Statistics Review - Mathematics and Statistics for the Environment - page 172

Now, we might ask, are these really different regressions? Or could the data be combined to pro-
duce a single regression that would be applicable to both groups? If there is no significant difference
between the mean square residuals for the two groups (this matter may be determined by Bartlett's
test, see Section 7.14.3), the test described below helps to answer the question.

7.14.4.1 Testing for the Common Regressions
Simple linear regressions may differ either in their slope or in their level. When testing for common
regressions the procedure is to test first for common slopes. If the slopes differ significantly, the
regressions are different and no further testing is needed. If the slopes are not significantly different,
the difference in level is tested. The analysis table is

Residuals

Line Group df Σ y 2 Σ xy Σ x 2 df SS MS

1 A 8 100.8889 80.5556 106.2222 7 39.7980 —

2 B 12 172.9231 151.3846 97.0769 11 56.6370 —

3 Pooled residuals 18 96.4350 5.3575

4 Difference for testing common slopes 1 0.0067 0.0067

5 Common slope 20 273.8120 231.9402 303.2991 19 5.0759 96.4417

6 Difference for testing trends 1 80.1954 80.1954

7 Single regression 21 322.7727 213.0455 310.5909 20 176.6371 —

The first two lines in this table contain the basic data for the two groups. To the left are the total
degrees of freedom for the groups (8 for A and 12 for B). In the center are the corrected sums of
squares and products. The right side of the table gives the residual sum of squares and degrees of
freedom. Since only simple linear regressions have been fitted, the residual degrees of freedom of
each group are one less than the total degrees of freedom. The residual sum of squares is obtained
by first computing the reduction sum of squares for each group.

= (

)

2

∑
∑

xy
x

Reduction SS

2

This reduction is then subtracted from the total sum of squares (Σ y 2 ) to give the residuals.
Line 3 is obtained by pooling the residual degrees of freedom and residual sums of squares for
the groups. Dividing the pooled sum of squares by the pooled degrees of freedom gives the pooled
mean square. The left side and center of line (we will skip line 4 for the moment) is obtained by
pooling the total degrees of freedom and the corrected sums of squares and products for the groups.
These are the values that are obtained under the assumption of no difference in the slopes of the
group regressions. If the assumption is wrong, the residuals about this common slope regression will
be considerably larger than the mean square residual about the separate regressions. The residual
degrees of freedom and sum of squares are obtained by fitting a straight line to these pooled data.
The residual degrees of freedom are, of course, one less than the total degrees of freedom. The
residual sum of squares is, as usual,

2

( 231 9402

303 2991

. )

Reduction SS = 273 8120 . − = 9 6 4417 .

.