1. 12e.1 ANOVA Within Subjects These notes are developed from “Approaching Multivariate Analysis: A Practical Introduction” by Pat Dugard, John Todman and Harry Staines.

2. 12e.2 An Example for Within Subjects This study is designed to determine whether choice reaction time (CRT) might serve as a marker of neurological function that could be used to assess the impact of experimental therapies on presymptomatic gene carriers for Huntington’s disease. Ten presymptomatic gene carriers, identified through family history investigations followed by genetic testing, are recruited. They complete a CRT task on three occasions at intervals of one year. The CRT task involves making a left or right touch response on a touch-sensitive screen to a visual or auditory stimulus presented to the individual’s left or right. The visual and auditory stimuli are presented in blocks, with order of modality balanced across participants.

3. 12e.3 An Example for Within Subjects So we have a 3 × 2 factorial design with two within-subjects (repeated measures) factors: year (year 1, year 2, year 3) and modality (visual, auditory). The dependent variable is mean CRT for 30 trials in each of the six conditions.

4. 12e.4 An Example for Within Subjects We will call our two within-subjects variables MODE (with two levels, 'visual' and 'auditory'), and YEAR with three levels, 1, 2 and 3. Notice that the two levels of MODE are both used within each level of YEAR, giving the two within- subjects variables a hierarchical structure. As always we need one row of the SPSS datasheet for each participant, and we need a variable name for each of the six columns of observations. The easiest way might be just to call the observations TIME1, TIME2, …, TIME6, or we might prefer to call them Y1M1, Y1M2, …, Y3M2 (where 'Y' stands for YEAR and 'M' stands for MODE), which reflects the meaning of the six conditions. This is what we have done, so the SPSS datasheet contains the six columns of data arranged as in the table, with the six variables named Y1M1, Y1M2, …..Y3M2.

5. 12e.5 Requesting The Analysis We select from the menu bar Analyze, General Linear Model and Repeated Measures, and we get the SPSS Dialog Box.

6. 12e.6 Requesting The Analysis We have two within-subjects factors to enter, and we need to do it in the correct order, with YEAR first, since that is the one that contains each level of MODE. So type YEAR in the Within-Subject Factor Name box, 3 in the Number of Levels box, and click Add. Then repeat the process with MODE and 2.

7. 12e.7 Requesting The Analysis We have two within-subjects factors to enter, and we need to do it in the correct order, with YEAR first, since that is the one that contains each level of MODE. So type YEAR in the Within-Subject Factor Name box, 3 in the Number of Levels box, and click Add. Then repeat the process with MODE and 2.

8. 12e.8 Requesting The Analysis There is no need to type anything in the Measure Name box, but if you like you could type TIME for the dependent variable to give us SPSS Dialog Box as shown. Click Define to get the SPSS Dialog Box.

9. 12e.9 Requesting The Analysis All we have to do is use the arrow to put the variables into the Within-Subjects Variables box in the correct order, so the result is as shown. You can check that the factors were entered in the correct order: see that the first column of the datasheet, Y1M1, is labelled (1,1), both factors at level 1. The second column, Y1M2 is labelled (1,2), factor 1 (YEAR) is at level 1 and factor 2 (MODE) is at level 2. This is correct for our first two columns of data. You can easily check that the entries are correct for the remaining four columns. We have no between- subjects factors or covariates.

10. 12e.10 Requesting The Analysis

11. 12e.11 Requesting The Analysis If we click the Model button we get a dialog box. We can either accept the default (Full Factorial) or click the Custom radio button and enter the main effects for MODE and YEAR and their interaction. The results are exactly the same so we may as well accept the default.

12. 12e.12 Requesting The Analysis In the Plots dialog box, put YEAR in the Horizontal Axis box and MODE in the Separate Lines box, don’t forget to select Add.

13. 12e.13 Requesting The Analysis In the Options dialog box, click Estimates of effect size and Observed power. Homogeneity tests are not available for within-subjects factors.

14. 12e.14 Requesting The Analysis You can get Residual plots but there will be one for each of the variables Y1M1 to Y3M2, each with ten points on it (one for each subject). The ten points on the Y1M1 plot all have YEAR at level 1 and MODE at level 1 so they all have the same predicted value. This means that, in a plot of predicted values on the horizontal axis against residual values on the vertical axis, all ten points will lie on a straight vertical line. All you can observe, therefore, is whether any of the points lie a long way from the others, indicating a poor fit for the corresponding observation. The same is true for each of the plots for the rest of the variables Y1M2 to Y3M2. None of the plots show such an effect and we don't reproduce them here.

15. 12e.15 Requesting The Analysis Now all that remains is to click Continue and OK to get the analysis.

16. 12e.16 Understanding The Output The first table in the output is a summary of the data, which is a useful check on the way the factors were defined (SPSS Output). Here you can see that we entered our two factors in the correct order, our six columns of data corresponding in pairs to the three levels of YEAR.

17. 12e.17 Understanding The Output First, we need to look at the result of the Mauchly sphericity test, which is in SPSS Output. We see that the result is nonsignificant for YEAR and for the interaction (YEAR*MODE). There is no test for MODE because it has only two levels.

18. 12e.18 Understanding The Output As the Mauchly test of sphericity was non-significant, we can look at the 'Sphericity Assumed' rows of the table, where we see that only the effect of YEAR is significant (F(2,18) = 14.71, p < 0.001) with partial η 2 = 0.62 and power = 1.00. Had the sphericity test been significant, we would have needed to look at one of the other rows, which make various adjustments to the dfs to allow for the violation of the sphericity assumption.

19. 12e.19 Understanding The Output We consider these in the next section on MANOVA. For YEAR, it makes sense to ask whether the linear trend was significant, since the time taken to reach a decision could depend on progress of the disease with time. That information is provided in the next (Tests of Within-Subjects Contrasts) table.

20. 12e.20 Understanding The Output We can report that the linear trend was significant (F(1,9) = 19.09, p < 0.01). The final table (Tests of Between-Subjects Effects) is of no interest when we have no between-subjects factors, as all it does is test the hypothesis that the overall mean is not zero, and it is not reproduced here.

21. 12e.21 Understanding The Output We can see that the decision TIME increases steadily with YEAR, for both 'visual' (MODE level 1) and 'auditory' (MODE level 2), with the values for 'auditory' similar to those for 'visual' at YEAR = 1 and 3 and somewhat lower than those for 'visual' at YEAR = 2 degrees. However, our ANOVA tells us that neither the difference between the two levels of MODE nor its interaction with YEAR are significant: with this number of observations a difference of this size could just be due to random variation.

22. 12e.22 Syntax GET FILE='12e.sav'. ← include your own directory structure c:\… DISPLAY DICTIONARY /VARIABLES y1m1 y1m2 y2m1 y2m2 y3m1 y3m2. GLM y1m1 y1m2 y2m1 y2m2 y3m1 y3m2 /WSFACTOR=YEAR 3 Polynomial MODE 2 Polynomial /MEASURE=TIME /METHOD=SSTYPE(3) /PLOT=PROFILE(YEAR*MODE) /PRINT=ETASQ OPOWER /PLOT=RESIDUALS /CRITERIA=ALPHA(.05) /WSDESIGN=YEAR MODE YEAR*MODE. The following commands may be employed to repeat the analysis.