# Uses of semi-Latin squares

An (n × n)/k semi-Latin square is a set of n2k objects with three partitions on it: the partition R into n rows; the partition C into n columns; and the partition L into nk letters. There is also the partition B into the n2 cells, or blocks, defined by R and C.

The different ways of using the semi-Latin square as a design for an experiment depend on which of these partitions correspond to nuisance factors and which to treatment factors.

All examples below use the same semi-Latin square. Note that this square is deliberately chosen to emphasise that a semi-Latin square is not the same thing as a set of mutually orthogonal Latin squares. This implies that the design given here is not best possible if B has any effect.

## Design Use 1

The nuisance factors are R and C; the treatment factor is L.

If B has no effect then all semi-Latin squares of the appropriate size are equally good. Otherwise the efficiency of the design depends on the relationship between B and L.

### Examples

1. Ten new vacuum cleaners are available for comparison during a five-week period. Five housewives volunteer to test them, each using two of the vacuum cleaners in her house each week.

Week Housewife
1 2 3 4 5
a A J G E I C F D HB
b B G A I E F H C J D
c C F B D A H J E I G
d I D H F B J A G C E
e H E C J D G I B A F

Before using such a design, randomly permute the rows (the way that the rows are assigned to actual weeks) and randomly permute the columns (the way that the columns are assigned to actual housewives). Where it makes sense (as in the following example), also randomly permute the order of the two letters in each cell.

2. Ten treatments are to be applied to sugar-beet, which is grown in a 5 × 10 rectangular array of plots. Each plot is a single long North-South row of sugar beet, so the 10 plots in a single row of the rectangle are close to each other and these rows must be regarded as a nuisance factor. The beet is sown from five seed-drills on an arm which protrudes from the right of the tractor. The tractor drives Northwards up the left-hand side of the array, sowing seed in the first five columns, then turns round and drives Southwards down the right-hand side of the array, sowing seed in the last five columns. Thus the first and last column are sown by the same drill, and drills form a second nuisance factor.

row drill
1 2 3 4 5 5 4 3 2 1
a A E C D B H F I G J
b B A E C D J H F I G
c F D H J I G E A B C
d I H J G E C A B F D
e H C G B A F I D J E

## Design Use 2

The nuisance factors are R and L; the treatment factor is C.

Now we simply have an orthogonal row-column design and B is assumed to have no effect.

### Example

Two vacuum cleaners of each of five new models are available for comparison during a five-week period. Ten housewives volunteer to test them, each using one of the vacuum cleaners in her house each week.

week housewife
A B C D E F G H I J
a 1 5 3 4 2 4 2 5 3 1
b 2 1 4 5 3 3 1 4 2 5
c 3 2 1 2 4 1 5 3 5 4
d 4 3 5 1 5 2 4 2 1 3
e 5 4 2 3 1 5 3 1 4 2

Before using such a design, randomly permute the rows (the way that the rows are assigned to actual weeks) and randomly permute the columns (the way that the columns are assigned to actual housewives).

## Design Use 3

The nuisance factor is C; the treatment factors are R and L. We have to assume that there is no interaction between R and L, because part of this interaction is completely confounded with C.

### Example

Ten varieties of tomato (A, B, ..., J) are to be compared in combination with five watering regimes (a, ..., e). There are five chambers available in the glasshouse, each with room for ten gro-bags.

chamber treatment combination
1 Aa Cc Fc Ja Bb Id He Gb Dd Ee
2 Bc Fd Ga Ab Je Ib Ea Dc Ce Hd
3 Hc Ia Jd Ca De Ac Bd Eb Ge Fb
4 Hb Gd Be Ie Cb Da Jc Ec Fa Ad
5 Ic Db Fe Gc Ha Ed Ae Cd Jb Ba

Before using such a design, randomly permute the way that the chambers are numbered and randomly permute the order of the treatment combinations within each chamber, doing this separately and independently for each chamber.

## Design Use 4

The nuisance factor is L; the treatment factors are R and C.

If there is no interaction between R and C then all semi-Latin squares are equally good. But we may assume that this interaction is non-zero, in which case the efficiency of the design depends upon the relationship between B and L.

### Example

Five varieties of tomato (1, ..., 5) are to be compared in combination with five watering regimes (a, ..., e). There are ten chambers available in the glasshouse, each with room for five gro-bags.

chamber treatment combination
A 1a 2b 3c 4d 5e
B 1b 4e 5a 3d 2c
C 2e 3a 5d 1c 4b
D 2c 4a 1d 5b 3e
E 1e 3b 5d 2a 4c
F 4a 5e 1c 2d 3b
G 5c 2a 4d 3e 1b
H 2d 3c 4b 1e 5a
I 5c 3a 1d 2b 4e
J 5b 2e 4c 1a 3d

Before using such a design, randomly permute the way that the chambers are labelled and randomly permute the order of the treatment combinations within each chamber, doing this separately and independently for each chamber.

## Design Use 5

There are no nuisance factors; the treatment factors are R, C and L.

We have to assume that there is no interaction between R and L or between C and L. If there is no interaction between R and C then all semi-Latin squares are equally good. But we may assume that this interaction is non-zero, in which case the efficiency of the design depends upon the relationship between B and L.

### Example

In bread-making, ten types of flour (A, ..., J) are to be compared in combination with five speeds of kneading (a, ..., e) and five baking temperatures (1, ..., 5). Fifty ovens are available for baking at a single time.

The fifty treatment combinations are Aa1, Ja1, ..., Fe5.

Before such a design is used, the order of all the treatment combinations must be completely randomized.

Page maintained by R. A. Bailey

Modified 24/6/00