Download - Using repeated measures anova with spss
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One-way Repeated Measures Design
Presented by
Dr.J.P.VermaMSc (Statistics), PhD, MA(Psychology), Masters(Computer Application)
Professor(Statistics)Director, Centre for Advanced Studies
Lakshmibai National Institute of Physical Education, Gwalior, India
(Deemed University)Email: [email protected]
This Presentation is based on the book titled Repeated Measures Designs for Empirical Researchers by Wiley USA
For Details Kindly click here
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An extension of paired t testFeatur
es Effect of one factor on some dependent variable is investigated
Example Effect of Time(morning, evening and evening) on the
memory retention
One-way Repeated Measures Design
Also known as within-group design or within-subjects design
Subjects are repeatedly tested in all the treatment conditions Subject receives treatment in a random fashion
Levels of the factor can be different treatments or different time durations
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Advantages of One-Way Repeated Measures Design
Pattern of behaviour due to intervention over the period of time can be detected.
Useful where getting more subjects is an issue Experimental error reduces as subjects serve their own control
Efficient than independently measured designs if subjects variability is significant. Design is sensitive in nature hence slight variation in dependent variable due to manipulation of independent variable can be detected.
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Due to carryover effect performance of the subjects may be affected in different treatment conditions.
Weaknesses of Repeated Measures Design
Since same subjects are tested in all treatment conditions hence large number of levels of a factor cannot be investigated.
The design will be inefficient if the subject’s variability is not significant
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In a clinical experiment the drug efficacy can be tested by taking hourly blood samples for 12 hours after its administration.
Application of Design
To compare recovery pattern of soccer players under light exercise, autogenic relaxation and underwater exercise A physiologist may study an intervention of pranayama in the relief of asthma
Pizza company may investigate the taste of different types of pizza on youngsters.
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When to use One-way RMD
To compare the taste of different pizza in a specific age category of the subjects. Six subjects participate in the study.
Example
Case I: Levels of within-subjects factor are different treatment conditions
Used in Two Types of Situations
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Layout Design
Issues in the Design
Carryover effect
controlled b
y
Keeping sufficient gap
between treatments
Order effect
controlled b
y
Counterbalancing
1. Divide sample into groups2. Randomized
treatments on these groups.
Designing procedure
S1
S2
S5
S6
S3
S4
Factor 1: Pizza
S3
S4
S1
S2
S5
S6
S5
S6
S3
S4
S1
S2
Testing protocol
First phase testing
Second phase testing
Third phase testing
ChickenPan Cheese
Subjects
Figure 4.1 Layout design
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When to use One-way RMD
To investigate the effect of time on efficacy of drug in 2 hours, 4 hours and 6 hours during an experiment. Five subjects participate in the study.
Example
Case II: Levels of within-subjects factor are different time durations
Used in Two Types of Situations
2 hours
S1
S2
S3
S4
S5
S1
S2
S3
S4
S5
S1
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S4
S5
4 hours 6 hours
Subjects
Before
S1
S2
S3
S4
S5
Factor 1: Time
Testing protocol
Figure 4.2 Layout design
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Steps in One Way RMDTest normality assumption
Describe layout design
Write research questions
Write H0 to be tested
Decide familywise error rates (α)
Use SPSS to generate outputs
Descriptive statistics
Mauchly's test of sphericity
F table for within-subjects effect
Pair-wise comparison of means
Means plot
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Steps in One Way RMD
Test Sphericity assumption
Is p<.0
5Test F ratio by
assuming sphericity N
Y
Check
<.75 Test F by using Huynh-
Feldt correctionNTest F by using
Greenhouse-Geisser correction
Y
If F is significant use Bonferroni correction for comparison of means
Report findings
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Solving One-way RMD with SPSS
To investigate the effect of time(two, four and six weeks) on the reasoning ability during an intervention of meditation programme on 10 sample.
Objective
Table 4.1 Data on reasoning ability ___________________________________Zero day2nd Week4th Week 6th Week___________________________________
31 36 35 3731 34 34 3532 31 37 3530 32 36 3534 33 37 3735 34 36 3836 31 31 3836 35 30 4032 31 35 3633 32 34 36
___________________________________
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Testing Assumptions
a. Data type The IV must be categorical having three or more levels
and DV should be on interval or ratio scale
IV : Time(Zero, Two, Four and Six Week)DV : Reasoning ability measured on interval scale
First assumption is satisfied
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Testing Assumptions
Sample has been randomly selected Observations have been independently obtained
Second assumption is also fulfilled
b. Independence of Observations The subjects are randomly selected and are independent
to each other
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Testing Assumptions c. Normality Assumption For each level of the independent variable the dependent variable must follow approximately normal distribution and should not have outlier.
