Hyper-Graeco-Latin square design
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In the design of experiments, hyper-Graeco-Latin squares are efficient designs to study the effect of one primary (treatment) factor in the presence of 4 blocking (nuisance) factors. They are restricted, however, to the case in which all the factors have the same number of levels.
Designs for 4- and 5-level factors are given in this article. These designs show what the treatment combinations should be for each run. When using any of these designs, the treatment units and trial order should be randomized as much as the design allows. For example, one recommendation is that a hyper-Graeco-Latin square design be randomly selected from those available, then randomize the run order.
There are no 3-level factor hyper-Graeco Latin square designs.
Contents |
[edit] Design for 4-level factors
For 4-level factors, the design consists of sixteen runs:
| X1: row blocking factor |
X2: column blocking factor |
X3: blocking factor |
X4: blocking factor |
X5: treatment factor |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 2 | 2 |
| 1 | 3 | 3 | 3 | 3 |
| 1 | 4 | 4 | 4 | 4 |
| 2 | 1 | 4 | 2 | 3 |
| 2 | 2 | 3 | 1 | 4 |
| 2 | 3 | 2 | 4 | 1 |
| 2 | 4 | 1 | 3 | 2 |
| 3 | 1 | 2 | 3 | 4 |
| 3 | 2 | 1 | 4 | 3 |
| 3 | 3 | 4 | 1 | 2 |
| 3 | 4 | 3 | 2 | 1 |
| 4 | 1 | 3 | 4 | 2 |
| 4 | 2 | 4 | 3 | 1 |
| 4 | 3 | 1 | 2 | 4 |
| 4 | 4 | 2 | 1 | 3 |
This can alternatively be represented as (A, B, C, and D represent the treatment factor and 1, 2, 3, and 4 represent the blocking factors):
| A11 | B22 | C33 | D44 |
| C42 | D31 | A24 | B13 |
| D23 | C14 | B41 | A32 |
| B34 | A43 | D12 | C21 |
[edit] Design for 5-level factors
For 5-level factors, the design consists of twenty-five runs:
| X1: row blocking factor |
X2: column blocking factor |
X3: blocking factor |
X4: blocking factor |
X5: treatment factor |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 2 | 2 |
| 1 | 3 | 3 | 3 | 3 |
| 1 | 4 | 4 | 4 | 4 |
| 1 | 5 | 5 | 5 | 5 |
| 2 | 1 | 2 | 3 | 4 |
| 2 | 2 | 3 | 4 | 5 |
| 2 | 3 | 4 | 5 | 1 |
| 2 | 4 | 5 | 1 | 2 |
| 2 | 5 | 1 | 2 | 3 |
| 3 | 1 | 3 | 5 | 2 |
| 3 | 2 | 4 | 1 | 3 |
| 3 | 3 | 5 | 2 | 4 |
| 3 | 4 | 1 | 3 | 5 |
| 3 | 5 | 2 | 4 | 1 |
| 4 | 1 | 4 | 2 | 5 |
| 4 | 2 | 5 | 3 | 1 |
| 4 | 3 | 1 | 4 | 2 |
| 4 | 4 | 2 | 5 | 3 |
| 4 | 5 | 3 | 1 | 4 |
| 5 | 1 | 5 | 4 | 3 |
| 5 | 2 | 1 | 5 | 4 |
| 5 | 3 | 2 | 1 | 5 |
| 5 | 4 | 3 | 2 | 1 |
| 5 | 5 | 4 | 3 | 2 |
This can alternatively be represented as (A, B, C, D, and E represent the treatment factor and 1, 2, 3, 4, and 5 represent the blocking factors):
| A11 | B22 | C33 | D44 | E55 |
| D23 | E34 | A45 | B51 | C12 |
| B35 | C41 | D52 | E31 | A24 |
| E42 | A53 | B14 | C25 | D31 |
| C54 | D15 | E21 | A32 | B43 |
[edit] See also
[edit] External links
[edit] References
- (1978) Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley and Sons.
This article incorporates text from a public domain publication of the National Institute of Standards and Technology, a U.S. government agency.
