Random and Mixed Effects Models
Random vs. Fixed
* Random and Fixed Variables. A "Fixed variable"
is one that is assumed to be measured without error.
It is also assumed that the values of a Fixed variable
in one study are the same as the values of the Fixed
variable in another study. "Random variables" are
assumed to be values that are drawn from a larger
population of values and thus will represent them.
You can think of the values of random variables as
representing a random sample of all possible values
of that variable. Thus, we expect to generalize the
results obtained with a random variable to all other
possible values of that random variable. Most of the
time in ANOVA and regression analysis we assume
the independent variables are Fixed.
* Random and Fixed Effects. The terms "random"
and "Fixed" are used in the context of ANOVA
1
and regression models, and refer to a certain type
of statistical model. Almost always, researchers
use Fixed effects regression or ANOVA and they
are rarely faced with a situation involving random
effects analyses. A Fixed effects ANOVA refers to
assumptions about the independent variable and
the error distribution for the variable. However,
if the researcher wants to make inferences beyond
the particular values of the independent variable
used in the study, a random effects model is used.
Random effects models are sometimes referred to
as "Model II" or "variance component models."
Analyses using both Fixed and random effects are
called "mixed models."
 Random of Fixed?
Example 1:
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.
Example 2:
a pharmaceutical company came up with a growth
hormone to be fed to cattle which would resulted in a
greater amount of edible meat on each cow. The
hormone had been tested for safety and for efficacy
before the study described here (i.e. it was known to
have some effect and believed to be safe). However,
concerns developed that some batches seemed to be
more effective than others. The company claimed that
the difference was due to other differences between
different herds.
The following study was set up to see if that was the
case. Five herds of cattle were randomly selected and
each herd split into four parts. Four batches of
hormone-enhanced feed were randomly selected and
some of each batch was sent to each of the farms in
the trial. On each farm, the herd was split into four
parts and each part fed a different batch of hormone.
From each herd/hormone combination, the weights of
the First 50 cows to be reared on the hormoneenhanced feed were recorded and a two-way ANOVA
carried out to determine whether there was
inconsistency between the batches.
* The cell means version of ANOVA model for
single-factor studies is as follows when all factor level
sample sizes are equal, ni = n
3
4
Questions of Interest
* There is no interest in inference about the particular
µi, but rather in the entire population of the µi,
specifically in center µ and variability .
The effect of variability of the ¹is is often measured
relative to the total variability, which is in fact is
the corelation between any two responses from the
same factor
* Consider the test whether all ¹i are equal:
H0 implies that all µ i are equal. Reject H0 if
ANOVA Model III-Mixed Factor Effects
* When one of the two factors has Fixed factor levels
which the other has random factor levels, a mixed
factor effects
applicable.
ANOVA
model
(model
III)
Important Features of the Model
* Yijk are normally distributed with
ANOVA tests for Model II and III Expected means
squares are given in Table 25.5, p.1052 and Test
Statistics for Balanced Two-Factor ANOVA Models
are given in Table 25.6, p.1053
is
Review
The distinction between a Fixed and random effect
factor often depends on what would happen if the
experiment were to be repeated. If the experiment
were to be repeated, would the same levels be chosen
(Fixed effects) or would a new set of levels be chosen
(random effects).
The distinction can also be made about the scope of
inference for the experiment. Is the experiment
interested in the effects of the levels that actually
occurred in the experiment (Fixed effects) or do you
wish to generalize to a large population of levels from
which you happened to choose a few for this
experiment (random effects).
It turns out that if a factor is a random effect, the
design of the experiment is affected very little, but
there can be dramatic changes in how the experiment
is analyzed. If a factor is a random effect, then the
only change in the design of the experiment has to do
with the selection of levels that will appear in the
experiment. As the name implies, levels of random
effects are to be chosen at RANDOM from the
population of levels of interest. To summarize, a factor
is a Fixed effect if: the same levels would be used if
the experiment were to be repeated; inference will be
limited ONLY to the levels used in the experiment;
A factor is a random effect if: the levels were chosen
at random from a larger set of levels new levels would
be chosen if the experiment were to be repeated
inference is about the entire set of potential levels - not
just the levels chosen in the experiment.
