Page 178 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

158 4 Parametric Tests of Hypotheses

variances, follows the F c−1,(c−1)(r−1) distribution. Similarly, and under the same
assumptions, the null hypothesis H 02 is tested using the ratio v r/v e and the
F r−1,(c−1)(r−1) distribution.
Let us now consider the more general situation where for each combination of
column and row categories, we have several values available. This repeated
measurements experiment allows us to analyse the data more fully. We assume that
the number of repeated measurements per table cell (combination of column and
row categories) is constant, n, corresponding to the so-called factorial experiment.
An example of this sort of experiment is shown in Figure 4.18.
Now, the breakdown of the total sum of squares expressed by the equation 4.40,
does not generally apply, and has to be rewritten as:

SST = SSC + SSR + SSI + SSE, 4.42

with:

c r n
2
1. SST = ∑∑∑ x( ijk − x ) .
...
= i 1 = j 1 = k 1
Total sum of squares computed for all n cases in every combination of the
c×r categories, characterising the dispersion of all cases around the global
mean. The cases are denoted x ijk, where k is the case index in each ij cell
(one of the c×r categories with n cases).
c
2
2. SSC = rn ∑ x( i.. − x ) .
...
= i 1
Sum of the squares representing the dispersion along the columns. The
variance along the columns is v c = SSC/(c – 1), has c – 1 degrees of freedom
2
2
and is the point estimate of σ + rnσ .
c
r
2
3. SSR = cn ∑ x ( . j. − x ) .
...
= j 1
Sum of the squares representing the dispersion along the rows. The variance
along the rows is v r = SSR/(r – 1), has r – 1 degrees of freedom and is the
2
2
point estimate of σ + cnσ .
r
4. Besides the dispersion along the columns and along the rows, one must also
consider the dispersion of the column-row combinations, i.e., one must
consider the following sum of squares, known as subtotal or model sum of
squares (similar to SSW in the one-way ANOVA):
c r
2
SSS = n ∑∑ x( ij. − x ) .
...
= i 1 = j 1
5. SSE = SST – SSS.