Biostatistics

8.2 THE COMPLETELY RANDOMIZED DESIGN 317
TABLE 8.2.2
Analysis of Variance Table for the Completely Randomized Design
Source of
Variation
Sum of Squares
Degrees of
Freedom
Mean Square
Variance
Ratio
Among samples
SSA ¼ Pk
j¼1
2
n j x j x :: k 1 MSA ¼ SSA= ðk 1Þ V:R ¼ MSA
MSW
Within samples
SSW ¼ Pk
P nj
j¼1 i¼1
2
x ij x j N k MSW ¼ SSW= ðN kÞ
Total
SST ¼ Pk
P nj
j¼1 i¼1
x ij x ::
2
N 1
As we have seen, we compute V.R. in situations of this type by placing the among
groups mean square in the numerator and the within groups mean square in the denominator,
so that the numerator degrees of freedom is equal to ðk 1Þ, the number of groups minus 1,
and the denominator degrees of freedom value is equal to
X k
j¼1
n j 1 !
¼ Xk
n j k ¼ N k
j¼1
The ANOVA Table The calculations that we perform may be summarized and
displayed in a table such as Table 8.2.2 , which is called the ANOVA table.
8. Statistical decision. To reach a decision we must compare our computed V.R.
with the critical value of F, which we obtain by entering Appendix Table G
with k 1 numerator degrees of freedom and N k denominator degrees of
freedom.
If the computed V.R. is equal to or greater than the critical value of F, we reject the null
hypothesis. If the computed value of V.R. is smaller than the critical value of F, we do not
reject the null hypothesis.
Explaining a Rejected Null Hypothesis There are two possible explanations
for a rejected null hypothesis. If the null hypothesis is true, that is, if the two sample
variances are estimates of a common variance, we know that the probability of getting a
value of V.R. as large as or larger than the critical F is equal to our chosen level of
significance. When we reject H 0 we may, if we wish, conclude that the null hypothesis is
true and assume that because of chance we got a set of data that gave rise to a rare event. On
the other hand, we may prefer to take the position that our large computed V.R. value does
not represent a rare event brought about by chance but, instead, reflects the fact that
something other than chance is operative. We then conclude that we have a false null
hypothesis.
It is this latter explanation that we usually give for computed values of V.R. that
exceed the critical value of F. In other words, if the computed value of V.R. is greater than
the critical value of F, we reject the null hypothesis.