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I have two different groups, Treatment A vs Treatment B, with measurements for each individual in four different time points. That is a 2 x 4 design. The dependent variable is a discrete scale from 1 to 4 where 1 is better than 4. The fact that it is a discrete scale means that computing means and standard deviations doesn't make sense (it can't be 2.7 for instance).

Thus, the problem I have is to understand if I could use a Mixed-Factorial ANOVA to analyze whether there are differences between groups and across individuals. A colleague has suggested me to use an Ordinal Logistic Mixed Model which seems a legit approach as well. However, given the nature of the data it looks to me that a Mixed-Factorial ANOVA would be perfectly fine (it is a numeric and ordered scale, although the scale has few levels).

Furthermore, it seems that there aren't good implementations of Mixed Ordered Logit in standard software like R or SPSS for instance.

I would really appreciate if you could give me your insights about how to tackle this problem and which model is better.

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    $\begingroup$ Have a look at the ordinal package, it has an ANOVA however I don't know how you are going to add random effects. $\endgroup$
    – user2974951
    Commented Oct 1, 2018 at 7:34
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    $\begingroup$ Random effects are possible in the ordinal package, do ?clmm after installing and loading it. $\endgroup$
    – mdewey
    Commented Oct 1, 2018 at 12:26
  • $\begingroup$ The ordinal package in R is relatively easy to use, and has pretty much everything one could ask for. It does handle mixed-effects designs. It also allows you to specify if the ordered categories should be assumed to equidistant, symmetrical, or neither. Group separations can be determined with the emmeans package. One thing to be aware of: (last I checked) absolutely don't use car::Anova with models from this package unless you use it with the RVAideMemoire package. $\endgroup$
    – Sal Mangiafico
    Commented Oct 2, 2018 at 14:28

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If your DV is ordinal, you probably don't want ANOVA as it assumes a continuous DV. It also assumes that the errors are normally distributed, which is unlikely when you have a four level DV.

Jade Schmidt wrote a paper "Ordinal Response Mixed Models" that both discusses this sort of problem and compares various R packages. I didn't read it in extensive detail, but it looked pretty good from a quick skim (but it seems not to be a peer reviewed paper).

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  • $\begingroup$ The Schmidt article is worth looking at. It's pretty approachable, but also mentions a few technical considerations. It mostly discusses the approach to ordinal regression used by the ordinal package in R. The vignette for this package is also very informative. $\endgroup$
    – Sal Mangiafico
    Commented Oct 2, 2018 at 14:20
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There are good implementations of mixed ordered logistic regression in R, check clmm() in the ordinal package. If your model will include additional predictors and some of them are continuous, then try clmm().

If the only predictors will be the cell membership of your factorial design, then a standard linear mixed model will give you relatively similar results to your ordinal mixed model. As an exercise, run both, your p-values will not be too dissimilar. And it will be much easier to interpret the linear mixed model results. Depending on the outlet/audience for your work, you may have flexibility in which you choose to report.

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  • $\begingroup$ Thank you for your answer! The problem with clmm is that when specifying the Random Effects variable, which is Treatment (treated vs control) shows an error saying that there should be 3 levels or more Error : all(sapply(gfl, nlevels) > 2) is not TRUE how would you overcome such problem? Thanks! $\endgroup$
    – adrian1121
    Commented Oct 2, 2018 at 11:43
  • $\begingroup$ @adrian1121 the random intercept should be on the individual. A quick review of mixed models might help here. $\endgroup$
    – Heteroskedastic Jim
    Commented Oct 2, 2018 at 11:47
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    $\begingroup$ With the caveat that this links to my own webpage, there is a relatively simple and complete example with ordinal::clmm here. $\endgroup$
    – Sal Mangiafico
    Commented Oct 2, 2018 at 14:34

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