Table 1.

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

			$\rho =0.98$			$\rho =0.5$
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
$M=10$	$n_x=2000$	Bias × 100	−0.01	−0.02	−0.03	0.13	0.07	0.26
		Standard deviation × 100	0.33	0.38	0.53	1.03	1.05	2.22
		Coverage (%)	96.1	97.5	97.2	95.8	95.5	94.5
	$n_x=4000$	Bias × 100	−0.01	−0.02	−0.02	0.11	0.09	0.25
		Standard deviation × 100	0.30	0.36	0.49	1.00	1.09	2.12
		Coverage (%)	95.4	96.5	96.8	95.4	95.4	95.7
$M=20$	$n_x=2000$	Bias × 100	−0.01	−0.02	−0.03	0.05	0.02	0.11
		Standard deviation × 100	0.27	0.31	0.39	0.64	0.71	1.39
		Coverage (%)	95.2	96.0	97.3	95.0	95.9	95.1
	$n_x=4000$	Bias × 100	−0.02	−0.02	−0.03	0.04	0.03	0.07
		Standard deviation × 100	0.24	0.27	0.34	0.63	0.73	1.36
		Coverage (%)	94.6	96.2	97.4	95.0	94.8	95.2
			Likelihood-based approach
$M=20$	$n_x=2000$	Bias × 100	0.03	0.02	0.04	0.11	0.08	0.20
		Standard deviation × 100	0.26	0.31	0.40	0.65	0.71	1.39
		Coverage (%)	95.5	95.4	94.4	94.5	94.2	93.7
	$n_x=4000$	Bias × 100	0.02	0.03	0.05	0.10	0.10	0.21
		Standard deviation × 100	0.22	0.26	0.35	0.64	0.71	1.42
		Coverage (%)	95.1	95.2	93.5	94.3	94.3	93.1