Statistical Methods in Medical Research 4ed

11.5 Analysis of covariance
11.5 Analysis of covariance 331
If, after an analysis of the type described in the last section, there is no strong
reason for postulating differences between the slopes of the regression lines in
the various groups, the following questions arise. What can be said about the
relative position of parallel regression lines? Is there good reason to believe
that the true lines differ in position, as in Fig. 11.4(b), or could they
coincide, as in Fig. 11.4(c)? What sampling error is to be attached to an
estimate of the difference in positions of lines for two particular groups? The
set of techniques associated with these questions is called the analysis of covariance.
Before describing technical details, it may be useful to note some important
differences in the purposes of the analysis of covariance and in the circumstances
in which it may be used.
1 Main purpose.
(a) To correct for bias. If it is known that changes in x affect the mean value
of y, and that the groups under comparison differ in their values of x, it will
follow that some of the differences between the values of y can be ascribed
partly to differences between the xs. We may want to remove this effect as far
as possible. For example, if y is forced expiratory volume (FEV) and x is
age, a comparison of mean FEVs for men in different occupational groups
may be affected by differences in their mean ages. A comparison would be
desirable of the mean FEVs at the same age. If the regressions are linear and
parallel, this means a comparison of the relative position of the regression
lines.
(b) To increase precision. Even if the groups have very similar values of x,
precision in the comparison of values of y can be increased by using the
residual variation of y about regression on x rather than by analysing the ys
alone.
2 Type of investigation.
(a) Uncontrolled study. In many situations the observations will be made on
units which fall naturally into the groups in questionÐwith no element of
controlled allocation. Indeed, it will often be this lack of control which leads
to the bias discussed in 1(a).
(b) Controlled study. In a planned experiment, in which experimental units
are allocated randomly to the different groups, the differences between values
of x in the various groups will be no greater in the long run than would be
expected by sampling theory. Of course, there will occasionally be large
fortuitous differences in the xs; it may then be just as important to correct
for their effect as it would be in an uncontrolled study. In any case, even with
very similar values of x, the extra precision referred to in 1(b) may well be
worth acquiring.