Geoscience Reference
In-Depth Information
Now, the difference between these residuals (line 4 = line 5 - line 3) provides a test of the
hypothesis of common slopes. The error term for this test is the pooled mean square from line 3:
0 0067
5 3575
.
.
Test of commonslopes:
F
=
113
/
df
The difference is not significant.
If the slopes differed significantly, the groups would have different regressions, and we would
stop here. Because the slops did not differ, we now go on to test for a difference in the levels of the
regression.
Line 7 is what we would have if we ignored the groups entirely, lumped all the original observa-
tions together, and fitted a single linear regression. The combined data are as follows:
n
=+ =
(
913 2sothe degrees of freedom =21
)
,
∑
∑
Y
=+=
(
82
79
)
161
2
Y
=+=
(
848
653
)
1501
2
(
161
22
)
∑
2
y
=
1501
−
=
322 7727
.
∑
∑
X
=+=
(
58
99
)
157
2
X
=
(
480
+
951
)
=
1431
2
(
157
22
)
∑
2
x
=
1431
−
=
310 5909
.
∑
XY
=
(
609
+
7
53
)
=
1362
(
157 161
22
)(
)
∑
xy
=
1362
−
=
213 0455
.
From this we obtain the residual values on the right side of line 7.
2
(
213 0455
310 5909
.
)
Residual
SS
=
322 7727
.
−
=
17
6 6371
.
.
If there is a real difference among the levels of the groups, the residuals about this single regres-
sion will be considerably larger than the mean square residual about the regression that assumed
the same slopes but different levels. This difference (line 6 = line 7 - line 5) is tested against the
residual mean square from line 5.
80 1954
5 0759
.
Test of levels:
F
=
=
15 80
.
118
/
df
.
As the levels differ significantly, the groups do not have the same regressions.
The test is easily extended to cover several groups, though there may be a problem in finding
which groups are likely to have separate regressions and which can be combined. The test can also
be extended to multiple regressions.
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