|Introduction to Analysis-of-Variance Procedures
Tests of Effects
Analysis of variance tests are constructed by comparing independent
mean squares. To test a particular null hypothesis, you
compute the ratio of two mean squares that have the same expected
value under that hypothesis; if the ratio is much larger than 1, then
that constitutes significant evidence against the null. In
particular, in an analysis-of-variance model with fixed effects only,
the expected value of each mean square has two components: quadratic
functions of fixed parameters and random variation. For example,
for a fixed effect called A, the expected value of its mean square is
Under the null hypothesis of no A effect, the fixed portion
Q() of the
expected mean square is zero. This mean square is then compared to
another mean square, say MS(E), that is independent of the first
and has expected value . The
ratio of the two mean squares
has the F distribution under the null hypothesis.
When the null hypothesis is false, the numerator term has
a larger expected value, but the expected value of the
denominator remains the same. Thus, large F values lead to
rejection of the null hypothesis. The probability of getting an
F value at least as large as the one observed given that the null
hypothesis is true is called the significance probability value
(or the p-value).
A p-value of less than 0.05, for example, indicates that
data with no real A effect will yield F values as large as
the one observed less than 5% of the time. This is usually
considered moderate evidence that there is a real A effect.
Smaller p-values constitute even stronger evidence.
Larger p-values indicate that the effect of interest is
less than random noise. In this case, you can conclude either that
there is no effect at all or that you do not have enough data to
detect the differences being tested.
F = [ MS(A)/ MS(E)]
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