A three-parameter logistic regression model

Figures & data

Table 1. D-optimal design for three-parameter logistic models.

λ	Dose levels
0.50	ED2 ED35 ED91
0.75	ED3 ED39 ED92
1.00	ED3 ED43 ED93
1.25	ED4 ED46 ED94
1.50	ED5 ED49 ED94
1.75	ED5 ED52 ED95
2.00	ED6 ED55 ED95

Figure 1. Dose–response curves in the simulation.

Table 2. RMSEs when fitting data from three-parameter logistic models under a number of designs.

Size	n = 30			n = 60			n = 120
Design	ED	Up-down	D-opt	ED	Up-down	D-opt	ED	Up-down	D-opt
	$\alpha =-6.265$, $\text{β}=0.055$, $\lambda =0.5$
Total	13.32	16.20	16.82	9.87	10.33	11.96	6.93	7.42	8.44
ED10	8.62	9.39	11.33	6.66	6.79	8.16	4.71	4.97	5.82
ED25	6.78	8.09	9.03	4.78	5.05	6.34	3.23	3.61	4.45
ED40	7.55	10.43	8.54	5.50	5.92	6.02	3.93	4.15	4.19
Total	14.02	15.68	16.97	10.49	10.85	11.95	7.47	7.69	8.46
ED25	8.20	8.00	9.03	6.15	5.83	6.33	4.46	4.38	4.45
ED50	6.72	7.68	8.72	4.83	5.06	6.16	3.48	3.51	4.29
ED75	9.17	11.09	11.42	7.00	7.62	8.05	4.88	5.25	5.77
Total	17.36	21.59	22.27	12.63	13.49	16.01	8.89	9.26	11.61
ED60	9.00	9.08	8.94	6.29	6.01	6.45	4.39	4.24	4.60
ED75	8.69	10.64	11.23	5.92	6.45	8.09	4.07	4.34	5.82
ED90	12.04	16.44	17.03	9.22	10.21	12.21	6.57	7.00	8.93
	$\alpha =-14.15$, $\text{β}=0.10$, $\lambda =2.0$
Total	16.76	18.61	21.35	12.26	12.98	15.84	8.92	9.09	11.05
ED10	12.01	12.16	15.58	9.03	9.00	11.63	6.61	6.66	8.22
ED25	8.34	9.44	11.23	5.79	6.38	8.24	4.12	4.45	5.66
ED40	8.21	10.46	9.32	5.93	6.85	6.91	4.35	4.30	4.75
Total	13.05	14.65	16.23	9.36	10.01	11.78	6.76	6.92	8.06
ED25	8.76	8.67	11.23	6.37	6.52	8.24	4.69	4.65	5.66
ED50	6.37	6.32	8.48	4.42	4.42	6.28	3.12	3.25	4.28
ED75	7.28	9.97	8.10	5.25	6.18	5.61	3.74	3.96	3.82
Total	12.32	15.42	16.30	8.67	9.98	11.54	6.23	6.66	8.30
ED60	6.84	7.22	7.93	4.81	4.99	5.77	3.46	3.52	3.89
ED75	6.24	7.63	8.10	4.14	4.96	5.61	2.83	3.27	3.82
ED90	8.12	11.28	11.72	5.90	7.08	8.28	4.35	4.61	6.26

Table 3. RMSEs when fitting three-parameter model to data from two-parameter model.

