APPENDIX 17
ANALYSIS OF VARIANCE FOR A RHIZOBIUM STRAIN SELECTION EXPERIMENT
The data in Table A.12 presents the dry weight (g) of plant tops
from a strain selection experiment for soybean (G. max var.
Jupiter).
The experiment was a Randomized Complete Block Design
(RCBD), with 3 blocks and 16 treatments (14 inoculated + 2
controls).
block.
Each treatment was replicated once within each
Each treatment-plot was a Leonard jar unit with two
soybean plants.
The plant tops were harvested at 32 days and
oven dried at 70C.
The strains of Bradyrhizobium japonicum have
been ranked according to dry weight.
Summary of calculations for the analysis of variance for the
strain selection experiment.
No. of treatments = k = 16
No. of blocks = b = 3
No. of replicates per treatment per block = n = 1
Calculate the Grand Total (GT) by adding up all the treatment
totals:
GT = T1 + T2
T3 ----- + Tk
= 31.09 + 28.85 + 28.04 ----- + 20.07
= 344.83
Table A.12.
Data from a strain selection experiment for soybean.
Dry Weight of Plant Tops (g)
BLOCKS
TREATMENTS
Treatment
Treatment
B1
B2
B3
Total (T)
Means (x)
TAL 102
9.66
10.60
10.83
31.09
10.36
TAL 379
9.36
9.00
10.49
28.85
9.62
TAL 206
8.41
9.44
10.19
28.04
9.35
TAL 435
8.61
9.23
8.22
26.06
8.69
TAL 411
9.20
8.19
8.46
25.85
8.62
Allen 527
8.11
8.82
8.62
25.55
8.52
TAL 211
8.83
6.32
9.14
24.29
8.10
TAL 487
6.27
8.67
8.35
23.29
7.76
CB 1795
6.79
8.17
5.70
20.66
6.89
TAL 650
6.95
5.83
6.83
19.61
6.54
TAL 649
6.55
4.82
8.10
19.47
6.49
TAL 860
6.00
4.83
6.54
17.37
5.79
TAL 183
6.11
3.46
5.51
15.08
5.03
TAL 378
5.39
4.46
5.07
14.92
4.97
Control*
1.53
1.30
1.80
4.63
1.54
Control**
8.41
7.83
5.83
20.07
6.36
114.18
110.97
119.68
344.83
116.36
* Uninoculated
** 70 ppm N
Calculate the Grand Mean (X) by adding up all the treatment
means:
X = x1 + x2 + x3 ----- xk
= 10.36 + 9.62 + 9.35 ----- + 6.36
= 116.13
Calculate the Correction Factor (CF)
CF =
=
(GT)2
bkn
=
(344.83)2
3 x 16 x 1
2477.2444
Calculate the total sum of Squares (SS)
SS =
Σx2 - CF
=
9.662 + 10.602 + 10.832 ----- + 5.832 - 2477.244
=
247.8507
Calculate the Treatment Sum of Squares (SST)
SST = ∑T2 – CF
bn
= 31.092 + 28.852...+ 4.632 + 20.072 – 2477.2444
3 X 1
= 217.4785
Calculate the Block Sum of Squares (SSB)
SSB = ∑B2 – CF
kn
= 114.182 + 110.972 + 119.682 – 2477.2444
16
= 2.4253
Calculate the Error Sum of Squares (SSE)
SSE = SS - (SST + SSB)
= 247.8507 - (217.4785 + 2.4253)
= 27.9469
Prepare the Analysis of Variance according to Table A.13.
Table A.13.
Analysis of Variance
Sources of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Squares
Treatments
SST
k-1
SST
(k-1)
SST
SSE
x
(bkn-k-b+1)
(k-1)
Blocks
SSB
b-1
SSB
(b-1)
SSB
SSE
x
(bkn-k-b+1)
(b-1)
Error
SSE
bkn-k-b+1
Total
F-Ratio
SSE
(bkn-k-b+1)
SS
bkn-1
Using the above formulations, substitute with actual figures from
the calculations and prepare the Table A.14.:
Table A.14.
Analysis of Variance
Sources of
Variation
Sum of
Squares
Degrees
of
Freedom
Mean
Squares
F-Ratio
(calculated)
Treatments
217.4785
16-1=15
217.4785= 14.4986
15
14.4986= 15.5
0.9316
2.01
1.2126= 1.30
0.9316
3.32
Blocks
2.4253
3-1=2
2.4253= 1.2126
2
F-Ratio
(tabular
5%)
Error
27.9469
Total
247.8507
48-163+1=30
27.9469= 0.9316
30
48-1=47
Use of the F-distribution
The statistic F is a ratio of two variances and these variances
are the 'mean squares'.
To identify the F-distribution, the
degrees of freedom (df) of each variance needs to be specified.
The degrees of freedom of two variances may be represented as df1
and df2, where df1 is the number of degrees of freedom in the
numerator and df2 is the number of degrees of freedom in the
denominator.
From the calculations in the table of analysis of variance for
Treatments, F(df1, df2) = F(15,30).
From a F - distribution
table, the critical value for F(15,30) with p = 0.05 is 2.01.
Enter this tabular value into the table.
Similarly for Blocks, the critical value for F(2,30) with p =
0.05 is 3.32.
Enter this tabular value into the table.
Since the calculated F-ratio for treatments is greater than the
tabular value of F at the 5% level, the results indicate significant differences between the strains of B. japonicum in their
nitrogen-fixing effectiveness.
The calculated F-ratio for blocks is less than the tabular value
indicating the "blocking" of the experiment did not create any
significant disuniformity in the aeration, light, or other
environmental factors in the greenhouse.
Calculate the Least Significant Different (LSD)
Where t0.05 = The tabular value of t for degrees of freedom for
error at the 5% probability level
s2 =
Mean square for error
n
Number of replications
=
= 2.042 x 0.79
= 1.60 g
The LSD is used to compare values of two adjacent means.
A pair
of means which differ by more than the LSD is considered significantly different at the probability level of t employed.
If
comparison between means not adjacent to each other in a ranked
array are made, the Duncan's Multiple Range test should be used.
However, this test requires the computation of the Bayes LSD
whose value may differ from the LSD as calculated above.
The
calculation of the Bayes LSD is not presented here but its use is
illustrated in Figure A.17.
Means not joined by the same line differ at p = 0.05 as given by
Duncan's New Multiple Range Test.
Figure A.17.
Effect of various strains of B. japonicum on the
dry weight of shoots of soybean (G. max var.
Jupiter)