Statistical Methods in Medical Research 4ed

there is none, or because a consideration of ordering is deferred until a later stage
of the analysis.
Suppose there are k groups of observations on a variable y, and that the ith
group contains ni observations. The numbering of the groups from 1 to k will be
quite arbitrary, although if there is a simple ordering of the groups it will be
natural to use this in the numbering. Further notation is as follows:
Group
Number of observations
1
n1
2 ... i
n2 ... ni
... k
... nk
All groups combined
N ˆ Pk iˆ1 ni
Mean of y
Sum of y
y1
T1
y2 ... yi ... yk
T2 ... Ti ... Tk
y ˆ T=N
T ˆ Pk iˆ1 Ti
Sum of y2 S1 S2 ... Si ... Sk S ˆ Pk iˆ1 Si
Note that the entries N, T and S in the final column are the sums along the
corresponding rows, but y is not the sum of the yi ( y will be the mean of the yi if
all the ni are equal; otherwise y is the weighted mean of the yi, P niyi= P ni). Let
the observations within each group be numbered in some arbitrary way, and
denote the jth observation in the ith group by yij.
When a summation is taken over all the N observations, each contributing
once to the summation, we shall use the summation sign
P
i, j
or, in the text, P
i, j . When the summation is taken over the k groups, each group
contributing once, we shall use the sign
P
i
or P
i , whilst summation over the members of a particular group will be denoted
by
P
j
8.1 One-way analysis of variance 209
or P
j .
The deviation of any observation from the grand mean, y, may be split into
two parts, as follows:
yij y ˆ…yij yi†‡…yi y†: …8:1†
The first term on the right of (8.1) is the deviation of yij from its group mean, yi,
and the second term is the deviation of the group mean from the grand mean. If
both sides of this equation are squared and summed over all N observations, a
similar result holds: