of looking at this structure is because sometimes we want to only conduct a fractional factorial. We sometimes can't afford 27 runs, certainly not 108 runs. Often we can only afford a fraction of the design. So, let's construct a \(3^{3-1}\) design which is a \(\frac{1}{3}\) fraction of a \(3^3\) design. In this case, \(N = 3^{3-1} = 3^2 = 9\), the total number of runs. This is a small, compact design. For the case where we use the \(L_{ABC}\) pseudo factor to create the design, we would use just one block of the design above, and below here is the alias structure:

Here A is confounded with part of the 3-way and part of the 2-way interaction, likewise for B and for C. This design only has 9 observations. It has A, B and C main effects estimable and if we look at the AOV we only have nine observations so we can only include the main effects:

Below is the \(3^3\) design where we partitioned the treatment combinations for one Rep of the experiment using the levels of \(L_{ABC}\). It is of interest to notice that a \(3^{3-1}\) fractional factorial design is also a design we previously discussed. Can you guess what it is?

If we look at the first light blue column, we can call A the row effect, B the column effect and C the Latin letters, or in this case 0, 1, 2. We would use this procedure to assign the treatments to the square. This is how we get a 3 × 3 Latin square. So, a one third fraction of a \(3^3\) design is the same as a 3 × 3 Latin square design that we saw earlier in this course. Watch the video below to see how this works.

It is important to see the connection here. We have three factors, A, B, C, and before when we talked about Latin squares, two of these were blocking factors and the third was the treatment factor. We could estimate all three main effects and we could not estimate any of the interactions. And now you should be able to see why. The interactions are all aliased with the main affects.

Let's look at another component \(L_{AB^2 C}\) of the three factor interaction: \(A\times B\times C\):

\([)L_{AB^2 C}=X_{1}+2X_{2}+X_{3}\ (mod 3)\)

We can now fill out the table by first plugging in the levels of \(X_{1}\), \(X_{2}\) and \(X_{3}\) from the levels of A, B and C to generate the column \(L_{AB^2 C}\). When you assign treatments to the level of \(L_{AB^2 C}=0\) you get an arrangement that follows (only the principle block filled in):

Then it also generates its own Latin square using the same process that we used above. You should be able to follow how this Latin square was assigned to the nine treatment combinations from the table above.

The benefit of doing this is to see that this one third fraction is also a Latin square. This is a Resolution III design, (it has a three letter word generator), and so it has the same properties that we saw at the two level designs, i.e. the main effects are clear of each other and estimable and aliased with higher order interactions including two-way. In fact, since the \(ABC\) and the \(AB^2 C\) are orthogonal to each other - they partition the \(A\times B\times C\) interaction - the two Latin squares we constructed are orthogonal Latin Squares.

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.

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.

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 \(2^2\).

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.

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.