1 Decreased susceptibility of marginal odds ratios to finite-sample bias
Supplemental Digital Content Table of contents
eMethods. Data generation
eFigure 1. Relative error of the log of the odds ratio, comparison of covariate-conditional and marginal estimates at of 2 and 15 events per parameter.
eTable 1. Sensitivity analysis: Results when covariate-conditional odds ratio set to 1.5.
eTable 2. Sensitivity analysis: Results when marginal exposure and outcome prevalence set to 0.15 and number of covariates is 14
2 eMethods. Data generation
Let ππ be a binary random variable representing treatment (where π = 1 is treated and π = 0 is untreated) for individual π, ππ represent a vector of standard normal covariates , ππ0 be a binary random variable representing the potential outcome when not treated (π = 0), ππ1 be a binary random variable representing the potential outcome when treated, and ππ represent the observed outcome given the observed value of ππ. We generated 5,000 trials of sample size π = 200 for EPV 2, and π = 1500 for EPV 15. Individuals (π = 1 to π) were independent.
Subscript π is suppressed for remaining eMethods.
The following parameters were set (input) to generate the data. Let ππ₯ and ππ¦ be the referent prevalence of π and π, respectively (i.e., prevalence when all other variables in the probability model are 0); and ππ ππ, ππ ππ, and ππ ππ be the conditional odds ratios of each π and π, each π and π, and π and π, respectively.
Let ππ₯πππ‘(π) = 1/(1 + πβb), where π is the ln(ππππ ).
Data were generated in the following order:
1. πβ²: 29 independent draws from a standard normal distribution, π(0,1).
2. π: drawn from a Bernoulli distribution with π = ππ₯πππ‘ (ln ( ππ₯
1βππ₯) + ln(ππ ππ) Γ πβ²).
3. π0: drawn from a Bernoulli distribution with π = ππ₯πππ‘ (ln ( ππ¦
1βππ¦) + ln(ππ ππ) Γ πβ²) 4. π1: drawn from a Bernoulli distribution with π = ππ₯πππ‘ (ln ( ππ¦
1βππ¦) + ln(ππ ππ) Γ πβ²+ lnβ‘(ππ ππ)) 5. π: set according to realized value of π (i.e., if π = 0, then π = π0, else π = π1).
In scenarios where covariates were independent of exposure, ππ ππ = 1.0. In scenarios where covariates were confounders, ππ ππ= 1.2. Throughout, ππ ππ= 1.2, andβ‘ππ ππ = 3.0.β‘β‘ππ₯ was selected to obtain a marginal treatment prevalence of 0.3 (i.e, πΈ[π] = 0.3). ππ¦ was selected to obtain a marginal outcome prevalence of 0.3 (i.e, πΈ[π] = 0.3).
In sensitivity analyses, (1) ππ ππ= 1.5 and (2) πΈ[π] = πΈ[π] = 0.15 and length of the covariate vector π was 14.
3 SAS Code for data generation
%macro sim(seed=,scenario=,nsim=,nobs=,ncov=,p_x=,covx=,p_y=,or=,covy=);
*DATA GENERATION;
data a;
call streaminit(&seed.);
scenario = &scenario;
n=&nobs;
do j=1 to ≁ *trial indicator;
do i=1 to &nobs; *obs within trial;
*Standard normal covariates;
array cov cov1-cov&ncov.;
do over cov;
cov=rand("normal");
end;
*Exposure;
x=rand("bernoulli",1/(1+exp(-1*(log(&p_x/(1-&p_x)) %do t=1 %to &ncov.; + log(&covx)*cov&t. %end; ))));
*Potential outcomes;
y0=rand("bernoulli",1/(1+exp(-1*(log(&p_y/(1-&p_y)) + log(&or)*0 %do t=1 %to &ncov.; + log(&covy)*cov&t.
%end; ))));
y1=rand("bernoulli",1/(1+exp(-1*(log(&p_y/(1-&p_y)) + log(&or)*1 %do t=1 %to &ncov.; + log(&covy)*cov&t.
%end; ))));
*Observed outcome;
if x=0 then y=y0; else y=y1;
output;
end;
end;
run;
%mend;
4 eFigure 1. Relative error of the log of the odds ratio, comparison of covariate-conditional and marginal estimates at of 2 and 15 events per parameter.
Abbreviations: OR, odds ratio; IPTW, inverse probability of treatment weighting; AIPW, augmented inverse probability weighting
Panel A are results from the scenario where the covariates were predictors of the outcome (OR=1.2) but were independent of the exposure.
