analysis of variance (anova) statistics for the social sciences psychology 340 spring 2010
TRANSCRIPT
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Analysis of Variance (ANOVA)
Statistics for the Social SciencesPsychology 340
Spring 2010
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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 )
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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
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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
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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
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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