In the following example, from Cochran and Cox (1957, p. 406), the data are yields (Yield
) in bushels per acre of 25 varieties (Treatment
) of soybeans. The data are collected in two replications (Group
) of 25 varieties in five blocks (Block
) containing five varieties each. This is an example of a partially balanced square lattice design.
data Soy(drop=plot); do Group = 1 to 2; do Block = 1 to 5; do Plot = 1 to 5; input Treatment Yield @@; output; end; end; end; datalines; 1 6 2 7 3 5 4 8 5 6 6 16 7 12 8 12 9 13 10 8 11 17 12 7 13 7 14 9 15 14 16 18 17 16 18 13 19 13 20 14 21 14 22 15 23 11 24 14 25 14 1 24 6 13 11 24 16 11 21 8 2 21 7 11 12 14 17 11 22 23 3 16 8 4 13 12 18 12 23 12 4 17 9 10 14 30 19 9 24 23 5 15 10 15 15 22 20 16 25 19 ;
proc lattice data=Soy; run;
The results from these statements are shown in Output 56.1.1.
Output 56.1.1: Displayed Output from PROC LATTICE
Analysis of Variance for Yield | |||
---|---|---|---|
Source | DF | Sum of Squares | Mean Square |
Replications | 1 | 212.18 | 212.18 |
Blocks within Replications (Adj.) | 8 | 501.84 | 62.7300 |
Component B | 8 | 501.84 | 62.7300 |
Treatments (Unadj.) | 24 | 559.28 | 23.3033 |
Intra Block Error | 16 | 218.48 | 13.6550 |
Randomized Complete Block Error | 24 | 720.32 | 30.0133 |
Total | 49 | 1491.78 | 30.4445 |
Adjusted Treatment Means for Yield |
|
---|---|
Treatment | Mean |
1 | 19.0681 |
2 | 16.9728 |
3 | 14.6463 |
4 | 14.7687 |
5 | 12.8470 |
6 | 13.1701 |
7 | 9.0748 |
8 | 6.7483 |
9 | 8.3707 |
10 | 8.4489 |
11 | 23.5511 |
12 | 12.4558 |
13 | 12.6293 |
14 | 20.7517 |
15 | 19.3299 |
16 | 12.6224 |
17 | 10.5272 |
18 | 10.7007 |
19 | 7.3231 |
20 | 11.4013 |
21 | 11.6259 |
22 | 18.5306 |
23 | 12.2041 |
24 | 17.3265 |
25 | 15.4048 |
The efficiency of the experiment relative to a randomized complete block design is 174.34%. Precision is gained using the lattice design via the recovery of intra-block error information, enabling more accurate estimates of the treatment effects. Variety 8 of soybean had the lowest adjusted treatment mean (6.7483 bushels per acre), while variety 11 of soybean had the highest adjusted treatment mean (23.5511 bushels per acre).