Repeated measures ANOVA

2008-08-14T00:36:21Z

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==Overview==
The '''repeated measures design''' is also known as a '''within-subject design'''. In this design, participants may present scores for:
# A measure repeated over '''''time''''' (e.g., [[w:Self-confidence|self-confidence]] before, after, and following-up a psycho-social intervention), and/or
# A measure repeated cross more than one '''''condition''''' (e.g., experimental and control conditions), and/or
# Several related, comparable measures (e.g., sub-scales of an [[IQ]] test).

Repeated-measures designs can also be understood as an extension of the '''paired-samples t-test''' (to include comparison between more than two repeated measures).

Repeated-measures designs may also be combined with between-subjects factors to create mixed-design ANOVA. Multiple repeated-measures designs can also be tested using '''MANOVAs'''.

==Why use?==
By collecting data from the same participants under repeated conditions:
* Individual differences can be reduced/eliminated as a source of between-groups differences (which helps to create a more powerful test).
* Inferential testing becomes more powerful (the sample size is not divided between conditions/groups)

==Possible designs==
# '''One-way repeated measures''' - repeated measures across one [[IV]]
# '''Two-way repeated measures''' - repeated measures across two IVs
# '''Two-way mixed split-plot design (SPANOVA)''' - repeated measures on one IV, independent groups on another IV

==Assumptions==
Most of the assumptions for between-subjects ANOVA design apply, however the key variation is '''Sphericity''':
# Instead of the '''homogeneity of variance assumption''' (part of one-way (between-subjects) and factorial ANOVA designs), repeated-measures designs have the assumption of sphericity.
# Means that the variance of the population difference scores for any two conditions should be the same as the variance of the population difference scores for any other two conditions.
# Tested by [[w:Mauchly's sphericity test|Mauchly's sphericity test]].
# When the significance level of Mauchly’s test is < 0.05 then sphericity cannot be assumed.
# Note: the sphericity assumption is only relevant to the univariate (one-way) RM ANOVA. This assumption is commonly violated and so it is generally not recommended to use the univariate analyses – the ''p''-values tend to be inaccurate to the extent that this assumption is violated. The alternative, a multivariate test, does not require the assumption of sphericity and is therefore recommended. The multivariate test is conducted on difference scores and evaluates whether the population means for the sets of difference scores are simultaneously equal to zero.

In addition, for SPANOVA, consider '''homogeneity of intercorrelations''':
# The pattern of intercorrelations across the levels of the repeated measures factor should be consistent from level to level of the between subjects factor
# Tested using Box's ''M'' statistic.

==Exercises==
# One-way repeated measures ANOVA:
#* [http://www.pearsoned.com.au/wpsBridge/francis5/files/AQUES.sav AQUES.sav] - Francis 3.3.7, p. 66 (5th ed.)
# Two-way repeated measures
#* [http://www.pearsoned.com.au/wpsBridge/francis5/files/Repmeas.sav Repmeas.sav] - Francis 3.3.8.2, p. 76 (5th ed.)
# Mixed ANOVA
#* [http://www.pearsoned.com.au/wpsBridge/francis5/files/AQUES.sav AQUES.sav] - Francis 3.3.8.3, p. 81 (5th ed.)

==Example write-up==

==Background==
The researcher wanted to determine whether the average husband wants to express his worries to his wife more or less the longer they are married. The researcher developed the Desire to Express Worry scale (DEW) and had 30 husbands answer the questionnaire when they initially got married, and then after 5, 10, and 15 years of marriage.

==Results==
A one-way within-subjects analysis of variance (ANOVA) was conducted with the within-subjects factor being Time (four levels indicating the number of years married) and the dependent variable being the Desire to Express Worry Scale (DEW) scores. The means and standard deviations for the DEW scors are presented in Table 1. The assumptions for ANOVA were met (explain in more detail).

The ANOVA indicated a significant time effect, Wilks’ <math>\lambda</math> = 0.62, ''F'' (3,27) = 5.57, ''p'' = .004, multivariate <math>\eta^2</math> =.38. Follow-up polynomial contrasts indicated a significant linear effect with means decreasing over time, ''F'' (1,29) = 11.56, ''p'' =.002, <math>\eta</math><math>_p^2</math> =.29. Higher-order polynomial contrasts were not significant. Men were increasingly less likely to desire to express worry to their wives with increasing years of marriage. It should be noted that there was little change in the means from 0 to 5 years, and therefore the the significant trend was due to changes after 5 years of marriage. These results suggest that men are more eager to express worry to their wives early in marriage, and that this desire decreases after 5 years of marrriage(also add a Figure and possibly pairwise, Cohen's ''d'' effect sizes).

