Advanced ANOVA/Repeated measures ANOVA

The purpose of this tutorial is teach the use of repeated measures ANOVAs, including one-way repeated measures, 2-way within-subjects ANOVA, and mixed designs. Practical exercises are based on using SPSS.

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

Repeated-measures designs can be thought of as an extension of the paired-samples t-test to include comparison between more than two repeated measures.

Repeated-measures designs can be combined with between-subject factors to create mixed-design ANOVAs. Multiple repeated-measures designs can also be tested using MANOVAs.

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

Possible designs

 * 1) One-way repeated measures - repeated measures across one IV
 * 2) Two-way repeated measures - repeated measures across two IVs
 * 3) 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 that instead of the homogeneity of variance assumption, repeated-measures designs have the assumption of Sphericity which 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.
 * 1) Tested by Mauchly's sphericity test.
 * 2) When the significance level of Mauchly’s test is < 0.05 then sphericity cannot be assumed.
 * 3) 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:
 * 1) The pattern of intercorrelations across the levels of the repeated measures factor should be consistent from level to level of the between subjects factor
 * 2) Tested using Box's M statistic.
 * 3) See also lecture slides

Exercises

 * 1) One-way repeated measures ANOVA:
 * 2) * AQUES.sav - Francis 3.3.7, p. 66 (5th ed.)
 * 3) Two-way repeated measures
 * 4) * Repmeas.sav - Francis 3.3.8.2, p. 76 (5th ed.)
 * 5) Mixed ANOVA
 * 6) * AQUES.sav - Francis 3.3.8.3, p. 81 (5th ed.)

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’ $$\lambda$$ = 0.62, F (3,27) = 5.57, p = .004, multivariate $$\eta^2$$ =.38. Follow-up polynomial contrasts indicated a significant linear effect with means decreasing over time, F (1,29) = 11.56, p =.002, $$\eta$$$$_p^2$$ =.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 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

Note: Skewness and kurtosis should be added to the table. Perhaps a marginal total would also be appropriate.

University of Canberra

 * Repeated measures ANOVA (ucspace)
 * Repeated measures ANOVA Notes (Handout)

Other

 * Chapter 14 Within-Subjects ANOVA (HyperStat Online)

Effect sizes

 * Cohensd.xls