This document provides an overview of one-way analysis of variance (ANOVA), which allows researchers to compare the means of three or more groups. It explains that ANOVA decomposes the total variability in a set of scores into two sources: variability between groups and variability within groups. The key metric in ANOVA is the F ratio, which compares the variability between groups to the variability within groups. If the between-groups variability is significantly greater than the within-groups variability, then the group means are significantly different from each other.

1. PSYCH 200: ANOVA PART 1 Decomposing variance One-way ANOVA Multiple comparisons

2. Beyond t tests T-tests only let us compare 2 groups A sample vs. a population A sample vs. another sample But what if we want to compare 3 groups or more?

3. For example… Does class level influence amount of study time? Do gender and education level interact in determining one’s susceptibility to sexual harassment? How do gender and marital status contribute to one’s level of anxiety? Which of three therapeutic methods are most effective at battling depression?

4. Comparing multiple groups Are there differences between these (3+) groups? If so, where do the differences occur? A ≠ B ≠ C A = B ≠ C Freshman > Seniors > Sophomores = Juniors Why not just conduct multiple t tests? It’s cumbersome and sloppy : How many tests must we perform across 3 groups? 5 groups? For 5 Groups (A,B,C,D,E) we would have to do 10 Tests It inflates our overall Pr (Type I error) for a given alpha For three t tests, with α = 0.05 each, Pr (Type I error) ≈ 0.15 We need a new method for comparing multiple groups

5. Basic terminology Factor An independent variable (or grouping variable) Level A particular value that a factor can possess Group mean The mean value of the DV across observations within a particular level of an IV Class Grand mean The mean value of the DV across observations in the experiment as a whole Freshman Sophomore Junior Senior

6. ANOVA One-way ANOVA An an alysis o f the va riance in a set of scores or observations, with the goal of determining whether the differences in means across levels of some factor is significantly greater than the differences among scores in general “ Difference in values” “ Natural variability” “ Difference in group means” One factor “ Variability across group means” We will start with a One-Way Between Subjects ANOVA . Independent Groups, just like the Independent Sample t-test

8. total treatment error error treatment

9. Group 1 Group 2 H0 vs. H1 - 2 Groups X X

10. Group 1 Group 2 H0 vs. H1 - 3 Groups Group 3 Group Mean Grand Mean X X X X

11. Decomposing variance “ Natural variability” “ Variability across group means” The essence of an ANOVA is to determine how the variability across group means (treatment effect) relates to the natural variability (or error in measurement). Specifically, we want to know the relative amount of total variability that is attributable to each of these sources. F =

12. F = 1 Variability due to groups = Natural variability Decomposing variance F > 1 Variability due to groups > Natural variability F > 1 Variability due to groups > Natural variability

13. Decomposing variance Variability between groups is called between groups variability Natural variability is the variability among groups or the variability within groups, and is called within groups variability (same as natural variability) The goal is to determine if the variability between groups is larger than the variability within groups.

14. Decomposing variance Group 1 Group 2 Group 3 X X X X

15. Implications of the F ratio F distribution differs from z and t in some key respects: Is always positive Is centered around 1, not 0 Distribution is skewed, not normal/symmetric Assumptions of the one-way ANOVA F ratio Normal sampling distribution (normal population and/or large N ) Homogeneity of variance Independent observations Interpretations of hypotheses and directionality In a one-way ANOVA, H 1 is always “ μ 1 , μ 2 , …, μ k are not all equal” In a one-way ANOVA, H 0 is always “ μ 1 = μ 2 = … = μ k ” We are always looking for F > 1, so it is always one-tailed NO MORE DIRECTIONAL OR NON-DIRECTIONAL!

