Biology Reference
In-Depth Information
Regression Analysis:Y versus X
The regression equation is
Y
¼
2.33
þ
0.606 X
Predictor
Coef
SE Coef
T
P
Constant
2.3292
0.9092
2.56
0.034
X
0.6062
0.1564
3.88
0.005
S
¼
0.189009 R- Sq
¼
65.2% R- Sq (adj )
¼
60.9%
Analysis of Variance
Source
DF
SS
MS
F
P
Regression
1
0.53646
0.53646
15.02
0.005
Residual Error 8
0.28579
0.03572
Total
9
0.82225
This output shows that for the dataset in Table 4-4, we find
SSR = 0.28579 and TSS = 0.82225. Thus, the sum of squares attributed to
the regression line is the difference TSS
53646. This number
presents the portion of the variance in the data explained by the
genotype. Therefore, the heritability ratio h
2
SSR
¼
0
:
V
P
, representing the
genetic variance divided by the entire variance of the phenotype,
translates, in our notation, to h
2
¼
V
A
=
TSS. Thus, the
proportion of the variance in the child's stature data explained by
parental stature data is h
2
¼
V
A
=
V
P
¼ð
TSS
SSR
Þ=
652. In this case, the
heritability ratio h
2
is exactly the coefficient of determination, R
2
, in the
MINITAB linear regression analysis output that gives the percentage of
variation of Y attributable to the approximate linear relationship
between X and Y.
¼
0
:
53646
=ð
0
:
82225
Þ¼
0
:
The next question is whether this result is statistically significant.
More specifically, we want to see whether the portion of the variance
explained by the regression (genotype) is statistically significant. We
need to formulate a null hypothesis H
0
about the regression line and
define a statistic that allows us to decide whether we can reject it. The
hypothesis that we would like to reject is:
6.4
Y
6.2
H
0
: There is no genotypic (inherited) component in the child's stature caused by
the parental stature.
6.0
Y
=
Y
5.8
5.6
Mathematically, this would correspond to a horizontal regression line.
Therefore, the hypothesis H
0
can be stated in mathematical terms as:
5.4
5.2
5
H
0
: The slope of the regression line is equal to zero.
5
5.2
5.4
5.6
5.8
6
6.2
6.4
X
FIGURE 4-8.
Deviation of individuals' heights from the mean.
The data are plotted as in Figure 4-7, with
parental data on the x-axis and child's data plotted
on the y-axis.
If H
0
were true, the mean square error of the residuals would be equal
to the total mean square error of the residuals and the variance in the
data explained by the regression would be zero (see Figure 4-8).
Furthermore, recall that the mean square error of the residuals would
