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.