Covariance among independent variables determines the overfitting and underfitting in variation partitioning methods: with a focus on the mixed co-variation

The effectiveness and validity of applying variation partitioning methods in community ecology has been questioned. Here, using mathematical deduction and numerical simulation, we made an attempt to uncover the underlying mechanisms determining the effectiveness of variation partitioning techniques. The covariance among independent variables determines the under-fitting and over-fitting problem with the variation partitioning process. Ideally, it is assumed that the covariance among independent variables will be zero (no correlation at all), however, typically there will be some colinearities. Therefore, we analyzed the role of slight covariance on influencing species variation partitioning. We concluded that when the covariance between spatial and environmental predictors is positive, all the three components-pure environmental, spatial variations and mixed covariation were over-fitted, with the sign of the true covariation being negative. In contrast, when the covariance is negative, all the three components were under-fitted with the sign of true covariation being positive. Other factors, including extra noise levels, the strengths of variable coefficients and the patterns of landscape gradients, could reduce the fitting problems caused by the covariance of variables. The conventional calculation of mixed covariation is incorrect and misleading, as the true and estimated covariations are always sign-opposite. In conclusion, I challenge the conventional three-step procedure of variation partitioning, suggesting that a full regression model with all variables together is robust enough to correctly partition variations