2^2 Factorial Designs

Two factors each with two levels.

Readings 6.1, Rmarkdown

1. A special case of general factorial design is the $2^K$ design. This arises when there are $K$ factors, but only 2 levels for each factor. We assume $n$ samples at each configuration of factor levels. With only two levels for each factor, these experiments aren’t useful for teasing out subtle variations across levels of a factor. However they are useful for determining which of a large number of factors might be worth investigating further (factor screening).

2. The simplest case of a $2^K$ design is when the number of factors $K = 2$. The experimental design can be represented by corners of a square.

Notation

3. It will be handy to define ways of indexing corners of the square. One approach is to write + or - for whether we are at a low or high level for that factor. Alternatively, we can represent the corner by all the letters that are at high levels.

A	B	label
-	-	(1)
+	-	a
-	+	b
+	+	ab

4. We abuse notation and write $a$ to represent the total of the response values at the corner +-, rather than just the index of that corner.

Estimating effects

5. Since there are only two levels for each factor, there are transparent formulas for estimating main and interaction effects.
   • The main effect for A summaries the average change in the response when A is activated. It is defined as $$\begin{align} A &= \frac{1}{2n}\left(\left(ab +a\right) - \left(b + (1)\right)\right) \end{align}$$ which is the average of the responses on the edge where A is active minus the average when A is inactive. The definition for B is analogous.
   • The interaction effect measures the degree to which the effect of A changes depending on whether or not B is active. It is defined as $$\begin{align} AB &= \frac{1}{2n}\left[\left(ab - b\right) - \left(a - \left(1\right)\right)\right] \end{align}$$ Notice that the role of A and B is symmetric — we could read the interaction as how the effect of B changes depending on whether A is active.

6. All these effect estimates can be summarized by our tabular notation,

label	effect A	effect B	effect AB
(1)	-	-	+
a	+	-	-
b	-	+	-
ab	+	+	+

Code Example

7. This code example shows how a plot can help detect an interaction effect. We read in a dataset on agricultural yield when varying two factors, A and B.

library(readr)
library(dplyr)
yield <- read_table2("https://uwmadison.box.com/shared/static/bfwd6us8xsii4uelzftg1azu2f7z77mk.txt")

8. We can either use facet_wrap in ggplot to split a plot of the data into separate panels or use the base R interaction.plot function. Both plots show how the effect of factor A varies as the value of factor B changes. In this case, the effect of factor A is slightly smaller (lower slope) when factor B isB inactive. In the next lecture though, we’ll see that this difference is not significant (it could very well happen by change under the model with no interaction term).

ggplot(yield) +
  geom_point(aes(A, Yield)) +
  facet_wrap(~B)

interaction.plot(yield$A, yield$B, yield$Yield)