Introduction to analysis of variance

Introduction to analysis of variance Chapter 13

A new research situation • You want to know if psychology majors, physics majors, and math majors differ in their happiness • You can’t use any of the tests we’ve discussed so far, since you have three levels of major (i.e., three different types of major people could have) • What to do?

Analyze the variance • Where does the difference lie? • Is it between all the majors? • Is it between one major and the other majors? •  Analysis of variance • ANOVA

Key question • Where is there more variability – between groups or within groups? • If the null hypothesis were true, these would be equal • If there is more variability between groups than within groups, this provides support for the research hypothesis

To calculate this • Need to calculate variance between groups and variance within groups • First, though, will calculate SS • Then, will divide by df to get variance

How to calculate this • Total SS = SS between groups + SS within groups • Total SS = computing SS for all data, regardless of group • Within SS = computing SS for each group of scores, and then adding those group SS’s together •  SS between groups = total SS – SS within groups

Getting to df • df total = total number of participants minus 1 • df within groups = sum of df within each group • df between groups = df total – df within

Putting it all together • Variance between groups = SS between/df between • AKA MS between (for mean squared between) • Variance within groups = SS within/df within • AKA MS within (for mean squared within)

Figuring out where there’s more variability • MSbetween/MSwithin: AKA F ratio • If this is 1, there is the same amount of variability between groups as within groups • As this gets greater than 1, there is more variability between groups than within groups •  less likely to get by chance if the null is true

How big is big enough for F? • Determined by critical F value • Found by using df for the numerator (df between) and df for the denominator (df within) • If calculated F > critical F, reject the null, since p < alpha

What about effect size? • Assessed with r2: how much of the outcome variable is explained by knowing which groups someone is from? • Calculated by SSbetween/SStotal • Referred to as eta squared (h2)

Telling the world in APA style • F (df numerator, df denominator) = calculated F value, p information, h2 = X

Where is the difference? • The result of the ANOVA test tells you there’s a difference somewhere between groups, but not where •  post hoc (after the fact) tests are used, if there’s a significant ANOVA, to figure out which groups are different from each other • (if multiple independent samples t-tests were used instead, there would be an inflated familywise type 1 error)

Post hoc option 1: Tukey • Gives a number that captures how big the difference between group means needs to be in order for that difference to be considered significant

Post hoc option 2: Scheffe • Recalculates a new F value for each comparison of two groups • Uses MS between from just those two groups • Is more conservative than an ANOVA with just those two groups, since it uses MS within from all groups, and uses df from all groups

Points to take away • If you’re comparing more than two independent groups, you cannot use independent samples t-tests • Must use an ANOVA • This tells if there’s a difference somewhere • To figure out where, need follow-up (post hoc) tests

Introduction to analysis of variance