Mixed models

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Mixed models
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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

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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.

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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.

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• 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.

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• 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.

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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.

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Properties of random effects in linear mixed
models

• Notation

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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

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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.

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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?

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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
• To increase sensitivity by making within-subject
  comparisons
• To study changes through time
• To use subject efficiently once they are enrolled
  in the study.

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• 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 (dose0) 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

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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

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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?