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) Compare faculty, staff, students’ attitudes about Blm-Normal.

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

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)

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)

–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)

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)

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 F obtained need to be? Use F table to find F critical value If F obtained > F critical  reject Null

Carrying out an ANOVA 1) Find population variance from the variation of scores within each group (Within-groups = S 2 within) –Will need to start w/estimates of each group’s variance (S 2 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 S 2 Within-groups variance a.k.a Mean Squares Within (MSwithin)

Between-Group variance 2a) Estimate Between-groups variance –focuses on diffs between group means –Estimate the variance of the distribution of means (S 2 M ) –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…

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

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

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.