Statistical Methods in Medical Research 4ed

352 Modelling continuous data
In the table on p.351, the SSq between intercepts and between slopes are obtained by
subtraction. Neither term is at all significant. Note that the test for differences between
slopes is approximately equivalent to x2 …4† ˆ 4 1 37 ˆ 5 48, rather less than the sum of
the two x2 …2† statistics given earlier, confirming the non-independence of the two previous
tests. Note also that even if the entire SSq between intercepts were ascribed to a trend
across the groups, with 1 DF, the variance ratio (VR) would be only 421 33/220 6 ˆ 1 91,
far from significant. The point can be made in another way, by comparing the intercepts,
in the model of (11.57), for the two extreme groups, A and C. These groups
differ in the values taken by z1 (1 and 0, respectively). In the analysis of the model (11.57),
the estimate d1, of the regression coefficient on z1 is 7 39 5 39, again not significant.
Of course, the selection of the two groups with the most extreme contrast biases the
analysis in favour of finding a significant difference, but, as we see, even this contrast is
not significantly large.
In summary, it appears that the overall multiple regression fitted initially to the data of
Table 11.6 can be taken to apply to all three groups of patients.
The between-slopes SSq was obtained as the difference between the residual
fitting (11.57) and the sum of the three separate residuals fitting regressions for
each group separately. It is often more convenient to do the computations as
analyses of the total data set, as follows.
Consider first two groups with a dummy variable for group 1, z, defined as
earlier. Now define new variables wj, for j ˆ 1top,bywj ˆ zxj. Since z is zero in
group 2, then all the wj are also zero in group 2; z ˆ 1 in group 1 and therefore
wj ˆ xj in group 2. Consider the following model
E…y† ˆb 0 ‡ dz ‡ b1x1 ‡ ...‡ bpxp ‡ g1w1 ‡ ...‡ gpwp: …11:64†
Because of the definitions of z and the wj, (11.64) is equivalent to
E…y† ˆ b0 ‡ d ‡…b1 ‡ g1†x1 ‡ ...‡…bp ‡ gp†xp for group 1
b0 ‡ b1x1 ‡ ...‡ bpxp
for group 2,
…11:65†
which gives lines of different slopes and intercepts for the two groups. The
coefficient g j is the difference between the slopes on xj in the two groups.
The overall significance test for the null hypothesis that g 1 ˆ g 2 ˆ ...ˆ g p ˆ 0
is tested by deleting the wj variables from the regression to give the F test on p
and n 2p 2 DF.
When k > 2, the above procedure is generalized by deriving p…k 1† variables,
wij ˆ zixj, and an overall test of parallel regressions in all the groups is
given by the composite test of the regression coefficients for all the wij. This is an
F test with p…k 1† and n …p ‡ 1†k DF.
In the above procedure the order of introducing (or deleting) the variables is
crucial. The wij should only be included in regressions in which the correspond-