Statistical Methods in Medical Research 4ed

Vital capacity (l)
6
5
4
3
20 30 40
Age (years)
B
A
A
50 60 70
B Separate slope
}
Parallel lines
Separate slope
Fig. 11.5 Scatter diagram showing age and vital capacity of 84 men working in the cadmium industry,
divided into three groups (Table 11.4). (d) Group A1; ( ) Group A2; (s) Group B.
the slope for group A may be partly or wholly due to a curvature in the regression: there is
a suggestion that the mean value of y at high values of x is lower than is predicted by
the linear regressions (see p. 330). Alternatively it may be that a linear regression is
appropriate for each group, but that for group A the vital capacity declines more rapidly
with age than for group B.
More than two groups
With any number of groups the pooled slope b is given by a generalization of
(11.21):
P
i b ˆ
Sxyi
P
i Sxxi
: …11:23†
Parallel lines may now be drawn through the mean points (xi, yi), each with
the same slope b. That for the ith group will have this equation:
Yci ˆ yi ‡ b…x xi†: …11:24†
The subscript c is used to indicate that the predicted value Yci is obtained using
the common slope, b.
The deviation of any observed value y from its group mean yi may be divided
as follows:
y yi ˆ…y Yi†‡…Yi Yci†‡…Yci yi†: …11:25†
g
11.4 Regression in groups 327