Principles of Plant Genetics and Breeding

152 CHAPTER 9
Covariance =−2.45
Correlation =−0.757
Intercept = 59.34
Slope =−1.087
Standard error = 0.332
Student’s t-value = 3.273; probability = 0.010
The results indicate a significant negative association
between seed oil and protein content. The breeding
implication is that as one selects for high seed oil, seed
protein will decrease.
Table 9.3 Data for oil and protein content of soybean
seed.
Oil content Protein content
(%) (%)
20.1 35.7
21.2 35.1
19.5 33.2
18.3 40.6
19.0 37.5
21.3 36.1
19.8 39.5
22.6 34.8
17.5 39.1
19.9 40.2
Mean 19.92 37.68
Simple linear regression
Unlike simple linear correlation, simple linear regression
is a relationship between two variables that involves
cause and effect. There is a dependent variable (Y ) and
an independent variable (X). For example, grain yield
depends on seed size, number of seeds per pod, etc. The
changes in the dependent variable (effect) are brought
about by the changes in the independent variable
(cause). Another way of looking at it is that regression is
a study of the relationship between variables with the
objective of identifying, estimating, and validating the
relationship.
Simple linear regression has the mathematical form of
the equation of a straight line:
Y = a + bX
where Y = dependent variable, X = independent variable,
b = slope of the regression line, and a = intercept on
the y axis.
Table 9.4 Data on plant yield and maturity of soybean.
Yield (bushels) Days to maturity
44 138
40 136
38 125
35 118
33 115
32 111
30 110
28 109
24 98
18 93
Mean 32.2 115.3
Figure 9.2 The linear regression line.
The regression coefficient is calculated as:
b = [N(∑XY ) − (∑X)(∑Y )]/N(∑X 2 ) − (∑X) 2
The data in Table 9.4 represent the yield of soybean
corresponding to various days to maturity of the crop.
The results of a regression analysis are as follows:
Covariance = 110.04
Correlation = 0.976
Intercept =−27.03
Slope (b) = 0.514
Standard error = 0.040
Student’s t-value = 12.794; probability = 0.000
The prediction equation is hence (Figure 9.2):
ˆY i = 127.03 + (0.514)X i
Y = a + bX