Statistical Methods in Medical Research 4ed

Between columns :
P …y:j y† 2 ˆ Pc
jˆ1
C 2 j =r T 2 =N:
The Total SSq is similarly calculated as
P …yij y† 2 ˆ P y 2 ij T 2 =N,
and the Residual SSq may be obtained by subtraction:
Residual SSq ˆ Total SSq Between-Rows SSq Between-Columns SSq:
…9:5†
The analysis so far is purely a consequence of algebraic identities. The relationships
given above are true irrespective of the validity of the model. We now
complete the analysis of variance by some steps which depend for their validity
on that of the model. First, the degrees of freedom (DF) are allotted as shown in
Table 9.2. Those for rows and columns follow from the one-way analysis; if the
only classification had been into rows, for example, the first line of Table 9.2 would
have been shown as Between groups and the SSq shown in Table 9.2 as Between
columns and Residual would have added to form the Within-Groups SSq. With
r 1 and c 1 as degrees of freedom for rows and columns, respectively, and
N 1 for the Total SSq, the DF for Residual SSq follow by subtraction:
…rc 1† …r 1† …c 1† ˆrc r c ‡ 1 ˆ…r 1†…c 1†:
The mean squares (MSq) for rows, columns and residual are obtained in each
case by the formula MSq ˆ SSq/DF, and those for rows and columns may each
be tested against the Residual MSq, s 2 , as shown in Table 9.2. The test for rows,
for instance, has the following justification. On the null hypothesis (which we
shall call HR) that all the row constants ai in (9.1) are equal (and therefore equal
to zero, since P ai ˆ 0), both s 2 R and s2 are unbiased estimates of s 2 .IfHR is not
true, so that the ai differ, s 2 R has expectation greater than s2 whereas s 2 is still an
unbiased estimate of s 2 . Hence FR tends to be greater than 1, and sufficiently
high values indicate a significant departure from HR. This test is valid whatever
values the b j take, since adding a constant on to all the readings in a particular
column has no effect on either s 2 R or s2 .
Table 9.2 Two-way analysis of variance table.
SSq DF MSq VR
Between rows
P
i R2i =c T 2 =N r 1 s2 R FR ˆ s2 R =s2
Between columns P
j C2 j =r T 2 =N c 1 s2 C FC ˆ s2 C =s2
Residual By subtraction …r 1†…c 1† s2 Total
P
y2 ij T 2 =N rc 1…ˆ N 1†
i, j
9.2 Two-way analysis of variance: randomized blocks 241