Does the factor affect the response?
We initially assume data from a completely randomised experiment with a categorical factor satisfying a normal model of the form:
yij = |
(explained by factor) µi |
+ |
(unexplained) εij |
for i = 1 to g and j = 1 to ni |
where εij ∼ normal (0, σ)
If the factor does not affect the response, the treatment means in the model will all be equal. A test for this therefore involves the hypotheses,
H0 : | µi = µj | for all i and j | |
HA : | µi ≠ µj | for at least some i, j |
However unexplained (random) variation will result in sample treatment means that are unlikely to be equal, even if the factor really has no effect.
How much variation in the observed treatment means is needed for us to conclude that the factor does affect the response?
Both variation between treatment means and variation within treatments must be used to answer this question.
Variation between treatment means
The jittered dot plots below show artificial data that should be considered to have arisen from a completely randomised experiment in which 4 levels of a factor are each applied to 10 experimental units.
Use the slider to alter the difference between the treatment means. Observe that:
Variation within treatments
The diagram below is similar, but the slider adjusts the spread of values observed within each factor level, leaving the treatment means unaltered.
Observe that ...
Are the underlying means equal?
The evidence for a difference between the treatment means depends on both the variation between and within treatments. It is strongest when: