Post on 19-Jan-2018
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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 Fobtained 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 = 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
Groups
2Last
22
21
Within2Within
...or N
SSSMSS
Within-groupsvariance a.k.a Mean SquaresWithin (MSwithin)
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…
Between-Group variance
Between
22M
)(df
GMMS
1GroupsBetween Ndf
Variance of distributionof means…will use tofind Betw-grp variance
Sum up squared deviations ofeach group mean – Grand mean
(cont.)– 2b) Take S2
M 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
))((or 2MBetween
2Between nSMSS
or Within
Between2Within
2Between
MSMS
SSF
n= group size,not total samplesize
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…
1GroupsBetween Ndf
Last21Within ... dfdfdfdf Df1 = n1 – 1,Df2 = n2 – 1,etc.