Education 257 HW2 Feb 2 2005 ("Due" Feb. 8 2005) A. Nested Designs 1. Group Decision Making Example (see course ex "cross-nested") NWK v4 Section 28.9; NWK v5 sec 26.9 For this design with crossed and nested factors use the data in course example (NWK Table 28.12 26.12) a. obtain cell means for the Nationality X Team Size 2x2 factorial design b. recreate the (Minitab) anova table shown in NWK Carry out tests for the Nationality and Team Size Main effects and the interaction between these two factors. How would these tests change if observer was treated as a fixed rather than random factor? ----------------- 2. Health awareness data, NWK ver4 p.1157 problem 28.9 onward, ver 5 prob 26.9; Three states (factor A) participated in a health awareness study. Each state independently devised a health awareness program. Three cities (factor B) within each state were selected for participation and 5 households within each city were randomly selected to evaluate the effectiveness of the program. All members of the selected households were interviewed before and after participation in the program and a composite index was formed for each household measuring the impact of the health awareness program (larger the index, greater the awareness) Data p.1157 (p.1121) exist in file haware.dat Do problems: 28.9 c (26.9 c) 28.10 parts a b c d (26.10 abcd) 28.11 c (26.11c) -------------------------------------- 3. Revisit Training Data from NWK Ch 28 (ch26) & Course Example a. For this data analysis, recreate the anova table for this nested design using only basic analysis of variance methods (i.e. not the nested capability of the anova command) by obtaining SS etc for the nested factor separately at each level of School. b. Recreate the construction of the Tukey pairwise comparisons on p.1135-6 (p1101) of NWK. ------------------------------------------------------- B. Repeated Measures Designs 4. Hearing Tests Story Hearing aids must be fit individually. A common way to test whether a particular hearing aid is right for a patient is to play a tape on which 25 words are pronounced clearly but at low volume, and ask the patient to repeat the words as heard. Different lists are available that are supposed to be of equal difficulty to understand correctly. However, a major problem for those wearing hearing aids is that the aids amplify background noise as well as the desired sounds. The question here is, Are the test lists still equally difficult to understand in the presence of background noise? In this experiment, 24 subjects with normal hearing listened to standard audiology tapes of English words at low volume, with a noisy background. They repeated the words and were scored correct or incorrect in their perception of the words. The order of list presentation was randomized. "Hearing" is the dependent variable and "List" and "Subject" as factors An objective of the data collection is to assess whether the different lists are equally difficult to understand . This design results in only one observation per cell since there are 96 observations, 24 subjects and 4 lists (24x4 = 96). In file hearing.dat, c1 is the subject indicator (1,...24), c2 is the list indicator (1,...4), and c3 is the hearing score. For these data a. carry out the repeated measures anova and test for list main effect b. do Tukey pairwise comparisons among the 4 lists (as was done in the wine-judging example NWK Section 29.2 (27.2) c. obtain an estimate of the variance component for subjects. -------------------------------------------------------------------- 5. REPEATED MEASURES "WALK-THROUGH" a. shoes from NWK , the example used in Sec 29.4 (sec27.4) of NWK; the series of class example files is shoes.* (i) Use these data to replicate the information in NWK Fig 29.8 (27.6) (ii) replicate the Minitab anova output in shoes.lis also shown in NWK Fig 29.9 (27.7). Note that there are two parts to this anova model: a crossing of the fixed effects ad-type and time and a nesting of the random factor sites (serving as subjects) within ad-type. (iii) The between-subjects portion of the analysis. verify that the test statistic for the between subjects factor ad-type usues as its error term sites nested within ad- type. Refer to NWK Tables 29.8 and 29.9 (27.5,27.6) to justify this error term. What are the relevant degrees of freedom for this test statistic? (iv) The within-subjects portion of the analysis. verify that test statistics for the repeated measures factor time and its interaction with ad-type use MSerror in the denominator of their test statistics and justify with Tables 29.8 and 29.9. What are the degrees of freedom for the test statistic for time if the compound symmetry assumptions of this mixed model are assumed to hold perfectly? In class, we discussed a conservative correction first suggested by Box to multiply degrees of freedom in tests of within-subjects factors by "epsilon", where the lower bound for epsilon is 1/(r - 1) where r is the number of levels of the repeated measures factor (here the number of time points). What would epsilon be and what would the degrees of freedom associated with the test statistics for time if the Box correction were used? (A less conservative epsilon