Geoscience Reference
In-Depth Information
b
Height
=+
a
Age
then we could let
Y
= height and
X
1
= 1/age and fit
Y
=
a
+
b
1
X
1
Similarly, if the relationship between
Y
and
X
is quadratic
Y
=
a
+
bX
+
cX
2
then we can let
X
=
X
1
and
X
2
=
X
2
and fit
Y
=
a
+
b
1
X
1
+
b
2
X
2
Functions such as
Y
=
aX
b
Y
=
a
(
b
x
)
10
Y
=
aX
b
which are nonlinear in the coefficients can sometimes be made linear by a logarithmic transforma-
tion. The equation
Y
=
aX
b
would become
log
Y
= log
a
+
b
(log
X
)
which could be fitted by
Y
′ =
a
+
b
1
X
1
where
Y
′ = log
Y
, and
X
1
= log
X
.
The second equation transforms to
log
Y
= log
a
+ (log
b
)
X
The third becomes
Y
= log
a
+
b
(log
X
)
Both can be fitted by the linear model.
When making these transformations, the effect on the assumption of homogeneous variance
must be considered. If
Y
has homogeneous variance, then log
Y
probably will not—and
vice versa
.
Some curvilinear models cannot be fitted by the methods that have been described, including
Y
=
a
+
b
x
Y
=
a
(
X
-
b
)
2
Y
=
a
(
X
1
-
b
)(
X
2
-
c
)
Fitting these models requires more cumbersome procedures.
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