Principles of Plant Genetics and Breeding

By plugging values for X i , corresponding Y values can
be predicted. The results indicate that the regression
line will be a good predictor of an unknown value of the
independent variable.
Chi-square test
The chi-square (χ 2 ) test is used by plant breeders to test
hypotheses related to categorical data such as would be
collected from inheritance studies. The statistic measures
the deviations of the observed frequencies of each
class from that of expected frequencies. Its values can be
zero or positive but not negative. As the number of
degrees of freedom increases, the chi-square distribution
approaches a normal distribution. It is defined
mathematically as:
χ 2 =∑[( f o − f e ) 2 ]/f e
where f o = observed sample frequency, and f e = expected
frequency of the null hypothesis (H 0 ), the hypothesis to
be “disproved”.
Chi-square test of goodness-of-fit
Suppose a breeder is studying the inheritance of a
trait. A cross is made and the following outcomes are
recorded:
Observed Expected
Character frequency frequency
Green cotyledon 78 75
Yellow cotyledon 22 25
If we assume that the trait is controlled by a single
gene pair exhibiting dominance, we expect to find
a phenotypic ratio of 3 : 1 in the F 2 (or 1 : 1 ratio in
a testcross). This is the null hypothesis (H 0 ). The
expected frequencies based on the 3 : 1 ratio are also
given. The chi-square value is calculated as follows:
χ 2 =∑[( f o − f e ) 2 ]/f e
= (78 − 75) 2 /75 + (22 − 25) 2 /25
= 0.12 + 0.36
= 0.48
The degrees of freedom (df ) = 2 − 1 = 1; the tabulated
t-value = 3.81 at probability = 0.05. The calculated chisquare
value is less than the tabulated value; therefore,
the discrepancy observed above is purely a chance event.
COMMON STATISTICAL METHODS IN PLANT BREEDING 153
The null hypothesis is hence accepted, and the cotyledon
color is declared to be controlled by a single gene
pair with complete dominance.
Chi-square test of independence
Also called a contingency chi test, the chi-square test
of independence may be applied to different situations.
For example, it is applicable where a breeder has made
one set of observations obtained under a particular set
of conditions, and wishes to compare it with a similar
set of observations under a different set of conditions.
The question being asked in contingency chi square is
whether the experimental results are dependent (contingent
upon) or independent of the conditions under
which they were observed. In general, whenever two or
more systems of classification are used, one can check for
independence of the system.
There is a cross-classification when one individual is
classified in multiple ways. For example, a cultivar may
be classified according to species and also according to
resistance to a disease. The question then is whether the
classification of one individual according to one system
is independent of its classification by the other system.
More specifically, if there is independence in this
species–infection classification, then the breeder would
interpret the results to mean that there is no difference
in infection rate between species.
The short cut method for solving contingency chisquare
problems is as follows:
Categories of observation
I II Total
A a b a + b
B c d c + d
Total a + c b + d a + b + c + d
χ 2 = [(ad − bc) 2 ]n/[(a + b)(c + d)(a + c)(b + d)]
where a, b, c, and d are the observed frequencies. This
is called a 2 × 2 contingency chi-square, but can be
extended to more complex problems (2 × 4, 4 × 6, etc.).
t-test
The t-test is used to make inferences about population
means. A breeder may wish to compare the yields of two
cultivars, for example. Assuming the sample observations
are drawn at random, the two population variances
are equal, and the populations from which the samples