# Is it correct to run a mixed model (2-ways anova) on a unbalanced and incomplete design?

I am studying the effect of land use on biomass. I am struggling to find the right model to correctly analyse my data.

Data description

I have surveyed land use changes in 20 fields for 3 years. I obtained various land uses which include, for instance, maize, rice, no crop (= fallow), young trees, mature trees, etc. Fields with no tree or young trees can be combined with any crop. In order to limit the number of factors, I have decided to focus on 2 qualitative variables: Tree "presence of trees" (no trees, young trees, mature trees) and Annual "type of crop" (no crop, rice, maize). As the data was obtained from observations the group sizes are very variable:

       Tree     No tree        Young trees        Mature trees
Annual
No crop          4                 9                   15
Rice             11                2                   0
Maize            13                5                   0


For instance, mature trees are always associated with "no crop", because it is biologically impossible to grow rice or maize under the canopy. Fields which belong to the "mature trees" group also remain in this group in all 3 years, while the land use can change in others.

Questions

I want to determine the effect of land use (crop AND trees) on biomass. I am using mixed models to account for the repeated measurements :

> mm = lmer(Biomass ~ Tree + Annual + (1|Field))

1. can I use pairwise comparisons tests (e.g. cld and emmeans)? It returns all combinations, so I don't know whether the estimates are right (and more precisely if I can test for significance).

> emmeans(mm, ~ Tree | Annual)

Tree = no tree     emmean         SE                lower.CL       upper.CL
No crop         1.225439      0.02068960          1.183148      1.267730
Rice            1.239332      0.02022856          1.197828      1.280836
Maize           1.234615      0.01994940          1.193585      1.275645

Tree = Young trees emmean         SE                lower.CL       upper.CL
No crop         1.211010      0.02123882          1.167735      1.254284
Rice            1.224903      0.02198322          1.180324      1.269482
Maize           1.220186      0.02220742          1.175199      1.265173

Tree = Mature trees  emmean         SE              lower.CL       upper.CL
No crop          1.270676      0.02369005         1.221675      1.319676
Rice             1.284569      0.02459490         1.234082      1.335056
Maize            1.279852      0.02453023         1.229470      1.330233

Degrees-of-freedom method: kenward-roger  Confidence level used: 0.95

2. Is it correct to use mixed models when some individuals (fields here) are associated only to 1 combination (mature trees + no crop) ?

• Looks ok to me, but note your model is additive so the comparisons of X1 will be identical at each X2. BTW, may I suggest using meaningful variable names rather than X1 and X2? You’re doing science, not math — right? – rvl Jan 8 at 1:06
• @rvl I am not sure I understand why the comparisons of X1 (Trees) will be identical on each X2 (Annuals). I do obtain different marginal effects (for instance using ggpredict). Also, I don't usually use X1/X2, I just hoped to make the question easier to read. I edited this. – Nausi Jan 9 at 11:47
• You fitted a model without interaction so the effect of Trees is modelled as the same for each level of Annuals and vice versa. Note that if you do add an interaction R will drop degree(s) of freedom because nothing grows under mature trees. – mdewey Jan 9 at 13:39
• emmeans()'s results are based on the model, and the model you show is additive, i.e., it presumes that the effects of X1 are independent of those of X2. Try pairs(emmeans(mm, ~ Tree | Annual)) and see for yourself. – rvl Jan 9 at 15:39
• Regarding Q2, again, that's OK based on the model, which also presumes the variation among fields is homogeneous. If you have no replications in one condition, then that condition simply doesn't contribute to the estimate of the common variance of fields. – rvl Jan 9 at 15:42