Typical Fixed effects are factors such as gender type,
species, dose amount, and chemical. Typical random
effects are subject, locations, sites, animals.
Failure to specify that a factor is a random effect will
lead to erroneous conclusions and inappropriate
analyses.
In designs where all factors are Fixed, the focus of the
ANOVA procedure started with hypothesis testing
(usually with hypotheses about interactions being
tested prior to hypotheses about main effects). Then
estimates of effect sizes were obtained along with
measures of precision (standard errors).
In design with Fixed and random factors, hypothesis
testing only makes sense for Fixed effect factors. The
focus on random effect factors is on estimating just
how variable the results could be when a new sample
of random levels are chosen.
Example - Rancid fat
A study was conducted to investigate the effects of
irradiating fat with gamma radiation to prevent it from
going rancid. (This is a proposed treatment for many
foods to kill many of the bacteria which cause foods to
spoil. The promoters of this treatment claim that it
doesn't affect taste or nutrition, and is perfectly safe).
There are of course, many machines available to
perform the irradiation. In this experiment, 12 batches
of fat were obtained and split between two machines.
Three of the samples were irradiated by each machine
and three of the samples were not treated and served
as controls. Note that the control was subject to the
same processing as the treated samples (e.g. passed
through the machine) except that they were not
irradiated. Then 12 rates (all aged 30 to 34 days) were
obtained, and randomly assigned to the 12 batches of
fat. The rats were allowed to feed ad libitum and the
total consumption of fat (grams) was noted over 73
days. What are the factors in this experiment?
Their levels? What is the response variable? What is
the treatment structure in this experiment? What are
the sources of random variation? What is the type of
design used?
Example - Mosquito repellent
Biting insects can be a real pest and a health hazard
- e.g. malaria and equine encephilitus are serious
diseases transmitted by mosquitoes. What is the best
method of deterring these pesky critters from biting?
There are a number of insect repellents available on
the market place. Some use the chemical DEET which
is quite effective but many people are reluctant to use
these sprays because they are quite strong - e.g., many
sprays containing DEET will soften paint. As an
alternative, there is a strong urban legend about an
Avon (a perfume and toiletry company) product called
Skin so soft that many people claim is also an
effective repellent. To investigate these claims,
twenty-four volunteers were recruited. These were
randomly assigned to 3 groups of 8 people which then
went to 3 locations on the University Campus. At each
location, half of the volunteers spread a DEET product
on their right arm; the other half used the Avon
product on their right arm. Each subject stood at least
10 m from any other subject. Then the subjects let
mosquitos bite their exposed arm, and after 15
minutes, the total number and severity of the bites was
scored using a standard scale for such studies (how
some one came up this scale I can only hazard a
guess!). The higher the score, the worse the biting
experience. What are the factors in this experiment?
Their levels? What is the response variable? What is
the treatment structure in this experiment? What are
the sources of random
variation? What is the type of design used?
Example - Effect of photo-period and temperature
on gonadosomatic index The Mirogrex terrau-sanctae
is a commercial sardine like Fish found in the Sea of
Galilee. A study was conducted to determine the effect
of light and temperature on the gonadosomatic index
(GSI), which is a measure of the growth of the ovary.
[It is the ratio of the gonad weight to the non-gonad
weight.]
Two photo-periods 14 hours of light, 10 hours of dark
and 9 hours of light, 15 hours of dark and two
temperature levels 16 and 27 C are used. In this way,
the experimenter can simulate both winter and
summer conditions in the region. Twenty females
were collected in June. This group was randomly
divided into four subgroups - each of size 5. Each Fish
was placed in an individual tank, and received one of
the four possible treatment combinations. At the end
of 3 months, the GSI was measured.