Page 164 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

144 4 Parametric Tests of Hypotheses

Let us now express these sums of squares, related by 4.24, in terms of variances:

SST = n ( − v ) 1 . 4.25a
c  c 
SSW ≡ SSE = ∑ n ( i − v ) 1 i = ∑ n ( i − ) 1  v W = n ( − c) v . 4.25b
W
i 1=  i 1= 
SSB = c ( − v ) 1 B . 4.25c

Note that:

1. The within-group variance, v W, is the pooled variance and corresponds to
the generalization of formula 4.9:
c
∑ ( n −1 v ) i
i
v W ≡ v = i=1 . 4.26
p
n − c
This variance represents the stochastic behaviour of the cases around their
2
group means. It is the point estimate of σ , the true variance of the
population, and has n – c degrees of freedom.
2. The within-group variance v W represents a mean square error, MSE, of the
observations:
SSE
MSE ≡ v W = n − c . 4.27

3. The between-group variance, v B, represents the stochastic behaviour of the
2
group means around the global mean. It is the point estimate of σ when the
null hypothesis is true, and has c – 1 degrees of freedom.
When the number of cases per group is constant and equal to n, we get:
c
∑ ( x − x) 2
i
v = n i=1 = nv , 4.28
B
c −1 X
which is the sample expression of formula 3.8, allowing us to estimate the
population variance, using the variance of the means.
4. The between-group variance, v B, can be interpreted as a mean between-
group or classification sum of squares, MSB:
SSB
MSB ≡ v B = . 4.29
c − 1
With the help of formula 4.24, we see that the total sample variance, v, can be
broken down into two parts:

n ( − v ) 1 = n ( − c) v W + c ( − v ) 1 B , 4.30