ANOVA for Independent

Measures

within

withinwithinwithin

df

SSsMS 2

between

betweenbetweenbetween

df

SSsMS 2

within

between

within

between

MS

MS

s

sF

2

2

ANOVA II

Repeated-Measures

and Two-Factor

ANOVA

Repeated-Measures ANOVA Independent-measures ANOVA uses multiple

participant samples to test the treatments

Participant samples may not be identical

If groups are different, what was responsible?

Treatment differences?

Participant group differences?

Repeated-measures solves this problem by

testing all treatments using one sample of

participants

Repeated-Measures ANOVA Repeated-Measures ANOVA used to

evaluate mean differences in two general

situations

In an experiment, compare two or more

manipulated treatment conditions using the same

participants in all conditions

In a nonexperimental study, compare a group of

participants at two or more different times

Before therapy; After therapy; 6-month follow-up

Compare vocabulary at age 3, 4 and 5

Individual differences

Participant characteristics may vary

considerably from one person to another

Participant characteristics can influence

measurements (Dependent Variable)

Repeated measures design allows control of

the effects of participant characteristics

Eliminated from the numerator by the research

design

Must be removed from the denominator

statistically

Numerator of F-ratio

Numerator of F-ratio

Denominator of F-ratio

Denominator of F-ratio

Repeated-Measures ANOVA

Logic

Numerator of the F ratio includes

Systematic differences caused by treatments

Unsystematic differences caused by random

factors are reduced because the same individuals

are in all treatments

Denominator estimates variance reasonable

to expect from unsystematic factors

Effect of individual differences is removed

Residual (error) variance remains

Structure of the F-Ratio for

Repeated-Measures ANOVA

ally)mathematic removed sdifference l(individua

effect treatmentno with expected es)(differenc variance

s)difference individual(without

eatmentsbetween tr es)(differenc variance

F

The biggest change in repeated-measures ANOVA is mathematically removing the individual differences variance component from the denominator of the F-ratio

Figure 13.1 Structure of the

Repeated-Measures ANOVA

Two Stages of the Repeated-

Measures ANOVA

First stage

Identical to independent samples ANOVA

Compute SStotal, SSbetween treatments and

SSwithin treatments

Second stage

Done to remove the individual differences from

the denominator

Compute SSbetween subjects and subtract it from

SSwithin treatments to find SSerror (also called residual)

Repeated-Measures ANOVA

Stage One Equations

N

GXSStotal

22

treatment each insidetreatmentswithin SSSS

N

G

n

TSS treatmentsbetween

22

Repeated-Measures ANOVA

Stage Two Equations

N

G

k

PSS subjectsbetween

22

_

bjectsbetween_suatmentswithin tre SSSSSSerror

P: Personal Total

the sum of the scores for the person in all treatments

Degrees of freedom for

Repeated-Measures ANOVA

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1

dfbetween subjects = n – 1

dferror = dfwithin treatments – dfbetween subjects

F Ratio

error

errorerror

error

treatmentsbetween

df

SSMS

MS

MSF

Ex. 13.1

Stage 1

12224

1681298

222

N

GXSStotal

62 Insidetreatmentswithin SSSS

6022

N

G

n

TSS treatmentsbetween

231 Ndftotal

20 kNdfdf treatmenttreatmentswithin

31 kdf treatmentsbetween

Stage 2

514822

ndfN

G

k

PSS subjectsbetweensubjectsbetween

Stage 2

15520

144862

5161

4822

subjectsbetweentreatmentswithinerror

subjectsbetweentreatmentswithinerror

subjectsbetween

subjectsbetween

dfdfdf

SSSSSS

ndf

N

G

n

PSS

Result

Exercise

Source SS df MS

Between treatments F=5.0

Within treatments 50

Between subjects

Error 2

Total

Treatments: before therapy, after therapy, three month after therapy

Sample: 10 patients

Effect size for the

Repeated-Measures ANOVA

or subjectsbetween total

eatmentsbetween tr2

SS SS

SS

errorSSSS

SS

eatmentsbetween tr

eatmentsbetween tr2

total

between2

SS

SSη

Repeated Measures ANOVA

post hoc tests

Significant F indicates that H0 (“all

populations means are equal”) is wrong in

some way

Use post hoc test to determine exactly where

significant differences exist among more than

two treatment means

Tukey’s HSD and Scheffé can be used

Substitute SSerror and dferror in the formulas

Repeated-Measures ANOVA

Advantages and Disadvantages

Advantages of repeated-measures designs

Individual differences among participants do not

influence outcomes

Smaller number of participants needed to test all

the treatments

Disadvantages of repeated-measures

designs

Some (unknown) factor other than the treatment

may cause participant’s scores to change

Practice or experience may affect scores

independently of the actual treatment effect

Two-Factor ANOVA

A study on self-esteem and being observed.

What Are We Interested?

Does Factor A have an impact on scores?

Does Factor B have an impact on scores?

