# Correlation between constrained and unconstrained ordination axes

I am comparing several attributes between two types of things. My goals are (1) hypothesis testing and (2) to summarize the information from multiple attributes.

PCA is an option, but this method is agnostic to the groups which I would like to directly compare. Discriminant analysis also seems not appropriate as this is geared towards classification. I think that constrained ordination, e.g. RDA or CCA should be appropriate, and so I am specifying the matrix of attributes as the response variables and a dummy variable for type as the explanatory variable.

My question is: how can there be a positive correlation between the constrained axis and the first residual axis? The code I am using is this:

> library(vegan)

> sample data

type   trait1   trait2    trait3
1     1 1.758808 2.435267 1.0193438
2     2 1.856269 2.531002 1.0926777
3     2 1.792604 2.470863 1.0203008
4     1 1.842596 2.419813 1.0078421
5     2 1.867928 2.540830 1.0499309
6     2 1.899811 2.530313 1.1007924
7     2 1.943430 2.432563 1.0344759
8     1 1.808498 2.465469 1.1102114
9     1 1.773386 2.349574 0.9659113
10    1 1.800390 2.444265 1.0380732
11    2 1.826511 2.533739 1.1153933
12    1 1.895856 2.539948 1.0701774
13    1 1.873971 2.543685 1.0622794
14    1 1.819450 2.508187 0.9820533
15    1 1.848569 2.512621 0.9966571
16    2 1.855092 2.490199 1.0482279
17    2 1.907723 2.500366 1.0839503

ord <- rda(sample.data[, -1] ~ sample.data[,1])
plot(ord)


There is a clear positive association between the constrained axis (called RDA by the output) and the first residual axis (called PCA1). I thought that a feature of constrained ordination was that residual variance should be uncorrelated with the constrained variance. Is this somehow an artifact of having a single explanatory variable and multiple response variables?

My understanding of constrained ordination is that, rather than performing a singular value decomposition as is the case in PCA and using a sparse representation of the Sigma matrix, "constrained ordination" displays the residuals from a regression analysis adjusting for the other $$p-2$$ variables in the covariance matrix. These factors appear as a point cloud if they are in fact conditionally independent. If there are other unmeasured variables that you have not accounted for (mediators or moderators), you will see a trend in the point cloud like what you have shown above.