Introduction
The purpose of the BRACE function is to estimate and
correct for certain forms of treatment selection bias arising in
observational competing risks data. The function will fit Cox models for
primary event(s), competing event(s), and the combined event, adjusting
for confounders, following the BRACE method described in Williamson,
Casey W., et al. (2022) to adjust for bias from unmeasured confounders.
Results are presented in the output of $Summary table. The
Cox models for primary, competing and combined endpoint, adjusting for
measured confounders, are given as “Adjusted Primary Event”, “Adjusted
Competing Event”, and “Adjusted Combined Event”. The proposed adjusted
estimate using the BRACE method is given as “BRACE Estimate”, with a
bootstrapped confidence interval.
The function allows for two alternative methods for adjusting for the
effects of measured confounders. One way is using multivariable Cox
proportional hazards regression models. The other way is using
propensity score (PS) models. For the PS method, we first estimate the
propensity scores by modeling the treatment assignment with all the
measured confounders through logistic regression. The propensity scores
are then applied as weights to the regression models to make the
patients in each group have similar distributions. Note that applying
the PS method may not be sufficient to adjust for residual unmeasured
confounding, which can be detected following competing event analysis.
Thus, the BRACE estimate can be called even after PS adjustment. To
apply BRACE method, the treatment should have an apparent benefit in
reducing the hazard for competing events. Thus, if the estimate of the
treatment effect on hazard for the competing event(s) is non-negative,
we do not suggest using this method.
The inference of the BRACE estimator is based on non-parametric
bootstrap. B denotes the bootstrap sample size, with a
default set to 1000. Users can reduce this number if it is
computationally inefficient, but we recommend a large enough
B to ensure the bootstrap consistency. The confidence
interval is obtained from \((2.5\%,
97.5\%)\) quantile of the bootstrap samples.
$BRACE HR Distribution outputs B bootstrap
estimates.
Estimates of the cumulative baseline omega (relative hazard)
function, omega_0, can be obtained by $Omega Estimate. This
is the relative cumulative hazards for primary event(s) vs. the combined
event at baseline. The approach assumes that omega_0 is approximately
invariant to time (including, if applicable, any time-dependent
covariates). For the convenience of assumption validity checking and
sensitivity analysis, we provide a time-dependent omega_0 curve given by
$omega curve.
The bias (epsilon) is given as (see Williamson, Casey W., et al
(2022) for details):
\(e = (1 - \omega_0) (\Theta_2 -
E(\widehat\Theta_2)) \geq (1 - \omega_0)(1 -
E(\widehat\Theta_2))\).
After BRACE adjustment, the estimate of epsilon is \((1 - \omega_0)(1 - E(\widehat\Theta_2))\),
and can be obtained from $epsilon.
Run a toy example of brace function
We generate a toy data by gendat2 function and run the
brace function to get our BRACE estimator.
The toy data is generated by
- the BRACE estimator based on adjusting for the effects of measured
confounders through multivariable Cox proportional hazards regression
models
braceoutput = brace(ftime, fstatus, covs, trt, PS=0, B=200)
braceoutput$Summary
#> HR Estimate Lower 95% CI Upper 95% CI
#> BRACE Estimate 0.7554990 0.6766368 0.8379061
#> Adjusted Combined Event 0.5804403 0.5022396 0.6708172
#> Adjusted Primary Event 0.5430074 0.4454809 0.6618848
#> Adjusted Competing Event 0.6254957 0.5058648 0.7734179
braceoutput$`Omega Curve`
- the BRACE estimator based on adjusting for the effects of measured
confounders through using propensity score (PS) models
braceoutput = brace(ftime, fstatus, covs, trt, PS=1, B=200)
braceoutput$Summary
#> HR Estimate Lower 95% CI Upper 95% CI
#> BRACE Estimate 0.7422218 0.6631261 0.8236102
#> Adjusted Combined Event 0.5798191 0.5053580 0.6652516
#> Adjusted Primary Event 0.5479526 0.4497555 0.6675894
#> Adjusted Competing Event 0.6162369 0.5015419 0.7571610
braceoutput$`Omega Curve`