Hyper-Graeco-Latin square design
Encyclopedia
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.
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):
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):
Design of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
, hyper-Graeco-Latin squares are efficient designs to study the effect of one primary (treatment) factor in the presence of 4 blocking
Blocking (statistics)
In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups that are similar to one another. For example, an experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male...
(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.
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 |
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 | E13 | A24 |
| E42 | A53 | B14 | C25 | D31 |
| C54 | D15 | E21 | A32 | B43 |