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Mixed models
Concepts
• We are often interested in attributing the
variability that is evident in data to the various
categories, or classifications, of the data.
• For example, in a study of basal cell
epithelioma sites, patients might be classified
by gender, age-group, and extent of exposure
to sunshine.
• Table:
• Another example:
Fixed and random effects
• First is the case of parameters being considered as fixed
constants, or we call them, fixed effects. These are the
effects attributable to a finite set of levels of a factor that
occur in the data and which are there because we are
interested in them.
• The second case corresponds to parameters being
considered random, we call them random effects. These
are attributable to a usually finite set of levels of a factor, of
which only a random sample are deemed to occur in the
data.
• For example, four loaves of bread are taken from each of six
batches of bread baked at three different temperatures.
Fixed effect model
• Example 1: Placebo and a drug
Diggle et al. (1994) describe a clinical trial to treat
epileptics with the drug Progabide. We consider a
response which is the number of seizures after
patients were randomly allocated to either the
placebo or the drug.
• Model:
• There are the only two treatment being used, and
in using them there is no thought for any other
treatments. This is the concept of fixed effects.
• Example 2: Comprehension of humor
• A recent study of the comprehension of humor
involved showing three types of cartoons (visual
only, linguistic only, and visual-linguistic
combined) to two groups of adolescent (normal
and learning disabled).
• Suppose the adolescents record scores of 1
through 9, with 9 representing extremely funny
and 1 representing not funny at all.
• Model:
• Because each of the same three cartoon types is
shown to each of the two adolescent groups, this
is an example of two crossed factors, cartoon
type and adolescent group.
• Example 3: Four dose levels of a drug
• Suppose we have a clinical trial in which a
drug is administered at four different dose
levels.
• Model:
• The four dose levels are fixed effects because
they are used in the clinical trial and are the
only dose levels being studied.
• They are the doses on which our attention is
fixed.
Random effect models
• Example 4: Clinics
• Suppose that the clinical trial of example 3 was conducted at 20
different clinics in New York City. Consider just the patients
receiving the dose level numbered 1.
• Model:
• It is not unreasonable to think of those clinics as a random sample
of clinics from some distribution of clinics, perhaps all the clinics in
New York City.
• Note: model here is essentially the same algebraically as in example
3. However, the underlying assumptions are different.
• Characteristic of random effects: they can be used as the basis for
making inferences about populations from which they have come.
• The random effect is a random variable and the data will be useful
for making inference about the variation among clinics; and for
predicting which clinic is likely to have the best reduction of
seizures.
Properties of random effects in linear
mixed models
• Notation:
Example 5: Ball bearings and calipers
• Consider the problem of manufacturing ball bearings
to a specified diameter that must be achieved with a
high degree of accuracy.
• Suppose that each of 100 bal bearings is measured
with each of 20 micrometer calipers, all of the same
brand.
• Model:
• Two random effects: 100 ball bearings being
considered as a random sample from the production
line and 20 calipers considered as a random sample
from some population of available calipers.
• An additional property:
Example 6: Medications and clinics
• Considering four dose levels of example 3 were used in
all 20 clinics of example 4, such that in each clinic each
patient was randomly assigned to one of the dose
levels.
• Model:
• Since the doses are the only doses considered, it is a
fixed effect.
• But the clinics that have been used were chosen
randomly, so it is a random effect.
• The interaction between a fixed effect and random
effect is still a random effect.
• So this is a mixed model.
Example 7: Drying methods and fabrics
• Devore and Peck (1993) report on a study for
assessing the smoothness of washed fabric after
drying.
• Each of nine different fabrics were subjected to
five methods of drying (line drying, line drying
after brief machine tumbling, line drying after
tumbling with softener, line drying with air
movement, and machine drying)
• Method of drying is a (fixed or random) effect?
• Fabric is a fixed or random effect?
Longitudinal data
• A common use of mixed models is in the analysis of
longitudinal data which are defined as data collected
on each subject on two or more occasions.
• Methods of analysis have typically been developed for
the situation where the number of occasions is small
compared to the number of subjects.
• Reasons for using longitudinal analysis:
(1) To increase sensitivity by making within-subject
comparisons
(2) To study changes through time
(3) To use subject efficiently once they are enrolled in the
study.
• The decision as to whether a factor should be fixed or
random in a longitudinal study is often made on the basis of
which effects vary with subjects.
• That is, subjects are regarded as a random sample of a larger
population of subjects and hence any effects that are not
constant for all subjects are regarded as random.
• For example: suppose we are testing a blood pressure drug at
each of two doses and a control dose (dose=0) for each
subject.
• Individuals clearly have different average blood pressures, so
our model should have a separate intercept for each subject.
• Similarly, the response of each subject to increasing dosage
of the drug might vary from subject to subject, so we model
the slope for dose separately for each subject.
• Also assume that blood pressure changes gradually with age.
• Model:
Fixed or random?
• A multicenter clinical trial is designed to judge the
effectiveness of a new surgical procedure.
• If the procedure will eventually become a widespread
procedure practiced at a number of clinics, the we
would like to select a representative collection clinics in
which to test the procedure and we would then regard
the clinics are a random effect.
• However, suppose we change the situation slightly.
Now assume that the surgical procedure is highly
specialized and will be performed mainly at a very few
referral hospitals (Assume that all of those referral
hospitals are enrolled in the trial). In such case…
Making a decision
• The decision as to whether certain effects are fixed or random is
not immediately obvious.
• An important question is: are the levels of the factor going to be
considered a random sample from a population of values which
have a distribution?
If “yes”, then the effects are to be considered as random effects; if
“no”, then fixed.
• When inferences will be made about a distribution of effects from
which those in the data are considered to be a random sample, the
effects are considered as random; when inferences are going to be
confined to the effects in the model, the effects are considered
fixed.
• Another way is to ask the questions: “Do the levels of a factor come
from a probability distribution?” and “Is there enough information
about a factor to decide that the levels of it in the data are like a
random sample?”