Statistical Methods in Medical Research 4ed

in group 1 and 0 for all observations in group 2. As a model for the data as a
whole, suppose that
E…y† ˆb 0 ‡ dz ‡ b 1x1 ‡ b 2x2 ‡ ...‡ b pxp: …11:55†
Because of the definition of z, (11.55) is equivalent to assuming that
8
< b0 ‡ d ‡ b1x1 ‡ b2x2 ‡ ...‡ bpxp for group 1
E…y† ˆ
:
b0 ‡ b1x1 ‡ b2x2 ‡ ...‡ bpxp for group 2,
…11:56†
which is precisely the model required for the analysis of covariance. According to
(11.56) the regression coefficients on the xs are the same for both groups, but
there is a difference, d, between the intercepts. The usual significance test in the
analysis of covariance tests the hypothesis that d ˆ 0. Since (11.55) and (11.56)
are equivalent, it follows from (11.55) that the whole analysis can be performed
by a single multiple regression of y on z, x1, x2, ..., xp. The new variable z is
called a dummy, orindicator, variable. The coefficient d is the partial regression
coefficient of y on z, and is estimated in the usual way by the multiple
regression analysis, giving an estimate d, say. The variance of d is estimated as
usual from (11.43) or (11.51), and the appropriate tests and confidence limits
follow by use of the t distribution. Note that the Residual MSq has n p 2DF
(since the introduction of z increases the number of predictor variables from p to
p ‡ 1), and that this agrees with (v) on p. 348 (putting k ˆ 2).
When k > 2, the procedure described above is generalized by the introduction
of k 1 dummy variables. These can be defined in many equivalent ways.
One convenient method is as follows. The table shows the values taken by each
of the dummy variables for all observations in each group.
Dummy variables
Group z1 z2 ... zk 1
1 1 0 ... 0
2
.
.
0
.
.
1
.
.
... 0
.
.
k 1 0 0 ... 1
k 0 0 ... 0
The model specifies that
E…y† ˆb 0 ‡ d1z1 ‡ ...‡ dk 1 zk 1 ‡ b 1x1 ‡ ...‡ bpxp …11:57†
and the fitted multiple regression equation is
11.7 Multiple regression in groups 349
Y ˆ b0 ‡ d1z1 ‡ ...‡ dk 1 zk 1 ‡ b1x1 ‡ ...‡ bpxp: …11:58†