	Simple logistic model						Three-parameter logistic model
Design	ED-design			D-optimal			ED-design			D-optimal
Size	30	60	120	30	60	120	30	60	120	30	60	120
Total	17.93	13.25	9.35	23.58	15.79	10.87	18.36	13.63	9.88	23.56	17.50	12.41
ED10	12.68	9.48	6.73	17.53	11.96	8.02	12.30	9.60	7.03	17.29	13.04	9.32
ED25	8.93	6.52	4.55	12.15	8.02	5.63	9.55	6.57	4.68	12.03	8.76	6.18
ED40	9.01	6.56	4.63	10.07	6.46	4.69	9.74	7.10	5.13	10.56	7.72	5.37
Total	17.14	12.54	8.74	20.18	12.63	8.87	17.52	12.33	8.86	19.99	14.52	10.22
ED25	10.55	7.80	5.46	12.48	7.70	5.49	10.64	7.80	5.66	12.03	8.76	6.18
ED50	8.12	5.89	4.09	9.89	6.15	4.41	8.45	5.48	4.04	10.33	7.56	5.23
ED75	10.79	7.85	5.46	12.40	7.90	5.39	11.06	7.81	5.48	12.17	8.77	6.23
Total	18.91	13.42	9.40	26.03	15.76	10.58	19.00	13.48	9.92	23.77	17.25	12.49
ED60	9.49	6.64	4.86	9.09	6.45	4.62	9.77	6.97	5.08	10.57	7.70	5.33
ED75	9.39	6.51	4.57	12.87	8.01	5.47	9.43	6.36	4.51	12.17	8.77	6.23
ED90	13.40	9.67	6.63	20.72	11.94	7.79	13.30	9.64	7.22	17.47	12.70	9.42

Table 4. RMSEs when fitting three-parameter model to data from three-parameter model.

	Simple logistic model						Three-parameter logistic model
Design	ED-design			D-optimal			ED-design			D-optimal
Size	30	60	120	30	60	120	30	60	120	30	60	120
	$\alpha =-6.265$, $\text{β}=0.055$, $\lambda =0.5$
Total	13.87	9.96	7.16	16.62	13.38	11.04	13.32	9.87	6.93	16.82	11.96	8.44
ED10	10.14	7.08	4.99	11.75	9.50	7.77	8.62	6.66	4.71	11.33	8.16	5.82
ED25	6.53	4.74	3.43	8.63	7.15	6.09	6.78	4.78	3.23	9.03	6.34	4.45
ED40	6.86	5.17	3.82	7.97	6.14	4.94	7.55	5.50	3.93	8.54	6.02	4.19
Total	15.57	11.14	7.65	15.72	11.46	8.69	14.02	10.49	7.47	16.97	11.95	8.46
ED25	9.45	6.63	4.30	8.63	7.15	6.09	8.20	6.15	4.46	9.03	6.34	4.45
ED50	6.64	5.06	3.57	8.12	5.82	4.30	6.72	4.83	3.48	8.72	6.16	4.29
ED75	10.44	7.39	5.22	10.34	6.81	4.47	9.17	7.00	4.88	11.42	8.05	5.77
Total	18.98	12.89	8.98	19.80	13.86	10.23	17.36	12.63	8.89	22.40	15.83	11.48
ED60	8.16	5.78	4.23	8.65	5.82	3.90	9.00	6.29	4.39	9.33	6.58	4.63
ED75	8.74	5.90	4.21	10.34	6.81	4.47	8.69	5.92	4.07	11.42	8.05	5.77
ED90	14.73	9.89	6.71	14.50	10.57	8.34	12.04	9.22	6.57	16.86	11.94	8.78
	$\alpha =-14.15$, $\text{β}=0.10$, $\lambda =2.0$
Total	17.43	12.39	8.99	24.82	15.78	12.04	16.76	12.26	8.92	21.35	15.84	11.05
ED10	14.00	9.78	6.98	19.42	12.80	10.33	12.01	9.03	6.61	15.58	11.63	8.22
ED25	7.50	5.40	3.95	12.51	7.47	5.11	8.34	5.79	4.12	11.23	8.24	5.66
ED40	7.19	5.35	4.06	9.07	5.42	3.50	8.21	5.93	4.35	9.32	6.91	4.75
Total	13.80	9.57	6.95	16.25	10.09	6.93	13.05	9.36	6.76	16.23	11.78	8.06
ED25	9.43	6.64	4.76	12.51	7.47	5.11	8.76	6.37	4.69	11.23	8.24	5.66
ED50	5.95	4.25	3.47	7.52	4.69	3.13	6.37	4.42	3.12	8.48	6.28	4.28
ED75	8.14	5.43	3.70	7.13	4.90	3.49	7.28	5.25	3.74	8.10	5.61	3.82
Total	13.06	8.75	6.35	15.63	10.30	7.10	12.32	8.67	6.23	16.30	11.54	8.30
ED60	6.27	4.57	3.49	6.59	4.37	3.06	6.84	4.81	3.46	7.93	5.77	3.89
ED75	6.22	4.08	2.96	7.13	4.90	3.49	6.24	4.14	2.83	8.10	5.61	3.82
ED90	9.63	6.25	4.40	12.25	7.93	5.37	8.12	5.90	4.35	11.72	8.28	6.26

Table 5. RMSEs when fitting three-parameter model to data from probit model.

