Partially Balanced Lattice Designs
The partially balanced lattice design is similar to the balanced lattice design but allows for more flexibility in the number of replications. While it requires the number of treatments to be a perfect square and the block size to equal the square root of the treatment number, the number of replications is not fixed. Any number of replications can be used. With two replications, it is called a simple lattice; with three, a triple lattice; with four, a quadruple lattice, and so on.
Design without Repetition
To illustrate the analysis of variance for a partially balanced lattice design without repetition, we use a 9 × 9 triple lattice design that evaluates the performance of 81 rice varieties. The yield data is rearranged according to the plan shown below.
Replication 1:
Blocks Treatments
B1 1 2 3 4 5 6 7 8 9
B2 10 11 12 13 14 15 16 17 18
B3 19 20 21 22 23 24 25 26 27
B4 28 29 30 31 32 33 34 35 36
B5 37 38 39 40 41 42 43 44 45
B6 46 47 48 49 50 51 52 53 54
B7 55 56 57 58 59 60 61 62 63
B8 64 65 66 67 68 69 70 71 72
B9 73 74 75 76 77 78 79 80 81
Replication 2:
Blocks Treatments
B1 1 10 19 28 37 46 55 64 73
B2 2 11 20 29 38 47 56 65 74
B3 3 12 21 30 39 48 57 66 75
B4 4 13 22 31 40 49 58 67 76
B5 5 14 23 32 41 50 59 68 77
B6 6 15 24 33 42 51 60 69 78
B7 7 16 25 34 43 52 61 70 79
B8 8 17 26 35 44 53 62 71 80
B9 9 18 27 36 45 54 63 72 81
Replication 3:
Blocks Treatments
B1 1 20 12 58 77 70 34 53 1
B2 10 2 21 67 59 78 43 35 10
B3 19 11 3 76 68 60 52 44 19
B4 28 47 39 4 23 15 61 80 28
B5 37 29 48 13 5 24 70 62 37
B6 46 38 30 22 14 6 79 71 46
B7 55 74 66 31 50 42 7 26 55
B8 64 56 75 40 32 51 16 8 64
B9 73 65 57 49 41 33 25 17 73
The analysis will be demonstrated with the following example:
Example: Grain Yield Data from a Trial of 81 Upland Rice Varieties
We will use data from a 9 × 9 Lattice Design to evaluate the yield performance of 81 rice varieties. Below is a portion of the data used for analysis:
Block No. Grain Yield t/ha
Replication I
1 (1) (2) (3) (4) (5) (6) (7) (8) (9)
2.7 1.6 4.45 2.91 2.78 3.32 1.7 4.72 4.79
2 (10) (11) (12) (13) (14) (15) (16) (17) (18)
4.2 5.22 3.96 1.51 3.48 4.69 1.57 2.61 3.16
3 (19) (20) (21) (22) (23) (24) (25) (26) (27)
4.63 3.33 6.31 6.08 1.86 4.1 5.72 5.87 4.2
4 (28) (29) (30) (31) (32) (33) (34) (35) (36)
3.74 3.05 5.16 4.76 3.75 3.66 4.52 4.64 5.36
5 (37) (38) (39) (40) (41) (42) (43) (44) (45)
4.76 4.43 5.36 4.73 5.3 3.93 3.37 3.74 4.06
6 (46) (47) (48) (49) (50) (51) (52) (53) (54)
3.45 2.56 2.39 2.3 3.54 3.66 1.2 3.34 4.04
7 (55) (56) (57) (58) (59) (60) (61) (62) (63)
3.99 4.48 2.69 3.95 2.59 3.99 4.37 4.24 3.7
8 (64) (65) (66) (67) (68) (69) (70) (71) (72)
5.29 3.58 2.14 5.54 5.14 5.73 3.38 3.63 5.08
9 (73) (74) (75) (76) (77) (78) (79) (80) (81)
3.76 6.45 3.96 3.64 4.42 6.57 6.39 3.39 4.89
Replication II
1 (1) (10) (19) (28) (37) (46) (55) (64) (73)
3.06 2.08 2.95 3.75 4.08 3.88 2.14 3.68 2.85
2 (2) (11) (20) (29) (38) (47) (56) (65) (74)
