Education 257 HW1 Jan 13 2005 ("Due" Jan 22 2005) ------------------------------------------------------------------------ Preliminaries: Due date indicates approximate date for posting solutions Data sets: unless otherwise indicated (or part of web examples) data reside in the class HW directory URL is http://www-stat.stanford.edu/~rag/ed257/hw/[file] ------------------------------------------------------------------------- 1. From Neter Wasserman Kutner A rehabililitation 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 sucessful rehabilitation. 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. In the data file knee.dat c1 contains the number of days required for sucessful completion of physical therapy and c2 contains an indicator of prior physical fitness status-- 1 = below average; 2 = average; 3 = above average. (So this data set is of the form of a time-to-mastery study.) ------- a) obtain mean and variance of time to recovery for each group b) present a graphical look at the scores for the three groups by constucting aligned dotplots for the three groups c) carry out an anova for this one-way classification and test the omnibus null hypothesis of no differences between the group means using Type I error rate .05. d) display residuals from the fit of the anova model for each group. e) carry out the post-hoc pairwise comparison procedure in order to obtain interval estimates of each pairwise comparison using experimentwise error rate .05. f) in planning a follow-up study which will have equal numbers of subjects in each group, how many subjects should there be in each group so that the interval estimate for these pairwise comparisons will have width of 5 days (again using experimentwise error rate .05)? ----------------------------------------------------------------- 2. An experiment was conducted to examine the effects of different levels of reinforcement and different levels of isolation on children's ability to recall. A single analyst was to work with a random sample of 30 children selected from a relatively homogeneous group of fourth-grade students. Two levels of reinforcement (none and verbal) and three levels of isolation (20, 40, and 60 minutes) were to be used. Students were randomly assigned to the six treatment groups, with a total of six students being assigned to each group. Each student was to spend a 30-minute session with the analyst. During this time the student was to memorize a specific passage, with reinforcement provided as dictated by the group to which the student was assigned. Following the 30-minute session, the student was isolated for the time specified for his or her group and then tested for recall of the memorized passage. These data appear in the accompanying table. Time of Isolation (Minutes) Level of Reinforcement 20 40 60 26 19 30 36 6 10 None 23 18 25 28 11 14 28 25 27 24 17 19 15 16 24 26 31 38 Verbal 24 22 29 27 29 34 25 21 23 21 35 30 Clearly, both factors are fixed factors. a. Construct a profile plot and comment. b. Write out the statistical model for this two-way classification c. Carry out the series of hypothesis tests for the two-way anova. Keep your overall error rate at or below .05 for the 3 tests. ------------------------------------------------------------------------ 3. Prior to conducting a clinical trial that involves a subjective evaluation of a patient's progress, the participating physicians are asked to agree on certain criteria for reaching an evaluation. To examine the consistency in their evaluations before the initiation of a particular clinical trial, a pilot study was conducted on four patients who had been treated with a drug that was to be included in the trial. Each of the five physicians who were to participate in the study was asked to evaluate (on a 0-to-l0-point scale) the degree of cure after a two-week treatment period. Since the clinical evaluations of a patient's cure were to be based on the results of a bacterial culture analysis, each physician analyzed two cultures from each patient. This feature was unknown to the physicians, who were merely told they would be analyzing eight separate bacterial cultures. The evaluations based on these cultures are recorded here. a. Treating physicians as fixed and patients as random, write an appropriate model. Identify all terms in the model. b. Construct the AOV table. Show the expected mean squares. Test hypotheses for main effects and interactions. c. For the fixed factor carry out pairwise comparisons using Tukey method. Patient physician 1 2 3 4 1 7.2 4.2 9.5 5.4 9.6 3.5 9.3 3.9 2 8.5 2.9 8.8 6.3 9.6 3.3 9.2 6.0 3 9.1 1.8 7.6 6.1 8.6 2.4 7.1 5.6 4 8.2 3.6 7.3 5.0 9.0 4.4 7.0 5.4 5 7.8 3.7 9.2 6.5 8.0 3.9 8.3 6.9 ------------------------------------------------------------------ 4. Unbalanced 2-way designs: Why pay 'em anything? A sociologist selected a random sample of 45 adjunct professors who teach in the evening division of a large metropolitan university for a study of special problems associated with teaching in the evening division. The data collected include the amount of payment received by the faculty member for teaching a course during the past semester. The sociologist classified the faculty members by subject matter of course (C2 factor A; i = 1,...4 {Humanities, Social Sciences, Engineering, Management}) and highest degree earned (C3 factor B; j = 1,2,3 {Bachelor, Master, Ph.D.