TABLE 2.

Mean values of 100 * NB(t) estimated using observed risks R in a population assuming that the model is well-calibrated; risk estimates in a case–control sampling when the disease prevalence μ is known, and based on observations of (R, Y) in the population

					Standard deviation of 100 * $\widehat{\text{NB}}$
		100 * $\widehat{\text{NB}}$			R		CC		(R, Y)		VarRatio
t	true	R	CC	(R, Y)	emp	infl	emp (emp($\widehat{\mu }$)a)	infl	emp	infl	CC/R	RY/R	RY/CC
α = 6.55, β = 124.45*
0.02	3.07	3.07	3.07	3.07	0.02	0.02	0.02 (0.23)	0.02	0.23	0.22	1.39	132.94	95.83
0.03	2.15	2.15	2.14	2.15	0.02	0.02	0.06 (0.22)	0.06	0.22	0.22	11.08	139.15	12.56
0.04	1.36	1.37	1.37	1.37	0.02	0.02	0.1 (0.21)	0.10	0.20	0.20	38.98	153.73	3.94
0.05	0.79	0.79	0.79	0.79	0.01	0.01	0.13 (0.19)	0.13	0.18	0.18	92.49	178.79	1.93
0.06	0.42	0.42	0.43	0.42	0.01	0.01	0.12 (0.15)	0.13	0.14	0.15	170.31	214.02	1.26
0.07	0.21	0.21	0.20	0.21	0.01	0.01	0.12 (0.13)	0.12	0.11	0.11	270.05	262.23	0.97
0.08	0.09	0.09	0.10	0.10	0.00	0.00	0.09 (0.1)	0.09	0.08	0.08	390.57	321.22	0.82
0.09	0.04	0.04	0.04	0.04	0.00	0.00	0.07 (0.07)	0.07	0.06	0.06	513.47	381.24	0.74
α = 1, β = 19**
0.02	3.41	3.41	3.41	3.42	0.04	0.05	0.06 (0.22)	0.06	0.22	0.21	1.60	22.20	13.84
0.03	2.80	2.81	2.81	2.82	0.04	0.04	0.08 (0.21)	0.08	0.21	0.21	3.52	23.28	6.61
0.04	2.30	2.30	2.29	2.30	0.04	0.04	0.10 (0.22)	0.10	0.22	0.20	6.21	24.45	3.94
0.05	1.89	1.89	1.89	1.88	0.04	0.04	0.11 (0.20)	0.12	0.19	0.19	9.40	25.91	2.76
0.06	1.54	1.54	1.54	1.54	0.04	0.04	0.12 (0.19)	0.13	0.19	0.18	13.24	27.59	2.08
0.07	1.26	1.26	1.26	1.27	0.03	0.03	0.13 (0.19)	0.13	0.17	0.17	17.52	29.34	1.68
0.08	1.03	1.02	1.02	1.02	0.03	0.03	0.14 (0.19)	0.14	0.17	0.16	22.34	31.04	1.39
0.09	0.83	0.83	0.84	0.83	0.03	0.03	0.13 (0.17)	0.14	0.15	0.15	27.72	33.13	1.20
α = 3, β = 5.7***
0.02	3.95	3.95	3.94	3.95	0.08	0.08	0.05 (0.21)	0.05	0.21	0.21	0.47	7.33	15.46
0.03	3.57	3.56	3.57	3.55	0.08	0.08	0.07 (0.22)	0.07	0.21	0.21	0.86	7.53	8.76
0.04	3.24	3.24	3.24	3.23	0.07	0.07	0.09 (0.22)	0.09	0.22	0.21	1.31	7.75	5.93
0.05	2.95	2.95	2.95	2.95	0.08	0.07	0.10(0.2)	0.10	0.19	0.21	1.82	8.00	4.40
0.06	2.69	2.69	2.68	2.70	0.07	0.07	0.11 (0.21)	0.11	0.20	0.20	2.39	8.22	3.44
0.07	2.46	2.46	2.46	2.46	0.07	0.07	0.11 (0.20)	0.12	0.19	0.20	3.03	8.47	2.79
0.08	2.25	2.25	2.25	2.25	0.07	0.07	0.13 (0.20)	0.13	0.20	0.19	3.73	8.72	2.34
0.09	2.06	2.07	2.06	2.06	0.07	0.06	0.14 (0.22)	0.13	0.20	0.19	4.47	8.97	2.01

Note. Results are based on 500 simulations for each set of parameters (α, β) for the beta distribution and values of threshold t. Each simulation has N = 10, 000 samples with μ =.05. For the case–control design, three controls were sampled for each case. Variance ratios (VarRatios) are computed as the ratio of the influence function-based variances.

^*AUC = 0.61;

^**AUC = 0.76;

^***AUC = 0.88.

Abbreviations: emp, empirical; infl, influence function-based; CC/R, var(NB_CC)/var(NB_R); RY/R, var(NB_RY)/var(NB_R); RY/CC, var(NB_RY)/var(NB_CC).

^aStandard deviations of ${\widehat{\text{NB}}}_{CC}$ with $\widehat{\mu }=\overline{Y}$ substituted for μ.