Statistical Methods in Medical Research 4ed

244 Experimental design
The sum of squares of the 32 residuals in the body of the table is 13 7744, in agreement
with the value found by subtraction in Table 9.3 apart from rounding errors. (These errors
also account for the fact that the residuals as shown do not add exactly to zero along the
rows and columns.) No particular pattern emerges from the table of residuals, nor does
the distribution appear to be grossly non-normal. There are 16 negative values and 16
positive values; the highest three in absolute value are positive (1 66, 1 33 and 1 22), which
suggests mildly that the random error distribution may have slight positive skewness.
If the linear model (9.1) is wrong, there is said to be an interaction between the
row and column effects. In the absence of an interaction the expected differences
between observations in different columns are the same for all rows (and the
statement is true if we interchange the words `columns' and `rows'). If there is an
interaction, the expected column differences vary from row to row (and, similarly,
expected row differences vary from column to column). With one observation in
each row/column cell, the effect of an interaction is inextricably mixed with the
residual variation. Suppose, however, that we have more than one observation per
cell. The variation between observations within the same cell provides direct
evidence about the random variance s 2 , and may therefore be used as a basis of
comparison for the between-cells residual. This is illustrated in the next example.
Example 9.2
In Table 9.4 we show some hypothetical data related to the data of Table 9.3. There are
three subjects and three treatments, and for each subject±treatment combination three replicate
observations are made. The mean of each group of three replicates will be seen to agree
with the value shown in Table 9.3 for the same subject and treatment. Under each group of
replicates is shown the total Tij and the sum of squares, Sij (as indicated for T11 and S11).
The Subjects and Treatments SSq are obtained straightforwardly, using the divisor 9
for the sums of squares of row (or column) totals, since there are nine observations in each
row (or column), and using a divisor 27 in the correction term. The Interaction SSq is
obtained in a similar way to the Residual in Table 9.3, but using the totals Tij as the basis
of calculation. Thus,
Interaction SSq = SSq for differences between the nine subject/treatment cells
± Subjects SSq ± Treatment SSq,
and the degrees of freedom are, correspondingly, 8 2 2 ˆ 4. The Total SSq is obtained
in the usual way and the Residual SSq follows by subtraction. The Residual SSq could
have been obtained directly as the sum over the nine cells of the sum of squares about the
mean of each triplet, i.e. as
…S11 T 2 11 =3†‡…S12 T 2 12 =3†‡...‡…S33 T 2 33 =3†:
The F tests show the effects of subjects and treatments to be highly significant. The
interaction term is not significant at the 5% level, but the variance ratio (VR) is nevertheless
rather high. It is due mainly to the mean value for subject 8 and treatment 4 being
higher than expected.