How would A and B affect the scores together?

Main effects

Mean differences among different levels of the same factor

Interactions

Mean differences between individual levels of the same

factors differ from what main effects would predict.

Main Effects

Mean differences among levels of one factor

Differences are tested for statistical significance

Each factor is evaluated independently of the

other factor(s) in the study

21

21

:

:

1

0

AA

AA

H

H

21

21

:

:

1

0

BB

BB

H

H

Interactions Between Factors

The mean differences between individuals

treatment conditions, or cells, are different

from what would be predicted from the overall

main effects of the factors

H0: There is no interaction between

Factors A and B

H1: There is an interaction between

Factors A and B

Interpreting Interactions

Dependence of factors

The effect of one factor depends on the level or

value of the other

Sometimes called “non-additive” effects because

the main effects do not “add” together predictably

Non-parallel lines (cross, converge or

diverge) in a graph indicate interaction is

occurring

Typically called the A x B interaction

For A 2x2 Design

Factor A and B

Two levels for each: yes/no

w/t A w/ A

w/ B

w/t B

Main Effect for B, No Interaction

w/t A w/ A

w/ B

w/t B

Main Effects for A and B, No Interaction

Figure 13.2 Group Means

Graphed without (a) and with

(b) Interaction

Structure of the Two-Factor

Analysis of Variance

Three distinct tests

Main effect of Factor A

Main effect of Factor B

Interaction of A and B

A separate F test is conducted for each

Results of one are independent of the others

effecttreatmentnoisthereifexpectedsdifferencemeanvariance

treatmentsbetweensdifferencemeanvarianceF

)(

)(

Structure of Two-Factor

ANOVA

Two Stages of the Two-Factor

Analysis of Variance

First stage

Identical to independent samples ANOVA

Compute SStotal, SSbetween treatments and

SSwithin treatments

Second stage

Partition the SSbetween treatments into three separate

components: differences attributable to Factor A;

to Factor B; and to the AxB interaction

Stage 2

BfactorAfactortreatmentsbetweenBA

BfactorAfactortreatmentsbetweenBA

B

COL

COLB

A

ROW

ROWA

dfdfdfdf

SSSSSSSS

columnsofnumberdf

N

G

n

TSS

rowsofnumberdf

N

G

n

TSS

1

1

22

22

Mean Squares and F-ratios

reatmentst within

reatmentst withinreatmentst within

df

SSMS

AxB

AxBAxB

B

BB

A

AA

df

SSMS

df

SSMS

df

SSMS

within

AxBAxB

within

BB

within

AA

MS

MSF

MS

MSF

MS

MSF

Ex. 13.4: Impact of Media and

Time Control on Learning

Two-Factor ANOVA Effect

Size

η2, is computed to show the percentage of

variability not explained by other factors

treatments withinA

A

AxBBtotal

AA

SSSS

SS

SSSSSS

SS

2

treatmentswithinB

B

AxBAtotal

BB

SSSS

SS

SSSSSS

SS

_

2

treatments withinAxB

AxB

BAtotal

AxBAxB

SSSS

SS

SSSSSS

SS

2

To Report ANOVA Result

Report mean and standard deviations

(usually in a table or graph due to the

complexity of the design)

Report results of hypothesis test for all three

terms (A & B main effects; A x B interaction)

For each term include F, df, p-value & η2

E.g., F (1, 20) = 6.33, p<.05, η2 = .478

Two-Factor ANOVA

Summary Table Example

Source SS df MS F

Between treatments 200 3

Factor A 40 1 40 4

Factor B 60 1 60 *6

A x B 100 1 100 **10

Within Treatments 300 20 10

Total 500 23

F.05 (1, 20) = 4.35*

F.01 (1, 20) = 8.10**

(N = 24; n = 6)

Interpreting the Results

Focus on the overall pattern of results

Significant interactions require particular

attention because even if you understand the

main effects, interactions go beyond what

main effects alone can explain.

Extensive practice is typically required to be

able to clearly articulate results which include

a significant interaction

Figure 13.4

Sample means for Example

13.4

Two-Factor ANOVA

Assumptions

The validity of the ANOVA presented in this

chapter depends on three assumptions

common to other hypothesis tests

The observations within each sample must be

independent of each other

The populations from which the samples are

selected must be normally distributed

The populations from which the samples are

selected must have equal variances

(homogeneity of variance)

Summary

Independent-measures ANOVA F-ration: between treatment variance vs. within treatment

variance

Repeated-measures ANOVA Removing the individual differences from the within-

treatments variance (the denominator of the F-ratio)

Two-factor ANOVA Three F-ratios

Factor A, Factor B, and Interaction (A X B)

Post hoc tests Only applicable when H0 is rejected in ANOVA.

Scheffé vs. Tukey’s HSD

Homework

Chapter 13: 14, 26

Next Week

Tuesday:

Experimental Design I: Simple Experiments;

Thursday

Midterm