analysis of variance and repeated measures design
TRANSCRIPT
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: [email protected]
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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
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Types of Experimental Design
Classification on the basis of subjects receive treatments
Independent measures designs
Repeated measures designs
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Independent measures design
Each subject receives one and only treatment.
Requires more subjects Error variance is more
Features
Also known as between-subjects design
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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
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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
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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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