ANOVAs Analysis of Variance (ANOVA) Difference in two or more average scores in different groups Each participant tested once Same outcome tested in each group Simplest is one-way ANOVA (one variable as predictor); but can include multiple predictors 1
Differences between the two 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 Factor: the variable that designates the groups to be compared Levels: the individual comparable parts of the IV Between vs. within Factorial designs have more than one variable as a predictor of an outcome 2
F is based on variance, not mean differences Partial out the between condition variance from the within condition variance F = MS between MS within MS between = SS between /df between MS within = SS within /df within 3
Total Variability Between-treatments variance Within-treatments variance Measures differences due to 1. Treatment effects 2. Chance Measures differences due to 1. Natural variability 1. State hypotheses Null hypothesis: there is no effect of treatment on spider phobia OR mean spider phobia does not differ among the three treatment groups Mean T1 = Mean T2 = Mean T3 Research hypothesis: at least one mean spider phobia differs among the three treatment groups OR there is an effect of at least one treatment on spider phobia Mean T1 Mean T2 Mean T1 Mean T3 Mean T2 Mean T3 Mean T1 Mean T2 Mean T3 4
Need to calculate 3 SOS scores Between ee subjects SOS Within subjects SOS Total SOS (sum of between & within group SOS) SS = Σ(ΣX) 2 /n (ΣΣX) 2 between ( ) /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) 5
SS = ΣΣ(X 2 Σ(ΣX) 2 within ( ) ( ) /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 p in each group Ss = ΣΣ(X 2 (ΣΣX) 2 total ( ) /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 p (in all groups) 6
Group 1 Group 2 Group 3 Score X 2 1 Score X 2 2 Score X 2 3 5 25 3 9 1 1 2 4 3 9 0 0 5 25 0 0 1 1 4 16 2 4 2 4 2 4 2 4 1 1 n 5 5 5 N = 15 ΣX 18 10 5 ΣΣX=33 X 3.6 2 1 (ΣΣX) 2 =1089 Σ(X 2 ) 74 26 7 ΣΣ(X 2 ) = 107 (ΣX 2 )/ n 324/5 64.8 100/5 20 25/5 5 Σ(ΣX 2 )/n = 89.8 F = MS between MS within MS between = SS between /df between Df between = k-1 (k=# of groups) = MS within = SS within /df within Df within = N-k (N=sample size) = 7
5. Determine the critical value for rejection of the null Look at table in book or other place (p. 336) At critical value at p <.05 is 6. Determine whether the statistic exceeds the critical value 7. If over the critical value, reject the null & conclude that there is a significant difference between at least one of the means 8
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 (that correct for increased Type I error rates) Here, we will use t-tests because they are already familiar In results 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. 9
Between Groups Within Groups Sum of Mean Squares df Square F Sig. 17.200 2 8.600 6.000.016 17.200 12 1.433 Total 34.400 14 Type III Sum Source of Squares df Mean Square F Sig. Corrected Model 17.200(a) 2 8.600 6.000.016 Intercept cond Error Total Corrected Total 72.600 1 72.600 50.651.000 17.200 2 8.600 6.000.016 17.200 12 1.433 107.000 15 34.400 14 10