}). The earnings per course (in thousand dollars) are given in C1 in these data. output given below. The full data are given in NWK. The data are also in file adjprof.dat a. Construct a profile (cell mean) plot for the cell means for this 4 x 3 data structure. Comment on the appearance of main effects and interactions. b. From the GLM output below (some entries obscured by &&&&) carry out a test of Ho: alpha(i) = 0 vs. Ha: not all alpha(i) = 0 using Type I error rate .01. c. These data reside in file adjprof.dat in the class directory. Replicate the full GLM analysis which is given in abbreviated form below. Compare the GLM results--e.g. the anova Table and test statistics-- with the approximate solution (unweighted means) described in class (adapted from Miller). MTB > table c2 c3; SUBC> mean c1; SUBC> count. ROWS: C2 COLUMNS: C3 1 2 3 ALL 1 1.8000 1.9500 2.7000 2.4250 2 2 8 12 2 2.4500 2.5200 3.4500 2.7846 4 5 4 13 3 2.7500 2.8500 3.7400 3.2364 2 4 5 11 4 2.5500 2.5500 3.4200 3.0333 2 2 5 9 ALL 2.4000 2.5385 3.2364 2.8489 10 13 22 45 CELL CONTENTS --C1:MEAN COUNT MTB > glm c1 = c2|c3 Factor Levels Values C2 4 1 2 3 4 C3 3 1 2 3 Analysis of Variance for C1 Source DF Seq SS Adj SS Adj MS F P C2 && 4.1676 4.2326 &&&&&& &&&&& &&&&& C3 && 8.3825 8.2287 &&&&&& &&&&&& &&&&& C2*C3 && 0.0444 0.0444 &&&&&& &&&& &&&&& Error && 0.7180 0.7180 &&&&&& Total 44 13.3124 ------------------------------------------------------ 5. In the rehabililitation center example in problem #1 Obtain the power of the test in 1(c) if the population group mean are 37, 35, 28 and the within-cell variance is 4.5 ---------------------------------------------------- Randomized Blocks Designs ------------------------------------------------------- 6. We revisit the in class example of the experiment on treating depression by the Imipramine, an anti-depressant drug. The example is taken from the text "The Design and Analysis of Clinical Experiments" by J L Fleiss. A total of 60 patients were paired on age sex time of entry in study to form 30 matched pairs or blocks. One member of each pair was randomly asssigned to receive Imipramine and the other to receive a placebo. The outcome measure was the score on the Hamilton rating scale for depression (higher score = more severe depression) after 5 weeks of treatment. The file depress.dat in the class directory contains the outcome scores for each of the 30 pairs, c1 Imipramine, c2 Placebo. a. For this randomized block design, carry out an anova to test the equality of the treatment outcomes. State your conclusion. b. Give a measure of the efficiency of the randomized block design relative to the completely randomzed design. What is the importance of this efficiency measure? c. Analyze these same data using the paired t-test. Compare your results. ------------------------------------------------------------------ 7. Randomized Block Designs Statistical training, Tax audits, scary stuff? These data come from the "Auditor Training" example, NWK An accounting firm, prior to introducing in the firm widespread training in statistical sampling for auditing, tested three training mathods: (1) study at home with programmed training materials (2) training sessions conducted at local offices conducted by local staff (3) training sessions in Chicago conducted by national staff Thirty auditors were grouped into 10 blocks of 3, according to time elapsed since college graduation, and the three auditors in each block were randomly assigned to the three training methods. At the end of the training, each auditor was asked to analyze a complex case involving statistical applications; a proficiency measure based on this analysis was obtained for each auditor. In file audit.dat the columns are proficiency measure; block; training method a. Write an appropriate linear statistical model for this randomized blocks design. List the assumptions and identify terms. b. Construct a profile plot by plotting proficiency by blocks. What does this plot suggest about the appropriateness of the no interaction assumption here? c. Obtain an anova table for this design; carry out a test of the null hypothesis that mean proficiency is the same for all 3 training methods. Use Type I error rate .01 and state your conclusion. d. Follow-up the testing of the null hypothesis of no difference between training methods by forming interval estimates for all the pairwise comparisons. Use the Tukey method with family-wise confidence coefficient .99. e. Obtain an indication of how useful blocking on 'time since college' was in increasing the precision of this study compared to a simple completely randomized (one-way) design. For example, this study uses a total of 30 subjects; how many subjects would be neccessary to achieve the same precision if a completely randomized (i.e. no blocking) design were employed?