Table 2 Model comparison of the fitted INLA models using linear and non-linear associations across three background buffer radii. The criteria for comparison are DIC Deviance Information Criterion, WAIC Watanabe-Akaike Information Criterion, LCPO Conditional Predictive Ordinate, Variables acronyms are Distw Distance to water, Ca Soil calcium, Elev Elevation, Swater Soil water, BIO4 Temperature Seasonality, \(\mu\) Spatial effect.

Buffer radius	Model option	Type	Variables	DIC	WAIC	CPO
50 km	1	Linear	Baseline	817	817	0.69
	2	Linear	Baseline + μ	187	181	0.15
	3	Linear	BIO4 + Distw + Elev + Ca + Swater + μ	85	85	0.07
	4	Linear	BIO4 + Distw + Elev + Ca + Swater	83	84	0.07
	5	Linear	Distw + Elev + Ca + Swater	81	81	0.07
	6	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater)	78	118	1.75
	7	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater) + μ	120	263	5.29
	8	Non-linear	Distw + f(Elev) + Ca + Swater	60	61	0.05
	9	Non-linear	Distw + f(Elev) + Ca + Swater + μ	63	64	0.06
75 km	10	Linear	Baseline	817	817	0.69
	11	Linear	Baseline + μ	187	182	0.15
	12	Linear	BIO4 + Distw + Elev + Ca + Swater + μ	60	75	1.93
	13	Linear	BIO4 + Distw + Elev + Ca + Swater	50	60	0.08
	14	Linear	Distw + Elev + Ca + Swater	49	53	0.07
	15	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater)	108	301	2.20
	16	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater) + μ	184	604	9.79
	17	Non-linear	Distw + f(Elev) + Ca + Swater	44	47	0.05
	18	Non-linear	Distw + f(Elev) + Ca + Swater + μ	53	63	2.95
100 km	19	Linear	Baseline	817	817	0.69
	20	Linear	Baseline + μ	176	170	0.14
	21	Linear	BIO4 + Distw + Elev + Ca + Swater + μ	95	96	0.08
	22	Linear	BIO4 + Distw + Elev + Ca + Swater	95	96	0.08
	23	Linear	Distw + Elev + Ca + Swater	92	92	0.08
	24	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater)	77	161	5.16
	25	Non-linear	f(Distw) + f(Elev) + f(Ca) + f(Swater) + μ	145	613	9.50
	26	Non-linear	Distw + f(Elev) + Ca + Swater	85	85	0.07
	27	Non-linear	Distw + f(Elev) + Ca + Swater + μ	86	86	0.08

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