When \(p\) is small, the test from the previous notes can have low power. If we can collect more than \(p^2\) samples, we should. But how exactly should the samples be collected, and how is the replicated design analyzed?
The design decision is context dependent,
- We can use the same row and column levels.
- We can keep column levels, but have different row levels.
- Equivalently, can keep rows, but different columns.
- We can have different row and different column levels.
Note that in each case, we use a different Latin square in each replicate.
Figure 1: Design where all the row and column levels are reused.
Figure 2: Design where only the column levels are reused. Pay attention to the difference in row labels from one group to the next.
Figure 3: Design where neither the row nor column levels are reused.
- Fortunately, the analysis is conceptually unified. We continue to have row, column, and treatment mean squares. Then, we have to add in replicate mean squares, to track variation from replicate-to-replicate. The code in each setting is the same,
lm(Y ~ Replicate + W1 + W2 + X)Graceo-Latin Squares
We motivated Latin squares by asking how we can block 2 factors simultaneously. What if we have 3? We could go on forever…. That said, the transition from 2 to 3 is not hard
Introduce \(p\) greek letters to represent the third blocking factor. When \(p=5\), we would have \(\alpha, \beta, \gamma, \delta, \epsilon\), for example.
A Graeco-Latin square is like two Latin Squares overlaid on one another,
Figure 4: An example graeco-latin square.
with one additional requirement: each Latin and each Greek letter must only appear together once. This last requirement is called orthogonality.
- A hypothesis test can be defined, by using the decomposition, \[ SS_{\text{Total}} = SS_{\text{Rows}} + SS_{\text{Columns}} + SS_{\text{Treatments}} + SS_{\text{Greek}} + SS_{E} \] and noting that,
\[ \frac{MS_{\text{Treatment}}}{MS_{E}} = \frac{\frac{1}{p - 1}SS_{\text{Treatment}}}{\frac{1}{\left(p - 1\right)\left(p - 3\right)}SS_{E}} \sim F\left(p - 1, \left(p - 1\right)\left(p - 3\right)\right). \]