Psychology 340 Spring 2010

Analysis of Variance (ANOVA)

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social SciencesOutline (for week)

• 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


Social SciencesOutline (for week)




Social SciencesExample

• Effect of knowledge of prior behavior on jury decisions– Dependent variable: rate how innocent/guilty– Independent variable: 3 levels

• Criminal record• Clean record• No information (no mention of a record)

Compare the means of these three groupsClean recordJurors

Guilt Rating

Criminal record

No Information

Guilt Rating

Guilt Rating

XC

XB

XA


Social Sciences

Statistical analysis follows design

• The 1 factor between groups ANOVA:– More than two– Independent & One

score per subject– 1 independent

variable


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

SSA =18.0

XB =4.0

SSB =20.0

XC =5.0

SSC =26.0

• More than two groups– Now we can’t just

compute a simple difference score since there are more than one difference

Generic test statistic

observed differencedifference expected by chance


Social Sciences Analysis of Variance

XB XAXC


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

– Need a measure that describes several difference scores

– Variance• Variance is essentially

an average squared difference

Observed variance

Variance from chanceF-ratio =

• More than two groups

test statistic

Tip: Many different groupings so use subscripts to keep things straight


Social Sciences Testing Hypotheses with ANOVA

– Step 1: State your hypotheses• Hypothesis testing: a five step program

• Null hypothesis (H0)– All of the populations all have same mean

• Alternative hypotheses (HA)– Not all of the populations all have same mean– There are several alternative hypotheses– We will return to this issue later

H 0 :μA =μB =μC


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• Reconciling our multiple alternative hypotheses


Social Sciences Step 4: Computing the F-ratio

• Analyzing the sources of variance– Describe the total variance in the dependent measure

• Why are these scores different?

XB XAXC

• Two sources of variability– Within groups– Between groups


Social Sciences Step 4: Computing the F-ratio

• Within-groups estimate of the population variance – Estimating population variance from variation from

within each sample• Not affected by whether the null hypothesis is true

XB XAXC

Different people within each group

give different ratings


Social Sciences

• Between-groups estimate of the population variance – Estimating population variance from variation between

the means of the samples• Is affected by whether the null hypothesis is true

Step 4: Computing the F-ratio

XB XAXC

There is an effectof the IV, so the

people in differentgroups give different

ratings


Social Sciences Partitioning the variance

Total variance

Stage 1

Between groups variance

Within groups variance

Note: we will start with SS, but willget to variance



Total varianceCriminal record Clean record No information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3

GM =X∑

N=8515

=5.67

SSTotal = X −GM( )∑ 2= 10 −5.67( )2 + ...+ 3−5.67( )2 =107.33

• Basically forgetting about separate groups– Compute the

Grand Mean (GM)

– Compute squared deviations from the Grand Mean

dfTotal =N−1=15 −1=14



Total varianceCriminal record Clean record No information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3

GM =X∑

N=8515

=5.67

SSTotal = X −GM( )∑ 2= 10 −5.67( )2 + ...+ 3−5.67( )2 =107.33

• Basically forgetting about separate groups– Compute the

Grand Mean (GM)

– Compute squared deviations from the Grand Mean

dfTotal =N−1=15 −1=14

Formula alert:

SSTotal = X∑ 2−

X∑( )2

N

= 102 + 72 + ...+ 32( ) −852

15= 107.33



Total variance

Stage 1



SSTotal = X−GM( )∑ 2

dfTotal =N−1



Within groups varianceCriminal record Clean record No information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3

SSWithin = SSeach group∑ =SSA + SSB + SSC =64

• Basically the variability in each group1. Add up of the SS

from all of the groups

dfWithin = dfeach group∑ =4 + 4 + 4 =12

XA =8.0 XB =4.0 XC =5.0

SSA =18.0 SSB =20.0 SSC =26.0

dfA =4 dfB =4 dfC =4



Total variance

Stage 1




dfTotal =N−1

SSWithin = SSeach group∑dfWithin = dfeach group∑



Between groups varianceCriminal record Clean record No information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3