	Simple logistic model						Three-parameter logistic model
Design	ED-design			D-optimal			ED-design			D-optimal
Size	30	60	120	30	60	120	30	60	120	30	60	120
	$\alpha =-6.265$, $\text{β}=0.055$, $\lambda =0.5$
Total	9.27	6.29	4.31	9.02	6.81	7.48	8.58	5.75	4.25	12.36	7.65	4.95
ED10	6.68	4.49	3.01	5.44	3.98	4.69	5.76	3.94	2.94	7.24	4.40	3.14
ED25	4.36	3.08	2.11	4.85	3.86	4.37	4.42	2.85	2.06	7.11	4.41	2.77
ED40	4.72	3.16	2.24	5.31	3.95	3.85	4.57	3.08	2.27	7.06	4.43	2.64
Total	9.73	6.22	4.38	10.29	7.28	6.40	8.57	6.00	4.25	12.09	7.56	4.83
ED25	6.22	3.91	2.72	4.85	3.86	4.37	5.17	3.70	2.68	7.11	4.41	2.77
ED50	4.18	2.71	2.00	5.72	4.06	3.52	4.27	2.85	2.00	6.99	4.41	2.65
ED75	6.21	4.01	2.80	7.05	4.65	3.07	5.34	3.77	2.61	6.84	4.28	2.94
Total	10.17	6.71	4.45	12.75	8.38	5.64	9.14	6.52	4.58	12.55	7.71	5.26
ED60	5.05	3.29	2.21	6.19	4.23	3.25	4.77	3.45	2.47	6.89	4.36	2.72
ED75	4.78	3.17	2.14	7.05	4.65	3.07	4.49	3.07	2.12	6.84	4.28	2.94
ED90	7.41	4.90	3.21	8.63	5.55	3.44	6.37	4.60	3.22	7.94	4.70	3.42
	$\alpha =-14.15$, $\text{β}=0.10$, $\lambda =2.0$
Total	8.52	5.65	3.70	8.75	5.95	4.31	8.39	5.43	3.13	12.89	7.62	4.33
ED10	6.33	4.13	2.67	6.24	4.19	3.05	5.81	3.80	2.13	9.13	5.16	2.80
ED25	3.97	2.69	1.79	4.63	3.10	2.16	4.24	2.67	1.57	6.67	4.02	2.39
ED40	4.09	2.75	1.83	4.04	2.87	2.14	4.32	2.82	1.68	6.19	3.91	2.28
Total	8.25	5.39	3.52	7.51	5.80	4.77	7.33	5.21	3.66	10.47	6.59	4.21
ED25	5.42	3.50	2.28	4.63	3.10	2.16	4.59	3.21	2.31	6.67	4.02	2.39
ED50	3.40	2.36	1.57	3.93	3.00	2.41	3.67	2.57	1.78	5.98	3.82	2.24
ED75	5.20	3.36	2.17	4.43	3.89	3.50	4.37	3.19	2.21	5.42	3.57	2.66
Total	8.29	5.30	3.55	8.29	7.05	6.24	7.12	5.06	3.65	9.63	6.39	4.99
ED60	3.97	2.66	1.79	3.99	3.27	2.80	3.97	2.77	2.05	5.76	3.70	2.29
ED75	3.87	2.51	1.71	4.43	3.89	3.50	3.76	2.51	1.71	5.42	3.57	2.66
ED90	6.17	3.84	2.55	5.76	4.88	4.33	4.56	3.41	2.48	5.48	3.79	3.54

Figure 2. Observed data and fitted curves for British coal miners.