Basic Ideas of
Variance Components Analysis  Properties of Random Effects
To illustrate some of the properties of random
effects, suppose you collected data on the amount of insect damage
done to different varieties of wheat. It is impractical to study insect
damage for every possible variety of wheat, so to conduct the experiment,
you randomly select four varieties of wheat to study. Plant damage is
rated for up to a maximum of four plots per variety. Ratings are on a
0 (no damage) to 10 (great damage) scale. The following data for this
example are presented in Milliken and Johnson (1992, p. 237); this data
set is also provided in the example data file Wheat.sta.
DATA: wheat.sta 3v 
VARIETY 
PLOT 
DAMAGE 
A 
1 
3.90 
A 
2 
4.05 
A 
3 
4.25 
B 
4 
3.60 
B 
5 
4.20 
B 
6 
4.05 
B 
7 
3.85 
C 
8 
4.15 
C 
9 
4.60 
C 
10 
4.15 
C 
11 
4.40 
D 
12 
3.35 
D 
13 
3.80 
To determine the components of variation in resistance to insect damage
for Variety and Plot,
an ANOVA can first be performed. Perhaps surprisingly, in the ANOVA, Variety can be treated as a fixed or
as a random factor without influencing the results (provided that the
Type I option button on the Variance
Components and Mixed Model ANOVA/ANCOVA Results dialog
 Advanced tab is
selected and that Variety is
always entered first in the model). The spreadsheet below shows the ANOVA
results of a mixed model analysis treating Variety
as a fixed effect and ignoring
Plot, i.e., treating the plottoplot
variation as a measure of random error.
ANOVA Results: DAMAGE (wheat.sta) 
Effect 
Effect
(F/R) 
df
Effect 
MS
Effect 
df
Error 
MS
Error 
F 
p 
{1}VARIETY 
Fixed 
3 
.270053 
9 
.056435 
4.785196 
.029275 
Another way to perform the same mixed model analysis is to treat Variety as a fixed
effects and Plot as a
random effect. The spreadsheet
below shows the ANOVA results for this mixed model analysis.
ANOVA Results for Synthesized Errors: DAMAGE
(wheat.sta) 

df error computed using Satterthwaite method 
Effect 
Effect
(F/R) 
df
Effect 
MS
Effect 
df
Error 
MS
Error 
F 
p 
{1}VARIETY 
Fixed 
3 
.270053 
9 
.056435 
4.785196 
.029275 
{2}PLOT 
Random 
9 
.056435 
 
 
 
 
The spreadsheet below shows the ANOVA results for a random
effect model treating Plot
as a random effect nested within
Variety, which is also treated
as a random effect.
ANOVA Results for Synthesized Errors: DAMAGE
(wheat.sta) 

df error computed using Satterthwaite method 
Effect 
Effect
(F/R) 
df
Effect 
MS
Effect 
df
Error 
MS
Error 
F 
p 
{1}VARIETY 
Random 
3 
.270053 
9 
.056435 
4.785196 
.029275 
{2}PLOT 
Random 
9 
.056435 
 
 
 
 
As can be seen, the tests of significance for the Variety
effect are identical in all three analyses (and in fact, there
are even more ways to produce the same result). When components of variance
are estimated, however, the difference between the mixed model (treating
Variety as fixed) and the random
model (treating Variety as random)
becomes apparent. The spreadsheet below shows the variance
component estimates for the mixed model treating Variety
as a fixed effect.
Components of Variance (wheat.sta) 

Mean Squares Type: 1 
Source 
DAMAGE 
{2}PLOT 
.056435 
Error 
0.000000 
The spreadsheet below shows the variance
component estimates for the random
effects model treating Variety
and Plot as random
effects.
Components of Variance (wheat.sta) 

Mean Squares Type: 1 
Source 
DAMAGE 
{1}VARIETY 
.067186 
{2}PLOT 
.056435 
Error 
0.000000 
As can be seen, the difference in the two sets of estimates is that
a variance component is estimated
for Variety only when it is considered
to be a random effect. This reflects
the basic distinction between fixed and random
effects. The variation in the levels of random factors is assumed
to be representative of the variation of the whole population of possible
levels. Thus, variation in the levels of a random factor can be used to
estimate the population variation. Even more importantly, covariation
between the levels of a random factor and responses on a dependent variable
can be used to estimate the population component of variance in the dependent
variable attributable to the random factor. The variation in the levels
of fixed factors is instead considered to be arbitrarily determined by
the experimenter (i.e., the experimenter can make the levels of a fixed
factor vary as little or as much as desired). Thus, the variation of a
fixed factor cannot be used to estimate its population variance, nor can
the population covariance with the dependent variable be meaningfully
estimated. With this basic distinction between fixed
effects and random effects
in mind, we now can look more closely at the properties of variance components.