Geoscience Reference
In-Depth Information
7.17. 3 . 2 I n t e r a c t i o n s
Suppose that there is a simple linear relationship between
Y
and
X
1
. If the slope (
b
) of this relation-
ship varies, depending on the level of some other independent variable (
X
2
), then
X
1
and
X
2
are said
to interact. Such interactions can sometimes be handled by introducing interaction variables. To
illustrate, suppose that we know that there is a linear relationship between
Y
and
X
1
:
Y
=
a
+
bX
1
Suppose further that we know or suspect that the slope (
b
) varies linearly with
Z
:
b
=
a
′ +
b
′
Z
This implies the relationship
Y
=
a
+ (
a
′ +
b
′
Z
)
X
1
or
Y
=
a
+
a
′
X
1
+
b
′
X
1
Z
where
X
2
=
X
1
Z
, an interaction variable.
If the
Y
-intercept is also a linear function of
Z
, then
a
=
a
′′ +
b
′′
Z
and the form of relationship is
Y
=
a
′′ +
b
′′
Z
+
a
′
X
1
+
b
′
X
1
Z
7.17.4 g
roup
r
egressions
Linear regressions of
Y
on
X
were fitted for each of two groups:
Group A
Sum
Mean
Y
3
7
9
6
8
13
10
12
14
82
9.111
X
1
4
7
7
2
9
10
6
12
58
6.444
where
∑
∑
∑
∑
2
2
2
n
=
9
,
Y
=
848
,
XY
=
609
,
X
=
480
,
y
=
100 8889
.
ˆ
∑
∑
2
xy
=
80 5556
.
,
x
=
106 2222
.
,
Y
=
4 224
.
+
0 7584
.
X
Residual
SS
= 39 7980
.
,
with7degrees of freedom
Group B
Sum
Mean
Y
4
6
12
2
8
7
0
5
9
2
11
3
10
79
6.077
X
4
9
14
6
9
12
2
7
5
5
11
2
13
99
7.616
where
∑
∑
∑
∑
2
2
2
n
=
13
,
Y
=
653
,
XY
=
753
,
X
=
951
,
y
=
172 9231
.
ˆ
∑
∑
2
xy
=
151 3846
.
,
x
=
197 0796
.
,
Y
=
0 228
.
+
0 7681
.
X
Residual
SS
= 56 6370
.
,
with11degrees of free
dom
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