Nonparametric Testing of the Covariate Significance for Spatial Point Patterns under the Presence of Nuisance Covariates

Figures & data

Fig. 1 The Barro Colorado Island dataset. From left to right, top to bottom: locations of trees, terrain elevation, terrain gradient, the soil contents of nitrogen, phosphorus, and potassium.

Fig. 2 Left to right: realization of a Poisson process with a quadratic intensity function, the nonparametric smoothed residual field, the parametric smoothed residual field with a log-linear model of the intensity function, the parametric smoothed residual field with a log-quadratic model of the intensity function.

Fig. 3 Left: plot of the correlation coefficient $\widehat{\tau }$ as a function of the parameter a for the example in Section 3.2. Right: plot of the correlation coefficient $\widehat{\tau }$ (black curve) and the partial correlation coefficient ${\widehat{\tau }}_p$ (red curve) as functions of the parameter a for the example in Section 3.3.

Table 1 Estimated sizes of the tests for experiments with independent covariates.

Test	Variant	P1	P2	L1	L2	S1	S2	H1	H2
${\widehat{\tau }}_p$	p, tor	0.053	0.058	0.059	0.065	0.056	0.062	0.063	0.061
${\widehat{\tau }}_p$	n, tor	0.049	0.057	0.059	0.063	0.058	0.060	0.061	0.062
${\widehat{\tau }}_p$	p, var	0.040	0.043	0.031	0.047	0.043	0.049	0.047	0.045
${\widehat{\tau }}_p$	n, var	0.043	0.046	0.035	0.047	0.046	0.047	0.047	0.050
CWR	p, tor	0.049	0.053	0.055	0.051	0.048	0.051	0.057	0.048
CWR	n, tor	0.049	0.050	0.054	0.050	0.045	0.045	0.056	0.051
CWR	p, var	0.050	0.049	0.050	0.054	0.047	0.050	0.056	0.048
CWR	n, var	0.048	0.051	0.049	0.048	0.048	0.045	0.055	0.049
ppm	Poisson	0.048	0.166	0.268	0.331	0.021	0.052	0.198	0.161
kppm	LGCP	0.020	0.023	0.080	0.027	–	–	–	–
kppm	MC	0.041	0.025	0.086	0.030	–	–	–	–
ppm	Str(0.02)	0.039	0.111	–	–	0.019	0.040	–	–
ppm	Str(0.05)	0.044	0.070	–	–	0.038	0.060	–	–
ppm	Str(0.10)	0.041	0.063	–	–	0.028	0.034	–	–
ppm	HC Str(0.02)	–	–	–	–	–	–	0.134	0.137
ppm	HC Str(0.05)	–	–	–	–	–	–	0.095	0.079
ppm	HC Str(0.10)	–	–	–	–	–	–	0.093	0.092

NOTE: Different rows correspond to different tests, either the nonparametric tests proposed here (${\widehat{\tau }}_p$, CWR) or the parametric tests based on different assumptions, denoted by the name of the spatstat function used to perform them (ppm, kppm). Different columns correspond to different models.

Table 2 Estimated powers of the tests for experiments with correlated covariates.

Test	Variant	$P_1^p$	$P_2^p$	$L_1^p$	$L_2^p$	$S_1^p$	$S_2^p$	$H_1^p$	$H_2^p$
${\widehat{\tau }}_p$	p, tor	0.198	0.126	0.153	0.155	0.127	0.114	0.101	0.128
${\widehat{\tau }}_p$	n, tor	0.215	0.214	0.164	0.199	0.137	0.152	0.104	0.134
${\widehat{\tau }}_p$	p, var	0.160	0.105	0.113	0.121	0.100	0.096	0.076	0.100
${\widehat{\tau }}_p$	n, var	0.177	0.178	0.125	0.164	0.111	0.122	0.080	0.109
CWR	p, tor	0.758	0.444	0.820	0.702	0.608	0.396	0.310	0.405
CWR	n, tor	0.793	0.677	0.846	0.791	0.690	0.514	0.331	0.418
CWR	p, var	0.753	0.441	0.808	0.698	0.604	0.392	0.304	0.402
CWR	n, var	0.789	0.675	0.835	0.783	0.688	0.511	0.324	0.416
ppm	Poisson	0.774	0.720	0.960	0.943	0.507	0.483	0.618	0.714
kppm	LGCP	0.644	0.254	0.841	0.477	–	–	–	–
kppm	MC	0.745	0.264	0.848	0.499	–	–	–	–
ppm	Str(0.02)	0.682	0.579	–	–	0.432	0.381	–	–
ppm	Str(0.05)	0.614	0.456	–	–	0.467	0.399	–	–
ppm	Str(0.10)	0.515	0.387	–	–	0.344	0.263	–	–
ppm	HC Str(0.02)	–	–	–	–	–	–	0.346	0.375
ppm	HC Str(0.05)	–	–	–	–	–	–	0.302	0.340
ppm	HC Str(0.10)	–	–	–	–	–	–	0.307	0.355

NOTE: Different rows correspond to different tests, either the nonparametric tests proposed here (${\widehat{\tau }}_p$, CWR) or the parametric tests based on different assumptions, denoted by the name of the spatstat function used to perform them (ppm, kppm). Different columns correspond to different models.

Table 3 Backward selection of covariates for the BCI dataset.

	Stage 1	Stage 2	Stage 3	Stage 4
Elevation	0.168	0.186	0.018	0.016
Gradient	0.228	0.240	0.232	–
Nitrogen	0.426	0.388	–	–
Phosphorus	0.066	0.128	0.220	0.048
Potassium	0.598	–	–	–

Individual cells show the p-values of the random shift tests. The row indicates the covariate of interest, while all other covariates considered in the given stage (column) are considered to be the nuisance covariates.

Fig. 4 The Barro Colorado Island dataset. Left to right: locations of trees, nonparametric estimate of the intensity function in the full model, nonparametric estimate in the final model with two covariates.