Statistical Methods in Medical Research 4ed

E…y† ˆa ‡ bx), or whether, as in (c), the lines coincide. In practice, the fitted
regression lines would rarely have precisely the same slope or position, and the
question is to what extent differences between the lines can be attributed to
random variation. Differences in position between parallel lines are discussed in
§11.5. In this section we concentrate on the question of differences between
slopes.
Suppose that there are k groups, with ni pairs of observations in the ith
group. Denote the mean values of x and y in the ith group by xi and yi, and the
regression line, calculated as in §7.2, by
Yi ˆ y ‡ bi…x xi†: …11:12†
If all the ni are reasonably large, a satisfactory approach is to estimate the
variance of each bi by (7.16) and to ignore the imprecision in these estimates of
variance. Changing the notation of (7.16) somewhat, we shall denote the residual
mean square for the ith group by s 2 i and the sum of squares of x about xi by
P
…i† …x xi† 2 . Note that the parenthesized suffix i attached to the summation
sign indicates summation only over the specified group i; that is,
P
…x xi†
…i†
2 ˆ Pni
…xij xi†
jˆ1
2 :
To simplify the notation, denote this sum of squares about the mean of x in
the ith group by
Sxxi, …11:13†
the sum of products of deviations by Sxyi, and so on. Then following the method
of §8.2, we write
and calculate
wi ˆ 1
var…bi†
G ˆ P wib 2 i
Sxxi
ˆ
s2 , …11:14†
i
P
… wibi†
2 = P wi: …11:15†
On the null hypothesis that the true slopes bi are all equal, G follows approximately
a x2 …k 1† distribution. High values of G indicate departures from the null
hypothesis, i.e. real differences between the bi.IfG is non-significant, and the
null hypothesis is tentatively accepted, the common value b of the bi is best
estimated by the weighted mean
with an estimated variance
11.4 Regression in groups 323
b ˆ P wibi= P wi, …11:16†