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Comparing Many Means Statistics 2126

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Introduction We have talked about comparing two means You know, like is group 1 different from group 2 So you do a 2 sample t test, or perhaps in special cases a dependent sample t test and you find out the answer

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However… What if we had a situation where we had more than 2 groups, say 4 groups How many t tests would you have to do? 1 v 2 1 v 3 1 v 4 2 v 3 2 v 4 3 v 4 Ok so that is a lot, what would it do to your Type I error rate?

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Consider… As noted the alpha level would increase, which would mean you would have to decrease it So you lose power Plus what about the overall pattern of differences? Is the pattern of differences different than chance?

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What do we need? Beer, lots of beer.. But before that we need a statistic that compares the overall amount of variation with the variation that you would expect due to chance (i.e., subject differences) The question is, did this pattern of differences come about simply due to chance variation?

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The hypotheses H 0 : μ 1 = μ 2 = μ 3 =…= μ k H a : at least two means differ Now think about this, we will look at two measures of variation, overall variance vs. individual differences, how would we compare them?

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How indeed… Use a ratio of the two Between group variation / within group variation If the variation due to groups was simply by change what would this ratio equal? 1 If the groups differ it would be > 1

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This ratio is called… The F ratio F is for Fisher So we can use variance estimates to see if group means differ, cool eh? E(F) = 1

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Pretty simple We divide between group variation by within group variation This is called analysis of variance or ANOVA Any score of any individual is made up of between group variation and within group variation

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How do we calculate this? We get sums of squares for between and within groups and divide by their degrees of freedom Then we divide MSBG / MSWG Gives us the F ratio which we compare to a critical value for F

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ANOVA summary table SourcedfSSMSF Between Groups k-1 Within Groups N-K TotalN-1

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The test itself Compare the obtained F value to the critical value The Critical value will have two sets of df, one for the numerator and one for the denominator

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An example Group 1Group 2Group 3Group 4 Mean102015 Standard deviation 5465 n10 SSBG = 50SSWG = 918

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ANOVA Summary Table svdfSSMSF Between groups 35016.67.65 Within Groups 3691825.5 TOTAL39968

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Compare to critical value F(3,36) ~ F(3,30) = 2.92 Well our obtained value is less than this so We fail to reject H 0 If we did reject the null we just know two means differ, not which two Post hoc tests take care of this

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Assumptions Normally distributed populations SRS Homogeneity of variance

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