Statistical Methods in Medical Research 4ed

332 Modelling continuous data
Two groups
If the t test based on (11.20) reveals no significant difference in slopes, two
parallel lines may be fitted with a common slope b given by (11.23). The
equations of the two parallel lines are (as in (11.24)),
and
Yc1 ˆ y1 ‡ b…x x1†
Yc2 ˆ y2 ‡ b…x x2†:
The difference between the values of Y at a given x is therefore
d ˆ Yc1 Yc2
ˆ y1 y2 b…x1 x2†
…11:32†
(see Fig. 11.7).
The sampling error of d is due partly to that of y1 y2 and partly to that of b
(the term x1 x2 has no sampling error as we are considering x to be a nonrandom
variable). The three variables, y1, y2 and b, are independent; consequently
which is estimated as
where s 2 c
var…d† ˆvar…y1†‡var…y2†‡…x1 x2† 2 var…b†
2 1
ˆ s ‡
n1
1
‡
n2
…x1 x2† 2
" #
,
Sxx1 ‡ Sxx2
s 2 c
1
‡
n1
1
‡
n2
…x1 x2† 2
" #
, …11:33†
Sxx1 ‡ Sxx2
is the residual mean square about the parallel lines:
s 2 c ˆ
Syy1 ‡ Syy2
…Sxy1 ‡ Sxy2† 2
n1 ‡ n2 3
Sxx1 ‡ Sxx2
: …11:34†
Note that s 2 c differs from the s2 of (11.18). The latter is the Residual mean square
(MSq) about separate lines, and is equivalent to the s2 of Table 11.5. The
Residual MSq s2 c in (11.34) is taken about parallel lines (since parallelism is an
initial assumption in the analysis of covariance), and would be obtained
from Table 11.5 by pooling the second and third lines of the analysis. The
resultant DF would be …k 1†‡…n 2k† ˆn k 1, which gives the
n1 ‡ n2 3 …ˆ n 3† of (11.34) when k ˆ 2.
The standard error (SE) of d, the square root of (11.33), may be used in a t
test. On the null hypothesis that the regression lines coincide, E…d† ˆ0, and