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Although this is five years late, I will post as an FYI for anyone else who happens to stumble upon this thread. The first part of the model is properly specified, i.e.

time ~ pres*fact*size

However, the error term is incorrectly specified. Note that in repeated measures ANOVA, "the error term for any within-subject effect is the interaction of that effect with subjects [or whatever entity that is repeatedly measured] (Keppel, G., & Wickens, T. D., 2004, p. 40)."

For this particular example, if using R's aov() function, the error term would be specified as such:

Error(subj/(pres*fact*size))

So in all, we have:

m2 <- aov(time ~ pres*fact*size + Error(subj/(pres*fact*size)), data = data)

WRT the interpretation of a three-way interaction: The implication is that a two-way interaction varied over the levels of a third factor. In other words, there was a two-way interaction (or interactions) that werewas only manifest at a specific level or levels of a third factor. To get to the heart of the matter, you would need to follow up your omnibus ANOVA with interaction contrasts. To determine what interaction contrasts you should test, it would be helpful to plot your data using a line graph. As it stands, there is not enough information available in your post to provide further guidance.

Edit Perhaps it goes without saying, but this is in the case that you obtain such a result with the model properly specified.

Although this is five years late, I will post as an FYI for anyone else who happens to stumble upon this thread. The first part of the model is properly specified, i.e.

time ~ pres*fact*size

However, the error term is incorrectly specified. Note that in repeated measures ANOVA, "the error term for any within-subject effect is the interaction of that effect with subjects [or whatever entity that is repeatedly measured] (Keppel, G., & Wickens, T. D., 2004, p. 40)."

For this particular example, if using R's aov() function, the error term would be specified as such:

Error(subj/(pres*fact*size))

So in all, we have:

m2 <- aov(time ~ pres*fact*size + Error(subj/(pres*fact*size)), data = data)

WRT the interpretation of a three-way interaction: The implication is that a two-way interaction varied over the levels of a third factor. In other words, there was a two-way interaction (or interactions) that were only manifest at a specific level or levels of a third factor. To get to the heart of the matter, you would need to follow up your omnibus ANOVA with interaction contrasts. To determine what interaction contrasts you should test, it would be helpful to plot your data using a line graph. As it stands, there is not enough information available in your post to provide further guidance.

Although this is five years late, I will post as an FYI for anyone else who happens to stumble upon this thread. The first part of the model is properly specified, i.e.

time ~ pres*fact*size

However, the error term is incorrectly specified. Note that in repeated measures ANOVA, "the error term for any within-subject effect is the interaction of that effect with subjects [or whatever entity that is repeatedly measured] (Keppel, G., & Wickens, T. D., 2004, p. 40)."

For this particular example, if using R's aov() function, the error term would be specified as such:

Error(subj/(pres*fact*size))

So in all, we have:

m2 <- aov(time ~ pres*fact*size + Error(subj/(pres*fact*size)), data = data)

WRT the interpretation of a three-way interaction: The implication is that a two-way interaction varied over the levels of a third factor. In other words, there was a two-way interaction that was only manifest at a specific level or levels of a third factor. To get to the heart of the matter, you would need to follow up your omnibus ANOVA with interaction contrasts. To determine what interaction contrasts you should test, it would be helpful to plot your data using a line graph. As it stands, there is not enough information available in your post to provide further guidance.

Edit Perhaps it goes without saying, but this is in the case that you obtain such a result with the model properly specified.

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Although this is five years late, I will post as an FYI for anyone else who happens to stumble upon this thread. The first part of the model is properly specified, i.e.

time ~ pres*fact*size

However, the error term is incorrectly specified. Note that in repeated measures ANOVA, "the error term for any within-subject effect is the interaction of that effect with subjects [or whatever entity that is repeatedly measured] (Keppel, G., & Wickens, T. D., 2004, p. 40)."

For this particular example, if using R's aov() function, the error term would be specified as such:

Error(subj/(pres*fact*size))

So in all, we have:

m2 <- aov(time ~ pres*fact*size + Error(subj/(pres*fact*size)), data = data)

WRT the interpretation of a three-way interaction: The implication is that a two-way interaction varied over the levels of a third factor. In other words, there was a two-way interaction (or interactions) that were only manifest at a specific level or levels of a third factor. To get to the heart of the matter, you would need to follow up your omnibus ANOVA with interaction contrasts. To determine what interaction contrasts you should test, it would be helpful to plot your data using a line graph. As it stands, there is not enough information available in your post to provide further guidance.