1. Click to edit Master title style t = df = = - ( + 1 1 ) df: df MS error pv =pv =

2. Click to edit Master title style Post Hoc Analysis: Which groups differ? If there are k groups, how many pairs (possible t tests) are there? If there are k groups, how many pairs (possible t tests) are there? Risk of at least one Type I error ( ‘family-wise error rate’ ): Risk of at least one Type I error ( ‘family-wise error rate’ ): k =k =k =k = mα m

3. Click to edit Master title style Post Hoc Analysis: Which groups differ? If there are k groups, how many pairs are there? If there are k groups, how many pairs are there? The Bonferroni Procedure The Bonferroni Procedure If you want the chance of 1 or more Type I errors to be less than 0.05, use 0.05/m for each post hoc comparison, where m is the number of comparisons to be made. k =k =k =k = m

4. Click to edit Master title style If you perform m tests using α, your family-wise risk is (approximately) m*α, α FW = m*α So, if you perform m tests, use for each test. Example: I performed 25 tests using α = 0.01 for each. My family-wise risk is (approximately) 25*0.01 = 0.25. Bonferroni Forwards and Backwards

5. Click to edit Master title style Assumptions of ANOVA (Dream Land) Normality The scores in e Normality The scores in each population (each level of the IV) have a normal distribution. H omogeneity of Variance H omogeneity of Variance The scores in each population have the same standard deviation.

6. Violations of Assumptions If enough observations are taken, the assumptions are safely ignored…unless: Really bad: Very unequal Really bad: Very unequal population variances (4 to 1 ratios or more) AND unequal sample sizes.

7. Types of ANOVA Designs

8. Click to edit Master title style One-way ANOVA Completely Randomized Design Goal: Compare three different fertilizers to see whether there is any difference in their effectiveness. Approach: Divide growing regions into 9 fields, randomly assign a fertilizer to each field: Fertilizer 2 Fertilizer 1 Fertilizer 3 Mojave Desert San Luis Obispo Montreal Canada

9. Click to edit Master title style Goal: Compare three different fertilizers to see whether there is any difference in their effectiveness. Approach: Divide farm into 3 fields that vary according to some measurable quantity (growing region). Subdivide these 3 fields into 3 parts, randomly assign a fertilizer to each part of the field. Growing Region is the "blocking variable". Two-Way ANOVA The Randomized Block Design (RBD) Purpose: Increase your ‘power’ to reject the null. Mojave Desert San Luis Obispo Montreal Canada Fertilizer 2 Fertilizer 1 Fertilizer 2 Fertilizer 1 Fertilizer 2 Fertilizer 3

10. Click to edit Master title style Example Problem A researcher wishes to determine whether the stress levels of students depends on their housing condition. It is known from previous research that the stress level of students depends strongly on their academic major. The researcher therefore decides to use the academic major as a blocking variable. Analyze the data below and determine whether the idea to block on major was successful (increased power). HousingMajorStress OCPSci 1.3 OCPHum 13.5 OWNASci 1.7 OWNAHum 11.9 OWNRSci 2.2 OWNRHum 12.2 DORMSci 9.2 DORMHum 19.2

11. Click to edit Master title style One-way ANOVA: Stress versus Housing Source DF SS MS F P Housing 3 75.3 25.1 0.44 0.735 Error 4 226.4 56.6 Total 7 301.7 Two-way ANOVA: Stress versus Housing, Major Source DF SS MS F P Housing 3 75.28 25.093 43.77 0.006 Major 1 224.72 224.720 391.95 0.000 Error 3 1.72 0.573 Total 7 301.72 It Worked!