The two level model for count variable analysis has assumed that the counts were all observed for the same number of days. However, this was not the case since the number of treatment days in the period (the 9th field of the input data file) did vary to some degree. Most of the counts were based on the full seven days in the week; however, some observations were made only for 1 day in the given week. To take this into account, we need to specify a so-called OFFSET variable. The offset variable indicates the amount of time that each count is based on. If OFFSET = no is specified, then SuperMix assumes that all counts are based on the same amount of time.

The data and the model without offset variable is available here .

• Analysis
• Discussion of results
• Graphical display

Data

The data for this example are taken from a paper by McKnight and Van Den Eeden (1993), who reported on the number of headaches in a two treatment, multiple period crossover trial. Specifically, the number of headaches per week was repeatedly measured for 27 patients. Following a seven-day placebo run-in period, subjects received either aspartame or placebo in four seven-day treatment periods according to the double-blind crossover treatment design. Each treatment period was separated by a washout day. The sample size is 122. Data for the first 10 observations of all the variables used in this section are shown below in the form of a SuperMix spreadsheet window for aspart.ss3.

The variables of interest are:

• ID is the patient ID (27 patients in total).
• HeadAche is the number of headaches during the week (from 0 to 7).
• Period1 is a period 1 treatment indicator (1 for the first treatment period and 0 otherwise).
• Period2 is a period 2 treatment indicator (1 for the second treatment period and 0 otherwise).
• Period3 is a period 3 treatment indicator (1 for the third treatment period and 0 otherwise).
• Period4 is a period 4 treatment indicator (1 for the fourth treatment period and 0 otherwise).
• DrugAsp indicates the type of drug being used for the treatment, (0 = placebo and 1 = aspartame). 75 observations used placebo and 47 used aspartame.
• Nperiods is the number of periods the individual was observed (from 2 to 5).
• NTDays is the number of treatment days in the period (from 1 to 7).

Analysis

To create the model specifications for this model, start by opening aspart.ss3 in a SuperMix spreadsheet window. Then, use the Open Existing Model Setup option on the File menu to load the Model Setup window for aspart1.mum. On the Configuration screen, extend the title in the Title 1 text box by adding the string "with Offset Variable". Next, click on the Advanced tab of the Model Setup window. Select yes from the Incorporate Time Offset drop-down list to activate the Offset Variable drop-down list box. Select the variable NTDays from the drop-down list of Offset Variable to produce the following Advanced screen.

Save the changes to the file aspart2.mum by using the Save As option on the File menu. To fit the revised model to the data, select the Run option on the Analysis menu to produce the output file aspart2.out. A portion of this output file is shown below.

Discussion of results

Here, we see a marginally significant positive relationship between drug treatment and number of headaches. All time effects are again non-significant.

As mentioned, the empirical Bayes estimates of the random effects are written to the file aspart2.ba2 as shown below at the conclusion of the SuperMix run. The first few lines of this file are shown below.

The file mixreg.ba2 contains four pieces of information per individual:

• the individual's ID,
• the number of repeated observations for that individual,
• the empirical Bayes estimate for that individual (which is the mean of the posterior distribution), and
• the associated posterior standard deviation, and
• the name of the relevant random coefficient.

Since they are estimates of for each individual, the empirical Bayes estimates are expressed on the standard normal scale. Inspection of these estimates indicates that subject 13 has a very high score. This person's estimate of 1.043 (with standard deviation .016) suggests a very high level of headaches. This agrees well with the raw data, which reveals that this person recorded 7 headaches on four occasions and 6 on the only other occasion.

Graphical display

Figure below is a comparison (represented by a dotted line) of the predicted average number of headaches reported by each patient when taking a placebo (left axis) as opposed to the predicted average number when the treatment is aspartame (right axis). From the graphical display, it appears as if all of the lines (each representing a patient) have a positive slope. The slopes become steeper as the number of headaches increases. This suggest an increase, albeit small, in the expected average number of headaches when aspartame is used. Note that patient 13, who reported a consistently high number of headaches at all occasions, was excluded from this graph.

Figure below is a graphical display of the fitted trajectory (solid line) and observed trajectory (dotted line) for a sample of 6 patients. These displays are ordered from a patient who reported a relatively small number of headaches at the different treatment occasions to one who reported a relatively high number of headaches at the treatment occasions. A study of the fitted and observed trajectories reveals that, in general, the model fit is best when the number of headaches is smaller and becomes less accurate as the number of headaches increases. For patient 13, who is not represented in the graphical display, the number of predicted headaches is almost twice the number observed.