Nonparametric constrained ordination

Background/Question/Methods

Constrained ordination in community ecology positions sample units in environmental space, according to the species relationships with environment. These relationships are typically fit by eigenanalysis using an underlying linear model (redundancy analysis, RDA) or a unimodal model (canonical correspondence analysis, CCA). Species relationships to environmental factors are, however, typically complex, non-linear, and interactive. We demonstrate a new method, nonparametric constrained ordination (NCO) that combines two techniques that excel at recovering structure from species abundance data, nonparametric multiplicative regression (NPMR) and nonmetric multidimensional scaling (NMS or NMDS). The techniques are combined in a way that is analagous to RDA: applying NMS in PC-ORD 6 to estimates based on response surfaces fitted with NPMR in HyperNiche 2. The nonparametric regression step provides rich, realistic, individualistic descriptions of species response surfaces to the predictors that are strongest for them. NCO automatically represents interactions among environmental factors, while in RDA and CCA those interactions must be explicitly specified; practitioners usually ignore them.

Results/Conclusions

We demonstrate the efficacy of NCO with simulated and real data sets. In analyses of several example real data sets we found that species response surfaces were usually nonlinear and often nonmonotonic and displayed interactions among predictors. In one example using vascular plant community relationships to heavy metals in soils, both NCO and CCA represented 14% of the variation in the communities. In a dataset on epiphyte distribution in a forest canopy, CCA represented 13% of the variation in the community matrix, based on 4 environmental predictors, while NCO represented 47% of the variation, using the same 4 predictors. The leave-one-out cross-validated R² values from the regression step of NCO ranged from 0.0 to 0.8 with most values between 0.1 and 0.6.  In a dataset examining the relationships between epiphytic lichen communities and six climate variables along with three forest variables, CCA represented 18% of the variation and NCO 30%. Cross-validated R² values for the regression step of NCO were between 0.0 and 0.6. We conclude that as a fully nonparametric alternative to CCA and RDA, NCO provides (1) effective summarization of community relationships to measured environmental variables and (2) information-rich, nonlinear, flexible representations of species relationships to environment that capture interactions among predictors.