Homework Assignments

Some assignments are from Fitzmaurice, Laird, and Ware; others are given in detail below.

• Problem 2.1 is essentially this: retrieve the file lead.txt (data for the succimer group) from the authors' website, and:

  1. Read the data and calculate sample means, standard deviations, and variances, by occasion.
  2. Plot and describe mean blood levels versus time in weeks.
  3. Calculate the (4x4) covariance and correlation matrices.
  4. Verify that the diagonal entries in the covariance matrix (from part 3) equal the variances (from part 1).
  5. Verify that the correlation between baseline and week 1 (from part 3) equals the covariance (from part 3) divided by the product of the standard deviations (from part 1).

• Univariate ANOVA of the TLC data can be carried out as follows: retrieve the file tlc.txt (data for the whole study) from the authors' website, and:

  1. Read the data, using SAS code like that shown in class.
  2. Convert to univariate form using something like this:

     data tlc_uni;
     /* Convert the data from multivariate form (one record per subject, with
      * 4 responses), to univariate form (one record per response, with a new
      * variable 'week' to identify the occasion. */
       set tlc;
       drop y0 -- y6;
       week = 0; y = y0; output;
       week = 1; y = y1; output;
       week = 4; y = y4; output;
       week = 6; y = y6; output;
     run;
     

  3. Use proc glm to fit a model with factors:
     • treatment;
     • child, nested within treatment;
     • week;
     • treatment by week interaction.
     and with child within treatment declared to be a random effect.
  4. If you compare the various Expected Mean Squares (Type III), you'll see that some of the default F-tests are appropriate, but one (for the main effect of treatment) is not.
  5. Add a test statement to get an appropriate test for treatment.

• Problem 5.1 is essentially this: retrieve the file cholesterol.txt (data for a gallstone study) from the authors' website, and:

  1. Read the data in multivariate form.
  2. Calculate sample means, standard deviations, and variances, by occasion and group.
  3. Plot mean serum cholesterol levels by group versus time in months.
  4. Convert to univariate format (5 records per subject, not 4, as it says in the book).
  5. Analyse reponse profiles and test for group x time interaction, assuming unstructured covariances.
  6. Write out the fitted model (baseline time and placebo group as reference groups).
  7. Find the weight matrix L to test the null hypothesis of no group effect on the change over time.
     Note: You can use proc mixed to do much of the work. Use the SAS listing of the estimated fixed effects to see the role of each component of the parameter vector, and then use those roles to construct a contrast statement to test the hypothesis of interest. If you exercise the E option of the contrast statement, SAS will print out the coefficients of the corresponding L matrix (transposed). See the online help to find out why SAS puts some coefficients on interactions when you specify a row with coefficients on main effects but not on the corresponding interactions.
  8. Use the fitted model to construct estimates of the mean response by group and time, and compare with sample means (part 2).
  9. Interpret each fitted regression coefficient.