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Chapter 9

Chapter 9. Introduction to the Analysis of Variance Part 1: Oct. 22, 2013. Analysis of Variance (ANOVA). Testing variation among the means of several groups One-way analysis of variance Compare 3 or more groups on 1 dimension (IV)

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Chapter 9

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  1. Chapter 9 Introduction to the Analysis of Variance Part 1: Oct. 22, 2013

  2. Analysis of Variance (ANOVA) • Testing variation among the means of several groups • One-way analysis of variance • Compare 3 or more groups on 1 dimension (IV) • Compare faculty, staff, students’ attitudes about Blm-Normal.

  3. Basic Logic of ANOVA • Null hypothesis • Several populations all have same mean • Do the means of the samples differ more than expected if the null hyp were true? • Analyze variances • Focus on variation among our 3 group means • Two different ways of estimating population variance

  4. Basic Logic of ANOVA • Estimating pop. variance from sample variances • Assume all 3 pop have the same variance  average the 3 sample variances into pooled estimate • Called “Within-groups estimate of the population variance” • Not affected by whether the null hypothesis is true and the 3 means are actually equal (or not)

  5. Basic Logic of ANOVA • Another way to estimate pop variance: • Use the variation between the means of the samples • When the null hypothesis is true, 3 samples come from pops w/same mean • Also assume all 3 pop have same variance, so if Null is true, all populations are identical (same mean & variance) • But sample means (and how much they differ) will depend on amount of variability of distribution • See examples on board (and see Fig 9-1)

  6. This is why the variation in the 3 means will tell us something about the pop variance • Called “Between-groups estimate of the population variance” • But… • When the null hypothesis is not true, the 3 populations have different means • Samples from those 3 pop will vary because of variation within each pop and because of variation between pop • See board for drawing (and see fig 9-2)

  7. Basic Logic of ANOVA • Sources of variation in within-groups and between-groups variance estimates (Table 9-2) • When Null is true, Within-groups and Between-groups estimates should be about = (their ratio = 1) • When Research hyp is true, Between-groups is > within-groups estimate (it has more variance; ratio > 1)

  8. F Ratio • The F ratio – (the concept)… • Ratio of the between-groups to within-groups population variance • If ratio > 1, reject Null  • there are signif differences between means • How much >1 does Fobtained need to be? • Use F table to find F critical value • If F obtained > F critical  reject Null

  9. Carrying out an ANOVA • 1) Find population variance from the variation of scores within each group (Within-groups = S2within) • Will need to start w/estimates of each group’s variance (S2 will be given in hwk, exam; or see Ch 2 for formula) • In this chapter, we assume equal group sizes, so just average the 3 estimates of S2 Within-groups variance a.k.a Mean Squares Within (MSwithin)

  10. Between-Group variance • 2a) Estimate Between-groups variance • focuses on diffs between group means • Estimate the variance of the distribution of means (S2M) • First, find “Grand Mean” (GM), the mean of the means (Add all means/# means) • Then, subtract GM from each mean, square that deviation • Finally, add all deviation scores…

  11. Between-Group variance Sum up squared deviations of each group mean – Grand mean Variance of distribution of means…will use to find Betw-grp variance

  12. (cont.) • 2b) Take S2M and multiply by group size (assuming equal group sizes…for Ch 9) • Gives you S2Between aka MSbetween (Mean Squares Between) 3) Figure F obtained (F Ratio) using 2 MS’s n= group size, not total sample size

  13. F Table • Need to use alpha, Between-groups df, & Within-groups df • Between-groups degrees of freedom • Within-groups degrees of freedom If F obtained > F critical, reject Null. Example… Df1 = n1 – 1, Df2 = n2 – 1, etc.

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