Analysis of Variance and Repeated Measures Design

Presented by

Dr.J.P.VermaMSc (Statistics), PhD, MA(Psychology), Masters(Computer Application)

Professor(Statistics)

Lakshmibai National Institute of Physical Education, Gwalior, India

(Deemed University)Email: vermajprakash@gmail.com

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Advantage of Experimental Research

Manipulate IV

To see the Impact on

DV in controlled environment

Benefit

Ensures change in DV is only due to the change in IV

More reliable findings in comparison to correlation studies

3

Types of Experimental Design

Classification on the basis of subjects receive treatments

Independent measures designs

Repeated measures designs

4

Independent measures design

Each subject receives one and only treatment.

Requires more subjects Error variance is more

Features

Also known as between-subjects design

5

This Presentation is based on

Chapter 2 of the book

Repeated Measures Design for Empirical Researchers

Published by Wiley, USA

Complete Presentation can be accessed on

Companion Website

of the Book

6

Repeated measures design

 Same subjects are tested in each treatment condition.

Requires less subjects. Error variance is reduced

Features

Also known as within-subjects design

Both types of designs are solved by using the concept of ANOVA

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Understanding Variance and Sum of Squares

Population variance is estimated by the sample mean square (S2)

22 xx1n

1S

dfSSS2

dfVariation

Thus Variation is measured by SS

8

Computation of Sum of Squares

___________________________________

___________________________________5 25 0 06 36 1 14 16 -1 18 64 3 92 4 -3 9

----------------------------------------------------25 145 20___________________________________

X 2X )XX( 2)XX(

25xG 145x 2 20xx 2

2xxSum of squares (SS)= = 20

2xxTSS NGx

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Two approaches of computing SS 20xxSS 2

205

25145N

GxSS22

2

1.2.

Second approach shall be used to compute various SS in different designs

Table 2.1 Computation of SS and MSS

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One way ANOVA for Independent Measures Design

A group of statistical techniques 

Comparing means of three or more groups

for

By comparing

How this comparison is made ?

(SS)B with (SS)w

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Role of Central Limit Theorem in Testing

w

b

MSSMSS

F As per Fisher

~ F-distribution with (r-1, N-r) df

If groups are from the same population, the MSSb should be lower than MSSw

r : number of groups N : total number of scores

Higher F indicates that the samples have come from different populations.

Conclusion

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Classification of ANOVA

Different levels of one factor are compared

One way ANOVA

Effect of two factors on criterion variable is investigated simultaneously

Two-way ANOVA

Effect of three factors on some criterion variable is investigated simultaneously difficult to explain interaction rarely used

Three-way ANOVA

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Assumptions for Independent Measures ANOVA

Samples have been drawn from a population which is normally distributed.

Observations are independent to each other. Populations from which the samples have

been drawn have equal variance.

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Understanding Independent Measures ANOVA

Objective: To compare the effect of strength training (low, medium and high) on pull-ups performance

Table 2.3 Computation in one-way ANOVA ____________________________ Strength Training

Low Medium High____________________________5 8 93 6 84 5 75 4 82 3 6

____________________________ Group Total R1= 19 R2= 26 R3= 38 G= R1+R2+R3 = 83Group Mean 3.8 5.2 7.6

____________________________

n = 5 r = 3 N = nr = 15

53.51583X

Computation

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Hypothesis testing by comparing variances

TSS = (SS)b + (SS)w

53452

86543

98786

8.3T1

2.5T2

6.7T3

53.5X

Different Types of Variations

w

b)MSS()MSS(

F

If Cal. F > Tab. F.05(k-1,N-k), Reject H0

And if Cal. F ≤ Tab. F.05(k-1,N-k), Accept H0

2.Gp_Within

2.Gp_Bet0 :H

highmediumlow0 :H

Partitioning of Variation in the Design

SSBet df=r-1

Total SS df = N-1

SSWithin df= N-r

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2 12

Fig. 2.1 Scheme of distributing SS and df

To compute F we need to compute SSbet and SSwithin

Computing Different Sum of Squares

5. SS within groups(SSw) = TSS - SSb = 63.73 – 36.93 = 26.8

27.4591583

NG 22

1. Correction factor (CF) 2. Raw Sum of Squares (RSS) = = (52+32+……22) + (82+62+……+32) + (92+82+……62) = 523

i j

2ijY

3. Total Sum of Square (TSS) = RSS – CF =

73.63N

GYi j

22ij

4. SS between treatment groups(SSb) =

NG

nR 2

i i

2i 93.3627.459

538

526

519 222

Data

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Computing SS within groups(SSw)

