Geoscience Reference
In-Depth Information
Substituting these estimates in the general equation gives
ˆ
Y
=
0 13606
.
+
0 005962
.
X
where
Y
is used to indicate that we are dealing with an estimated value of
Y.
With this equation we can estimate the basal area growth for the past 10 years (
Y
) from the mea-
surements of the crown volume
X.
Because
Y
is estimated from a known value of
X
, it is called the
dependent variable and
X
is the independent variable. In plotting on graph paper, the values of
Y
are
usually (purely by convention) plotted along the vertical axis (ordinate) and the values of x along the
horizontal axis (abscissa).
7.17.1.1 How Well Does the Regression Line Fit the Data?
A regression line can be thought of as a moving average. It gives an average value of
Y
associated
with a particular value of
X
. Of course, some values of
Y
will be above the regression line (moving
average) and some below, just as some values of
Y
are above or below the general average of
Y
. The
corrected sum of squares for
Y
(i.e., Σ
y
2
) estimates the amount of variation of individual values of
Y
about the mean value of
Y
. A regression equation is a statement that part of the observed variation in
Y
(estimated by Σ
y
2
) is associated with the relationship of
Y
to
X
. The amount of variation in
Y
that
is associated with the regression on
X
is the reduction or regression sum of squares:
=
(
)
2
∑
∑
xy
x
2
354 1477
59 397 67
(
.
)
Reduction
SS
=
= .
2 1115
2
,
.
75
As noted above, the total variation in
Y
is estimated by Σ
y
2
= 2.7826 (as previously calculated).
The part of the total variation in
Y
that is not associated with the regression is the residual sum of
squares:
∑
2
Residual
SS
=
y
−
Reduction
SS
=
2 7826
.
−
2 1115
.
=
0 6711
.
In analysis of variance we used the unexplained variation as a standard for testing the amount of
variation attributable to treatments. We can do the same in regression. What's more, the familiar
F
test will serve.
Source of Variation
Degrees of Freedom
a
Sums of Squares
Mean Squares
b
2
(
Σ
Σ
xy
x
)
Due to regression
=
1
2.1115
2.1115
2
Residual (unexplained)
60
0.6711
0.01118
61
2.7826
Total (= Σ
y
2
)
a
As there are 62 values of
Y
, the total sum of squares has 61 degrees of freedom. The regression of
Y
on
X
has one degree of freedom. The residual degrees of freedom are obtained by subtraction.
b
Mean square is, as always, equal to sum of squares/degrees of freedom.
The regression is tested by
F
=
Regression mean square
Residualmean square
2 1115
0 01118
.
.
=
=
188 86
.
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