Latin Square
The Latin square designs have a wide variety of applications in the experimental work. Besides agriculture, they are used in laboratory field, industrial, greenhouse, medical, educational, marketing, and sociological experimentation from comparing a group of varieties, from comparing worker differences in the laboratory to comparing weaving processes, and from testing tea to comparing patients in a hospital.
The merits of the Latin square design over other designs are as follows:
(i) With a two-way stratification or grouping, the Latin square controls more of the variation than completely randomized design or the randomized completely block design. The two-way elimination or variation frequently results in a small error mean square.
(ii) The analysis is simple. It is slightly more complicated than that for the randomized complete block design.
(iii) The analysis remains relatively simple even with missing data and the procedure are valuable for omitting one or more treatment of rows, or columns.
However, the number of treatment is fixed to the number of rowed and columns-except in some cases. For more than ten treatments, the Latin square is seldom used.
Steps in constructing a Latin square
The construction of Latin square includes the following steps:
1. At first compute the correction factor by squaring the grand total and dividing it by the number of observations.
2. Compute the total sum of squares, by adding the squares of the individual observations and subtracting the correction factor.
3. Compute the row sum of squares by adding the squares of the row sums, dividing it by the number of items in a row, and subtracting the correction factor.
4. Compute the column sum of squares by adding the squares of the column sums, dividing it by the number of items in a column, and subtracting the correction factor.
5. Compute the treatment sum of squares by summing the squares of the treatment sums dividing by the number of treatments, & subtracting the correction factor.
6. Compute the remainders sum of squares by subtracting the sum of 3, 4 and 5 from 2.
7. Enter these sums of squares in an analysis of the variance table and then compute the various mean squares
Analysis of variance table:
|
Source of variance
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S.S.
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Degree of freedom
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Mean of square
|
|
Rows
|
SSR
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(n - 1)
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MSR = SSR/n - 1
|
|
Columns
|
SSC
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(n - 1)
|
MSC = SSC/n -1
|
|
Treatments
|
SST
|
(n - 1)
|
MST = SSE/n - 1
|
|
Residual or error
|
SSE
|
(n - 2) (n - 20)
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MSE = SSE (n -1) (n -2)
|
|
total
|
TSS
|
n2 - 1
|
|
8. The last step is to calculate F by comparing the treatment mean square with the remainder square.