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

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Final Exam Spring 1999 1. Grizzle and Allen (1969) describe ramus height data (measured in mm) on a cohort of boys at ages 8, 8.5, 9, and 9.5 years. The objective is to establish a normal growth curve for use with orthodontists. The authors state that in the age range considered, a straight line should fit the data. a. Write a descriptive report that summarizes these data including profiles of ramus growth for individuals. b. Fit a variety of possbily appropriate mixed models using alternative variance structures to these data, including a compound symmetric model, a 1st order autoregressive model, a combination of compound symmetry and auto-regression, a multivariate model and a random coefficient model. Include a copy of your computer program in an appendix. c. Prepare a table that summarizes the results of these models. Let the columns of the table be defined by the following: Model number Type of Variance Random Effect Variance component estimates (or sub-matrix) Autocorrelation (estimate) Response Error (estimate) Number of variance parameters -2 Log(likelihood) Akaike's Information Criterion Schwartz's Bayesian Criterion d. Using likelihood ratio tests, AIC or SBC criterion, which model appears to be best? e. Develop a set of predicted ramus lengths (and contours that are 1 SD above and below the predicted length) based on model that you select. Use PROC IML to form the predicted values, and the contours of the predicted values. (Ask if you need help.) Produce a plot of these predicted ramus lengths on the same scale as your plot in a). f. Discuss your results. g. Include an appendix with a paper copy of all programs used.  2. Data are described by Littell et al (1996) for a study of weight gain among steers with a food addititive. The goal of the study is to determine the opimal food additive dose. Data are given by Littell et al.. As an introduction, read the description of the problem given on pages 177-187. a. Write a 2-3 page report that introduces and describes the study, and presents descriptive results. Include with your summary profiles of weight for animals over time by treatment. Also include the variance matrix for (initial, final) weight for animals in each level of treatment. b. Littell et al describe the design as a randomized block design. The blocks in this description are barns, and units in barns are steers. Steers are randomly assigned to levels of treatment, with two steers assigned to each treatment. The model used considers barns to be random effects. Write a summary that can be used to explain in what sense Barns are random effects, and what is meant by a barn effect. Define a potentially observable population, where a parameter for a barn effect can be calculated, and use this in your description. c. Fit the randomized block model to gain in weight. Ignore initial weight in the model. Summarize the results including estimates of variance components (and their SD), estimates of means for each treatment group (and their SE), and goodness of fit statistics. d. Repeat analysis c) using pretest weight as a covariate. Summarize these results, and discuss the difference between these results and the results in c). e. Rearrange your data so that you have two measures per subject represeting initial and final weight. Also create a time variable having value 0 for the initial weight and 1 for the final weight. Fit a nested mixed model to these data using a treatment by time interaction to test for treatment effects, and treating steers in barns as nested random effects. Compare the results of this model with the model that you fit in part d). f. Rescale treatment to represent a continuous variable, and include both a linear and quadratic effect. Using the model from d) and e), estimate the gain in weight by dose, and a 95% confidence interval for your estimates. Plot these estimates with the 95% CI band.