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Analysis of Variance (ANOVA)
Statistics for the Social SciencesPsychology 340
Spring 2010
PSY 340Statistics for the
Social SciencesOutline
• Basics of ANOVA• Why• Computations• Post-hoc and planned comparisons• Power and effect size for ANOVA• Assumptions • SPSS
– 1 factor between groups ANOVA
– Post-hoc and planned comparisons
PSY 340Statistics for the
Social SciencesOutline
• Basics of ANOVA• Why• Computations• Post-hoc and planned comparisons• Power and effect size for ANOVA• Assumptions • SPSS
– 1 factor between groups ANOVA
– Post-hoc and planned comparisons
PSY 340Statistics for the
Social SciencesExample
• Effect of knowledge of prior behavior on jury decisions– Dependent variable: rate how innocent/guilty
– Independent variable: 3 levels
Compare the means of these three groupsClean recordJurors
Guilt Rating
Criminal record
No Information
Guilt Rating
Guilt Rating
XC
XB
XA
PSY 340Statistics for the
Social Sciences Analysis of Variance
XB XAXC
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0 XB =4.0 XC =5.0
– Need a measure that describes several difference scores
• Variance
Test statistic
Observed variance
Variance from chanceF-ratio =
• More than two groups
PSY 340Statistics for the
Social Sciences Testing Hypotheses with ANOVA
– Step 2: Set your decision criteria
– Step 3: Collect your data
– Step 4: Compute your test statistics • Compute your estimated variances
• Compute your F-ratio
• Compute your degrees of freedom (there are several)
– Step 5: Make a decision about your null hypothesis
• Hypothesis testing: a five step program– Step 1: State your hypotheses
– Additional tests: Planned comparisons & Post hoc tests• Reconciling our multiple alternative hypotheses
PSY 340Statistics for the
Social Sciences
• Null hypothesis: H0: all the groups are equal
XB XAXC
H0 : μA =μB =μC
– Step 1: State your hypotheses
• Hypothesis testing: a five step program
• Alternative hypotheses (HA)
– Not all of the populations all have same mean
H A : μA ≠μB ≠μC
H A : μA =μB ≠μC
H A : μA ≠μB =μC
The ANOVA tests this one!!
The ANOVA tests this one!!
Testing Hypotheses with ANOVA
Choosing between these requires additional test
Choosing between these requires additional test
H0 : μA =μC ≠μB
PSY 340Statistics for the
Social Sciences 1 factor ANOVA
XB XAXC
H A : μA ≠μB ≠μC
H A : μA =μB ≠μC
H A : μA ≠μB =μC
H0 : μA =μC ≠μB
• Alternative hypotheses (HA)
– Not all of the populations all have same mean
• Planned contrasts and Post-hoc tests:– Further tests used to rule out the different alternative
hypotheses
Test1 H0 : μA =μB
Test2 H 0 : μA =μC
Test3 H0 : μB =μC
– reject
– reject
– fail to reject
PSY 340Statistics for the
Social Sciences Why do the ANOVA?
• What’s the big deal? Why not just run a bunch of t-tests instead of doing an ANOVA?– Experiment-wise error (see pg 398, Box 13.1 for discussion)
– The type I error rate of the family (the entire set) of comparisons
» αEW = 1 - (1 - α)c where c = # of comparisons
» e.g., If you conduct two t-tests, each with an alpha level of 0.05, the combined chance of making a type I error is nearly 10 in 100 (rather than 5 in 100)
– Planned comparisons and post hoc tests are procedures designed to reduce experiment-wise error
PSY 340Statistics for the
Social Sciences Which follow-up test?
