1. Analyze Of VAriance

2. Application fields ◦ Comparing means for more than two independent samples = examining relationship between categorical->metric variables ◦ Regression models’ fit, testing multiple correlation coefficient (later)

3. Comparing means for more than two independent samples (groups) Based on the number of grouping variables: one-way, two-way or multiple-way ANOVA can be differentiated Generalization of independent two-samples t- tests Why do not apply we „simple” two-samples t- tests?

4. Probability of type-1 error and the number of comparisons

5. One-way ANOVA From the aspect of comparing means H0: means of groups are considered to be equal H 0 :  a =  b =...=  k (=  From the aspect of examining a relationship H 0 : the grouping (independent) variable does not influence significantly the metric (dependent variable) OR there is no significant relationship between the examined variables H 1 : the grouping (independent) variable influences significantly the metric (dependent variable) OR there is a significant relationship between the examined variables Basis: variance decomposition (SS treatment, SS Error, SS Total )

6. Sum of Squares

7. F-test n: number of observations k: number of groups Retain H0 if F (computed) < F 1-  (k-1,n-k))

8. ANOVA requires the following conditions  Concerning sample size of each groups (nj) ANOVA can be applied if:  nj>100  or 30<nj<100 but the distribution is not skewed to the right strongly (skewness<+ 1 )  or nj<30 and there is a normal distribution  The populations have equal standard deviations: testing by Levene-test. If variance homogeneity is not assumed  applying Welch-test

9. Describing the relationship Variance coefficient Shows the proportion of variance in metric variable explained by categorical variable. Measuring the strength of the relationship H-measure (  )

10. Steps of executing ANOVA Identifying the problem, describe H0 Checking application conditions Computing the test If the relationship is significant: ◦ What is the strength of the relationship (H)? What is the value of explained variance (H2)? ◦ Why was the H0 rejected? Post Hoc test

11. Post Hoc test Why was H0 rejected? Pairwise comparisons for means concerning the number of comparisons H0: there is not significant difference If equal variances are assumed : Tukey- test If equal variances are NOT assumed: T2- test

12. ANOVA in SPSS Analyse/Compare Means/One-way Anova

13. Options Descriptive statistics Variance homogeneity Welch-test Means Plots Post Hoc test window

14. The output of one-way ANOVA in SPSS can contain the following parts: 1.Descriptive statistics 2.Test for variance homogeneity 3.ANOVA-table 4.Welch-test 5.Means Plots 6.Post Hoc test table (If H0 is rejected) 7.Showing H and H2 in Analyze/Means/Means  Option  Eta and Eta Square (If H0 is rejected)