171:162 Design and Analysis of Biomedical Studies, Summer 2011 Exam #3, July 16th

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1 Name 171:162 Design and Analysis of Biomedical Studies, Summer 2011 Exam #3, July 16th Use the selected SAS output to help you answer the questions. The SAS output is all at the back of the exam on pages Feel free to remove the output when answering the questions. Use α =.05 on all tests of hypothesis unless explicitly stated otherwise. 1. (8 pts) The skin response (Y) in rats to different concentrations (X, conc) of a newly developed vaccine was measured in an experiment, resulting in the data below. The investigators wish to know whether or not they should fit a quadratic model or straight-line regression model to the data. Using the SAS output on page 8, explain whether or not a quadratic model should be used for this data. Be sure to support your conclusion statistically. Conc Skin Response (8 pts) Now consider this question via a modeling framework. Compute the overall R 2 for both the quadratic model and the straight-line regression model. Based on the R 2 values, would you choose the same regression equation as you did in question (2)? Why or why not? Page 1 of 13

2 3. (6 pts) The scatterplot below depicts the relationship between skin response (Y) and concentration level (X). Given this scatterplot and your responses to questions (1) and (2) above, provide a reasonable model choice or alternative strategy for the researchers to consider. 4. Suppose 10 healthy men and 11 healthy women were studied during a maximal exercise treadmill test. While on the treadmill, they had a catheter (tube) inserted into the pulmonary artery and a short tube into the brachial artery. This allowed sampling and observation of arterial pressures and the oxygen content of the blood. From this, several parameters as described below were measured or calculated. Variable Description weight Weight in kilograms height Height in cm VO2max Oxygen used in 1 min at maximum exercise FAI Functional aerobic impairment HRmax Heart Rate in BPM at maximum exercise Pressure_AS Average pressure in arterial system mmhg Pressure_PA Average pressure in the pulmonary artery mmhg Gender Male (0) or Female (1) Age Age in years Page 2 of 13

3 a. (6 pts) A stepwise selection routine was used to determine the best fitting model for these data (α=0.15). All main effects were considered for the maximum model (and no interactions). SAS output (portions omitted) is provided on pages Write out the final fitted regression equation and explain how it was chosen (i.e. which variables were entered/removed from the model at each step?) b. (4 pts) Name one advantage and one disadvantage when using this selection routine. 5. After carefully considering the results of a few model selection routines and clinical relevance of variables in the dataset, the investigators chose to fit the model found in part (a) of problem 4. The regression diagnostics for this model are provided in a SAS diagnostics panel on page 11. Use this panel to answer the following questions. Page 3 of 13

4 a. (10 pts) Do the assumptions of linearity, homoscedasticity or normality appear to be violated? Be sure to address each assumption separately and defend your answer with specific information from the diagnostics panel. b. (8 pts) Do any of the observations exhibit high leverage and/or high influence? Defend your response with necessary calculations and evidence from the panel. c. (5 pts) When conducting a residual analysis, why would an investigator choose to consider jackknife residuals instead of studentized residuals? Page 4 of 13

5 6. (8 pts) Suppose an investigator was interested in studying the risk factors in patients undergoing coronary bypass surgery for coronary artery disease. The authors wish to look for an association between cholesterol level (a risk factor) and the number of diseased blood vessels (1, 2, 3 or 4). The researcher planned to use an ordinary regression model to estimate the mean cholesterol level for each number of diseased blood vessels (while considering the number of diseased blood vessels as a continuous variable). However, the researcher is undecided between fitting a linear or a quadratic regression model, and the data do not give clear evidence in favor of one model or the other. A colleague suggests: For your purposes you might simply use an ANOVA model. Is this a useful suggestion? Explain. 7. Twenty-two young asthmatic volunteers were studied to assess the short-term effects of sulfur dioxide (SO 2 ) exposure under various conditions. The asthmatic volunteers were stratified into three groups based on lung function (Group A (<75%), group B (75-84%), and group C (>84%)) Bronchial reactivity to SO 2 was measured for each asthmatic volunteer at the initial screening. You will find the SAS output on pages helpful. a. (5 pts) Would a one-way fixed effects ANOVA or one-way random effects ANOVA be more appropriate here? Explain. Page 5 of 13

6 b. (5 pts) Write out the ANOVA model for the method you chose in part (a). Be sure to include any necessary distributional assumptions. c. (6 pts) Carry out the test you specified in part (a) to test whether there is a difference in mean bronchial reactivity among the three lung-function groups. d. (6 pts) If you find a difference in part (c), compare the means using the Tukey-Kramer approach. Or explain why comparing means is unnecessary. Page 6 of 13

7 e. (5 pts) Suppose you wished to use a Bonferroni correction instead of the Tukey-Kramer approach. What would your alpha level have to be so as to fix the overall type I error at α=0.05? f. (10 pts) Assuming reference cell coding, write this ANOVA model as a regression model for the investigators (be sure to include your definition of dummy variables). Page 7 of 13

8 SAS OUTPUT QUESTIONS #1 and #2 The GLM Procedure Dependent Variable: skin_resp Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F conc conc Standard Parameter Estimate Error t Value Pr > t Intercept <.0001 conc conc Page 8 of 13

9 SAS OUTPUT QUESTION #4 The REG Procedure Model: MODEL1 Dependent Variable: VO2max VO2max Number of Observations Read 21 Number of Observations Used 21 Stepwise Selection: Step 1 Variable Age Entered: R-Square = and C(p) = Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept <.0001 Age Bounds on condition number: 1, Stepwise Selection: Step 2 Variable Gender Entered: R-Square = and C(p) = Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Page 9 of 13

10 The REG Procedure Model: MODEL1 Dependent Variable: VO2max VO2max Stepwise Selection: Step 2 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept <.0001 Gender Age Bounds on condition number: , Stepwise Selection: Step 3 Variable FAI Entered: R-Square = and C(p) = Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept <.0001 Gender <.0001 Age <.0001 FAI <.0001 Bounds on condition number: 1.313, All variables left in the model are significant at the level. No other variable met the significance level for entry into the model. Page 10 of 13

11 SAS OUTPUT (Diagnostics Panel) QUESTION #5 Page 11 of 13

12 The MEANS Procedure Analysis Variable : reactivity N group Obs N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ A B C ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Dependent Variable: reactivity The GLM Procedure Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE reactivity Mean Source DF Type I SS Mean Square F Value Pr > F group Source DF Type III SS Mean Square F Value Pr > F group Page 12 of 13

13 The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Tukey-Kramer reactivity LSMEAN group LSMEAN Number A B C Least Squares Means for effect group Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: reactivity i/j Page 13 of 13