Constrained ordination Regression is the key to understanding

Constrained ordination Regression is the key to understanding

Studying community means studying individual species and comparing them Moisture Manure (nutrients)

Linear Regression: the model Y = b 0 + b 1*X + e

Linear Regression: the quality • Total sum of squares (TSS): • Model sum of squares (MSS): • Residual sum of squares (RSS):

Multiple multiple responses, predictors

The best predictors ever: principal components

Comparing two regressions with first two PCA axes PCA: l 1 + l 2 = 0. 51

PCA ordination diagram

What to do with measured environmental factors? - I

What to do with measured environmental factors? - II • Predicting species values using PCA 1 and PCA 2: yik = b 1 k * PCA 1 i + b 2 k * PCA 2 i + e • Constraining scores definition PCA-> RDA 1 i = c 11*Moisturei + c 12*Manurei • Similarly: RDA 2 i = c 21*Moisturei + c 22*Manurei • Consequently: yik = b 1 k*c 11*Moisturei+b 1 k*c 12*Manurei+ b 2 k*c 21*Moisturei+b 2 k*c 22*Manurei

The boiled-down predictors: constrained axes

Definition of constrained ordination axes

Comparing regressions, PCA axes, and RDA axes RDA: l 1 + l 2 = 0. 37

RDA: alternative interpretation

When linearity is not a good idea • Weighted regression on proportional data leads to weighted averaging approach: yik (yik/ y+k)/(yi+/ y++) case weights are yi+ , variable weights are y+k • Roughly: Ø resulting gradients are best predictors for unimodal response model Ø species scores represent optima

Species scores vs. optima