Table 4.2 Tests of normality for the data on reasoning ability__________________________________________________________________ Kolmogorov-Smirnov Shapiro-Wilk
Statistics df Sig. Statistic df Sig.(p value) (p value)
_____________________ _____________________________________________ Zero day .178 10 .200* .924 10 .3932nd Week .192 10 .200* .905 10 .2464th Week .216 10 .200* .879 10 .1286th Week .166 10 .200* .902 10 .228__________________________________________________________________
Since no p-value is significant for S-W statistic hence data is normal
Normality assumption is satisfied
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Testing Assumptions Sphericity Assumption The sphericity should not exist among the data
Variances of the differences between all combinations of related groups must be equal.
or
All correlations among the repeated measures are equal.
Meaning of Sphericity Assumption
This assumption shall be tested while using
the outputs of SPSS later
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Solving RMD with SPSS
Data on reasoning ability_________________________Zero Two Four Sixweek week week week
_________________________31 36 35 3731 34 34 3532 31 37 3530 32 36 3534 33 37 3735 34 36 3836 31 31 3836 35 30 4032 31 35 3633 32 34 36________________________
Layout of Design
Hypothesis to be Tested
Factor 1: Time
Testing protocol
Subjects
Zero day
S1
S2
.
.
.
S9
S10
2nd Week 4th Week 6th Week
S1
S2
.
.
.
S9
S10
S1
S2
.
.
.
S9
S10
S1
S2
.
.
.
S9
S10
Week6Week4Week2day_Zero0 ththnd:H
H1: At least one group mean differs
Figure 4.3 Layout of the design for the study
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Distribution of SS in Repeated Measures Design
Data on reasoning ability_________________________Zero Two Four Sixweek week week week
_________________________31 36 35 3731 34 34 3532 31 37 3530 32 36 3534 33 37 3735 34 36 3836 31 31 3836 35 30 4032 31 35 3633 32 34 36________________________
SSTime df=r-1
Total SS df = N-1
SSWithin df= nr-r
39
3 36
SSError df= (n-1)(r-1)SSSubjects df= n-1 9 27
Figure 4.4 Scheme of distributing total SS and df
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Level of SignificanceThe level of significance = .05
No Post hoc test in RMD hence
t test used for group comparisonsIt inflates
αTo control error rate
Bonferroni correction is usedWhat correction it does? t is tested at new α (=α/k)k: no of group comparisons
This correction is automatically taken care of by increased
P-value if Bonferroni correction is used in SPSS.
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Steps in SPSS Screenshot 1
Analyze General Linear Model Repeated Measures
Figure 4.5 Screen for initiating commands for one-way rANOVA
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Steps in SPSS Screenshot 2
Figure 4.6 Options for defining dependent and independent variables
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Steps in SPSS Screenshot 3
Write repeated factor ‘Time’Write levels of the repeated factor ‘4’ and click on AddWrite name of the dependent variable
Click on Add to define this variables
Figure 4.7 Options for adding independent and dependent variables
1
2
3
4
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Figure 4.8 Option for selecting within subjects variables (Time) and obtaining means plot
Steps in SPSS Screenshot 4
Bring these variables from left panel to this location
Click on Plots for means plotBring ‘Time’ factor at this location
Click on Continue
1
2
3
4
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Steps in SPSS Screenshot 5
Figure 4.9 Option for descriptive statistics and pair wise Comparison of means using Bonferroni correction
Click on Options
Bring ‘Time’ factor at this location
Check these options
1
23
4
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SPSS outputs and Interpretation
Descriptive Statistics Mauchly's Test of Sphericity F Table for testing within-subjects
effects Table for pair-wise comparison of
means Marginal means plot
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Output of Repeated Measures in SPSS
Output 1: Descriptive Statistics
Table 4.3 Descriptive statistics_____________________________________
Mean SD N_____________________________________Zero_day 33.0000 2.16025 10Week_two 32.9000 1.79196 10Week_four 34.5000 2.36878 10Week_six 36.7000 1.63639 10_____________________________________
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Output of Repeated Measures in SPSS
