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 print data=Soy; id Treatment; run;
proc lattice data=Soy; run;
The results from these statements are shown in Output 49.1.1 and Output 49.1.2.
Treatment | Group | Block | Yield |
---|---|---|---|
1 | 1 | 1 | 6 |
2 | 1 | 1 | 7 |
3 | 1 | 1 | 5 |
4 | 1 | 1 | 8 |
5 | 1 | 1 | 6 |
6 | 1 | 2 | 16 |
7 | 1 | 2 | 12 |
8 | 1 | 2 | 12 |
9 | 1 | 2 | 13 |
10 | 1 | 2 | 8 |
11 | 1 | 3 | 17 |
12 | 1 | 3 | 7 |
13 | 1 | 3 | 7 |
14 | 1 | 3 | 9 |
15 | 1 | 3 | 14 |
16 | 1 | 4 | 18 |
17 | 1 | 4 | 16 |
18 | 1 | 4 | 13 |
19 | 1 | 4 | 13 |
20 | 1 | 4 | 14 |
21 | 1 | 5 | 14 |
22 | 1 | 5 | 15 |
23 | 1 | 5 | 11 |
24 | 1 | 5 | 14 |
25 | 1 | 5 | 14 |
1 | 2 | 1 | 24 |
6 | 2 | 1 | 13 |
11 | 2 | 1 | 24 |
16 | 2 | 1 | 11 |
21 | 2 | 1 | 8 |
2 | 2 | 2 | 21 |
7 | 2 | 2 | 11 |
12 | 2 | 2 | 14 |
17 | 2 | 2 | 11 |
22 | 2 | 2 | 23 |
3 | 2 | 3 | 16 |
8 | 2 | 3 | 4 |
13 | 2 | 3 | 12 |
18 | 2 | 3 | 12 |
23 | 2 | 3 | 12 |
4 | 2 | 4 | 17 |
9 | 2 | 4 | 10 |
14 | 2 | 4 | 30 |
19 | 2 | 4 | 9 |
24 | 2 | 4 | 23 |
5 | 2 | 5 | 15 |
10 | 2 | 5 | 15 |
15 | 2 | 5 | 22 |
20 | 2 | 5 | 16 |
25 | 2 | 5 | 19 |
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 |
Additional Statistics for Yield | |
---|---|
Variance of Means in Same Block | 15.7915 |
Variance of Means in Different Bloc | 17.9280 |
Average of Variance | 17.2159 |
LSD at .01 Level | 12.1189 |
LSD at .05 Level | 8.7959 |
Efficiency Relative to RCBD | 174.34 |
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).