SS within Low treatment group = (52+32+42+52+22) - = 79 – 72.2 = 6.8 5

19 2

SS within Medium treatment group = (82+62+52+42+32) - = 150- 135.2 = 14.8

SS within High treatment group = (92+82+72+82+62) - = 294 – 288.8 = 5.2

5262

5382

Thus, SS within group = 6.8 + 14.8 + 5.2 = 26.8

Thus, SS within group = 6.8 + 14.8 + 5.2 = 26.8

This is same as = TSS - SSb = 63.73 – 36.93 = 26.8 See Da

ta

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Computing Different Sum of Squares

Table 2.4 ANOVA Table for the data on pull-ups ____________________________________________________________________Source of Variation df SS MSS F-value Tab.F____________________________________________________________________Bet. Groups r-1 = 2 36.93 8.28* F.05(2,12)= 3.88

Within Groups N- r = 12 26.8 ____________________________________________________________________ Total N-1=14 63.73____________________________________________________________________ *Significant at .05 level

47.18293.36

23.212

8.26

From Table A.10 in the Appendix,

88.3F 12,2,05.

Since Calculated F(=8.28)>3.88, null hypothesis is rejected

What next?

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Post-hoc AnalysisFor understanding the differences among means

Critical difference (CD) =

Tukey’s HSD test useful for equal

sample size

nMSS

q wrN,r, 52.2

523.277.3

r : number of groupsn : number of scores in each group N : total number of scores. q : Studentized Range statistic

77.3q rN,r, from Table A.16,

For r= 3, N-r=12 the value of q at .05 level of significance can be obtained

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Post-hoc Test for Comparing Means

__________________________________High Medium Low Mean CD at

Diff. 5% level

______________________________________7.6 5.2 2.4 2.527.6 3.8 3.8* 2.52

5.2 3.8 1.4 2.52__________________________________*Significant at .05 level

Table 2.5 Post-hoc comparison of means

Means Plot

Result: High intensity is the best in comparison to that of low intensity program in improving pull-ups performance.

Figure 2.2 Means plot for the data on pull ups

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Repeated Measures DesignAn extension of paired t test

When to use Repeated Measures ANOVA(rANOVA)?

a. When treatment observations obtained on the subjects /objects needs to be investigated in three or more time durations Effect of an aerobic program on muscular

endurance after four, six and eight weeks. Example

4 weeks

S1

S2

S3

S4

S5

S1

S2

S3

S4

S5

S1

S2

S3

S4

S5

6 weeks 8 weeks

Treatment levels

Subjects

Figure 2.3 Layout design

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When to use rANOVA?b. When performance of the subjects are compared

under three or more treatment conditions.

Two Issues in the Design

Carryover effect Order effect

Controlled by

Controlled by

Keeping sufficient gap between treatments

Counterbalancing

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Computing Different Sum of Squares

To investigate the effect of sleep deprivation (24 hours, 30 hours and 36 hours) on EEG.

Example

 - Divide sample into groups- These groups are randomized on treatments

Designing procedure

24 hours

S1

S2

S5

S6

S3

S4

Treatment Conditions Sleep deprivation

First testing S3

S4

S1

S2

S5

S6

S5

S6

S3

S4

S1

S2

Second testing

Third testing

Test

ing

pro

toco

l

30 hours 36 hours

Figure 2.4 Layout design

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Assumptions in Repeated Measures Designs

1. IV- categorical, DV- metric 2. No outliers in the differences between any two related groups3. Normality in the difference of any two set of scores4. No sphericity in the data

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