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Analysis of RT distributions with R

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

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Analysis of RT distributions with R

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  1. Analysis of RT distributionswith 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

  3. Analysis of Variances

  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

  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?

  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

  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).

  8. ONE-way ANOVA

  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 Xi1, Xi2, ..., Xinidesignate the sample values

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

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

  12. Hypotheses • Suppose we have k independent, iid samples from populations with N(μi, σ) distributions, i = 1, ... k. A significance test of • Under H0, F has the F-distribution with k-1 and n-k degrees-of-freedom.

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

  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 (n1 = 10) Group 2:64, 75, 95, 56, 44, 130, 106, 80, 87, 115 (n2 = 10)

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

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

  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  H0 can be rejected

  18. ANOvA (cont.)

  19. Effect size η2 • The effect size describes the ratio of variance explained in the dependant variable by a predictor while controlling for other predictors

  20. Power Analysis • is often applied in order to assess the probability of successfully rejecting H0 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 H0when H1 is true.

  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

  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

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

  24. ANOVAs with R

  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

  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

  27. And now to

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