Introduction to analysis of variance

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# Introduction to analysis of variance - PowerPoint PPT Presentation

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

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### 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
• 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