Table 2.

Bias, standard deviation, and empirical coverage for estimates of prevention efficacy (PE) based on $M$ external cohorts used to estimate the association between HIV and an exposure biomarker with correlation $\rho$. A total of $n_x=2000$ person-years follow-up accrue in the active arm of the trial. Counterfactual placebo HIV incidence and true PE vary. Performance is shown for working model and likelihood-based estimation approaches, assuming log link function for marginal incidences.

	$M$	PE estimate	$\rho =0.98$			$\rho =0.5$
HIV incidence (cases per 100 person-years)			3	4.5	6	3	4.5	6
Exposure marker incidence (cases per 100 person-years)			7.1	11.8	17.0	4.8	13.2	26.7
			Working model approach
$PE=0.3$	10	Bias × 100	−1.42	−0.83	−0.96	−3.72	−2.28	−4.88
$n_x=2000$		Standard deviation × 100	14.49	11.45	10.20	28.78	19.50	28.72
		Coverage (%)	93.0	93.4	93.2	92.2	92.7	90.1
	20	Bias × 100	−1.09	−0.77	−0.63	−1.93	−1.20	−1.88
		Standard deviation × 100	13.55	10.58	9.20	19.89	14.38	18.24
		Coverage (%)	92.5	93.4	93.1	93.0	94.4	92.7
$PE=0.6$	10	Bias × 100	−0.70	−0.23	−0.53	−2.08	−1.11	−2.81
$n_x=2000$		Standard deviation × 100	10.03	7.97	7.03	17.51	12.04	16.99
		Coverage (%)	93.3	93.1	93.8	92.6	93.6	90.8
	20	Bias × 100	−0.46	−0.33	−0.42	−1.04	−0.44	−1.05
		Standard deviation × 100	9.49	7.47	6.56	12.63	9.45	11.1
		Coverage (%)	93.3	93.8	93.9	93.8	93.7	93.3
$PE=0.75$	10	Bias × 100	−0.33	−0.21	−0.22	−1.14	−0.62	−1.19
$n_x=2000$		Standard deviation × 100	7.48	5.95	5.29	11.80	8.30	10.38
		Coverage (%)	93.3	93.9	93.6	93.0	92.9	91.9
	20	Bias × 100	−0.17	−0.17	−0.20	−0.53	−0.36	−0.48
		Standard deviation × 100	7.18	5.80	5.03	8.91	6.65	7.49
		Coverage (%)	93.4	93.7	93.7	94.0	94.3	93.6
			Likelihood-based approach
$PE=0.3$	20	Bias × 100	0.08	−0.11	0.04	−0.62	−0.64	−1.04
$n_x=2000$		Standard deviation × 100	12.36	9.90	8.90	18.55	13.83	18.02
		Coverage (%)	94.9	95.0	93.9	93.8	94.3	92.8
$PE=0.6$	20	Bias × 100	−0.20	−0.29	0.10	−0.31	0.04	−0.26
$n_x=2000$		Standard deviation × 100	8.83	7.27	6.30	11.80	9.06	10.92
		Coverage (%)	95.3	94.8	94.5	94.3	94.6	93.0
$PE=0.75$	20	Bias × 100	−0.16	0.00	−0.05	−0.20	−0.03	−0.54
$n_x=2000$		Standard deviation × 100	6.81	5.48	4.86	8.44	6.55	7.38
		Coverage (%)	95.3	95.5	95.1	94.7	94.3	94.4