Principles of Plant Genetics and Breeding

154 CHAPTER 9
were drawn follow the normal distributions (these are
assumptions made in order to use this test), the hypothesis
to be tested is:
H 0 : µ 1 =µ 2 (no difference between the two means)
The alternative hypothesis to the null is:
H 1 : µ 1 ≠µ 2 (the two populations are not equal)
This may be tested as follows (for a small sample size):
t = [X ¯ 1 − X¯ 2 ]/s p √[1/n 1 + 1/n 2 ]
where:
s p =√{[(n 1 − 1)s 1 2 + (n2 − 1)s 2 2 ]/n1 + n 2 − 2}
= pooled variance
and X¯ 1 and X¯ 2 are the means of samples 1 and 2,
respectively.
Example A plant breeder wishes to compare the seed
size of two navy bean cultivars, A and B. Samples are
drawn and the 100 seed weight obtained. The following
data were compiled:
H 0 : µ 1 =µ 2
H 1 : µ 1 ≠µ 2
Cultivar
A B
n 10 8
X ¯ 21.2 19.5 (g/100 seeds)
s 1.3 1.1 (g/100 seeds)
where:
t = [X ¯ 1 − X¯ 2 ]/s p √[1/n 1 + 1/n 2 ]
s p = [(10 − 1)(1.3) 2 + (8 − 1)(1.1) 2 ]/(10 + 8 − 2)
= [9(1.69) + 7(1.21)]/16
= [15.21 + 8.47]/16
= 23.68/16
= 1.48
t = 10 − 8
= 2/(1.48 × 0.47)
= 0.70
= 2/0.70
t (calculated) = 2.857
at α 0.05 :
df = 10 + 8 − 2 = 16
t (tabulated) = 1.746
Since calculated t exceeds tabulated t, we declare a
significant difference between the two cultivars for seed
size (measured as 100 seed weight).
Analysis of variance
Frequently, the breeder needs to compare more than
two cultivars. In yield trails, several advanced genotypes
are evaluated at different locations and in different years.
The t-test is not applicable in this circumstance but its
extension, the analysis of variance (ANOVA), is used
instead. ANOVA allows the breeder to analyze measurements
that depend on several kinds of effects, and which
operate simultaneously, in order to decide which kinds
of effects are important, and to estimate these effects. As
a statistical technique, ANOVA is used to obtain and
partition the total variation in a data set according to
the sources of variation present and then to determine
which ones are important. The test of significance of an
effect is accomplished by the F-test. The results of an
analysis of variance are presented in the ANOVA table,
the simplest form being as follows:
Source of
variation df SS MS F
Treatment k − 1 SS Tr MS Tr = SS Tr /k − 1 MS Tr /MS E
Error N − k SS E MS E = SS E /N − k
Total N − 1 SS T
“Treatment” is the most important source of variance
caused by the applied treatments (e.g., different cultivars,
locations, years, etc.). The error is unaccounted
variation.
In more detailed analysis, interaction effects between
treatments are accounted for in the analysis. Examples
of ANOVA for G × E interaction analysis are provided
in Chapter 23. ANOVA is usually done on the computer
using software such as MSTAT and SAS. As
previously stated, ANOVA permits the breeder to
handle more than two genotypes (or variables) in one
analysis. The t-test is not efficient for comparing more
than two means. Commonly used tests to separate means
under such conditions include the least significant
difference (LSD) and Duncan’s multiple range test
(DMRT).