Homework #3 STAT 506 Spring 2013

1. Use these data. The experimental units are 12 thirsty albino rats who are trained to press a lever to get water prior to the experiment. Their pre-experiment pressing rate is recorded as low (1), medium (2), or high (3). They are then injected with one of four levels of a drug where 0 is a control saline solution, the other values are mg per kg of the rat's weight. This was a cross-over design replicated twice, so each rat has 8 measurements of postRate (number of lever presses per second), two at each of four drug levels, and the order of treatments was randomized for each rat. Time intervals between treatments were long enough to remove any carryover effects.
   1. Begin with appropriate plot(s) to examine how postRate changes with the two factors. Describe what you see.
   2. Write out a model for these data using preRate as a three-level factor and drug as a four-level factor. Include distributions for all random components (assuming normality throughout).
   3. Omit This part Fit the above model to the data using PROC GLM in SAS using a repeated statement for ratID. Explain the results.
   4. Fit the above model to the data using PROC MIXED in SAS using a random term for each rat. Explain results and compare to those just above.
   5. Compare results with those we got last semester in HW5 using R.
2. Load the Sitka data (from the MASS library in R) on the growth of 79 sitka spruce trees.
   1. Either include a nice R plot from last semester, or build a similar plot in SAS. Discuss the relationship between time and volume.
   2. Use PROC GLM to fit a quadratic model across all the data. Update the model adding treatment effects which allow the intercept, slope, or quadratic coefficients to depend on treatment.
   3. Using PROC MIXED
      1. Add AR1 correlation structure (within a tree).
      2. Add compound symmetric correlation (within a tree).
      3. Add symmetric correlation (within a tree).
      Compare the four models using AIC.
   4. Reduce the model one term at a time until all terms have small p-values in the t-tests.
   5. Plot the residuals versus fitted and normal quantile plots. Discuss any problems.
   6. How does the ozone treatment affect growth of these trees? Does SAS give results just like those of R?
3. In a study of soil properties, samples were taken on a 10 point by 25 point grid. Download the data here We'll work with two variables: response Ca (calcium concentration) and predictor pH (low numbers are acidic, high numbers basic, 7 is neither).
   1. Make a scatterplot of the two variables and fit a model for Ca based on pH. (Choose the form of the model based on the scatterplot.) Print the estimated coefficients and discuss the relationship.
   2. Fit the five forms of spatial correlation available in the nlme library. Correction: I haven't found rational quadratic in SAS. You can skip it. Compare them with each other and with the original model. Do any of the spatial correlation fits improve AIC by more than 2 units? Which is the best of the 5 fits?
      BTW, one way to get AIC is to run the original (no correlation) model in Proc Mixed.
   3. Compare results with those we got last semester in HW6 using R.