# What is the best approach for this set-up: RM ANOVA / MANOVA / Mixed-Models?

I have a simple dataset from a within-subject design. Each participant provided a verbal description of 3 stimuli. The descriptions were coded so that they consist from objects each belonging to 1 out of 3 classes. I am interested in the proportion of these objects between the classes. I want to know if some classes were occurring more often than others and if some stimuli caused more often occurrence than others. So we have a following dataset of counts:

Participant  stimulus  class_a class_b  class_c
1            stim1     8       0        3
1            stim2     7       9        5
1            stim3     1       2        4
2            stim1     3       4        6
...


The simplest way to analyse this data would be a Repeated-Measures ANOVA. However, I would have to run them 3 times - once for each 'class' as a dependent variable. And it doesn't give me the relations between the classes.

Another solution might be MANOVA. But I am a bit confused whether it can be used when there is no between-subject independent variable (like an experimental condition). And how does it deal with the within-subject factor? I searched for R tutorials on that, but they either miss the repeated-measures component or deal with more than 1 experimental condition.

One more idea that I had was to build a mixed-effect model. Since mixed-effect models require a single output variable though, I would need to transform the data to the following format:

Participant  stimulus  class  response
1            stim1     a      8
1            stim1     b      0
1            stim1     c      3
1            stim2     a      7
...


My concern is, that any model I construct from such a dataset will treat both 'stimulus' and 'class' as Independent Variables / Predictors. In other words, they will lie on the right hand side of the formula: response ~ stimulus*class + (1|Participant).

At the same time, mixed-models seem to offer the highest power, require less assumptions, and provide the most flexibility. Can they be used for this multivariate analysis? Or can multiple models be constructed (one for each class) and somehow related to each other to show the differences across classes?

• Is it right to say that participant1 described stim1 in the following way: 8 objects A, 0 objects B and 3 objects C ? Mar 12, 2015 at 18:26
• yes, that's correct. Mar 13, 2015 at 7:44

Your choice of analysis depends on what you want to focus on. It seems like you have a count dependent variable. In general they are better addressed with log-linear models (like poisson regression). However, if you want to analyse it like it was normally distributed than:

1) If your focus is the comparison of stimuli and you're having doubts about treating the class counts as one variable a MANOVA approach is probably superior.

a within-subject MANOVA would be adequate to test if the two stimuli result in a statistically significant difference in class counts. Given your first data frame the R syntax would be:

manova(cbind(class_a, class_b, class_c) ~ stimulus + Error(Participant))


This can be your omnibus test for differences in stimuli.

If you want to compare specific class counts across stimuli than you have to do post-hoc tests (treat the response count variable as the dependant and do, for example, paired t tests with the bonferroni correction).

2) If your focus is on comparing counts across classes and stimuli, than use mixed modelling. One thing that worries me here is the error structure due to the possible interdependence of the class counts.

• In case (1) what is then the advantage of running a MANOVA over running three RM ANOVAs? MANOVA gives me a single value showing there is an effect of stimulus (or a difference between at least 2 of stimuli). Whereas 3x RM ANOVA would tell me the same thing but within each class separately (so I'd have more detail without going into post-hocs for the start). Mar 13, 2015 at 10:34
• @KubaKrukar RM ANOVA needs more assumptions about the covariance structure of your data, such as sphericity. Furthermore from what you describe you have a multivariate dependant variable corresponding to the class counts that naturally lends it self to the MANOVA. I would be weary of posing it as a 3x3 within subject design that a RM ANOVA would require because of the error structure which I mentioned, but if you choose to go this way I don't see why you shouldn't just go with mixed modelling. Mar 13, 2015 at 12:31
• @KubaKrukar if, however, you would run three separate RM ANOVA's (each class separately) you are essentially running post-hoc tests without an omnibus test. It would be similar to using sets of $t$ tests to compare the means in 3 groups instead of the usual ANOVA with post-hoc t test approach. One of the problems is that you lose testing power due to using conservative corrections for multiple testing like the Bonferroni. Mar 13, 2015 at 12:52
• Exactly, separate ANOVAs do feel like post-hocs in this case. I guess for a simple analysis with only a few comparisons being of interest, manova+t.tests and a nice boxplot will do. Since I started using mixed-effect models I just can't get rid of the feeling everything else is so limited (and yes, I realise the random-effect structure is probably incorrect in the example above). Thanks for valuable input! Mar 13, 2015 at 17:03