16. One-way ANOVA: Examples H 1 : Amount of study time varies by class level μ freshman , μ sophomore , μ junior , μ senior are not all equal H 0 : Amount of study time does not vary by class level μ freshman = μ sophomore = μ junior = μ senior H 1 : Three therapeutic methods have differing degrees of effectiveness in treating depression μ cognitive , μ psychodynamic , μ biomedical , are not all equal H 0 : Three therapeutic methods have the same degree of effectiveness in treating depression μ cognitive = μ psychodynamic = μ biomedical

17. One Way ANOVA Example Imagine you are performing a study in which you are interested in the effect of magnetism on moral reasoning. You believe that a magnetic wave pointed at a certain part of the brain can affect our moral decision making. You have 16 people come into the lab. 5 of them are in the control condition and are not exposed to any magnetic wave (control 1), 6 are in the magnetic wave condition at the part of the brain responsible for moral decision making (experimental condition), and 5 are also exposed to a magnetic wave, but at a part of the brain not responsible for moral reasoning (control 2). After the manipulation, everyone takes a test of moral reasoning on a scale of 1-10.

18. One Way ANOVA Example H 1 : Magnetic Waves can affect moral reasoning μ control 1 , μ control 2 , μ experimental , are not all equal H 0 : Magnetic Waves cannot affect moral reasoning μ control 1 , μ control 2 , μ experimental , are all equal Factor ? Magnetic Wave Level Levels ? 3: Control 1, Control 2, Experimental DV ? Moral Reasoning Test (1-10 scale) STEP 1: Null and Alternative Hypotheses STEP 2: Identity Factor, Levels, and DV

19. Example = 6.38 Control 1 Control 2 Experimental X 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X

20. 3 4 5 6 7 8 Control 2 Control 1 Experimental Example X

21. Decomposing variance “ Natural variability” “ Variability across group means” F = “ Estimate of population variance” “ Average deviation from grand mean”

22. Decomposing variance STEP 3: We need to identify the two sources of variance (Between and Within/Natural/Error) We need equations to do that… Well… let’s think about what variance is.

23. Decomposing variance F = “ Average deviation from grand mean” “ Estimate of population variance” General formula for variance of a set of numbers: SS df MS B MS W Σ ( X – X ) 2

24. Variance within-groups a.k.a. Natural Variability or Error variance As always, we are trying to obtain the best estimate of the (common) population variance, σ 2 Recall the independent-samples t , where we pooled the variance across samples to estimate σ 2 Similarly, because we also assume homogeneity of variance in the ANOVA, we use a pooled estimate So what is that pooled estimate equation?

25. Variance within-groups a.k.a. Natural Variability or Error variance A bit of notation first… X i,j refers to the some score X in group J X j refers to the average of group J

26. Variance within-groups Mean squared error (or within-groups), MS W SS df N - 1 N - k MS W = SS 1 + SS 2 + … + SS k Number of groups Σ ( X i,1 – X 1 ) 2 Σ ( X i,2 – X 2 ) 2 Σ ( X i,k – X k ) 2 … Σ ( X i,j – X j ) 2 s p 2 = SS 1 SS 2 + df 2 df 1 + + … + …

27. Variance within-groups Mean squared error (or within-groups), MS W 3 4 5 6 7 8 Control 2 Control 1 Experimental X

28. Variance within-groups Mean squared error (or within-groups), MS W So the equation is what?!? Well, the equation for MS w ( pooled variance ) for a One Way Between Subjects ANOVA is… Σ ( X i,j – X j ) N - k MS W = Sum of Squares Within (SSw) Degrees of Freedom Within (dfw) k = number of groups/levels in IV 2

29. Back to our Example… = 6.38 Control 1 Control 2 Experimental X 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X

30. Back to our Example… = 6.38 = 13.20 2 X j 7.0 7.4 5.0 s 1.00 1.14 .89 n 5 5 6 X Control 1 Control 2 Experimental X i 7,7,8,8,6 7,7,8,9,6 4,4,6,6,5,5 SS w = Σ ( X i,j – X j )

31. Back to our Example… SS w = 13.20 df w = N-k Well, in our example, we had an N of 16. And we had 3 groups in our IV (control 1, control 2, experimental) So our df w is 16 - 3 = 13

32. Decomposing variance General formula for variance of a set of numbers: SS df MS B MS W MS W = SS w /df w MS W = 13.20/13 = 1.105 Next step… we need to find MS B (Mean Square Between)

33. The End of Part 1