Panel B are results from the scenario where the covariates were predictors of both the outcome (OR=1.2) and the exposure (OR 1.2) making them confounders. The events per parameter are 2 and 15, on average, for the outcome regression model used for estimation of the covariate- conditional effect and the g-computation and AIPW marginal effects. The events/parameter of the exposure regression models used for estimation of the IPTW and AIPW marginal effects were, on average, 2.1 and 15.5, respectively. Relative error was calculated using the correct covariate-conditional OR (3.0) for covariate-conditional estimates and the correct marginal OR (2.53) for marginal estimates. Filled circle marks mean relative error (a.k.a. relative bias). Whiskers represent Β±1.5 times the intra-quartile range. N=5000.
5 eTable 1. Sensitivity analysis: Results when covariate-conditional odds ratio set to 1.5.
Covariates independent of exposure Covariates were confounders Events per
parameter Parameter Estimator Relative bias Empirical SE root MSE Relative bias Empirical SE root MSE
15
Conditional MLE 0.026 0.135 0.135 0.026 0.139 0.139
Conditional Firth 0.004 0.132 0.132 0.002 0.135 0.135
Marginal G-computation 0.001 0.111 0.111 0.002 0.117 0.117
Marginal IPTW -0.001 0.112 0.112 0.002 0.129 0.129
Marginal AIPW 0.000 0.112 0.112 -0.002 0.127 0.127
2
Conditional MLE 0.254 0.517 0.527 0.256 0.536 0.546
Conditional Firth MLE 0.021 0.412 0.412 0.025 0.425 0.425
Marginal G-computation 0.009 0.350 0.350 0.029 0.369 0.369
Marginal IPTW -0.017 0.400 0.400 0.069 0.465 0.465
Marginal AIPW -0.003 0.395 0.395 -0.004 0.452 0.452
Abbreviations: SE, standard error; MSE, mean squared error; MLE, maximum likelihood estimation; IPTW, inverse probability of treatment weighting; AIPW, augmented inverse probability weighting
True marginal odds ratio = 1.41. The events per parameter are 2 and 15, on average, for the outcome regression model used for estimation of the covariate-conditional effect and the g-computation and AIPW marginal effects. The events/parameter of the exposure regression models used for estimation of the IPTW and AIPW marginal effects were, on average, 2.1 and 16.1, respectively. Relative error was calculated using the correct covariate-conditional OR for covariate-conditional estimates and the correct marginal OR for marginal estimates. N=5000.
6 eTable 2. Sensitivity analysis: Results when marginal exposure and outcome prevalence set to 0.15 and number of covariates is 14
Covariates independent of exposure Covariates were confounders Events per
parameter Parameter Estimator Relative bias Empirical SE root MSE Relative bias Empirical SE root MSE
15
Conditional MLE 0.012 0.182 0.182 0.015 0.179 0.179
Conditional Firth -0.001 0.178 0.178 0.002 0.175 0.175
Marginal G-computation -0.001 0.165 0.165 0.002 0.167 0.167
Marginal IPTW -0.002 0.169 0.169 -0.001 0.183 0.183
Marginal AIPW -0.001 0.168 0.168 -0.003 0.182 0.182
2
Conditional MLE 0.120 0.642 0.655 0.136 0.625 0.643
Conditional Firth -0.002 0.541 0.541 0.003 0.528 0.528
Marginal G-computation -0.010 0.512 0.513 0.009 0.510 0.510
Marginal IPTW -0.048 0.599 0.601 -0.020 0.628 0.629
Marginal AIPW -0.032 0.593 0.594 -0.029 0.613 0.614
Abbreviations: SE, standard error; MSE, mean squared error; MLE, maximum likelihood estimation; IPTW, inverse probability of treatment weighting; AIPW, augmented inverse probability weighting
True covariate-conditional odds ratio = 3.0. True marginal odds ratio = 2.79. The events per parameter are 2 and 15, on average, for the outcome regression model used for estimation of the covariate-conditional effect and the g-computation and AIPW marginal effects. The
events/parameter of the exposure regression models used for estimation of the IPTW and AIPW marginal effects were, on average, 2.0 and 16.7, respectively. Relative error was calculated using the correct covariate-conditional OR for covariate-conditional estimates and the correct
marginal OR for marginal estimates.
N=5000 except for following:
Covariates independent of exposure at 2 events/parameter without Firth, N=4998 because of non-converged outcome models
Covariates were confounders at 2 events/parameter without Firth, N=4999 because of non-converged outcome model. N=4998 for AIPW estimate because additional estimate removed with risk <0.
