Table A.2.

Population log odds ratio (PLOR) and IV estimand compared to two-stage estimate of log odds ratio for unit increase in risk factor from model of confounded association with IV–covariate interaction (α₃), risk factor–covariate interaction (β₃), and skewed distribution of risk factor (ψ) (Scenario 4). Median estimates across 250 000 simulations

			α3 = 0			α3 = −0.1			α3 = +0.1
		β3 =	0	−0.2	+0.2	0	−0.2	+0.2	0	−0.2	+0.2
No skew: ψ = 1	β2 = −0.6	PLOR	0.393	0.421	0.364	0.412	0.497	0.300	0.377	0.350	0.432
		IV estimand	0.392	0.420	0.364	0.412	0.496	0.298	0.376	0.349	0.431
		Two-stage method	0.389	0.421	0.371	0.389	0.422	0.370	0.390	0.419	0.372
	β2 = −0.2	PLOR	0.392	0.380	0.399	0.378	0.424	0.311	0.412	0.338	0.491
		IV estimand	0.392	0.378	0.400	0.377	0.424	0.310	0.412	0.337	0.491
		Two-stage method	0.389	0.385	0.395	0.390	0.387	0.396	0.389	0.383	0.393
	β2 = 0	PLOR	0.386	0.355	0.406	0.356	0.386	0.310	0.423	0.327	0.502
		IV estimand	0.387	0.354	0.406	0.354	0.385	0.310	0.422	0.326	0.502
		Two-stage method	0.383	0.363	0.396	0.384	0.365	0.398	0.382	0.361	0.393
	β2 = 0.2	PLOR	0.376	0.329	0.404	0.330	0.349	0.305	0.424	0.313	0.501
		IV estimand	0.377	0.329	0.404	0.332	0.348	0.305	0.423	0.313	0.502
		Two-stage method	0.372	0.339	0.389	0.373	0.341	0.392	0.371	0.337	0.386
	β2 = 0.6	PLOR	0.345	0.280	0.380	0.287	0.281	0.287	0.403	0.279	0.473
		IV estimand	0.346	0.280	0.382	0.288	0.281>	0.286	0.403	0.280	0.472
		Two-stage method	0.340	0.291	0.362	0.342	0.293	0.366	0.339	0.290	0.359
Positive skew: ψ = 1.5	β2 = −0.6	PLOR	0.390	0.417	0.362	0.411	0.492	0.299	0.377	0.347	0.429
		IV estimand	0.393	0.421	0.364	0.412	0.496	0.298	0.377	0.349	0.431
		Two-stage method	0.388	0.418	0.369	0.388	0.419	0.368	0.388	0.416	0.369
	β2 = −0.2	PLOR	0.391	0.377	0.397	0.374	0.422	0.309	0.411	0.336	0.488
		IV estimand	0.392	0.378	0.401	0.376	0.423	0.311	0.412	0.337	0.492
		Two-stage method	0.388	0.383	0.393	0.388	0.385	0.394	0.388	0.381	0.392
	β2 = 0	PLOR	0.385	0.352	0.405	0.355	0.384	0.310	0.421	0.326	0.501
		IV estimand	0.387	0.354	0.407	0.354	0.385	0.311	0.421	0.327	0.502
		Two-stage method	0.382	0.361	0.395	0.383	0.364	0.397	0.381	0.359	0.392
	β2 = 0.2	PLOR	0.376	0.329	0.404	0.332	0.348	0.306	0.422	0.312	0.500
		IV estimand	0.377	0.329	0.404	0.331	0.348	0.306	0.423	0.313	0.502
		Two-stage method	0.371	0.338	0.389	0.372	0.340	0.392	0.370	0.336	0.386
	β2 = 0.6	PLOR	0.345	0.279	0.380	0.289	0.282	0.287	0.403	0.281	0.472
		IV estimand	0.345	0.279	0.381	0.287	0.281	0.288	0.402	0.280	0.473
		Two-stage method	0.340	0.291	0.362	0.342	0.293	0.366	0.338	0.289	0.359
Negative skew: ψ = 1.5−1	β2 = −0.6	PLOR	0.394	0.422	0.366	0.413	0.499	0.298	0.377	0.350	0.434
		IV estimand	0.392	0.420	0.364	0.412	0.496	0.299	0.376	0.349	0.432
		Two-stage method	0.390	0.423	0.373	0.390	0.425	0.372	0.391	0.422	0.374
	β2 = −0.2	PLOR	0.394	0.380	0.403	0.377	0.425	0.312	0.415	0.338	0.493
		IV estimand	0.392	0.378	0.401	0.377	0.423	0.310	0.412	0.337	0.492
		Two-stage method	0.390	0.387	0.397	0.391	0.389	0.398	0.390	0.385	0.395
	β2 = 0	PLOR	0.388	0.354	0.409	0.355	0.385	0.311	0.425	0.327	0.506
		IV estimand	0.386	0.354	0.405	0.355	0.385	0.309	0.422	0.326	0.504
		Two-stage method	0.384	0.364	0.397	0.385	0.366	0.399	0.383	0.362	0.394
	β2 = 0.2	PLOR	0.378	0.328	0.406	0.332	0.350	0.304	0.426	0.315	0.503
		IV estimand	0.377	0.328	0.404	0.331	0.349	0.307	0.422	0.312	0.501
		Two-stage method	0.373	0.340	0.390	0.374	0.342	0.393	0.371	0.338	0.387
	β2 = 0.6	PLOR	0.345	0.280	0.381	0.289	0.282	0.288	0.403	0.280	0.473
		IV estimand	0.346	0.280	0.381	0.289	0.281	0.288	0.403	0.280	0.472
		Two-stage method	0.341	0.292	0.362	0.343	0.293	0.366	0.339	0.290	0.359