Dept. of Biostatistics and Epidemiology at the : BioEpi 740: Mixed Models and Analysis of Repeated Measures/Longitudinal Data

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Solution for Assignment 1 ( To Print with WP: be740e8.wpd) A. Use the notation from the first reading to define the population(s), factors, and parameters in the population. Also define parameters corresponding to an average of population means, and a factor level effect, defined as a deviation from the average of population means. See: be740e8a.pdf B. Evaluate the values of the parameters in the populations corresponding to the mean and variance. Also evaluate parameters corresponding to the average of population means, and factor level effect parameters. Verify that the factor level effect parameters sum to zero. The program D40P2.SAS reads in data in file HW1A.TXT and evaluates population parameters, with the following output.   C. Suppose a study is conducted as described by Searle et al. with only "after" treatment measures. Describe a random sampling plan that would correspond to the study design. List the population and assign subject numbers from 1 to 24. Select four consecutive simple random samples without replacement of size 6 each from the population. Assign the 1st selected sample treatment 1, the second treatment 2, etc.. D. Define indices, random variables, and a model that represents response for a randomly selected subject. See be740e8d.pdf E. Conduct the experiment according to the sampling plan that is described in D. Describe how you conducted the experiment, and list the realized values for random variables that you obtain in four columns, with one column representing responses for each factor level. The experiment can be conducted in several ways. We discuss four ways the selection can be made. A. You can go to a random numbers table, and manually randomly selected (using two columns of single digit numbers) numbers without replacement until you get a random permutation of the 24 digits. This process may be faster if you label 24 identical checkers from 1 to 24, and select them in random order from a "hat". Once you have a random permutation, assign the first 6 selected to treatment 1, the second 6 to treatment 2, etc.. B. Use PROC PLAN from SAS, as illustrated in D40P3A.SAS. This program randomly permutes numbers from 1 to 24. Then, treatment 1 is assigned to the first 6 numbers in the permutation, treatment 2 to the second six numbers, etc. These treatment assignments are then combined with the original potentially observable data, as show in the output. C. Use the simple random sampling without replacement program from SAS Usage Manual to obtain a random ordering of subjects. This strategy will not work. We attempt it, and illustrate the problem in D40P3.SAS, with the following output. D. Generate a random permutation of the population listing numbers using a SAS data step, and then use groups of listing numbers (as they appear in the permutation) to identify treatment groups. We illustrate this in four programs. 1. D40P4.SAS creates a random permutation of listing numbers, and illustrates (see output) the details of the SAS code to produce the permutations. 2. d40p4a.sas is an operational version of the permutation program in SAS. Starting with a population of TOTAL units, a random permutation of TSAMP units is selected. The random permutation of unit numbers in the sample is contained in variables PID1 ... up to PID (TSAMP). 3. d40p5.sas illustrates that the permutation problem works correctly by generating 10,000 random permutations, and then showing that the relative frequency of each possible permutation is approximately equal. Since there are 24 possible permutations, we expect each to occur 417 times (see output). 4. d40p6.sas uses the random permutation program to randomly allocate 24 subjects to four treatment groups, with 6 subjects per group (see output). F. Can you tell in your sample whether there is a subject by treatment interaction? Why or why not? In the sample, it is not possible to see whether of not there is a subject by treatment interaction, since one can not disentangle the subject effect from the treatment effect. G. In the potentially observable population in Table 1., can you tell whether there is a subject by treatment interaction. Why or why not? In Table 1, we can see that there is no subject by treatment interaction. This is evident since for each subject, the difference in response between treatments is constant.