Refer to Rehabilitation therapy Problems 16.9 and 22.11. Assume that covariance model (22.3) is applicable.
a. Prepare a symbolic scatter plot of the data. Does it appear that there are effects of physical fitness status on the mean number of days required for therapy? Discuss.
b. State the regression model equivalent to covariance model (22.3) for this case; use 1, -1, indicator variables. Also state the reduced regression model for testing for treatment effects.
c. Fit the full and reduced regression models and test for treatment effects; use α = .01. State the alternatives, decision rule, and conclusion. What is the P-value of the test?
d. Is MSE(F) for the covariance model substantially smaller than MSE for the analysis of variance model in Problem 16.9d? Does this affect the conclusion reached about treatment effects? Does it affect the P-value?
e. Estimate the mean number of days required for therapy for patients of average physical fitness and age 24 years; use a 99 percent confidence interval.
f. Make all pairwise comparisons between the treatment effects; use either the Bonferroni or the Scheffe procedure with a 95 percent family confidence coefficient, whichever is more efficient. State your findings.
Problems 22.11
Refer to Rehabilitation therapy Problem 16.9. The rehabilitation researcher wishes to use age of patient as a concomitant variable. The ages (Xij) of patients in the study follow.
a. Obtain the residuals for covariance model (22.3).
b. For each treatment, plot the residuals against the fitted values. Also prepare a normal probability plot of the residuals and calculate the coefficient of correlation between the ordered residuals and their expected values under normality. What do you conclude from your analysis?
c. State the generalized regression model to be employed for testing whether or not the
treatment regression lines have the same slope. Conduct this test using α = .05. State the
alternatives, decision rule, and conclusion. What is the P-value of the test?
d. Could you conduct a formal test here as to whether the regression functions are linear?
Explain.
Problem 16.9
Rehabilitation therapy. A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and time required in physical therapy until successful rehabilitation. Patient records in the rehabilitation center were examined, and 24 male subjects ranging in age from 18 to 30 years who had undergone similar corrective knee surgery during the past year were selected for the study. The number of days required tor successful completion of physical therapy and the prior physical fitness status
Assume that ANOYA model (16.2) is appropriate.
a. Prepare aligned dot plots of the data. Do the factor level means appear to differ? Does the variability of the observations within each factor level appear to be approximately the same for all factor levels?
b. Obtain the fitted values.
c. Obtain the residuals. Do they sum to zero in accord with (16.21)?
d. Obtain the analysis of variance table.
e. Test whether or not the mean number of days required for successful rehabilitation is the same for the three fitness groups. Control the α risk at .01. State the alternatives, decision rule, and conclusion.
f. Obtain the P-value for the test in part (e). Explain how the same conclusion reached in part (e) can be obtained by knowing the P-value.
g. What appears to be the nature of the relationship between physical fitness status and duration of required physical therapy?