value is given by the Greenhouse-Geisser factor ) b. Winer dial example. Now that we have revisited the shoes example, consider a famous (though not scintillating) example from Winer's traditional text (Chap 7 esp p.525). The data description (p.525) "Consider a factorial experiment in which the levels of factor A [between-subjects factor] are two methods for calibrating dials and the levels of factor B are four shapes for the dials. Outcome measure are accuracy scores on a series of trials on each of the dials." For each calibration method, three randomly chosen subjects complete the trial for each of the four dial shapes. To summarize: dial is a 2 (calibration method) x 4 (shape of dial) design with repeated measues on the second factor. Outcome variable is accuracy in callibrating dials veiwed as a function of calibration method (between subjects) and dial shape (within subjects). The data are presented in the following form in Winer's text dial.data: subj|dial 1 dial 2 dial 3 dial 4 1 | 0 0 5 3 method 1 2 | 3 1 5 4 3 | 4 3 6 2 ----|-------------------------- 4 | 4 2 7 8 method 2 5 | 5 4 6 6 6 | 7 5 8 9 (i) create a data file dial.dat that mimics shoes.dat in structure (ii) use minitab to run the repeated measures anova to mimic the analysis in shoes.lis (iii) carry out tests for the between subjects and within-subjects factors. State the values of the test statistics and their associated degrees of freedom. For test statistics for within-subjects factors use the Box correction to the degrees of freedom. c. Implementation in SAS. the following set of commands will produce the repeated measures analysis for the shoes data ----------- data shoes; input ad m1 m2 m3; datalines; 1 958.00 1047.00 933.00 1 1005.00 1122.00 986.00 1 351.00 436.00 339.00 1 549.00 632.00 512.00 1 730.00 784.00 707.00 2 780.00 897.00 718.00 2 229.00 275.00 202.00 2 883.00 964.00 817.00 2 624.00 695.00 599.00 2 375.00 436.00 351.00 ; proc glm data=shoes; class ad; model m1--m3 = ad /nouni; repeated Time 3 (1 2 3) /summary printe; run; ------------ you can execute one of two ways. with SAS on a PC just paste these commands into the editor window and run on leland system make these commands into a shoes.sas text file then simply and the command line: sas shoes.sas will create a shoes.lst file containg the output From SAS output use the value for the Huynh-Feldt Epsilon to adjust the critical values for the within 'subjects' tests. ---------------- 6. Polynomial trends (picking up from class 1/31) Consider the bock text reading data from class handout The vocabulary learning data bock*.* comes from Bock, MSMBR, p.454. "Data are drawn from test results on file in the Records Office of the Laboratory School of the University of Chicago. They consist of scores, obtained from a cohort of pupils at the eigth through eleventh gade level on alternative forms of the vocabulary section of the Cooperative Reading Tests." There are 64 students in all, 36 male, 28 female (ordered). In bock.dat the first column is ID, the next four the vocab scores and the last gender (Male = 1). A repeated measures anova (as in the handout) will give an occasions effect with 3 degress of freedom. As discussed in class the shape of the overall growth curve is often examined via orthogonal polynomials (here 1df for each of linear, quadratic, cubic trends). This is a common example of the use of planned orthogonal contrasts. The SS for each component is given by n*(SUM[Y-bar(j)*c(j)])^2/SUM[c(j)^2] where c(j) are the components of the contrast and Y-bar(j) components of the mean vector. Reproduce the 1df decomposition of the occasions effect for these data (ref handout). Alternatively have SAS or other program compute the decomposition (eg /polynomial) ----------------------- 7. Student supplied repeated measures experimental data (discussed in class) Description The Expectation Study This experiment was designed to study the effects of expectation of item format on reading comprehension performance. Students were randomly placed into treatment groups. They were either told to expect multiple-choice questions after reading a passage or to expect constructed-response questions. After reading the passage, both groups answered the same 12 questions. Questions were classified by type (factual or relational) and format (multiple-choice or constructed response). questions.dat data file c1 number of questions correct (0-3) c2 treatment group : question expectation (1 for multiple-choice, 2 for constructed-response) c3 student number (nested within treatment) c4 question type (1 for factual, 2 for relational) c5 question format (1 for multiple-choice, 2 for constructed-response) Carry out a repeated measures anova analysis for these data. You can either view the within-subjects 2x2 structure as a 1x4 to match prior examples, or much better, use the within subjects factorial structure. --------------------------------------------------------------------- C Nonparametric Alternatives NWK ver 5 sec 18.7, p.900, 1138-9 Nonparametrics A good review of basic nonparametric procedures is in Minitab student handbook/primer Chaps 12 or 13 (depending on version) 8. 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. END