1. PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 13 Two-factor Analysis of Variance (Independent Measures)

2. Two-factor ANOVA ANOVA is a hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations) A research study with two independent variables: The effects of two different teaching methods and three different class size are evaluated. DV: math achievement test score IV 1: Class Size Small class Medium Class Large Class IV 2: Teaching Methods Method ASample 1Sample 2Sample 3 Method BSample 4Sample 5Sample 6

3. Two-factor ANOVA A two-by-three factorial design 2X3= 6 different treatments IV 1: Class Size Small class Medium Class Large Class IV 2: Teaching Methods Method ASample 1Sample 2Sample 3 Method BSample 4Sample 5Sample 6 levels

4. Two-factor ANOVA Two factor ANOVA will allow researcher to test for mean differences in the experiment: 1. Mean difference between teaching methods. 2. Mean differences between the three class sizes. 3. Any other mean differences that may result for unique combinations of a specific teaching method and a specific class size.

5. Two-factor ANOVA Main effect: The mean differences among the levels of one factor are referred to as main effect of that factor. Main effect for class size (factor B) Main effect for methods (factor A)

6. Two-factor ANOVA

7. Interaction effect: There is an interaction between factors if the effect of one factor depends on the levels of the second factor. The interaction is identified as the AXB interaction.

8. Two-factor ANOVA

9. There is no interaction between factors A and B. The effect of factor A does not depend on the levels of factor B (and B does not depend on A)

10. Two-factor ANOVA It is composed of three distinct hypothesis tests: 1. The main effect of factor A (The A-effect) 2. The main effect of factor B (The B-effect) 3. The interaction (AXB interaction)

11. Two-factor ANOVA Total variability Between-treatments variability 1.Treatment effects 2. Individual differences 3. Experimental error Within-treatments variability 1.Individual differences 2.Experimental error

12. Two-factor ANOVA 1.Treatment effect (factor A, factor B and AXB) 2.Individual differences (there are different subjects for each trwatment condition) 3.Experimental error

13. Two-factor ANOVA df total N-1 df between ab-1 Factor A df=a-1 Factor B df=b-1 Interaction df=df of A X df of B df within N-ab

14. Two-factor ANOVA SS total SS between SS within

15. Distribution of F-ratios Table B.4 The F-Distribution

16. Example (Do these data indicate that the size of the class and /or programs has a significant effect on test performance?) Class size 18 Students24 students30 students programsProgram 1 5338653386 9 13 6 8 3833338333 Program 2 0200302003 0005000050 0375503755

17. Assumptions 1. The observations within each sample must be independent. 2. The populations from which the samples are selected must be normal. 3. The populations from which the samples are selected must have equal variances.

18. Example In 1968, Schachter published an article in Science reporting a series of experience on obesity and eating behavior. One of these studies examined the hypothesis. One of these studies examined the hypothesis that these individuals do not respond to internal, biological signals of hunger. In simple terms, this hypothesis says that obese individuals tend to eat whether or not their bodies are actually hungery. In Shachter’s study, subjects were led to believe that they were taking part in a “taste test.” All subjects were told to come to the experiment wthout eating for several hours beforehand.

19. Example The study used two indepedent variables or factors: 1.Weights (obese versus normal subjects) 2.Full stomach versus empty stomach All subjects were then invited to taste and rate five different types crackers. The dependent variables was the number of crackers eaten by each subject.

20. Example Factor B: Fullness Empty StomachFull Stomach Factor A: Weight Normaln=20 Mean=22 Cell Sum =440 SS=1540 n=20 Mean=15 Cell Sum =300 SS=1270 Obesen=20 Mean=17 Cell Sum =340 SS=1320 n=20 Mean=18 Cell Sum =360 SS=1266