Specifying
Fixed and Random Factors in Mixed or Multi-Level Models
Since SAS introduced
Proc Mixed about fifteen years ago, S-Plus, Stata and SPSS have
implemented procedures to analyze mixed models, greatly broadening the
options available to researchers. These programs require correctly
specifying the fixed and random factors of the model to obtain accurate
analyses. The definitions in many texts often do not help with
decisions to specify factors as fixed or random, since textbook
examples are often artificial and hard to apply. Furthermore, the same
factor can often be considered fixed or random, depending on the
objective; This article outlines a different way to think about
fixed and random factors.
Consider an experiment that
examines beetle damage on cucumbers. The experiment is replicated at
five farms and on four fields at each farm. There are two varieties of
cucumbers, and beetle damage is assessed on each of 50 plants at the
end of the season. The researcher is interested in comparing
differences in how much damage the two varieties sustain. The
experiment then has the following factors: VARIETY, FARM, and FIELD.
Fixed factors can be thought of
in terms of differences. The effect of a categorical fixed factor is
defined by differences from the overall mean and the effect of a
continuous fixed factor is defined by its slope--how the mean of the
dependent variable differs with alternate values of the factor. The
output for fixed factors provides estimates for mean-differences or
slopes. Conclusions regarding fixed factors are particular to the
values of these factors. For example, if one variety of cucumber is
found to suffer significantly less damage than the other, this says
nothing about cucumber varieties that were not tested.
Random factors, on the other
hand, are defined by a distribution and not by differences. The values
of a random factor are assumed to be chosen from a population with a
normal distribution with a certain variance. The output for a random
factor is an estimate of this variance and not a set of differences
from a mean. Conclusions regarding random factors should be expressed
in terms of variance. For example, we may find that the variability
among fields makes up a certain percentage of the overall variability
in beetle damage.
Situations that indicate fixed
factors:
The factor is the
primary treatment that the researcher wants to compare. In
our example, VARIETY is definitely fixed as the researcher wants to
compare the mean beetle damage on the two varieties.
The factor is a
secondary covariate that might be confounded with the treatment, and
the researcher wants to control for differences in this covariate. If
these farms were specifically chosen for some feature they had, such as
specific soil types or topographies that may affect beetle damage, and
if the researcher would like to compare the farms as representatives of
those soil types, then FARM should be fixed.
The factor has only
two values. Even if everything else indicates that a factor
should be random, if it has only two values, the variance cannot be
calculated, and it should be fixed.
Situations that indicate
random factors:
The researcher is
interested in quantifying how much of the overall variation to
attribute to this factor. If the researcher was
interested in how much of the variation in beetle damage was
attributable to the farm at which the damage took place, FARM would be
random.
The researcher is
not interested in knowing which means differ, but wants to account for
the variation in this factor. If the farms were chosen
at random, not for a specific feature, but because the researcher
suspected that there is some variation in their soil types, which is
representative of the variation across all farms, FARM should be random.
The researcher would
like to generalize the conclusions about this factor to the whole
population. There is nothing about comparing these
specific fields that is of interest to the researcher. Rather, the
researcher wants to generalize the results of this experiment to all
fields, so FIELD is random.
Any interaction with
a random factor is also random.
How the factors of a model are
specified can have great influence on the results of the analysis and
on the conclusions drawn.
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