Statistics for the Social Sciences Psychology 340 Spring 2005 Within Groups ANOVA

Statistics for the Social Sciences

Psychology 340Spring 2005

Within Groups ANOVA


Outline

• Basics of within groups ANOVA– Repeated measures– Matched samples

• Computations• Within groups ANOVA in SPSS


Example

• Suppose that you want to compare three brand name pain relievers. – Give each person a drug, wait 15 minutes, then ask them to keep their hand in a bucket of cold water as long as they can. The next day, repeat (with a different drug)• Dependent variable: time in ice water• Independent variable: 4 levels, within groups

– Placebo– Drug A– Drug B– Drug C


Statistical analysis follows design

• The 1 factor within groups ANOVA:

– One group

– Repeated measures– More than 2 scores per subject


Statistical analysis follows design

• The 1 factor within groups ANOVA:

– One group

– Repeated measures– More than 2 scores per subject

– More than 2 groups

– Matched samples– Matched groups

- OR -


Within-subjects ANOVA

Placebo Drug A Drug B Drug C

3 4 6 7

0 3 3 6

2 1 4 5

0 1 3 4

0 1 4 3

XA =2.0

SSA =8.0

XB =4.0

SSB =6.0

XC =5.0

SSC =10.0

XP =1.0

SSP =8.0

XBXA XCXP

• n = 5 participants• Each participates in every condition (4 of these)



– Step 2: Set your decision criteria– Step 3: Collect your data – Step 4: Compute your test statistics

• Compute your estimated variances (2 steps of partitioning used)

• Compute your F-ratio• Compute your degrees of freedom (there are even more now)

– Step 5: Make a decision about your null hypothesis

• Hypothesis testing: a five step program

– Step 1: State your hypotheses


Step 4: Computing the F-ratio

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

• Why are these scores different? • Sources of

variability– Between groups– Within groups

XBXA XCXP

• Individual differences• Left over variance (error)

Because we use the same people in each

condition, we can figure out how much of the

variability comes from the individuals and remove it from the

analysis


condition, we can figure out how much of the

variability comes from the individuals and remove it from the

analysis


Partitioning the variance

Total variance

Stage 1

Between groups variance Within groups variance



Total variance

Stage 1


Stage 2Between subjects varianceError variance



Total variance

Stage 1



1) Treatment effect

2) Error or chance (without individual differences)

1) Individual differences

2) Other error

1) Other error (without individual differences)



condition, none of this variability comes from having different people in different conditions


condition, none of this variability comes from having different people in different conditions


• The F ratio– Ratio of the between-groups variance estimate to the population error variance estimate

Step 4: Computing the F-ratio

Observed variance

Variance from chanceF-ratio = =

MSBetweenMSError



Total variance

Stage 1



1) Treatment effect

2) Error or chance (without individual differences)


2) Other error

1) Other error (without individual differences)


F =MSBetweenMSError



Total variance

Stage 1


SSTotal = X−GM( )∑ 2

dfTotal =N−1

SSWithin = SSeach group∑dfWithin = dfeach group∑

SSBetween = n X−GM( )∑ 2

dfbetween =#groups−1




3 4 6 7

0 3 3 6

2 1 4 5

0 1 3 4

0 1 4 3

XA =2.0SSA =8.0

XB =4.0SSB =6.0

XC =5.0SSC =10.0

XP =1.0SSP =8.0

GM =3.0

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

dfTotal =N−1=20 −1=19SSTotal =82


dfbetween =#groups−1=4 −1=3

=5 1 − 3( )2

+ 5 2 − 3( )2

+ 5 4 − 3( )2

+ 5 5 − 3( )2

=50.0

SSWithin = SSeach group∑ =SSP +SSA +SSB +SSC =32

dfWithin = dfeach group∑ =4 + 4 + 4 + 4 =16



Total variance

Stage 1




dfTotal =N−1





PWhat is ?



3 4 6 7

0 3 3 6

2 1 4 5

0 1 3 4

0 1 4 3

XA =2.0

SSA =8.0

XB =4.0

SSB =6.0

XC =5.0

SSC =10.0

XP =1.0

SSP =8.0

The average score for each person

GM =3.0

Between subjects variance

SSBetweenSubs = ngroups P −GM( )∑ 2




3 4 6 7

0 3 3 6

2 1 4 5

0 1 3 4

0 1 4 3

XA =2.0

SSA =8.0

XB =4.0

SSB =6.0

XC =5.0

SSC =10.0

XP =1.0

SSP =8.0SSBetweenSubs = ngroups P −GM( )∑ 2

dfbetweenSubs =nsubjects −1=5−1=4

PWhat is ?The average score for each person

P20

4 =5.012

4 =3.012

4 =3.08

4 =2.08

4 =2.0

GM =3.0

=4 5 − 3( )2

+ 4 3 − 3( )2

+ 4 3 − 3( )2

+

4 2 −3( )2 + 4 2 −3( )2

=24

Between subjects variance



Total variance

Stage 1




dfTotal =N−1




dfbetweenSubs =nsubjects −1





3 4 6 7

0 3 3 6

2 1 4 5

0 1 3 4

0 1 4 3

XA =2.0

SSA =8.0

XB =4.0

SSB =6.0

XC =5.0

SSC =10.0

XP =1.0

SSP =8.0SSError =SSWithin −SSBetweenSubsGM =3.0

Error variance

SSError =32 −24 =8

dferror = Nscores −ngroups( )− nsubjects −1( )

dferror = 20 −4( )− 5 −1( ) =12



Total variance

Stage 1




dfTotal =N−1




SSError =SSWithin −SSBetweenSubs

dfbetweenSubs =nsubjects −1


dferror = Nscores −ngroups( )− nsubjects −1( )



Mean Squares (Variance)

SSBetween =50.0

dfbetween =3

MSBetween =50.03

=16.67 MSError =812

=0.67

Between groups variance

Error variance

SSError =8

dferror =12

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

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

Recall:Recall:

variance =SSdf



Total variance

Stage 1



dfTotal =19SSTotal =82

SSWithin =32dfWithin =16

SSBetween =50dfbetween =3

SSError =8dferror =12

SSBetweenSubs =24

dfbetweenSubs =4

MSBetween =6.0

MSError =0.67



• The F table– Need two df’s

• dfbetween (numerator)

• dferror (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

– Separate tables for 0.05 & 0.01

Do we reject or failto reject the H0?Do we reject or failto reject the H0?

– From the table (assuming 0.05) with 3 and 12 degrees of freedom the critical F = 3.89.

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

F =MSBetweenMSError

=16.67

0.67= 24.89


Within-subjects ANOVA in SPSS

– Setting up the file– Running the analysis– Looking at the output

Documents

Statistics for the Social Sciences Psychology 340 Spring 2005 Within Groups ANOVA