1.61 5.3 2.75 4.06 3.89 2.6 4.19 3.14 4.82
3 (3) (12) (21) (30) (39) (48) (57) (66) (75)
4.19 3.33 4.67 4.99 4.58 3.17 2.69 2.57 3.82
4 (4) (13) (22) (31) (40) (49) (58) (67) (76)
2.99 2.5 4.87 3.71 4.85 2.87 3.79 5.28 3.32
5 (5) (14) (23) (32) (41) (50) (59) (68) (77)
3.81 3.48 1.87 4.34 4.36 3.24 3.62 4.49 3.62
6 (6) (15) (24) (33) (42) (51) (60) (69) (78)
3.34 3.3 3.68 3.84 4.25 3.9 3.64 5.09 6.1
7 (7) (16) (25) (34) (43) (52) (61) (70) (79)
2.98 2.69 5.55 3.52 4.03 1.2 4.36 3.18 6.77
8 (8) (17) (26) (35) (44) (53) (62) (71) (80)
4.2 2.69 5.14 4.32 3.47 3.41 3.74 3.67 2.27
9 (9) (18) (27) (36) (45) (54) (63) (72) (81)
4.75 2.59 3.94 4.51 3.1 3.59 2.7 4.4 4.86
Replication III
1 (1) (12) (20) (34) (45) (53) (58) (70) (77)
3.52 2.18 3.5 3.3 3.88 2.45 3.75 4.45 4.14
2 (2) (10) (21) (35) (43) (54) (59) (67) (78)
0.79 3.58 4.83 3.63 3.02 4.2 3.59 5.06 6.51
3 (3) (11) (19) (36) (44) (52) (60) (68) (76)
4.69 5.33 4.43 5.31 4.13 1.98 4.66 4.5 4.5
4 (4) (15) (23) (28) (39) (47) (61) (72) (80)
3.06 4.3 2.02 3.57 5.8 2.58 4.27 4.84 2.74
5 (5) (13) (24) (29) (37) (48) (62) (70) (81)
3.79 0.88 3.4 4.92 2.12 1.89 3.73 3.51 3.5
6 (6) (14) (22) (30) (38) (46) (63) (71) (79)
3.34 3.94 5.72 5.34 4.47 4.18 2.7 3.96 3.48
7 (7) (18) (26) (31) (42) (50) (55) (66) (74)
2.35 2.87 5.5 2.72 4.2 2.87 2.99 1.62 5.33
8 (8) (16) (27) (32) (40) (51) (56) (64) (75)
4.51 1.26 4.2 3.19 4.76 3.35 3.61 4.52 3.38
9 (9) (17) (25) (33) (41) (49) (57) (65) (73)
4.21 3.17 5.03 3.34 5.31 3.05 3.19 2.63 4.06
Arrangement of Data
First, arrange the data in an Excel sheet. The first column contains the replication number, the second column contains the block code, and the third column contains the treatment code. The observed values for various characters (such as plant height, number of tillers, and yield) are entered in the subsequent columns. The arrangement of the data for the above example will look like this in Excel:
Fig 1: Arrangement of Data in Excel Sheet
The final data for 9 X 9 triple lattice having 81 treatment, 3 replications and 1 character should look like
1 1 1 2.7 1 1 2 1.6 1 1 3 4.45 1 1 4 2.91 1 1 5 2.78 1 1 6 3.32 1 1 7 1.7 1 1 8 4.72 1 1 9 4.79 1 2 10 4.2 1 2 11 5.22 1 2 12 3.96 1 2 13 1.51 1 2 14 3.48 1 2 15 4.69 1 2 16 1.57 1 2 17 2.61 1 2 18 3.16 1 3 19 4.63 1 3 20 3.33 1 3 21 6.31 1 3 22 6.08 1 3 23 1.86 1 3 24 4.1 1 3 25 5.72 1 3 26 5.87 1 3 27 4.2 1 4 28 3.74 1 4 29 3.05 1 4 30 5.16 1 4 31 4.76 1 4 32 3.75 1 4 33 3.66 1 4 34 4.52 1 4 35 4.64 1 4 36 5.36 1 5 37 4.76 1 5 38 4.43 1 5 39 5.36 1 5 40 4.73 1 5 41 5.3 1 5 42 3.93 1 5 43 3.37 1 5 44 3.74 1 5 45 4.06 1 6 46 3.45 1 6 47 2.56 1 6 48 2.39 1 6 49 2.3 1 6 50 3.54 1 6 51 3.66 1 6 52 1.2 1 6 53 3.34 1 6 54 4.04 1 7 55 3.99 1 7 56 4.48 1 7 57 2.69 1 7 58 3.95 1 7 59 2.59 1 7 60 3.99 1 7 61 4.37 1 7 62 4.24 1 7 63 3.7 1 8 64 5.29 1 8 65 3.58 1 8 66 2.14 1 8 67 5.54 1 8 68 5.14 1 8 69 5.73 1 8 70 3.38 1 8 71 3.63 1 8 72 5.08 1 9 73 3.76 1 9 