Prepared by:

Prof. Dr Bahaman Abu SamahDepartment of Professional Development and Continuing Education

Faculty of Educational StudiesUniversiti Putra Malaysia

Serdang

Use in experiment, quasi-experiment and field studies in

which the same subject is measured under all levels of

one or more trials

The dependent variable is interval or ratio

Trials is referred to as repeated-measure factor or within-

subjects factor

Test the effect of within-subjects factor (trial), between-

subjects factor (treatment) and interaction on the

dependent variable

1. Treatment main effect: Is there any significant difference

in sentence construction scores among the three

treatment groups?

2. Trial main effect: Is there any significant difference in

sentence construction scores among the three trials?

3. Interaction between treatment and trial: Do the

differences in means for the sentence construction scores

among the treatment groups vary depending on the

trials?

1 DV is normally distributed in the population for each level of the within-subjects factor (trial)

2 The population variances of the difference scores computed between any two levels of a within-subjects factor are the same value regardless of which two levels are chosen.This assumption is also known as the sphericity assumption or homogeneity of variance of differences assumption

3 The cases represent a random samples from the populations, and the scores are independent of each other

Set Alpha

State

HO and HA

(3 Hypotheses)

Report

F & sig-F

Decision

Conclusion

Next ►

- Post-Hoc Comparison

- Effect size (Partial Eta2)

Criteria Decision

sig-F> α Reject HO

sig-F ≥ α Fail to reject HO

Steps in:

State HO and HA

Set confidence level (α)

Run analysis & report F and sig-F

Decision

Conclusion

1

2

3

4

5

Next ►

1. Trial Main Effect (Within-Subjects)

HO: μt1 = μt2 = μt3

HA: Not all means are equal

2. Treatment Main Effect (Between-Subjects)

HO: μ1 = μ2 = μ2

HA: Not all means are equal

3. Interaction (I*J)

HO: μij = μˈijHA: μij ≠ μˈij

Next ►

α = .05

F and sig-F

Reject HO: sig-F ≤ α

Fail to reject HO: sig-F > α

– Only two (2) possible decisions.

– Reject or Fail to Reject HO

Criteria Decision

sig-F ≤ α Reject HO

sig-F > α Fail to reject HO

Treatment (Group) Main effect

Reject HO There is a significant treatment (group)

main effect on the DV

Fail to reject HO There is no significant treatment (group)

main effect on the DV

Decision criteria

Criteria Decision

sig-F ≤ α Reject HO

sig-F > α Fail to reject HO

Trial Main effect

Reject HO There is a significant trial main effect on the DV

Fail to reject HO There is no significant trial main effect on the

DV

Interaction: Treatment x Trial

Reject HO There is a significant treatment-by-trial

interaction effect on the DV

Fail to reject HO There is no significant treatment-by-trial

interaction effect on the DV

– Use partial eta squared (η2) as a measure of effect size

– Formula to calculate partial η2

– Partial η2 indicated relationship between repeated-measures factor

and the dependent variable; ranges between 0 to 1

– 0 indicates no relationship; 1 constitutes highest possible relationship

between repeated measures factor and the dependent variable

SSESSB

SSBPartial

ninteractioorMain

ninteractioorMain

ninteractioorMain

2

Effect size Conventions:< .10 Trivial.10 Small.25 Medium.40 Large

Cohen, 1992

In a study, a researcher is interested to

access the effectiveness of a training

program to improve students’ thinking skill.

Students were randomly assigned into

three groups based on their academic

achievement (low, moderate, and high).

Data were collected at pre, post1, and

post2.

16 17 25

9 17 22

10 18 26

6 18 25

8 17 24

17 19 28

16 18 27

10 17 26

9 15 24

10 14 23

9 13 22

8 12 22

9 13 21

8 13 21

9 13 21

8 12 22

7 8 9

2 3 4

1 7 9

4 10 20

8 10 12

9 11 13

5 10 14

4 11 14

1

2

3

4

5

6

7

8

1

1

2

3

4

5

6

7

8

2

1

2

3

4

5

6

7

8

3

ACHIEVE

PRE POST1 POST2

Data set: D8 Twowar Repeated Measure ANOVA THINKING SKILL

1. Treatment main effect: Is there any significant difference in

thinking skill scores among the three treatment groups?

2. Trial main effect: Is there any significant difference in thinking

skill scores among the three trials?

3. Interaction between treatment and trial: Do the differences in

means for the thinking skill scores among the treatment groups

vary depending on the trials?

HO: μ1 = μ2 = μ3

HA: Not all means are equal

1. Treatment main effect (Between group)

HO: μ1 = μ2 = μ3

HA: Not all means are equal

2. Trial main effect (Within-Subjects factor)

HO: μij = μ’ij

HA: μij ≠ μ’ij

3. Interaction treatment x trial

α = .05

Multivariate Testsc

.940 157.836a 2.000 20.000 .000

.060 157.836a 2.000 20.000 .000

15.784 157.836a 2.000 20.000 .000

15.784 157.836a 2.000 20.000 .000

.749 6.288 4.000 42.000 .000

.313 7.879a 4.000 40.000 .000

1.999 9.494 4.000 38.000 .000

1.894 19.890b 2.000 21.000 .000

Pillai's Trace

Wilks' Lambda

Hotelling's Trace

Roy's Largest Root

Pillai's Trace

Wilks' Lambda

Hotelling's Trace

Roy's Largest Root

Effect

TRIAL

TRIAL * ACHIEVE

Value F Hypothesis df Error df Sig.

