1. ANOVAs

2.  Analysis of Variance (ANOVA)  Difference in two or more average scores in different groups  Simplest is one-way ANOVA (one variable as predictor); but can include multiple predictors

3.  Differences between the groups are separated into two sources of variance ◦ Variance from within the group ◦ Variance from between the groups  The variance between groups is typically of interest

4.  Use when: ◦ you are examining differences between groups on one or more variables, ◦ the participants in the study were tested only once and ◦ you are comparing more than two groups

5.  Factor: the variable that designates the groups to be compared  Levels: the individual comparable parts of the factor  Factorial designs have more than one variable as a predictor of an outcome

6.  F is based on variance, not mean differences  Partial out the between condition variance from the within condition variance

7. F = MS between MS within MS between = SS between /df between MS within = SS within / df within

8.  Therapist wants to examine the effectiveness of 3 techniques for treating phobias. Subjects are randomly assigned to one of three treatment groups. Below are the rated fear of spiders after therapy.  X 1 : 5 2 5 4 2  X 2 : 3 3 0 2 2  X 3 : 1 0 1 2 1

9.  1. State hypotheses  Null hypothesis: spider phobia does not differ among the three treatment groups ◦ μ Treatement1 = μ Treatment2 = μ Treatment3  Research hypothesis: spider phobia differs in at least one treatment group compared to others OR there is an effect of at least one treatment on spider phobia ◦ X Treatment1 ≠ X Treatment2 ◦ X Treatment1 ≠ X Treatment3 ◦ X Treatment2 ≠ X Treatment3 ◦ X Treatment1 ≠ X Treatment2 ≠ X Treatment3 (just write this one for ease, but all are made)

10. F = MS between MS within MS between = SS between /df between MS within = SS within / df within

11. SS between = Σ(ΣX) 2 /n – (ΣΣX) 2 /N ΣX = sum of scores in each group ΣΣX = sum of all the scores across groups n = number of participants in each group N = number of participants (total)

12. SS within = ΣΣ(X 2 ) – Σ(ΣX) 2 /n ΣΣ(X 2 ) = sum of all the sums of squared scores Σ(ΣX) 2 = sum of the sum of each group’s scores squared n = number of participants in each group

13. Ss total = ΣΣ(X 2 ) – (ΣΣX) 2 /N ΣΣ(X 2 ) = sum of all the sums of squared scores (ΣΣX) 2 = sum of all the scores across groups squared N = total number of participants (in all groups)

14. F = MS between MS within MS between = SS between /df between  Df between = k-1 (k=# of groups)

15.  6. Determine whether the statistic exceeds the critical value ◦ 6.01 > 3.89 ◦ So it does exceed the critical value  7. If over the critical value, reject the null  & conclude that there is a significant difference in at least one of the groups

16.  For an ANOVA, the test statistic only tells you that there is a difference  It does not tell you which groups were different from other groups  There are numerous post-hoc tests that you can use to tell the difference  Here, we will use Bonferroni corrected post- hoc tests because they are already familiar (similar to t-tests, but with corrected critical value levels to reduce Type 1 error rates)

17.  In results ◦ There was a significant effect of type of treatment on spider phobia, F(2, 12) = 6.01, p <.05.  With post-hoc tests ◦ There was a significant effect of type of treatment on spider phobia, F(2, 12) = 6.01, p <.05. Participants who received treatment X 3 were less afraid of spiders (M = 1.00, SD = 0.71) than participants who received treatment X 1 (M = 3.60, SD = 1.52), t(8) = 3.47, p =.008, but did not differ from participants who received treatment X 2 (M = 2.00, SD = 1.22, t(8) = 1.58, n.s. Participants who received treatments X 1 and X 2 did not significantly differ, t(8) = 1.84, n.s.  If it had not been significant: ◦ There was no significant effect of type of treatment on spider phobia, F(2, 12) = 2.22, n.s.

18. Source Type III Sum of SquaresdfMean SquareFSig. Corrected Model 17.200(a)28.6006.000.016 Intercept 72.6001 50.651.000 cond 17.20028.6006.000.016 Error 17.200121.433 Total 107.00015 Corrected Total 34.40014