Table 1.
Basis sets for the partial independence constraints implied by each of the five alternative models.
| Model A | Model B | Model C | Model D | ||||
|---|---|---|---|---|---|---|---|
| Basis set | r (P) | Basis set | r (P) | Basis set | r (P) | Basis set | r (P) |
| (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) |
| (S,R)|{T,P} | −0.02 (0.851) | (S,R)|{T,N} | 0.17 (0.090) | (S,R)|{T,P,N} | −0.02 (0.869) | (S,R)|{T,P} | −0.02 (0.851) |
| (T,P)|{S} | −0.03 (0.792) | (T,P)|{S} | −0.03 (0.792) | (T,P)|{S} | −0.03 (0.792) | (T,P)|{S,N} | −0.07 (0.468) |
| (T,N)|{S} | −0.11 (0.301) | (T,N)|{S} | −0.11 (0.301) | (T,N)|{S} | −0.11 (0.301) | (T,N)|{S} | −0.11 (0.301) |
| (P,N)|{S} | −0.39 (<0.001) | (P,N)|{S} | −0.39 (<0.001) | (P,N)|{S} | −0.39 (<0.001) | (N,R)|{S,T,P} | 0.01 (0.947) |
| (N,R)|{S,T,P} | 0.01 (0.947) | (P,R)|{S,T,N} | 0.42 (<0.001) | ||||
| 12 df, C = 23.69 (0.022) | 12 df, C = 50.29 (<0.001) | 10 df, C = 23.54 (0.009) | 10 df, C = 5.01 (0.890) | ||||
| Model E | Model F | Model G | Model H | ||||
|---|---|---|---|---|---|---|---|
| Basis set | r (P) | Basis set | r (P) | Basis set | r (P) | Basis set | r (P) |
| (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) | (S,T)|{Ø} | −0.03 (0.721) |
| (S,R)|{T,N} | 0.17 (0.090) | (S,R)|{T,P,N} | −0.02 (0.869) | (T,P)|{S,N} | −0.07 (0.468) | (T,P)|{S} | −0.03 (0.792) |
| (T,P)|{S,N} | −0.07 (0.468) | (T,P)|{S,N} | −0.07 (0.468) | (T,N)|{S} | −0.11 (0.301) | (T,N)|{S} | −0.11 (0.301) |
| (T,N)|{S} | −0.11 (0.301) | (T,N)|{S} | −0.11 (0.301) | (P,R)|{S,T,N} | 0.42 (<0.001) | (P,N)|{S} | −0.39 (<0.001) |
| (P,R)|{S,T,N} | 0.42 (<0.001) | (N,R)|{S,T} | −0.17 (0.092) | (P,R)|{S,T} | 0.45 (<0.001) | ||
| (N,R)|{S,T} | −0.17 (0.092) | ||||||
| 10 df, C = 31.62 (<0.001) | 8 df, C = 4.86 (0.772) | 10 df, C = 31.59 (<0.001) | 12 df, C = 53.64 (<0.001) | ||||
The hypothesized relationship among variables can be seen in Fig. 1. S Species richness, T soil temperature, P biomass, N average community leaf N concentration, R soil respiration
The notation (X,Y)|{A,B,…} means that variables x and y are hypothesized to be probabilistically independent conditional on the set of variables {A,B,…}. Ø represents the null (empty) set. Pearson’s partial correlation coefficient (r), and P (assuming the null hypothesis) in parentheses, are given for each conditional independence claim; the overall model is tested with Fisher’s C statistic
P < 0.05 indicated in bold, rejecting the hypothesized independence relationship