74 6.45 1 9 75 3.96 1 9 76 3.64 1 9 77 4.42 1 9 78 6.57 1 9 79 6.39 1 9 80 3.39 1 9 81 4.89 2 1 1 3.06 2 1 10 2.08 2 1 19 2.95 2 1 28 3.75 2 1 37 4.08 2 1 46 3.88 2 1 55 2.14 2 1 64 3.68 2 1 73 2.85 2 2 2 1.61 2 2 11 5.3 2 2 20 2.75 2 2 29 4.06 2 2 38 3.89 2 2 47 2.6 2 2 56 4.19 2 2 65 3.14 2 2 74 4.82 2 3 3 4.19 2 3 12 3.33 2 3 21 4.67 2 3 30 4.99 2 3 39 4.58 2 3 48 3.17 2 3 57 2.69 2 3 66 2.57 2 3 75 3.82 2 4 4 2.99 2 4 13 2.5 2 4 22 4.87 2 4 31 3.71 2 4 40 4.85 2 4 49 2.87 2 4 58 3.79 2 4 67 5.28 2 4 76 3.32 2 5 5 3.81 2 5 14 3.48 2 5 23 1.87 2 5 32 4.34 2 5 41 4.36 2 5 50 3.24 2 5 59 3.62 2 5 68 4.49 2 5 77 3.62 2 6 6 3.34 2 6 15 3.3 2 6 24 3.68 2 6 33 3.84 2 6 42 4.25 2 6 51 3.9 2 6 60 3.64 2 6 69 5.09 2 6 78 6.1 2 7 7 2.98 2 7 16 2.69 2 7 25 5.55 2 7 34 3.52 2 7 43 4.03 2 7 52 1.2 2 7 61 4.36 2 7 70 3.18 2 7 79 6.77 2 8 8 4.2 2 8 17 2.69 2 8 26 5.14 2 8 35 4.32 2 8 44 3.47 2 8 53 3.41 2 8 62 3.74 2 8 71 3.67 2 8 80 2.27 2 9 9 4.75 2 9 18 2.59 2 9 27 3.94 2 9 36 4.51 2 9 45 3.1 2 9 54 3.59 2 9 63 2.7 2 9 72 4.4 2 9 81 4.86 3 1 1 3.52 3 1 12 2.18 3 1 20 3.5 3 1 34 3.3 3 1 45 3.88 3 1 53 2.45 3 1 58 3.75 3 1 69 4.45 3 1 77 4.14 3 2 2 0.79 3 2 10 3.58 3 2 21 4.83 3 2 35 3.63 3 2 43 3.02 3 2 54 4.2 3 2 59 3.59 3 2 67 5.06 3 2 78 6.51 3 3 3 4.69 3 3 11 5.33 3 3 19 4.43 3 3 36 5.31 3 3 44 4.13 3 3 52 1.98 3 3 60 4.66 3 3 68 4.5 3 3 76 4.5 3 4 4 3.06 3 4 15 4.3 3 4 23 2.02 3 4 28 3.57 3 4 39 5.8 3 4 47 2.58 3 4 61 4.27 3 4 72 4.84 3 4 80 2.74 3 5 5 3.79 3 5 13 0.88 3 5 24 3.4 3 5 29 4.92 3 5 37 2.12 3 5 48 1.89 3 5 62 3.73 3 5 70 3.51 3 5 81 3.5 3 6 6 3.34 3 6 14 3.94 3 6 22 5.72 3 6 30 5.34 3 6 38 4.47 3 6 46 4.18 3 6 63 2.7 3 6 71 3.96 3 6 79 3.48 3 7 7 2.35 3 7 18 2.87 3 7 26 5.5 3 7 31 2.72 3 7 42 4.2 3 7 50 2.87 3 7 55 2.99 3 7 66 1.62 3 7 74 5.33 3 8 8 4.51 3 8 16 1.26 3 8 27 4.2 3 8 32 3.19 3 8 40 4.76 3 8 51 3.35 3 8 56 3.61 3 8 64 4.52 3 8 75 3.38 3 9 9 4.21 3 9 17 3.17 3 9 25 5.03 3 9 33 3.34 3 9 41 5.31 3 9 49 3.05 3 9 57 3.19 3 9 65 2.63 3 9 73 4.06
Next, copy only the numeric data (excluding column names) and paste it into the text area provided on the left side of this webpage, as shown in the figure below:
Fig 2: Data Entry Interface for Partially Balanced Lattice Design
Entering Design Parameters
After entering the data, specify the name of the character in the text area on the right side of the data entry area. The character name should not contain spaces and should be entered on separate lines. Once you have entered the character names, press the submit button. A new page will appear, as shown below:
Fig 3: Form for Entering Parameters of the Design
In this form, you will be asked to specify the total number of characters to be analyzed, the number of treatments (which must be a perfect square), and the number of replications. Enter these values in the appropriate text boxes and then press the 'Analyze' button. The results of the analysis will be displayed on a separate page.