1. Analysis of RT distributions with R Emil Ratko-Dehnert WS 2010/ 2011 Session 07 – 21.12.2010

2. Last time... Introduced significance tests (most notably the t-test) – Test statistic and p-value – Confusion matrix – Prerequisites and necessary steps – Students t-test – Implementation in R 2

3. ANALYSIS OF VARIANCES 3

4. ANOVA The Analysis of Variance is a collection of statis- tical test Their aim is to explain the variance of a DV (metric) by one or more (categorial) factors/ IVs Each factor has different factor levels 4

5. Main idea Are the means of different groups (by factors) different from each other? Is the variance of a group bigger than of the whole data? 5

6. ANOVA designs One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be done by a t-test 6

7. ANOVA designs (cont.) Factorial ANOVA is used when the experimenter wants to study the interaction effects among the treatments. Repeated measures ANOVA is used when the same subjects are used for each treatment (e.g., in a longitudinal study). 7

8. ONE-WAY ANOVA 8

9. One-way ANOVA A one-way ANOVA is a generalization of the t-test for more than two independent samples Suppose we have k populations of interest From each we take a random sample, for the ith sample, let X i1, X i2,..., X ini designate the sample values 9

10. Prerequisites The data should... 1)be independent 2)be normally distributed 3)have equal Variances (homoscedasticity) 10

11. Mathematical model 11 X ij =dependant variable i=group (i in 1,..., k) j=elements of group i (j in 1,..., n i ) n i =sample size of group i ε ij =error term; ε ~ N(0, σ)

12. Hypotheses Suppose we have k independent, iid samples from populations with N(μ i, σ) distributions, i = 1,... k. A significance test of Under H 0, F has the F-distribution with k-1 and n-k degrees-of-freedom. 12

13. F k-1, n-k with k = amount of factors and n = sample size 13

14. Example Two groups of animals receive different diets The weights of animals after the diet are: Group 1: 45, 23, 55, 32, 51, 91, 74, 53, 70, 84 (n 1 = 10) Group 2: 64, 75, 95, 56, 44, 130, 106, 80, 87, 115 (n 2 = 10) 14

15. Example (cont.) Do the different diets have an effect on the weight? Means differ, but this might be due to natural variance 15

16. Example (cont.) Global variance Test statistic 16

17. Example (cont.) To assess difference of means, we need to compare this F-value with the one we would get for the for alpha = 0.05  F = 4.41  6.21 > 4.41  H 0 can be rejected 17

18. ANOVA (CONT.) 18

19. Effect size η 2 The effect size describes the ratio of variance explained in the dependant variable by a predictor while controlling for other predictors 19

20. Power Analysis is often applied in order to assess the probability of successfully rejecting H 0 for specific designs, effect sizes, sample size and α-level. can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the H 0 when H 1 is true. 20

21. A priori vs. post hoc analysis A priori analysis (before data collection) is used to determine the appropriate sample size to achieve adequate power Post hoc analysis (after data collection) uses obtained sample size and effect size to determine power of the study 21

22. Follow-up tests ANOVA only decides whether (at least) one pair of means is different, one commonly conducts follow-up tests to assess which groups are different: Bonferroni-Test Scheffé-Test Tuckey‘s Range Test 22

23. Visualisation of ANOVAs http://www.psych.utah.edu/stat/introstats/anov aflash.html 23

24. ANOVAS WITH R 24

25. oneway.test() The R function oneway.test() will perform the one-way ANOVA One can use the model notation oneway.test(values ~ ind, data = data) to assign values to groups 25

26. aov() Alternatively one can use the more general aov() command for the one-way ANOVA fit <- aov(y ~ A, data = mydataframe) plot(fit)# diagnostic plots summary(fit)# display ANOVA table 26

27. AND NOW TO 27