Statistical Methods in Medical Research 4ed

Equations (11.21) and (11.22) can easily be seen to be equivalent to (11.16) and
(11.17) if, in the calculation of wi in (11.14), the separate estimates of residual
variance s 2 i are replaced by the common estimate s2 . For tests or the calculation
of confidence limits for b using (11.22), the t distribution on n1 ‡ n2 4DF
should be used. Where a common slope is accepted it would be more usual to
estimate s 2 as the residual mean square about the parallel lines (11.34), which
would have n1 ‡ n2 3 DF.
Example 11.3
Table 11.4 gives age and vital capacity (litres) for each of 84 men working in the cadmium
industry. They are divided into three groups: A1, exposed to cadmium fumes for at least
10 years; A2, exposed to fumes for less than 10 years; B, not exposed to fumes. The main
purpose of the study was to see whether exposure to fumes was associated with a change in
respiratory function. However, those in group A1 must be expected to be older on the
average than those in groups A2 or B, and it is well known that respiratory test performance
declines with age. A comparison is therefore needed which corrects for discrepancies
between the mean ages of the different groups.
We shall first illustrate the calculations for two groups by amalgamating groups A1
and A2 (denoting the pooled group by A) and comparing groups A and B.
The sums of squares and products of deviations about the mean, and the separate
slopes bi are as follows:
Group i ni Sxxi Sxyi Syyi bi
A 1 40 4397 38 236 385 26 5812 0 0538
B 2 44 6197 16 189 712 20 6067 0 0306
Total 10594 54 426 097 47 1879 ( 0 0402)
The SSq about the regressions are
P
…1† …y Y1† 2 ˆ 26 5812 … 236 385† 2 =4397 38 ˆ 13 8741
and
P
…2† …y Y2† 2 ˆ 20 6067 … 189 712† 2 =6197 16 ˆ 14 7991:
Thus,
s 2 ˆ…13 8741 ‡ 14 7991†=…40 ‡ 44 4† ˆ0 3584,
and, for the difference between b1 and b2 using (11.19) and (11.20),
0 0538 … 0 0306†
t ˆ
1
…0 3584†
4397 38 ‡
r
1
6197 16
ˆ 0 0232=0 0118
ˆ 1 97 on 80 DF:
11.4 Regression in groups 325