Education 161 Winter 2000 Assignment 2 Due Feb 1, 2000 Note data files are available in one of two locations: path: /usr/class/ed161/[data file] or using web-services at URL http://www.stanford.edu/class/ed161/hw/[data file] 1. 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. Create a file with the 36 times from the table above (C1). Use Mintab SET command (or your editor) to add the factorial design designations in C2 and C3. 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. ---------------------------- 2. NOTE: In HW1 #5 you were asked to do parts a-e of the problem below with data in knee.dat. For HW2 continue with these data and complete part(f) 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)? ---------------------------------------------- 3. Nonparametrics A good review of basic nonparametric procedures is in Minitab student handbook/primer Chaps 12 or 13 (depending on version) The story below is kind of boring but you can think of the data as the number of errors (e.g. reading or writing) made under 5 different conditions/protocols: An experiment was conducted to compare the number of major defectives observed along each of five production lines in which changes were being instituted. Producton was monitored continuously during the period of changes, and the number of major defectives was recorded per day for each line. The data for the five lines are shown here (you can cut-and-paste Production line 1 2 3 4 5 34 54 75 44 8O 44 41 62 43 52 32 38 45 30 41 36 32 10 32 35 51 56 68 55 58 (i). Does the standard anova assumption of equal within-group variances appear to hold here? Does it matter? (ii). Conduct a standard one-way anova and test the omnibus null hypothesis of equal group means. Type I error rate .05. (iii). An alternative approach would be to turn nonparametric. Try a Kruskal-Wallis procedure on these data. ------------------------------------------------------------------ 4. "Contrast Statistics Class with Deprivation Torture" The data in file animal.dat represent outcomes in an animal learning task. There are five animals in each of the five conditions. Conditions 1 and 2 represent control conditions in which the animal received "ad lib" food and water (data in C1) or food and water twice a day (data in C2). In Condition 3 animals were food deprived (C3). In Condition 4 animals were water deprived (in C4) and in Condition 5 animals were deprived of both food and water (in C5). The planned comparisons (contrasts) proposed for use in the analysis of the one-way classification are: 1. (1,2) vs (3,4,5) 2. (1) vs (2) 3. (3,4) vs (5) 4. (3) vs (4) a. give a short interpretation for the question each contrast is addressing b. verify that these 4 contrasts are orthogonal or show that they are not c. construct point estimates of each contrast d. construct a set of interval estimates for these 4 contrasts. use overall error rate .10 and use alphatot.tab entry to identify the error rate (confidence coefficient) for use with each interval estimate. what do you conclude? ---------------------------------- 5. More on interactions for factorial designs. A question that was raised and discussed at length in a previous incarnation of this course centered on a question like, What more can be done about describing (or drawing inferences) about the interaction terms beyond just (rejecting or not) the omnibus null hypothesis of no interaction? We had extensively discussed the importance of the profile plot as the major descriptive technique for looking at interactions. There is more that can be done inferentially in addition. In lecture Wednesday the college Mathematics learning example (a 2x3 design) brings up the question of estimating the row effects separately for each level of the column factor. One of the TA's at that time said that in psychological journals inferences about the contrasts between methods at each level of math ability would be carried out by separate two-sample t procedures (which I very much hope have the total error rate divided up for the multiple inferences). An alternative is constructing either the Tukey multiple comparisons on all (axb) six cells to obtain the inferences about the desired pairwise comparisons or use a Bonferroni multiple comparison procedure which may offer some improvement over Tukey since not all pairwise comps. are of interest. To pursue this issue I constructed the following exercise. I constructed data for a 2 x 3 design (two rows, 3 columns as in the class discussion) with 5 replications per cell. Consider both factors to be fixed (you could adapt the setting from the college mathematics example). Carry out the standard analysis for the two-way design-- construct the profile plot from the cell means, obtain the anova table and construct the test statistics for main effects and interaction. Compare the method I suggested using a Tukey multiple comparison procedure on the 1x6 array of cells to estimate the contrasts of fiffences between the rows at each level of the column with the "journal method" of separate two-sample t based inferences. The data exist in file extraint.dat. In c1 are the outcome scores; in c3 and c4 are the row and column indicators for the 2x3 array. In c2 are cell indicators (1,2,...6) useful in one-way adaptation for this design. ----------------------------------- 6. Glass& Hopkins, Chap 18, page 527, problems 1 and 2 (parts b,c). You are encouraged to do this on the computer. note: for p.2 , the pairwise multiple comparisons can either be done by hand, as in the in-class Castle bakery illustration, or for the computationally adventurous, use the Minitab glm command in the same way you would use anova. For inexplicable reasons the glm graphical menu allows you to implement tukey in a two-way design, whereas anova does not. This will be illustrated in solutions and in section discussion.