Repeated Measures ANOVA Randomized Blocks Repeated Measures ANOVA
Repeated Measures ANOVA & Randomized Blocks
Repeated Measures ANOVA • In the traditional approach, subjects is treated as an additional classification variable. • A one-way RM ANOVA is really a two-way ANOVA, with subjects being the second factor. • This analysis assumes sphericity.
Sphericity • Suppose we have five levels of repeated factor A. • Find the standard error for the difference between level j and level k. • We assume that standard error is constant across jk pairs. • This assumption is frequently violated with behavioral data.
Corrections • There are procedures that correct for violation of the assumption of sphericity. • They reduce the degrees of freedom, much like done in the Welch ANOVA. • Greenhouse-Geisser is the more conservative procedure. • Huynh-Feldt is the less conservative procedure.
The Multivariate Approach • Suppose you have a one-way RM design with five levels of the grouping variable (G). • You treat the scores at any one level of G as one variable, so you now have five variables (G 1 through G 5), not two variables (G and Y).
Orthogonal Contrasts • Behind the scenes, your statistical program creates a complete set of orthogonal contrasts for the RM factor. • It then tests the null that every one of those contrasts has a mean of zero. • If that null is rejected, you conclude the RM factor has a significant effect. • There is no sphericity assumption with this analysis.
Doubly Multivariate Analysis • Suppose that you have a design with one or more RM factor(s) • And you also have multiple dependent variables. • If you take the multivariate approach to analysis of the RM factor(s), then you have a doubly multivariate analysis.
Effects of Cross-Species Rearing • Wuensch (1992) • Newborn Mus fostered onto Mus, Peromyscus or Rattus. • Tested in tunnel where could visit four tunnels which smelled like – Clean pine shavings – Mus – Peromyscus – Rattus
Mus musculus
Peromyscus maniculatus
Rattus norwegicus
The Design • Dependent variables were – Latency to first visit of each tunnel – Number of visits to each tunnel – Cumulative time spent in each tunnel • Independent variables were – Scent of tunnel (4 levels, within-subjects) – Foster species (3 levels, between-subjects)
Doubly Multivariate Results • There were significant results of Foster Species, Scent of Tunnel, and the Interaction. • This was followed by univariate ANOVA, Foster Species x Scent of Tunnel, on each of the three dependent variables.
Results of the Univariate ANOVAs • The interaction was significant for each dependent variable. • Conducted simple main effects analysis. • Mus reared by Rattus had significantly more visits to and cumulative time in the rat-scented tunnel that did the other groups, and shorter latencies as well. • The other groups avoided the rat-scented tunnel.
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