statistics for the social sciences psychology 340 spring 2009 analysis of variance (anova)
DESCRIPTION
Statistics for the Social Sciences Outline 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 comparisonsTRANSCRIPT
Statistics for the Social SciencesPsychology 340
Spring 2009
Analysis of Variance (ANOVA)
Statistics for the Social Sciences
Outline
• Basics of ANOVA• Why• Computations• ANOVA in SPSS• Post-hoc and planned comparisons• Assumptions • Power and effect size for ANOVA
Statistics for the Social Sciences
Outline
• 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
Statistics for the Social Sciences
Example
• 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
Statistics 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 3XA =8.0 XB =4.0 XC =5.0
– Need a measure that describes several difference scores
• Variance
Test statisticObserved variance
Variance from chanceF-ratio =
• More than two groups
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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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• 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!!
Testing Hypotheses with ANOVA
Choosing between these requires additional test H0 :μA =μC ≠μB
Statistics 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
hypothesesTest1 H0 :μA =μB
Test2 H0 :μA =μC
Test3 H0 :μB =μC
– reject
– reject– fail to reject
Statistics 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
– 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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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
Statistics 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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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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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 3XA =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 =SSBetweendfBetween
=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
Statistics for the Social Sciences
Planned Comparisons
• Example: compare criminal record & no info grps
Criminal record Clean record No information
10 5 4
7 1 65 3 9
10 7 3
8 4 3XA =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 =SSBetweendfBetween
=22.5
1= 22.5
GM =X∑
N=6510
=6.5
3) Figure F in usual way
F =MSBetweenMSWithin
=22.55.33
= 4.22 Fcrit (1,12) = 4.75α = 0.05
Fail to reject H0: Criminal record and no info are not statistically different
XB XAXC
Statistics 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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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 =SSBetweenSSTotal
=(MS2
Between )(dfBetween )(MS2
Between )(dfBetween ) + (MS2Within )(dfWithin )
Recall:S2 =MS=
SSdf
SStotal =SSbetween + SSwithin
=(F)(dfBetween )
(F)(dfBetween ) + (dfWithin )
Statistics 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– Size of effect depends, in part, on degrees of freedom
• See tables 9-9& 9-10 in textbook for what is considered “small” “medium” and “large”
Statistics 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
Statistics 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• Comparisons are entered with 1, 0, & -1