Within- and between-group regression for improving the robustness of causal claims in cross-sectional analysis

Table 1 Results of within- and between-regression modelling of PFOA on immune biomarkers (the C8-Health project, N = 25 817)

		Between-district effect		Within-district effect		Total effect		Model fit
Outcome	model	beta^a	SE^b	beta^a	SE^b	beta^a	SE^b	R²o^c	R²^d	dR²^e	R²w^f	R²b^g
CRP	log-linear	−0.232	0.0525	−0.113	0.0403	−0.157	0.0328	21.0 %	21.1 %	0.1 %	20.7 %	90.4 %
	log-log	−31.021	6.3289	−36.605	7.9959	−33.170	6.3668	21.0 %	21.2 %	0.2 %	20.8 %	92.8 %
Lymphocytes	log-linear	−0.012	0.0166	0.065	0.0127	0.036	0.0097	11.4 %	11.4 %	0.1 %	11.2 %	88.2 %
	log-log	−1.365	1.9997	23.091	2.5265	8.045	3.8012	11.4 %	11.5 %	0.1 %	11.4 %	88.4 %

^a: regression coefficient of PFOA * 1000
^b: standard error of regression coefficient * 1000
^c: coefficient of determination of a model only including covariates without PFOA
^d: coefficient of determination of a model additionally including PFOA
^e: difference in R² due to inclusion of PFOA
^f: coefficient of determination within districts
^g: coefficient of determination between districts

ISSN: 1476-069X