Quantifying the Impact of Unobserved Heterogeneity on Inference from the Logistic Model

Figures & data

Table 1 Theoretical contribution of first-, second-, and third-order terms to the approximation of bias due to unobserved heterogeneity, with variance

p 0, p 1	δ1	δ2	δ3
0.73, 0.82	−0.08	0.02	0.0
0.88, 0.99	−0.28	0.13	−0.03
0.73, 0.88	−0.15	0.04	−0.01
0.50, 0.38	−0.08	0.0	0.0
0.50, 0.73	−0.18	0.0	0.0
0.50, 0.82	−0.26	0.02	−0.01
0.50, 0.92	−0.39	0.09	−0.02
0.38, 0.62	−0.20	0.0	0.0
0.27, 0.73	−0.36	0.01	0.0
0.12, 0.88	−0.67	0.10	−0.03

Table 2 A comparison between theoretical bias and numerical (simulation) bias, for the (MLH) estimator , of the logistic model, for various probabilities, and for a range of variance (0.01–1.0) for the unobserved heterogeneity term ϵ

	Theoretical bias				Simulation bias
Variance
	δ1	δ2	δ3	Total	Bias  − β	95% C.I	Rej%	(p 0, p 1)
1.0	−0.12	0.0	0.0	−0.12	−0.12	−0.15, − 0.09	13	0.62, 0.73
0.75	−0.07	0.01	0.0	−0.06	−0.07	−0.10, − 0.04	12
0.5	−0.04	0.0	0.0	−0.05	−0.05	−0.06, 0.0	7
0.1	−0.01	0.0	0.0	−0.01	−0.01	−0.04, 0.02	6
0.01	0.0	0.0	0.0	0.0	0.01	−0.03, 0.03	3
1.0	−0.39	0.11	−0.03	−0.31	−0.33	−0.36, − 0.30	36	0.62, 0.95
0.75	−0.29	0.06	−0.01	−0.24	−0.23	−0.27, − 0.19	17
0.5	−0.19	0.02	0.0	−0.17	−0.15	−0.19, − 0.11	15
0.10	−0.03	0.0	0.0	−0.03	−0.04	−0.09, 0.01	6
0.01	0.0	0.0	0.0	0.0	0.05	0.0, 0.10	4
1.0	−0.09	0	0	−0.10	−0.08	−0.10, − 0.06	12	0.5, 0.62
0.75	−0.07	0	0	−0.07	−0.06	−0.08, − 0.04	8
0.5	−0.05	0	0	−0.05	−0.05	−0.07, − 0.03	5
0.1	−0.01	0	0	−0.01	0	−0.03, 0.03	4
0.01	0.0	0	0	0.0	0.01	−0.02, 0.02	2
1.0	−0.34	0.05	−0.01	−0.30	−0.29	−0.32, − 0.26	49	0.5, 0.88
0.75	−0.27	0.03	−0.01	−0.24	−0.23	−0.27, − 0.21	27
0.5	−0.18	0.01	0	−0.17	−0.16	−0.19, − 0.13	14
0.1	−0.04	0	0	−0.04	−0.03	−0.06, 0.0	4
0.01	0	0	0	0.0	−0.01	−0.02, 0.02	3
1.0	−0.48	0.11	−0.03	−0.40	−0.39	−0.43, − 0.35	49	0.5, 0.95
0.75	−0.36	0.04	−0.01	−0.32	−0.31	−0.35, − 0.27	32
0.5	−0.25	0.04	0	−0.21	−0.20	−0.24, − 0.16	21
0.1	−0.05	0	0	−0.04	−0.03	−0.08, 0.02	7
0.01	0.0	0	0	0	0.06	0.01, 0.11	5
1.0	−0.20	0	0	−0.20	−0.17	−0.19, − 0.15	20	0.38, 0.62
0.75	−0.15	0	0	−0.15	−0.14	−0.17, − 0.11	18
0.5	−0.10	0	0	−0.10	−0.10	−0.13, − 0.07	13
0.10	−0.02	0	0	−0.02	−0.03	−0.06, 0.0	10
0.01	0	0	0	0	.03	−0.01, 0.04	3