Output 2: Mauchly's Test
Measure: Reasoning_ability Table 4.4 Mauchly's test of Sphericitya
_________________________________________________________________________________
Epsilona( )Within Subjects Mauchly's W Approx. Chi- df Sig. Greenhouse- Huynh- Lower- Effect Square Geisser Feldt bound_________________________________________________________________________________ Time .062 21.441 5 .001 .546.650 .333
_________________________________________________________________________________Assumption of Sphericity is violated because chi-square is significant
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Output of Repeated Measures in SPSS
Table 4.5 F-Table for testing significance of Within-Subjects Effects Measure: Reasoning_ability_________________________________________________________________________________________
Source Type III df Mean Square F Sig. Partial SS Eta Squared
_________________________________________________________________________________________Time Sphericity Assumed 94.475 3 31.4927.281 .001 .447
Greenhouse-Geisser 94.475 1.637 57.7257.281.009 .447Huynh-Feldt 94.475 1.951 48.423 7.281 .005 .447Lower-bound 94.475 1.000 94.475 7.281 .024 .447
Error(Time)Sphericity Assumed 116.775 27 4.325
Greenhouse-Geisser 116.775 14.730 7.928Huynh-Feldt 116.775 17.559 6.650Lower-bound 116.775 9.000 12.975
__________________________________________________________________________________________
Output 3: rANOVA Table for testing within-subjects effects
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Why F Remains Same After Applying Correction
Greenhouse Geisser: df for Treatment = ε × 2= 0.517 × 2= 1.03 df for Error = ε × 8 = 0.517 × 8 = = 4.14Huynh- Feldt: df for Treatment = ε ×2 = 0.535 × 2 = 1.07
df for Error = ε × 8 = 0.535 × 8 = 4.28Due to correction in degrees of freedom p values
increases.
)1n)(1r(SS
)1r(SS
FError
Time
)1n)(1r(SS
)1r(SS
FError
Time
)1n)(1r(SS
)1r(SS
Error
Time
a. If sphericity is assumed
b. If sphericity exists the modified degrees of freedom for SStime and SSError gets modified by multiplying them by F remains same irrespective of the fact whether sphericity exists or not.
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Testing Significance of Within-Subjects Effect
After Greenhouse-Geisser correction the F is significant p=.009(<.05)
Partial Eta Square is .447, indicates very high effect size
The effect of time is meaningful to enhance reasoning ability with meditation intervention.
Conclusion
What Next ?Apply t test with Bonferroni correction for
pair-wise comparison of marginal means
Eta square Value .02 .13 .26Status Small Medium
Large
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Pair-wise Comparison of Marginal Means
Table 4.6 Pair wise Comparison of marginal means Measure: Reasoning_ability_____________________________________________________________________________
Mean Diff. 95% CI for Differencea
(I) Time (J) Time (I-J) Std. Error Siga Lower BoundUpper Bound_____________________________________________________________________________Zero_day Week_two .100 .875 1.000 -1.879 2.079
Week_four -1.500 1.267 1.000 -4.366 1.366Week_six -3.700* .367 .000 -4.529 -2.871
Week_twoZero_day -.100 .875 1.000 -2.079 1.879Week_four -1.600 1.013 .893 -3.892 .692Week_six -3.800* .573 .001 -5.097 -2.503
Week_four Zero_day 1.500 1.267 1.000 -1.366 4.366
Week_two 1.600 1.013 .893 -.692 3.892Week_six -2.200 1.153 .532 -4.808 .408
Week_six Zero_day 3.700* .367 .000 2.871 4.529Week_two 3.800* .573 .001 2.503 5.097Week_four 2.200 1.153 .532 -.408 4.808
_____________________________________________________________________________Based on estimated marginal meansa. Adjustment for multiple comparisons: Bonferroni*. The mean difference is significant at the .05 level.
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Marginal means plot
P>.05
P=.00
1
Zero_day Week_two Week_four Week_six
Time
P>.05
P=.000
Estim
ated
mar
gina
l mea
ns o
f rea
soni
ng a
bilit
y
Variable: Reasoning ability
Marginal means plot
Meditation intervention program significantly affects the reasoning ability of the subjects.
The significant effect was observed only after the six weeks of the intervention program.
Inference
Figure 4.10 Marginal means plot
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Thank You
This Presentation was based on the book titled Repeated Measures Designs for Empirical Researchers by Wiley USA
For Details Kindly click here