SSBetween = n X −GM( )∑ 2

• Basically how much each group differs from the Grand Mean1. Subtract the GM

from each group mean

2. Square the diffs3. Weight by number

of scores

dfbetween =#groups−1=3−1=2

XA =8.0 XB =4.0 XC =5.0

SSA =18.0 SSB =20.0 SSC =26.0

=5 8 − 5.67( )2 + 5 4 − 5.67( )2 + 5 5 − 5.67( )2

=43.3



Between groups varianceCriminal record Clean record No information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3


• Basically how much each group differs from the Grand Mean1. Subtract the GM

from each group mean

2. Square the diffs3. Weight by number

of scores


XA =8.0 XB =4.0 XC =5.0

SSA =18.0 SSB =20.0 SSC =26.0

=5 8 − 5.67( )2 + 5 4 − 5.67( )2 + 5 5 − 5.67( )2

=43.3

Formula alert:

SSBetween =T 2

n∑ −G 2

N

=402

5+

202

5+

252

5⎛⎝⎜

⎞⎠⎟

−852

15= 43.3

T=treatment totalN=#scores in treatment

G=grand total



Total variance

Stage 1




dfTotal =N−1



dfbetween =#groups−1



Total variance

Stage 1




dfTotal =N−1



dfbetween =#groups−1

Now we return to variance. But, we call it Means Square (MS)

MSWithin =SSWithin

dfWithinMSBetween =

SSBetween

dfBetween

Recall:variance =

SSdf



Mean Squares (Variance)

SSBetween =43.3

dfbetween =2SSWithin =64dfWithin =12

MSBetween =43.32

=21.67 MSWithin =6412

=5.33




Social Sciences

• The F ratio– Ratio of the between-groups to the within-groups

population variance estimate

Step 4: Computing the F-ratio

• The F distribution• The F table

Observed variance

Variance from chanceF-ratio = =

MSBetween

MSWithin

=21.675.33

= 4.07

Do we reject or failto reject the H0?


Social Sciences Carrying out an ANOVA

• The F distribution • The F table– Need two df’s

• dfbetween (numerator)

• dfwithin (denominator)

– Values in the table correspond to critical F’s

• Reject the H0 if your computed value is greater than or equal to the critical F

– Often separate tables for 0.05 & 0.01

http://www.psychology.ilstu.edu/psy138/resources/tables.html#fdist



• The F distribution • The F table– Need two df’s






Denominator df

1 2 3 4 5 6 …

1 1624,052

2005,000

2165,404

2255,625

2305,764

2345,859

2 18.5198.50

19.099.0

19.1799.17

19.2599.25

19.3099.30

19.3399.33

3 10.1334.12

9.5530.82

9.2829.46

9.1228.71

9.0128.24

8.9427.91

4 7.7121.20

6.9518.0

6.5916.7

6.3915.98

6.2615.52

6.1615.21

5 6.6116.26

5.7913.27

5.4112.06

5.1911.39

5.0510.97

4.9510.67

6 5.9913.75

5.1410.93

4.769.78

4.539.15

4.398.75

4.288.47

…

∞

Numerator df

Table B-4, pg 731-733Lightface type are Fcrits for α = 0.05

Boldface type are Fcrits for α = 0.01



• The F table– Need two df’s






F =MSBetween

MSWithin

=21.675.33

= 4.07

Do we reject or failto reject the H0?

– From the table (assuming 0.05) with 2 and 12 degrees of freedom the critical F = 3.89.

– So we reject H0 and conclude that not all groups are the same


Social Sciences Summary of Example ANOVA


10 5 4

7 1 6

5 3 9

10 7 3

8 4 3XA =8.0

SSA =18.0

XB =4.0

SSB =20.0

XC =5.0

SSC =26.0

GM =X∑

N=8515

=5.67

SSTotal = X −GM( )∑ 2=107.33

dfTotal =N−1=15 −1=14

SSWithin = SSeach group∑ =64

dfWithin = dfeach group∑ =4 + 4 + 4 =12

SSBetween = n X −GM( )∑ 2=43.3


F =MSBetween

MSWithin

=21.675.33

= 4.07

MSWithin =SSWithin

dfWithin

=5.33

MSBetween =SSBetween

dfBetween=21.67

Fcrit(2,12) = 3.89, so we reject H0

dfA =4 dfB =4 dfC =4


Social SciencesNext time



Documents

Psychology 340 Spring 2010