Statistical Methods in Medical Research 4ed

314 Modelling continuous data
Example 11.1
The analysis of variance of y, from the data of Example 7.1 (pp. 192 and 199), is as
follows:
SSq DF MSq VR
Due to regression 7 666 39 1 7666 39 24 2 (P < 0 001)
About regression 9 502 08 30 316 74 1 00
Total 17 168 47 31
The SSq have already been obtained on pp. 192 and 199. The value of t obtained
previously was 4 92; note that …4 92† 2 ˆ 24 2, the value of F.
Test of linearity
It is often important to know not only whether the slope of an assumed linear
regression is significant, but also whether there is any reason to doubt the basic
assumption of the linearity of the regression.
If the data provide a number of replicate readings of y for certain values of x,
a test of linearity is easily obtained. Suppose that, at the value xi of x, there are ni
observations on y, with a mean yi. Each such group of replicates is called an
array. Figure 11.2 illustrates three different situations. In (a), a linear regression
seems to be consistent with the observed data in that the array means yi are
reasonably close to the regression line. In (b) and (c), however, the array means
deviate from the line by more than can easily be explained by the within-arrays
variation. In (b) the deviations seem to be systematic, suggesting that a curved
regression line is required. In (c) the deviations seem to lack any pattern,
suggesting perhaps an extra source of variation associated with each array; for
example, if each array referred to observations on animals in a single cage, the
positioning of the cage in the laboratory might affect the whole array.
In discussing Fig. 11.2 we have made a rough comparison between the
magnitude of deviations of array means from the regression line and the
within-arrays variation. The comparison is made formally as follows. For any
value y in the array corresponding to xi, the residual y Yi may be divided into
two parts:
y Yi ˆ…y yi†‡…yi Yi†: …11:5†
When both sides are squared and summed over all observations, the sum of
products of the two terms on the right vanishes, and we have a partition of the
Residual SSq:
P
…y Yi† 2 ˆ P …y yi† 2 ‡ P …yi Yi† 2 , …11:6†