Analysis of Variance: repeated measures

Analysis of Variance: repeated measures Tests for comparing three or more groups or conditions: (a) Nonparametric tests: Independent measures: Kruskal-Wallis. Repeated measures: Friedman s. (b) Parametric tests: One-way independent-measures Analysis of Variance (ANOVA). One-way repeated-measures ANOVA. 1

Logic behind ANOVA: ANOVA compares the amount of systematic variation (from our experimental manipulations) to the amount of random variation (from the participants themselves) to produce an F-ratio: F = systematic variation random variation ( error ) F = systematic variation random variation ( error ) Large value of F: a lot of the overall variation in scores is due to the experimental manipulation, rather than to random variation between participants. Small value of F: the variation in scores produced by the experimental manipulation is small, compared to random variation between participants.

ANOVA is based on the variance of the scores. The variance is the standard deviation squared: variance = ( X X ) N In practice, we use only the top line of the variance formula (the "Sum of Squares", or "SS"): ( X X ) Sum of Squares = We divide this by the appropriate "degrees of freedom" (usually the number of groups or participants minus 1). One-way Repeated-Measures ANOVA: Use this where you have: (a) one independent variable; (b) one dependent variable; (c) each participant participates in every condition in the experiment. A one-way repeated-measures ANOVA is equivalent to a repeated-measures t-test, except that you have more than two conditions in the study.

Effects of sleep-deprivation on vigilance in air-traffic controllers: No deprivation vs. 1 hours' deprivation: One IV, levels - use repeated-measures t-test. No deprivation vs. 1 hours vs. 4 hours: One IV, levels (differing quantitatively) - use one-way repeated-measures ANOVA. Effects of sleep deprivation on vigilance: IV: length of sleep deprivation (0, 1 hours and 4 hours). DV: 1 hour vigilance test (number of planes missed). Each participant does all conditions, in a random order. Participant 1 0 hours 5 1 hours 1 15 4 hours 1 14 0 hours: Mean = 4.6, s.d. = 1.4. 4 5 6 4 7 16 11 1 16 1 11 1 hours: Mean = 1.0, s.d. =.1. 6 7 4 1 17 14 16 4 hours: 8 9 5 6 11 10 1 11 Mean = 1.0, s.d. = 1.8. 10 1 14 4

"Partitioning the variance" in a one-way repeated-measures ANOVA: Total SS Between Subjects SS Within Subjects SS (usually uninteresting: if it's large, it just shows that subjects differ from each other overall) SS Experimental (systematic within-subjects variation that reflects our experimental manipulation) SS Error (unsystematic within-subjects variation that's not due to our experimental manipulation) The ANOVA summary table: Source: SS df MS F Between subjects 48.97 9 5.44 Total within subjects 54.5 0 Between conditions 487.00 4.90 9.6 Within subjects 47.5 18.64 Total 584.0 9 Total SS: reflects the total amount of variation amongst all the scores. Between-condtions SS: a measure of the amount of systematic variation between the conditions. (This is due to our experimental manipulation). Within-subjects SS: a measure of the amount of unsystematic variation within each participant's set of scores. (This cannot be due to our experimental manipulation, because we did the same thing to everyone within each condition). (Total SS) = (Between-subjects SS) + (total within-subjects SS) 5

Assessing the significance of the F-ratio (by hand): The bigger the F-ratio, the less likely it is to have arisen merely by chance. Use the between-conditions and within-subjects d.f. to find the critical value of F. Your F is significant if it is equal to or larger than the critical value in the table. Here, look up the critical F-value for and 18 d.f. Columns correspond to between-groups d.f.; rows correspond to within-groups d.f. Here, go along and down 18: critical F is at the intersection. Our obtained F, 9.6, is bigger than.55; it is therefore significant at p<.05. (Actually it s bigger than the critical value for a p of 0.0001). 6

Interpreting the Results: A significant F-ratio merely tells us is that there is a statistically-significant difference between our experimental conditions; it does not say where the difference comes from. In our example, it tells us that sleep deprivation affects vigilance performance. To pinpoint the source of the difference: (a) planned comparisons - comparisons between groups which you decide to make in advance of collecting the data. (b) post hoc tests - comparisons between groups which you decide to make after collecting the data: Many different types - e.g. Newman-Keuls, Scheffé, Bonferroni. 7

Using SPSS for a one-way repeated-measures ANOVA on effects of fatigue on vigilance: (Analyze > General Linear Model > Repeated Measures...) Tell SPSS about your within-subjects IV (i.e. which columns correspond to your experimental conditions): 8

Ask for some planned comparisons: Various options here allow you to make different types of comparisons between conditions: I've picked "simple", to compare (a) no sleep with 1 hours' deprivation, and (b) no sleep with 4 hours' deprivation. 9

The SPSS output (ignore everything except what's shown here!): No sleep deprivation 1 hours' deprivation 4 hours' deprivation Descriptive Statistics Mean Std. Deviation N 4.6000 1.4984 10 1.0000.0940 10 1.000 1.8878 10 Similar to Levene's test - if significant, shows inhomogeneity of variance. Mauchly's Test of Sphericity b Measure: MEASURE_1 Within Subjects Effect deprivation Epsilon a Approx. Greenhous Mauchly's W Chi-Square df Sig. e-geisser Huynh-Feldt Lower-bound.06 9.475.009.590.67.500 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b. Design: Intercept Within Subjects Design: deprivation SPSS ANOVA results: Tests of Within-Subjects Effects Measure: MEASURE_1 Source deprivation Error(deprivation) Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Type III Sum of Squares df Mean Square F Sig. 487.800 4.900 9.60.000 487.800 1.181 41.186 9.60.000 487.800 1.54 88.985 9.60.000 487.800 1.000 487.800 9.60.000 47.5 18.641 47.5 10.65 4.474 47.5 11.86 4.1 47.5 9.000 5.81 Use Greenhouse-Geisser d.f. and F-ratio if Mauchly's test was significant. Significant effect of sleep deprivation (F, 18 = 9.6, p<.0001; OR, if using Grrenhouse-Geisser, F 1.18, 10.6 = 9.6, p<.0001). 10

Tests of Within-Subjects Contrasts Measure: MEASURE_1 Source deprivation Error(deprivation) deprivation Level vs. Level 1 Level vs. Level 1 Level vs. Level 1 Level vs. Level 1 Type III Sum of Squares df Mean Square F Sig. 705.600 1 705.600 90.05.000 756.900 1 756.900 106.7.000 70.400 9 7.8 64.100 9 7.1 Conclusions: sleep deprivation significantly affects vigilance (F, 18 = 9.6, p<.0001). Planned contrasts show that participants perform worse after 1 hours'deprivation (F 1,9 = 90.1, p<.0001) and 4 hours'deprivation (F 1, 9 = 106.7, p<.0001). Repeated measures ANOVA post hoc tests: 1. Click on Options.... Move factor ("deprivation") into "Display Means for.." box.. Click on "Compare main effects". 4. Change test to "Bonferroni". Measure: MEASURE_1 (I) deprivation 1 (J) deprivation 1 1 Mean Difference Pairwise Comparisons Based on estimated marginal means *. The mean difference is significant at the.05 level. a. Adjustment for multiple comparisons: Bonferroni. 95% Confidence Interval for Difference a (I-J) Std. Error Sig. a Lower Bound Upper Bound -8.400*.884.000-10.994-5.806-8.700*.844.000-11.176-6.4 8.400*.884.000 5.806 10.994 -.00.00 1.000-1.180.580 8.700*.844.000 6.4 11.176.00.00 1.000 -.580 1.180 11