Why adonis function DF changes with different factors combination? - r

Question

> data(dune)
> data(dune.env)
> str(dune.env)
'data.frame': 20 obs. of 5 variables:
$ A1 : num 2.8 3.5 4.3 4.2 6.3 4.3 2.8 4.2 3.7 3.3 ...
$ Moisture : Ord.factor w/ 4 levels "1"<"2"<"4"<"5": 1 1 2 2 1 1 1 4 3 2 ...
$ Management: Factor w/ 4 levels "BF","HF","NM",..: 4 1 4 4 2 2 2 2 2 1 ...
$ Use : Ord.factor w/ 3 levels "Hayfield"<"Haypastu"<..: 2 2 2 2 1 2 3 3 1 1 ...
$ Manure : Ord.factor w/ 5 levels "0"<"1"<"2"<"3"<..: 5 3 5 5 3 3 4 4 2 2 ...
As shown above, Moisture has four groups and Management has four groups, Manure has five groups when I run:
adonis(dune ~ Manure*Management*A1*Moisture, data=dune.env, permutations=99)
Call:
adonis(formula = dune ~ Manure * Management * A1 * Moisture, data = dune.env, permutations = 99)
Permutation: free
Number of permutations: 99
Terms added sequentially (first to last)
Df SumsOfSqs MeanSqs F.Model R2 Pr(>F)
Manure 4 1.5239 0.38097 2.03088 0.35447 0.13
Management 2 0.6118 0.30592 1.63081 0.14232 0.16
A1 1 0.3674 0.36743 1.95872 0.08547 0.21
Moisture 3 0.6929 0.23095 1.23116 0.16116 0.33
Manure:Management 1 0.1091 0.10906 0.58138 0.02537 0.75
Manure:A1 4 0.3964 0.09909 0.52826 0.09220 0.91
Management:A1 1 0.1828 0.18277 0.97431 0.04251 0.50
Manure:Moisture 1 0.0396 0.03963 0.21126 0.00922 0.93
Residuals 2 0.3752 0.18759 0.08727
Total 19 4.2990 1.00000
Why is DF of Management not 3(4-1)?

Answer 1

This is a general, rather than a specific answer.
Your formula Moisture*Management*A1*Manure corresponds to a linear model with 160 (!) predictors (2*4*4*5):
dim(model.matrix(~Moisture*Management*A1*Manure, dune.env))
adonis builds this model matrix internally and uses it to construct the machinery for calculating the permutation statistics. When there are multicollinear combinations of predictors, it drops enough columns to make the problem well-defined again. The detailed rules for which columns get dropped depends on the order of the columns; if you reorder the factors in your question you'll see the reported Df change.
For what it's worth, I don't think the df calculations change the statistical outcomes at all — the statistics are based on the distributions derived from permutations, not from an analytical calculation that depends on the df.

Answer 2

Ben Bolker got it right. If you only look at Management and Manure and forget all other variables, you will see this:
> with(dune.env, table(Management, Manure))
Manure
Management 0 1 2 3 4
BF 0 2 1 0 0
HF 0 1 2 2 0
NM 6 0 0 0 0
SF 0 0 1 2 3
Look at row Management NM and column Manure 0 that only have one non-zero case. This means that Management NM and Manure 0 are synonyms, the same thing (or "aliased"). After you have had Manure in your model, Management only has three new levels, and hence 2 d.f. If you do it in reversed order and first have Management then you only have four levels Manure that you do not yet know, and that would give you 3 d.f. of Manure.
Although you really have overparametrized your model, you would also get the same result with only these two variables. Compare models:
adonis2(dune ~ Manure + Management, data=dune.env)
adonis2(dune ~ Management + Manure, data=dune.env)