Table 1 
''Means and Standard Deviations for DEW Scores''
<center>
{|
|-
! Number of years married
! ''M''
! ''SD''
|-
| 0 years
| 65.8
| 9.23
|-
| 5 years
| 65.43
| 10.69
|-
| 10 years
| 63.1
| 10.68
|-
| 15 years
| 61.93
| 12.57
|}
</center>
Note: Skewness and kurtosis should be added to the table. Perhaps a marginal total would also be appropriate.

==See also==
* [[b:Research Methods/Mixed-model design|Mixed-model design]] (Wikibooks)
* [[w:Repeated measures design|Repeated measures design]] (Wikipedia)
* [[w:Sphericity#Sphericity in statistics|Sphericity]] (Wikipedia)

==External links==
===University of Canberra===
* [http://ucspace.canberra.edu.au/display/RMPE/Repeated+measures+ANOVA Repeated measures ANOVA] (ucspace)
** [http://ucspace.canberra.edu.au/download/attachments/45090985/RM+ANOVA+Notes.doc Repeated measures ANOVA Notes] (Handout)

===Other===
* [http://davidmlane.com/hyperstat/within-subjects.html Chapter 14 Within-Subjects ANOVA] (HyperStat Online)

===Effect sizes===
[http://wilderdom.com/301/Cohensd.xls Cohensd.xls]
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Archive:Advanced ANOVA/Factorial ANOVA

2008-08-07T00:44:17Z

137.92.97.111: /* SPSS tip */

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'''Factorial ANOVA''' involves testing differences between group means based on two or more independent variables.

One of the keys to understanding Factorial ANOVA is being able to intepret interactions.

==SPSS tips==
To show the syntax in the output:
* Edit - Options - Viewer - Display commands in log
To show only variables names (not labels) in dialog boxes:
* Edit - Options - General
** Display names (often easier than labels)
** File order or alphabetical order?

==Overview==
* Design
* 2 or more between subjects categorical/ordinal IVs
* 1 interval/ratio DV
* e.g., what is the effect of Gender (2) and Degree Type (3) on Overall Satisfaction?
This would be a 2 x 3 Factorial ANOVA (or 2 x 3 Between-Subjects ANOVA)
* Results of interest are:
** Main effect of IV1
** Main effect of IV2
** Interaction b/w IV1 and IV2
* If significant effects are found and more than 2 levels of an IV are involved, then specific contrasts are required, either:
** A priori (planned) contrasts
** Post-hoc contrasts
* Effect sizes should also be reported

==General steps==
# Establish [[hypothesis]]/hypotheses
#* Make sure you have separate hypotheses for:
#** Main effect for each IV
#** Interactions between IVs
#** Planned contrasts (if warranted)
# Examine assumptions:
# IVs (categorical; between-subjects) and DV (at least interval)
#* The data in each cell is normally distributed
#* Homogeneity of variance (the variance in each cell is similar)
#* Cells are independent
# Examine descriptive statistics, particularly the four moments (''M'', ''SD'', Skewness, Kurtosis) overall, and also for each cell
# Examine graphs
# Conduct inferential test (ANOVA) and interpret significance of ''F'' scores
# Conduct follow-up tests (planned contrasts or post-hoc tests) if ''F'' is significant
# Interpret interactions
# Calculate and interpret effect sizes
#* Eta-square (omnibus - equivalent to ''R''2)
#* Standardised mean effect size (difference b/w two means)

==Example SPSS outputs==
* [http://www.slideshare.net/jtneill/anova-part-ii/52 Factorial ANOVA] (example) - Are there differences in Unniversity Student Satisfaction levels between Gender and Age?
* [http://www.slideshare.net/jtneill/anova-part-ii/57 Factorial ANOVA] (example) - Are there differences in Unniversity Student Satisfaction levels between Gender and Age? - Are there differences in Locus of Control between Gender and Age?

==Understanding interactions==
A recommended experiential exercise for learning about interactions is to fabricate a dataset which can be used to demonstrate factorial ANOVAs in which there are:
# No effects
# Main effect A, no main effect B, no interaction
# Main effect A, no main effect B, interaction
# No main effect A, main effect B, no interaction
# No main effect A, main effect B, interaction
# Main effect A, main effect B, no interaction
# Main effect A, main effect B, interaction
# Interaction, no main effects

==Francis exercises==
* 3.3 (Analysing Differences)
** 3.3.8 Factorial ANOVA
*** [http://www.pearsoned.com.au/wpsBridge/francis5/files/AQUES.sav AQUES.sav]
*** [http://www.pearsoned.com.au/wpsBridge/francis5/files/motiv.sav Motiv.sav]

==Effect sizes==
* [[Eta-squared]]

==External links==
* [http://ucspace.canberra.edu.au/display/RMPE/Factorial+ANOVA Factorial ANOVA] (ucspace)
** [http://ucspace.canberra.edu.au/download/attachments/45090983/Factorial+ANOVA+Notes.doc Factorial ANOVA Notes] (Handout)
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Repeated measures ANOVA

Archive:Advanced ANOVA/Factorial ANOVA