• Planned comparisons– A set of specific comparisons that you “planned” to do
in advance of conducting the overall ANOVA
• Post-hoc tests– A set of comparisons that you decided to examine only
after you find a significant (reject H0) ANOVA
– Often end up looking at all possible pair-wise comparisons
PSY 340Statistics for the
Social Sciences Planned Comparisons
• General Rule of Thumb– Don’t plan more contrasts than (# of conditions – 1)
• Different types– Simple comparisons - testing two groups– Complex comparisons - testing combined groups– Bonferroni procedure (Dunn’s test)
• Use more stringent significance level for each comparison– Divide your desired α-level by the number of planned contrasts
PSY 340Statistics for the
Social Sciences Planned Comparisons
• Basic procedure:1. Within-groups population variance estimate
(denominator)2. Between-groups population variance estimate of the
two groups of interest (numerator)3. Figure F in usual way
PSY 340Statistics for the
Social Sciences Planned Comparisons
• Example: compare criminal record & no info grps
XB XAXC
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
SSWithin =64dfWithin =12
MSWithin =6412
=5.33
SSBetween =43.3dfbetween =2
MSBetween =43.32
=21.67
1) Within-groups population variance estimate (denominator)
MSWithin =6412
=5.33
2) Between-groups population variance estimate of the two groups of interest (numerator)
SSBetween = n X −GM( )∑ 2
dfbetween =#groups−1
MSBetween =SSBetween
dfBetween
=2 −1 = 1
=22.5
1= 22.5
=5 8 − 6.5( )2
+ 5 5 − 6.5( )2
GM =X∑
N=
6510
=6.5
=22.5
PSY 340Statistics for the
Social Sciences Planned Comparisons
• Example: compare criminal record & no info grps
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
SSWithin =64dfWithin =12
MSWithin =6412
=5.33
SSBetween =43.3dfbetween =2
MSBetween =43.32
=21.67
1) Within-groups population variance estimate (denominator)
MSWithin =6412
=5.33
2) Between-groups population variance estimate of the two groups of interest (numerator)
MSBetween =SSBetween
dfBetween
=22.5
1= 22.5
GM =X∑
N=
6510
=6.5
3) Figure F in usual way
F =MSBetween
MSWithin
=22.5
5.33= 4.22 Fcrit (1,12) = 4.75
α = 0.05
Fail to reject H0: Criminal record and no info are not statistically different
XB XAXC
PSY 340Statistics for the
Social Sciences Post-hoc tests
• Generally, you are testing all of the possible comparisons (rather than just a specific few)– Different types
• Tukey’s HSD test (only with equal sample sizes)
• Scheffe test (unequal sample sizes okay, very conservative)
• Others (Fisher’s LSD, Neuman-Keuls test, Duncan test)
– Generally they differ with respect to how conservative they are.
PSY 340Statistics for the
Social Sciences Effect sizes in ANOVA
• The effect size for ANOVA is r2
– Sometimes called η2 (“eta squared”)
– The percent of the variance in the dependent variable that is accounted for by the independent variable
r2 =SSBetween
SSTotal
=(MS2
Between )(dfBetween )
(MS2Between )(dfBetween ) + (MS2
Within )(dfWithin )
Recall:
S2 =MS=SSdf
SStotal =SSbetween + SSwithin
=(F)(dfBetween )
(F)(dfBetween ) + (dfWithin )
PSY 340Statistics for the
Social Sciences Effect sizes in ANOVA
• The effect size for ANOVA is r2
– Sometimes called η2 (“eta squared”)
– The percent of the variance in the dependent variable that is accounted for by the independent variable
r2 =SSBetween
SSTotal
=43.3
107.33= .404
SSTotal = X −GM( )∑ 2=107.33
SSBetween = n X −GM( )∑ 2=43.3
PSY 340Statistics for the
Social Sciences ANOVA Assumptions
• Basically the same as with T-tests– Assumes that the distributions are Normal– Assumes that the distributions have equal
variances
– In both cases ANOVA analyses are generally robust against violations of these assumptions
PSY 340Statistics for the
Social Sciences ANOVA in SPSS
• Let’s see how to do a between groups 1-factor ANOVA in SPSS (and the other tests too)– Enter the data: similar to independent samples t-test,
observations in one column, a second column for group assignment
– Analyze: compare means, 1-way ANOVA• Observations -> Dependent list
• Group assignment -> factor
– specify any comparisons or post hocs at this time too• Planned Comparisons (contrasts): are entered with 1, 0, & -1
• Post-hoc tests: make sure that you enter your α-level
PSY 340Statistics for the
Social Sciences Analysis of Variance
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0 XB =4.0 XC =5.0