Analysis of Variance (ANOVA)

1. Analysis of Variance (ANOVA) The F distribution Good for two or more groups

2. The F distribution • F is a ratio of two independent estimates of the variance of the population • Consequently, it depends on the analysis (separating into parts) of the variance in a set of scores. • We have already analyzed the variance in a set of scores when we did t tests

3. The analysis of variance • The numerator of the t test is the difference between two means. Since the variance is the average of the differences from a mean, the difference between two means drawn from the same set of scores is one estimate of variance. • The denominator of the t test, although expressed as a standard deviation, is also an estimate of variance.

4. The F ratio • A ratio of two estimates of variance drawn from the same set of scores is called an F ratio. • F = larger estimate of variance smaller estimate of variance • F = variance estimate 1 of s2 variance estimate 2 of s2

5. Why use ANOVA? • We sometimes have independent variables with more than two groups: • Grape Kool-Aid, Lemon Kool-Aid, and water • One marijuana cigarette, two, three, four… • TQM in place for one month, two, three… • Four styles of government • With more than two groups, multiple t tests would be necessary. • Multiple t tests inflate the Type I error rate.

6. Logic of ANOVA • 1. Find the total variance in a set of data. • 2. Analyze the variance into • a. The part due to the treatment (plus people) • b. The part due to people without treatment (individual differences and error) • 3. Form an F ratio of the two parts

7. The ANOVA summary table • To keep track of the ANOVA process, start with a summary table: Source SS df MS ( s2) F p Decision Between Within Total

8. Notation summary • N is the number of scores altogether. • n is the number of scores in a single group. • k is the number of groups. • g is a particular group. • The variance, s2 , is also symbolized MS, which stands for Mean Square.

9. SS total • To get SStotal, simply group all of the scores for all of the groups together and find the SS as if there were only one group: SStotal = SX2 - (SX)2 / N Or SStotal = SX2 – (G2 / N)

10. + + + n1 n2 nk SS within groups • SS within groups is due to individual differences and error. • To compute SSwithin,calculate SS normally for each group separately, and add them up. SSwithin = S(SX2 - (SX)2/ n) =SS1+SS2+…+SSk = SX2 - (SX1)2 (SX2)2 … (SXk)2

11. + + + n1 n2 nk SS between groups • To get SSbetween, treat the sum of each group as a score, and apply the be-bop version of the sum of squares song: • SSB = • = (SX1)2 (SX2)2 … (SXk)2 - (SX)2 N

12. Completing the one-way ANOVA Source SS dfMS ( s2) F p Between k-1 SSB/dfB MSB/MSW Within N-k SSW/dfW Total N-1 To obtain p, either use the table to find the critical value of F, or SPSS to find the probability of the obtained value of F.

13. Come-By- Chance Pop: 6,284 Making your decision To decide whether the obtained F ratio is significant, compare it to the table value (the critical value). If the obtained F is equal to or larger than the critical F, reject H0. Computed value: Length of walk Fishy zone: Reject H0 Table value: Length of pier .05 level

14. Decision rules for ANOVA • Find the critical values for F from the table. Use dfB for the (between) column, and dfW for the (within) row in the table. • The first table has the critical value at the .05 level, and the second table for the .01 level. • In the p column of the ANOVA summary table, enter <.01 if the obtained F ratio is greater than the value in the second table; <.05 if it is greater than the value in the first table but less than the value in the second table; and n.s. (for not significant) if it is less than the value in the first table.

15. More ANOVA items • For two group studies, t2 = F • Two-way factorial ANOVA • Main effects: Combines two (or more) studies • Fully crossed factors: A x B • Interaction effect: The effect of one independent variable depends on the level of the other independent variable

16. Try this one • Here are the number of errors on an analysis of variance problem for each of nine students after drinking grape Kool-Aid®, Frankenberry Punch®, or water. • GKAFBPH2O 5 8 10 3 7 11 4 7 12

17. Effect size measures • Significant t and F ratios show that there is a real effect of the treatment, a real difference between the groups that cannot be explained by chance. • Effect size measures show how big the effect of the treatment is. • C.O.D. is one effect size measure: C.O.D. = r2 = SSB /SSTotal = h2

18. Effect sizes in samples vs. populations • h2 is a sample estimate of the proportion of the variance in the dependent variable that is accounted for by the independent variable. • For population estimates of effect size, use a different statistic, w2

19. Computing w2 • For the t test for independent samples, w2 = t2 – 1 t2 + N – 1 • For the analysis of variance, w2 = SSB – (k – 1)MSW SSTotal + MSW

20. Concluding details • ANOVA assumptions are the same as the assumptions for the t-test for independent samples. • Try to have each group/sample be the same size, that is, have equal ns. However, the ANOVA computations will work with unequal sample sizes.

21. Reporting ANOVA in APA format • Report the mean and standard deviation for each sample/group in a table, and refer to it in a sentence reporting the F ratio: • Table 1 contains the means and standard deviations for the IQs, which were significantly different, F(3, 36) = 9.47, p < .05. Table 1. Tested IQ.