Variations of One Factor ANOVA Designs - Agricultural Statistical Data Analysis Using Stata - page 84

Table 5.1 Source of variation and degrees of freedom for two Latin square experiments
conducted simultaneously

SOURCE OF VARIATION DEGREES OF FREEDOM (DF) RESULTING DF

Experiments ( e ) e - 1 2 - 1 = 1

Replications within experiments e ( r - 1) 2(3 - 1) = 4

Hoppers within experiments e ( h - 1) 2(3 - 1) = 4

Treatments ( t ) t - 1 3 - 1 = 2

Error ( et - e - 1)( t - 1) ((2)(3) - 2 - 1)(3 - 1) = 6

Total et 2 - 1 2(3)2 - 1 = 17

This will result in the following output:

Number of obs = 18 R-squared = 0.8246
Root MSE = 2.4226 Adj R-squared = 0.5030

Source | Partial SS df MS F Prob > F
-------+----------------------------------------------------
Model | 165.543835 11 15.0494395 2.56 0.1294
|
exp| 15.5558083 1 15.5558083 2.65 0.1546
rep|exp| 114.810878 4 28.7027195 4.89 0.0426
hop|exp| 24.581224 4 6.145306 1.05 0.4561
trt| 10.5959244 2 5.29796218 0.90 0.4542
|
Residual | 35.2139522 6 5.86899204
---------+----------------------------------------------------
Total | 200.757787 17 11.8092816

The vertical bars ( | ) indicate the variable preceding it is nested
in the variable just after the bar. This is best illustrated with a table
showing the sources of the degrees of freedom (Table 5.1).
Wherever you see a source of variation nested within another vari-
able, such as [ e ( r - 1)], reverse their position and place the bar character
between as indicated above (rep|exp). In this experiment, there were
no treatment effects with an F value of 0.90 and Prob>F of 0.4542.
The replication within an experiment was significant, while the hop-
per within an experiment was not and we would expect the relative
efficiency to have been increased by replications, but not by the hopper
as a source of variation.

Balanced Incomplete Block Designs
Balanced incomplete block designs are marked by having less experi-
mental units than there are treatments to be tested within a block.