Between-Groups ANOVA Chapter 12 > When to use an F distribution • Working with more than two samples > ANOVA • Used with two or more nominal independent variables and an interval dependent variable Why not use multiple t-tests? > The problem of too many t tests • Fishing for a finding • Problem of Type I error The F Distribution > Analyzing variability to compare means • F = variance between groups variance within groups > That is, the difference among the sample means divided by the average of the sample variances Types of Variance > Between groups: estimate of the population variance based on differences among group means > Within groups: estimate of population variance based on differences within (3 or more) sample distributions Check Your Learning > If between-groups variance is 8 and within-groups variance is 2, what would F be? Types of ANOVA > One-Way: hypothesis test including one nominal variable with more than two levels and a scale DV > Within-Groups: more than two samples, with the same participants; also called repeated-measures > Between-Groups: more than two samples, with different participants in each sample Assumptions of ANOVAs > Random selection of samples > Normally distributed sample > Homoscedasticity: samples come from populations with the same variance One-Way Between-Groups ANOVA > Everything about ANOVA but the calculations > 1. Identify the populations, distribution, and assumptions. > 2. State the null and research hypotheses. > 3. Determine the characteristics of the comparison distribution. > 4. Determine the critical value, or cutoff. > 5. Calculate the test statistic. > 6. Make a decision. Step 3. Characteristics df between N groups 1 df within df1 df 2 df 3 ...df last df1 n1 1 • What are the degrees of freedom? > If there are three levels of the independent variable? > If there are a total of 20 participants in each of the three levels? > Step 4: Critical Values Determine Cutoffs for an F Distribution (Step 4) Formulae SStotal ( X GM )2 SSwithin ( X M )2 SSbetween ( X GM ) 2 SStotal SS within SSbetween MS within SS within df within MS between MS between F MS within SS between df between Logic behind the F Statistic > Quantifies overlap > Two ways to estimate population variance • Between-groups variability • Within-groups variability The Logic of ANOVA The Source Table > Presents important calculations and final results in a consistent, easy-toread format Bringing it All Together > What is the ANOVA telling us to do about the null hypothesis? > Do we reject or accept the null hypothesis? An F Distribution Here the F statistic is 8.27 while the cutoff is 3.86. Do we reject the null hypothesis? Making a Decision > Step 1. Compare the variance (MS) by diving the sum squares by the degrees of freedom. > Step 2. Divide the between-groups MS by the within-groups MS value. > Step 3. Compare the calculated F to the critical F (in Appendix B). • If calculated is bigger than critical, we have a significant difference between means Calculating Effect Size > R2 is a common measure of effect size for ANOVAs. SS between R SS total 2 Post-Hoc Tests to Determine Which Groups Are Different > When you have three groups, and F is significant, how do you know where the difference(s) are? • Tukey HSD • Bonferonni > A priori (planned) comparisons Tukey HSD Test > Widely used post hoc test that uses means and standard error M1 M 2 HSD sM MS within sM N The Bonferroni Test > A post-hoc test that provides a more strict critical value for every comparison of means. > We use a smaller critical region to make it more difficult to reject the null hypothesis. • Determine the number of comparisons we plan to make. > Divide the p level by the number of comparisons.