Table 3.

Residual maximum likelihood (REML) analysis of soil organic carbon (SOC), nitrogen (N), C:N ratio, the stable isotope ratios δ ¹³C and δ ¹⁵N, and total odd‐chained n‐alkane concentration (C₂₃–C₃₃) of soil under long‐term grass, arable (1949–) or fallow (1959–) treatments in 2008 to test for effects of the experiment (E) and treatment (T) factors and the depth (D_v) variate

Property	Spline model	σ 2	SE	Deviance	d.f	Significant improvement with spline?	Significant interaction	F statistic	d.d.f	P	SED
SOC / % (g 100 g−1)	None	0.144	0.033	−1.21	35	–	–	–	–	–	–
	Dv / T	0.030	0.008	−39.46	33	Yes	E × T × Dv	128.99	29.0	< 0.001	0.135
N / % (g 100 g−1)	None	0.0014	0.0003	−178.06	35	–	–	–	–	–	–
	Dv / T	0.0003	0.0001	−218.47	33	Yes	E × T × Dv	153.03	23.8	< 0.001	0.013
C:N ratio	None	1.092	0.250	75.80	35	–	E / T	11.17	38.0	0.002	0.381
	Dv / T	0.866	0.208	72.46	33	No	–	–	–	–	–
δ 13C / ‰	None	0.151	0.038	9.56	35	–	E × T × Dv	9.75	31.9	0.004	0.279
	Dv / T	0.118	0.032	6.31	33	No	–	–	–	–	–
δ 15N / ‰	None	0.602	0.138	53.16	35	–	E × T × Dv	7.60	38.0	0.009	0.367
	Dv / T	0.560	0.151	52.63	33	No	–	–	–	–	–
n‐alkane / µg g−1	None	1.633	0.402	97.57	36	–	–	–	–	–	–
	Dv / T	0.819	0.225	84.11	34	Yes	E × T × Dv	4.07	26.6	0.054	0.790

The estimate and standard error (SE) of the residual variance (σ ²), the deviance (−2 × log‐likelihood) and degrees of freedom (d.f.) are given for REML analyses without and with a fitted spline model, and the significance of the change in deviance was checked with reference to the critical χ ² value. The variance ratio (F) statistic with the denominator d.f. (d.d.f.), the probability level associated with the variance ratio (P) and the average standard error of a difference of means (SED) are given for the interaction of factors and the variate. Note that the Wald statistic is identical to the F statistic because the numerator d.f. was 1. The REML structures are outlined in models (4), (5), (6) and the appropriate models are shown in Figures 2, 3, 4.