Example 35.1: Analysis of Variance through PROC LATTICE

The LATTICE Procedure

Example 35.1: Analysis of Variance through PROC LATTICE

In the following example, from Cochran and Cox (1957, p. 406), the data are yields (Yield) in bushels per acre of 25 varieties (Treatmnt) 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;
     do Group = 1 to 2;
       do Block = 1 to 5;
         do Plot = 1 to 5;
            input Treatmnt Yield @@;
            output;
         end;
       end;
     end;
     drop Plot;
     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 Treatmnt;
   run;

   proc lattice data=Soy;
   run;

The results from these statements are shown in Output 35.1.1 and Output 35.1.2.

Output 35.1.1: Displayed Output from PROC PRINT

Treatmnt	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

Output 35.1.2: Displayed Output from PROC LATTICE

The Lattice Procedure

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

The Lattice Procedure

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 percent. Precision is gained using the lattice design via the recovery of intrablock 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).

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