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

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Solutions 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. Solution: We read the data into SAS with the program d40p53.sas., and calculate simple means at each time. To display the data, we plot the profile of ramus height over time for each child, super-imposing the average height (see d40p56.sas). From Figure 1, a straight line appears to be able to fit the average profile. However, some subjects have growth curves that appear more closely to fit a quadratic curve. 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. See d50p57.sas. c. Prepare a table that summarizes the results of these models. Let the columns of the table be defined by the following: # Var Random Effects AutoCorr Response Error # Var Parm. -2log(L) Akaike Schwartz 1a cs (cat age) 6.1 - 0.70 2 268.7 -136 -139 1b cs (lin age) 6.1 - 0.68 2 267.4 -136 -138 1c cs (mean) 5.7 - 2.1 2 334.7 -169 -172 2 ar(1) - 0.95 6.89 2 238.6 -121 -124 3 cs+ar(1) 0 0.953 6.89 3 238.6 -122 -126 4 mv 6.3 6.2 5.8 5.6 - 6.4 6.2 5.9 - - 6.9 7.0 - - - 7.5 10 229.0 -124 -136 5a Linear, RC Using Age (8, 8.5, 9, 9.5) 92.2 -10.2 - 1.2 .19 4 234.3 -121 -126 5b Linear, RC Using Time (1,2,3,4) 7.2 -0.6 - 0.3 0.19 4 235.7 -122 -127 5c Linear, RC Using Age (8, 8.5, 9, 9.5) Cell Mean 91.6 -10.1 - 1.2 0.20 4 238.0 -123 -128 5d Linear, RC Using Time (1,2,3,4) 7.2 -0.6 - 0.3 0.20 4 238.0 -123 -128 6 Quadratic,RC 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. Produce a plot of these predicted ramus lengths on the same scale as your plot in a). f. Discuss your results.