Understanding and extending the geographical detector model under a linear regression framework

Figures & data

Figure 1. Topology of counties in North Carolina.

Figure 2. (a) The empirical relationship between strength of spatial auto-correlation ($\rho$) and percentage bias of q-statistic in GDM from three spatial econometrics models (true data generating processes). To avoid overlapping error bars, the curve of SEM is shifted 0.01 units to the left and the curve of SDM is shifted 0.01 units to the right; (b) The empirical relationship between Moran’s I and percentage bias of q-statistic in GDM under three spatial econometrics models. The same value of $\rho$ corresponds to different Moran’s I under three models. As a result, the starting and ending points of the three curves exhibit tiny differences.

Figure 3. Variables examined in the study: (a) Desertification index; (b) Nighttime lights intensity; (c) Population density; (d) Total precipitation; and (e) Soil moisture.

Table 1. Statistical summaries of variables used in this study’s models.

Factors	Unit	Mean	Variance	Data sources
Desertification index ($DI$)	1	171.507	7076.201	Google Earth Engine
Nighttime light intensity ($\mathit{ntl}$)	nW/cm^2/sr	4.971	182.744	NOAA-NGDC
Population density ($\mathit{pop}$)	Person/km^2	50.177	16495.87	ERA5
Total Precipitation ($\mathit{pre}$)	m	12.202	290.300	ERA5
Soil moisture ($sm$)	m^3/m^3	100.534	12136.690	ERA5

Note: The mean and variance of variables are calculated before the discretization procedure; NOAA-NGDC refers to National Oceanic and Atmospheric Administration’s National Geophysical Data Center.

Table 2. Estimation results on the LM-test.

Test	Value	p-value
LM-error	4478.1	0.000
LM-lag	4689.5	0.000
Robust LM-error	362.4	0.000
Robust LM-lag	573.7	0.000

Table 3. Estimation results on the Akaike information criterion and variable importance.

	Model specifications		AIC	R-squared values
Models	Main effects	Interaction effects	SDM	GDM	SDM
1	$\mathit{sm}+\mathit{ntl}$	$\stackrel{⃛}{\cap }(\mathit{sm},\mathit{ntl})$	29336.4	0.627	0.737
2	$pr+sm$	$\stackrel{⃛}{\cap }(pr,sm)$	29336.7	0.644	0.726
3	$pr+sm+\mathit{ntl}$	$\stackrel{⃛}{\cap }(pr,sm,\mathit{ntl})$	29346.6	0.661	0.747
4	$sm+\mathit{pop}$	$\stackrel{⃛}{\cap }(sm,\mathit{pop})$	29354.3	0.613	0.721
5	$sm+\mathit{pop}+\mathit{ntl}$	$\stackrel{⃛}{\cap }(sm,\mathit{pop},\mathit{ntl})$	29361.3	0.627	0.738
6	$pr+\mathit{ntl}$	$\stackrel{⃛}{\cap }(pr,\mathit{ntl})$	29361.8	0.554	0.686
7	$pr+sm+\mathit{pop}$	$\stackrel{⃛}{\cap }(pr,sm,\mathit{pop})$	29367.6	0.648	0.730
8	$pr+sm+\mathit{pop}+\mathit{ntl}$	$\stackrel{⃛}{\cap }(pr,sm,\mathit{pop},\mathit{ntl}$)	29387.6	0.663	0.748
9	$pr+\mathit{pop}$	$\stackrel{⃛}{\cap }(pr,\mathit{pop})$	29391.5	0.524	0.642
10	$pr+\mathit{pop}+\mathit{ntl}$	$\stackrel{⃛}{\cap }(pr,\mathit{pop},\mathit{ntl})$	29396.2	0.555	0.687
11	$\mathit{pop}+\mathit{ntl}$	$\stackrel{⃛}{\cap }(\mathit{pop},\mathit{ntl})$	29457.5	0.185	0.449

The symbol $\stackrel{⃛}{\cap }(·)$ denotes all possible interactions of variables, such as $\stackrel{⃛}{\cap }\left(a,b,c\right)=a\cap b+a\cap c+b\cap c+a\cap b\cap c.$

Table 4. Decomposition results for variable importance.

Discretization	Variables	Shapley values			Contribution share (%)
		$S(\mathrm{OLS})$	$S(\mathrm{SLM})$	$S(\mathrm{SDM})$	$S_{\%}(\mathrm{OLS})$	$S_{\%}(\mathrm{SLM})$	$S_{\%}(\mathrm{SDM})$
Natural breaks	$\mathit{ntl}$	0.054	0.099	0.145	8.672	13.616	19.737
	$sm$	0.488	0.510	0.408	77.813	70.398	55.329
	$sm\cap \mathit{ntl}$	0.085	0.116	0.184	13.515	15.986	24.934
	total	0.627	0.725	0.737	100	100	100
Quantile	$\mathit{ntl}$	0.050	0.094	0.148	8.261	13.628	21.139
	$sm$	0.477	0.471	0.360	78.167	68.295	51.452
	$sm\cap \mathit{ntl}$	0.083	0.125	0.192	13.572	18.077	27.409
	total	0.611	0.690	0.700	100	100	100

OLS refers to ordinary least squares model and the R-squared (0.627) from the OLS model is equivalent to the q-statistic from the GDM.

Data and codes availability statement

Data used in the empirical study and the R code for implementing the Monte Carlo simulation experiment are available for download at Figshare, https://doi.org/10.6084/m9.figshare.24196608.