Fixed effects Completely Randomized Design model
In this model,
yij =µ+τi+εij,
the tvalues of τi,τ1, τ2, ..., τt,are the only effects of interest, and εij is the
only random term, εij has a normal distribution with mean 0 and variance
σ2
e(εij ∼N(0, σ2
e) ).
Random effects Completely Randomized Design model
In this model,
yij =µ+τi+εij,
The τiterms are now random, τi∼N(0 , σ2
t) , and the τiand εij are
independent.
Now the τi‘s are viewed as a random sample from a population of τi‘s,
and the ANOVA H0is H0:σ2
τ= 0 vs. Ha:σ2
τ>0 .
How do you know if an effect is fixed or random?
1. How were the levels for the factor in question chosen?
2. Is it desired to generalize the results to levels that weren’t used?
Implications for random effects
1. The denominator of the F statistic may change.
2. Generally you are not interested in doing multiple comparisons. There
may be interest in estimating σ2
τ,σ2
e, etc.
3. Estimating σ2
tmay still be of interest even if we reject H0:σ2
τβ = 0
Random effects model for Randomized Block Design
The model is:
yij =µ+τi+βj+εij,
Where τi∼N(0, σ2
τ), βj∼N(0, σ2
β), εij ∼N(0, σ2
e),and τi,βj, and
εij are mutually independent. For the RB design (no replication) we assume
that σ2
τβ = 0.
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