Statistics for the Social Sciences Psychology 340 Spring 2009 Analysis of Variance (ANOVA)

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


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


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


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


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


• 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


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


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


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


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


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


Planned Comparisons

• Example: compare criminal record & no info grps

XB XAXC


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


Planned Comparisons

• Example: compare criminal record & no info grps


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


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.


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 )


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”


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


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

Statistics for the Social Sciences Psychology 340 Spring 2009 Analysis of Variance (ANOVA)

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