Projection and Blocking in 2^{K - p} Designs

Special considerations designing fractional factorials.

Readings 8.3, 8.4, Rmarkdown

library(ggplot2)
theme424 <- theme_minimal() + 
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theme_set(theme424)

include_graphics("https://uwmadison.box.com/shared/static/b94hfeyugeun1xywlm2ccls9am5l38xg.jpg")

Figure 1: Our earlier picture of projection from 3 to 2 factors in a complete factorial design.

1. To be fluent in working with fractional factorial designs, it’s worth knowing how to project and block them. They provide useful alternative perspectives of the designs we have already constructed.

Projection

2. Projection makes it possible to turn a fractional factorial screening experiment into a full factorial experiment. Effectively, we delete the columns of our design matrix involving the factors outside the ones we want to project down to. Depending on what design we project, a \(2^{K - p}\) design will become either (1) a full factorial design or (2) a \(2^{K - r}\) fractional factorial design, where \(r < p\).

3. Beware that conclusions will be somewhat tentative, because effects in the full factorial could be aliased with high-order effects from factors that have been projected out.

4. Warm-up: (Projecting a \(2 ^ {4 - 1}\) Design). Consider the \(2 ^ {4 - 1}\) design with defining relation \(I = ABCD\).

A	B	C	D
-	-	-	-
+	-	-	+
-	+	-	+
-	-	+	+
+	+	-	-
+	-	+	-
-	+	+	-
+	+	+	+

What happens when we project out A?

B	C	D
-	-	-
-	-	+
+	-	+
-	+	+
+	-	-
-	+	-
+	+	-
+	+	+

This design includes all corners of the cube B-C-D, so it’s a full \(2^{3}\) design.

5. Example (Different projections of \(2 ^ {6 - 2}\) design). Let’s consider the example from the previous notes.

A	B	C	D	E = ABC	F = BCD
-	-	-	-	-	-
+	-	-	-	+	-
-	+	-	-	+	+
-	-	+	-	+	+
-	-	-	+	-	+
+	+	-	-	-	+
+	-	+	-	-	+
+	-	-	+	+	+
-	+	+	-	-	-
-	+	-	+	+	-
-	-	+	+	+	-
+	+	+	-	+	-
+	+	-	+	-	-
+	-	+	+	-	-
-	+	+	+	-	+
+	+	+	+	+	+

This study examined 6 factors over 16 runs. Its complete defining relations are \(I = ABCE = BCDF = ADEF\). What happens when we project out \(A\)?

B	C	D	E	F
-	-	-	-	-
-	-	-	+	-
+	-	-	+	+
-	+	-	+	+
-	-	+	-	+
+	-	-	-	+
-	+	-	-	+
-	-	+	+	+
+	+	-	-	-
+	-	+	+	-
-	+	+	+	-
+	+	-	+	-
+	-	+	-	-
-	+	+	-	-
+	+	+	-	+
+	+	+	+	+

6. Each configuration appears once. Since we have 16 rows still, this is evidently a \(2^{5 - 1}\) fractional factorial design. What happens if we project out \(E\) and \(F\) instead?

• This is exactly the original “basic design.” I.e., the the full \(2 ^ 4\) factorial before using the generators.

7. We could try to project out two factors that don’t have such an obvious answer. So let’s project out \(A\) and \(B\).

• It turns out we still get a \(2 ^ 4\) full factorial, but now on \(C\) through \(F\).

C	D	E	F
-	-	-	-
-	-	+	-
-	-	+	+
+	-	+	+
-	+	-	+
-	-	-	+
+	-	-	+
-	+	+	+
+	-	-	-
-	+	+	-
+	+	+	-
+	-	+	-
-	+	-	-
+	+	-	-
+	+	-	+
+	+	+	+

8. We might start suspecting that whenever we project out two variables, we will end up with a full \(2^{4}\) factorial. But this is not the case!

• Consider projecting out \(A\) and \(E\).
• Now we have a \(2 ^ {4 - 1}\) fractional factorial with 2 replicates.
• Only half of the corners are present, but each one that appears is present twice.

B	C	D	F
-	-	-	-
-	-	-	-
+	-	-	+
-	+	-	+
-	-	+	+
+	-	-	+
-	+	-	+
-	-	+	+
+	+	-	-
+	-	+	-
-	+	+	-
+	+	-	-
+	-	+	-
-	+	+	-
+	+	+	+
+	+	+	+

9. This ends up being a general rule. If none of the defining words contain all the factors that you’re projecting out, then you’re left with a full factorial design; otherwise you get a fractional factorial with replicates.

• E.g., \(AE\) is part of the first defining relation, so we got a fractional factorial.

10. Exercise: Try projecting onto \(A, B, C\). Is it a fractional or full factorial? How many replicates each? Could you have determined this, just by looking at the defining relations?

11. A design is said to have projectivity \(v\) if collapsing to any subset of \(v\) factors will result in a full factorial design. The example above has projectivity 3.

Blocking

12. Let’s say we’ve subsetted to a set of runs corresponding to a fractional factorial design. But, we still have to break the runs into batches, we have too many runs to be able to reasonably run them in one block. How can we appropriately block a fractional factorial design?

13. The typical solution is to choose a high-order effect and alias it with the batches. This divides the runs into two blocks, those where the effect is + and those where it is -.

14. We should ensure that the alias group doesn’t contain any low-order effects of interest (e.g., we don’t want the blocks to be confounded with any main effects). We effectively sacrifice the alias group that is confounded with the block.

15. Example: For our \(2 ^{6 - 2}\) design above, we can confound \(ACD\) with the block, to avoid aliasing any main effects with the block effect. This let us divide our runs into two blocks of size 8.

A	B	C	D	E	F	Block = ACD
-	-	-	-	-	-	- (B1)
+	-	-	-	+	-	+ (B2)
-	+	-	-	+	+	- (B1)
-	-	+	-	+	+	+ (B2)
-	-	-	+	-	+	+ (B2)
+	+	-	-	-	+	+ (B2)
+	-	+	-	-	+	- (B1)
+	-	-	+	+	+	- (B1)
-	+	+	-	-	-	+ (B2)
-	+	-	+	+	-	+ (B2)
-	-	+	+	+	-	- (B1)
+	+	+	-	+	-	- (B1)
+	+	-	+	-	-	- (B1)
+	-	+	+	-	-	+ (B2)
-	+	+	+	-	+	- (B1)
+	+	+	+	+	+	+ (B2)