1. Mixed models

2. 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:

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. Properties of random effects in linear mixed models Notation:

9. 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:

10. 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.

11. 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?

12. 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.

13. 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:

14. 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…

15. 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?”