Exact statistica.

The statistic is an upper bound on F that yields a lower bound on the significance level.b.

Design: Intercept+ACHIEVE

Within Subjects Design: TRIAL

c.

Use of Multivariate tests does not require the assumption of sphericity

Report F-ratio. Decision is based on sig-F

Sig-F (.000) < .05; Significant trial main effect

Multivariate Tests

Mauchly's Test of Sphericityb

Measure: MEASURE_1

.628 9.316 2 .009 .729 .844 .500

Within Subjects Effect

TRIAL

Mauchly's W

Approx.

Chi-Square df Sig.

Greenhous

e-Geisser Huynh-Feldt Lower-bound

Epsilona

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is

proportional to an identity matrix.

May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the

Tests of Within-Subjects Effects table.

a.

Design: Intercept+ACHIEVE

Within Subjects Design: TRIAL

b.

Sig-value < α indicates violation of sphericityassumption

Tests of Sphericity

Tests of Within-Subjects Effects

Measure: MEASURE_1

1554.778 2 777.389 200.411 .000

1554.778 1.457 1066.857 200.411 .000

1554.778 1.687 921.405 200.411 .000

1554.778 1.000 1554.778 200.411 .000

137.639 4 34.410 8.871 .000

137.639 2.915 47.223 8.871 .000

137.639 3.375 40.784 8.871 .000

137.639 2.000 68.819 8.871 .002

162.917 42 3.879

162.917 30.604 5.323

162.917 35.435 4.598

162.917 21.000 7.758

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Source

TRIAL

TRIAL * ACHIEVE

Error(TRIAL)

Type III Sum

of Squares df Mean Square F Sig.

Tests of Within-Subjects Effects Use this value if the test meets the sphericity assumption

Use any of the other three values if the sphericity assumption is violated

Tests of Between-Subjects Effects

Measure: MEASURE_1

Transformed Variable: Average

13667.556 1 13667.556 1064.675 .000 .981

1137.528 2 568.764 44.306 .000 .808

269.583 21 12.837

Source

Intercept

ACHIEVE

Error

Type III Sum

of Squares df Mean Square F Sig.

Partial Eta

Squared

Report the F-value. However, decision is based on sig-F

Sig-F (.000) < .05; reject the null hypothesis. Significant treatment effect

Effect size

Tests of Between-Subjects Effects

Decision criteria

Treatment (Group) Main effect

F = 44.306, sig –F = .000

sig-F (.000) is smaller than α (.05)

Reject HO

There is a significant treatment (group) main effect on

sentence construction scores at .05 level of significance

Criteria Decision

sig-F> α Reject HO

sig-F ≥ α Fail to reject HO

Trial Main effect

F = 200.411, sig-F = .000

sig-F (.000) is smaller than α (.05)

Reject HO

There is a significant trial main effect on sentence

construction scores at .05 level of significance

Interaction: Treatment x Trial

F = 8.871, sig-F = .000

sig-F (.000) is smaller than α (.05)

Reject HO

There is a significant treatment-by-trial interaction effect on

sentence construction scores at .05 level of significance

If the ANOVA reveals a significant result, proceed with the pairwise

comparisons to assess which means differ from each other

Pairwise Comparisons

Measure: MEASURE_1

3.542* 1.034 .003 1.391 5.693

9.625* 1.034 .000 7.474 11.776

-3.542* 1.034 .003 -5.693 -1.391

6.083* 1.034 .000 3.932 8.234

-9.625* 1.034 .000 -11.776 -7.474

-6.083* 1.034 .000 -8.234 -3.932

(J) Academic

achievement

2

3

1

3

1

2

(I) Academic achievement

1

2

3

Mean

Difference

(I-J) Std. Error Sig.a

Lower Bound Upper Bound

95% Confidence Interval for

Differencea

Based on estimated marginal means

The mean difference is significant at the .05 level.*.

Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).a.

Pairwise comparison: Treatment

There are significant differences for the following pairs of groups:

1. 1 and 2

2. 1 and 3

3. 2 and 3

Pairwise Comparisons

Measure: MEASURE_1

-4.750* .560 .000 -5.915 -3.585

-11.333* .706 .000 -12.802 -9.865

4.750* .560 .000 3.585 5.915

-6.583* .397 .000 -7.408 -5.759

11.333* .706 .000 9.865 12.802

6.583* .397 .000 5.759 7.408

(J) TRIAL

2

3

1

3

1

2

(I) TRIAL

1

2

3

Mean

Difference

(I-J) Std. Error Sig.a

Lower Bound Upper Bound

95% Confidence Interval for

Differencea

Based on estimated marginal means

The mean difference is significant at the .05 level.*.

Adjustment for multiple comparisons: Least Significant Difference (equivalent to no

adjustments).

a.

Pairwise comparison: Trial

There are significant differences for the following pairs of trials:

1. 1 and 2

2. 1 and 3

3. 2 and 3

1. Click Analyze | General Linear Model | Repeated Measures

2. At the dialog box below, type ‘trial’ as within-subject factor name.

and type 3 for the number of levels.

- Click Add button.

- Then click Define button.

3. Block all the within-subjects factors (Pre, Post1 and Post2),

click the right arrow button.

4. Click the Academic achievement and enter into Between-

Subject Factor(s) box. Then click the Option button

5. In the following Option dialog box, tick and select the following

options. Click the Continue button

6. In the following Option dialog box, click OK