Analysis of Variance: Repeated measures

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Andrea Holland
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1 Repeated-Measures ANOVA: Analysis of Variance: Repeated measures Each subject participates in all conditions in the experiment (which is why it is called repeated measures). A repeated-measures ANOVA is equivalent to a repeated-measures t-test, except that you have more than two treatment conditions. 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. From last week: Analysis of variance implies analyzing or breaking down variance. We start by breaking down Sum of Squares or SS. We saw these first when we calculated SD. 2 sum of squares = (X X ) Small value of F: the variation in scores produced by the experimental manipulation is small, compared to random variation between participants. We divide SS by the appropriate "degrees of freedom" (usually the number of treatments or subjects minus 1) to get variance. 1

One-way Repeated-Measures ANOVA: Effects of sleep-deprivation on vigilance in air-traffic controllers: Use this where you have: (a) one independent variable (with 2 or more levels); No deprivation vs.

(c) each participant participates in every condition in the experiment (repeated measures).

2 One-way Repeated-Measures ANOVA: Effects of sleep-deprivation on vigilance in air-traffic controllers: Use this where you have: (a) one independent variable (with 2 or more levels); No deprivation vs. 12 hours' deprivation: One Independent Variable, 2 levels use (b) one dependent variable; repeated-measures t-test. (c) each participant participates in every condition in the experiment (repeated measures). 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: Independent Variable: length of sleep deprivation (0, 12 hours and 24 hours). Dependent Variable: 1 hour vigilance test (number of planes missed). Each participant does all conditions, in a random order. Participant 0 hours 12 hours 24 hours 0 hours: Mean = standard deviation = hours: Mean = standard deviation = hours: Mean = standard deviation = 1.8. No deprivation vs. 12 hours vs. 24 hours: One Independent Variable, levels (differing quantitatively) use one-way repeated-measures ANOVA. "Partitioning the variance" in a one-way repeated-measures ANOVA: 2

3 within treatments variability between treatments variability step 1 The null hypothesis: Participant 0 hours 12 hours 24 hours hours: Mean = hours: Mean = hours: Mean = 1.0 H 0 : µ 1 = µ 2 = µ = µ 4 No treatment effect α =.05 steps 2,, 4, 5 & 6 Calculate 5 SS values: 1) Total 2) Between treatments ) Within treatments 4) Between subjects 5) Error step 2 SS total Total SS Participant 0 hours 12 hours 24 hours = (X i G ) 2 SS Total = 584. G =10. step Between treatments SS P# 0 hours 12 hours 24 hours X 1 = 4.6 X 2 =1 X 2 =1. SS between treatments = n [(X 1 G ) 2 + (X 2 G ) 2 + (X G ) 2 ] SS between treatments = 487.8

4 step 4 Within treatments SS P# 0 hours 12 hours 24 hours X 1 = 4.6 X 2 =1 X 2 =1. SS 1 = (X i X 1 ) 2 SS 2 = (X i X 2 ) 2 SS = (X i X ) 2 step 5 Between subjects SS P# 0 hours 12 hours 24 hours SS between subjects P 1 = 9. P 2 =11. P =12.67 P 4 = 9 P 5 =10 P 6 =10 P 7 =12. P 8 = 9. P 9 = 9 P 10 =10 [ ] = n (P 1 G ) 2 + (P 2 G ) 2 + (P G ) (P 10 G ) 2 SS within treatments = SS 1 + SS 2 +SS = 96.5 SS between subjects = step 6 Error SS SS error = SS within treatments - SS between subjects = = 47.5 step 7 Calculating df df total = All scores 1 = 29 df between treatments = Number of treatments 1 = 2 df within treatments = df 1 + df 2 + df = 27 df between subjects = Number of subjects 1 = 9 df error = df within treatments df between subjects = 27 9 = 18 4

step 8 The ANOVA summary table: Source: SS df MS F Total Between treatments Within treatments 584. 487.8 96.5 29 2 27 20.15 24.9.57 92.6 Between subjects Error 48.97 47.5 9 18 5.44 2.

5 step 8 The ANOVA summary table: Source: SS df MS F Total Between treatments Within treatments Between subjects Error F = MS between treatments MS error = = 92.6 Total SS: reflects the total amount of variation amongst all the scores. Between treatments SS: a measure of the amount of systematic variation between the treatments. Within treatments SS: a measure of the amount of unsystematic variation inside each treatment Between subjects SS: a measure of the amount of unsystematic variation between the subjects. (This is not due to our experimental manipulation). Error SS: a measure of the amount of unsystematic variation within each subject s set of scores. Total SS = Between subjects SS + Within subjects SS 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-treatments and error degrees of freedom 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 2 and 18 degrees of freedom Columns correspond to TREATMENTS degrees of freedom Rows correspond to ERROR degrees of freedom Here, go along 2 and down 18: critical F is at the intersection Our obtained F, 92.6, is bigger than.55; it is therefore significant at p<.05. (Actually it s bigger than the critical value for a p of ) 5

Interpreting the Results: A significant F-ratio merely tells us that there is a statistically-significant difference between our experimental conditions; it does not say where the difference comes

6 Interpreting the Results: A significant F-ratio merely tells us 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. Data entry Using SPSS for a one-way repeated-measures ANOVA on effects of fatigue on vigilance 6

within-subjects Independent Variable (i.

7 Go to: Analyze > General Linear Model > Repeated Measures Tell SPSS about your within-subjects Independent Variable (i.e. number of levels; and which columns the levels of the independent variable are in): Move VAR 4, VAR 5 and VAR 6 into the Within-Subjects Variables box by pressing the top arrow; then press options button 7

8 The SPSS output (ignore everything except what's shown here!): Similar to Levene's test - if significant, shows inhomogeneity of variance. Then click continue and OK SPSS ANOVA results: This is not too interesting; this just tells us that the subjects are significantly different from each other. Use Sphericity Assumed F-ratio if Mauchly's test was NOT significant. Significant effect of sleep deprivation (F 2, 18 = 92.6, p<.0001) OR, (if Mauchly s test was significant) use Greenhouse-Geisser (F 1.18, 10.6 = 92.6, p<.0001). 8

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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) Two types of ANOVA tests: Independent measures and Repeated measures Comparing 2 means: X 1 = 20 t - test X 2 = 30 How can we Compare 3 means?: X 1 = 20 X 2 = 30 X 3 = 35 ANOVA