. Author manuscript; available in PMC: 2024 Mar 13.

Published in final edited form as: Stat Med. 2023 Jan 11;42(5):632–655. doi: 10.1002/sim.9635

TABLE 1.

Summary of distributions used to generate all population variables for simulations to evaluate the statistical performance of the proposed methods

Variable	Description	Distribution

$X$	Predictor	$X_{i j} ~ N (1, 0.5)$
$Z$	Auxiliary Variable	$Z_{i j} ~ N (2, 9)$
$T_{V}$	Follow-up Time	Truncated $N (6, 1)$ , bounded between 3 and 9
$Y_{0}$	Baseline Biomarker	Normal distribution: $110 + a_{i} + η_{i j}$
		$a_{i} ~ N (0, 4), η_{i j} ~ N (0, 4)$
$B_{V}$	Disease at Follow-up	Bernoulli distribution with the following probability: ${l o g i t}^{- 1} (c_{0} + b_{i})$
		$b_{i} ~ N (0, 0.09)$
$T_{M}$	Time-to-treatment	If $B_{V i j} = 1$ : Weibull distribution with the following hazard function (varied by stage 1 and stage 2 sampling strata):
		Stage 1 Stratum 1, Stage 2 Stratum A: $2 λ t * e x p \{- 0.2 X_{i j} - 0.2 Z_{i j} - 0.001 Y_{0 i j} + c_{i}\}$
		Stage 1 Stratum 1, Stage 2 Stratum B: $2 λ t * e x p \{0.1 - 0.15 X_{i j} - 0.2 Z_{i j} - 0.001 Y_{0 i j} + c_{i}\}$
		Stage 1 Stratum 2, Stage 2 Stratum A: $2 λ t * e x p \{- 0.1 - 0.2 X_{i j} - 0.2 Z_{i j} - 0.001 Y_{0 i j} + c_{i}\}$
		Stage 1 Stratum 2, Stage 2 Stratum B: $2 λ t * e x p \{- 0.15 X_{i j} - 0.2 Z_{i j} - 0.001 Y_{0 i j} + c_{i}\}$
		$c_{i} ~ N (0, 0.09)$
$Y_{V}$	Untreated Biomarker at Follow-up	If $B_{V i j} = 0$ : The following truncated normal distribution bounded from above by 130 (varied by stage 1 and stage 2 sampling stratum):
		Stage 1 Stratum 1, Stage 2 Stratum A: $Y_{0 i j} + 0.2 X_{i j} - 1.6 Z_{i j} + 0.2 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 1, Stage 2 Stratum B: $Y_{0 i j} + 0.1 + 0.25 X_{i j} - 1.6 Z_{i j} + 0.2 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum A: $Y_{0 i j} - 0.1 + 0.2 X_{i j} - 1.6 Z_{i j} + 0.2 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum B: $Y_{0 i j} + 0.25 X_{i j} - 1.6 Z_{i j} + 0.2 T_{V i j} + d_{i} + ϵ_{i j}$
		If $B_{V i j} = 1$ : The following truncated normal distribution bounded from below by 130 (varied by stage 1 and stage 2 sampling stratum):
		Stage 1 Stratum 1, Stage 2 Stratum A: $Y_{0 i j} + 20 + 1.8 X_{i j} - 1.1 Z_{i j} + 4 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 1, Stage 2 Stratum B: $Y_{0 i j} + 20.1 + 1.85 X_{i j} - 1.1 Z_{i j} + 4 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum A: $Y_{0 i j} + 19.9 + 1.8 X_{i j} - 1.1 Z_{i j} + 4 T_{V i j} + d_{i} + ϵ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum B: $Y_{0 i j} + 20 + 1.85 X_{i j} - 1.1 Z_{i j} + 4 T_{V i j} + d_{i} + ϵ_{i j}$
		$d_{i} ~ N (0, 4), ϵ_{i j} ~ N (0, 4)$ ¹
$Y_{V}^{trt}$	Treated Biomarker at Follow-up	$(Y_{V i j} - Y_{V i j}^{t r t})$ generated from the following normal distribution (varied by stage 1 and stage 2 sampling stratum):
		Stage 1 Stratum 1, Stage 2 Stratum A: $- 11 + 0.1 Y_{0 i j} - 3 X_{i j} + 1.5 Z_{i j} + 1.1 T_{V i j} + e_{i} + κ_{i j}$
		Stage 1 Stratum 1, Stage 2 Stratum B: $- 10.9 + 0.1 Y_{0 i j} - 2.95 X_{i j} + 1.5 Z_{i j} + 1.1 T_{V i j} + e_{i} + κ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum A: $- 11.1 + 0.1 Y_{0 i j} - 3 X_{i j} + 1.5 Z_{i j} + 1.1 T_{V i j} + e_{i} + κ_{i j}$
		Stage 1 Stratum 2, Stage 2 Stratum B: $- 11 + 0.1 Y_{0 i j} - 2.95 X_{i j} + 1.5 Z_{i j} + 1.1 T_{V i j} + e_{i} + κ_{i j}$
		$e_{i} ~ N (0, 4), κ_{i j} ~ N (0, 4)$

NOTE: Subscript $i$ denotes the PSU, and subscript $j$ denotes the individual.

NOTE: The first 79 PSUs were assigned to Stage 1 Stratum 1, and the last 297 PSUs were assigned to Stage 1 Stratum 2. 60% of the individuals were randomly assigned to stage 2 stratum A, and the remaining 40% of individuals were assigned to participant stratum B.

$V a r (d_{i}) / [V a r (d_{i}) + V a r (ϵ_{i j})]$ equals the intra-class correlation of $Y_{Vij}$ at the PSU level.