Algorithm 1.

Posterior predictive odds ratios.

Input:
Odds ratio OR for a Δ unit increase in the reference biomarker.
Outcome prevalence π̄ at reference biomarker value b̄.
Sample size N at which odds ratios are to be estimated.
Output: S simulated odds ratios ORm and geometric standard errors SEm for quantification methods m = 1, … ,M.
1:	β1 ← log(OR)	▻ Reference logistic slope
2:	${\beta }_0\leftarrow \text{logit}(\overline{\pi })-{\beta }_1\frac{\overline{b}}{\mathrm{\Delta }}$	▻ and intercept.
3:	for s=1 to S do
4:	$\text{Draw}({\mathit{\mu }}^{(s)},{\mathrm{\sum }}_{\iota }^{(s)},{\mathit{\sigma }}_{\omega }^{2(s)},{\mathit{\sigma }}_{(\iota \omega )}^{2(s)},{\mathit{\sigma }}_{\epsilon }^{2(s)})~p(\mathit{\theta }\mid \mathit{b})$
5:	for i=1 to N do
6:	$\text{Draw}{({\stackrel{\sim }{\iota }}_{1i},\dots ,{\stackrel{\sim }{\iota }}_{Mi})}^{\top }~N_M({\mathit{\mu }}^{(s)},{\mathrm{\sum }}_{\iota }^{(s)})$
7:	for m=1 to M do
8:	$\text{Draw}{\stackrel{\sim }{\omega }}_{mi}~N(0,{\sigma }_{{\omega }_m}^{2(s)})$
9:	$\text{Draw}{(\stackrel{\sim }{\iota \omega })}_{mi}~N(0,{\sigma }_{{(\iota \omega )}_m}^{2(s)})$
10:	$\text{Draw}{\stackrel{\sim }{\epsilon }}_{mi}~N(0,{\sigma }_{\epsilon m}^{2(s)})$
11:	${\stackrel{\sim }{b}}_{mi}\leftarrow {\stackrel{\sim }{\iota }}_{mi}+{\stackrel{\sim }{\omega }}_{mi}+{(\stackrel{\sim }{\iota \omega })}_{mi}+{\stackrel{\sim }{\epsilon }}_{mi}$
12:	end for
13:	b̃i ← ι̃1i	▻ Reference biomarker value,
14:	${\stackrel{\sim }{\pi }}_i\leftarrow \text{invlogit}({\beta }_0+{\beta }_1\frac{{\stackrel{\sim }{b}}_i}{\mathrm{\Delta }})$	▻ outcome probability,
15:	Draw ỹi ~ Bernoulli(π̃i)	▻ and simulated outcome.
16:	end for
17:	for m=1 to M do
18:	$({\widehat{\beta }}_{1m},\widehat{\text{se}}({\widehat{\beta }}_{1m}))\leftarrow \text{estimates}\text{from}\text{logistic}\text{regression}$ $$\begin{gathered} {\stackrel{\sim }{y}}_i~\text{Bernoulli}({\pi }_{mi});i=1,\dots ,N \\ \text{logit}({\pi }_{mi})={\beta }_{0m}+{\beta }_{1m}\frac{{\stackrel{\sim }{b}}_{mi}}{\mathrm{\Delta }} \end{gathered}$$
19:	$O{R}_m^{(s)}\leftarrow exp\{{\widehat{\beta }}_{1m}\}$
20:	$S{E}_m^{(s)}\leftarrow exp\{\widehat{\text{se}}({\beta }_{1m})\}$
21:	end for
22:	end for