The simulations examined the effect of empirical calibration in terms of the bias-variance trade-off for each type of bias. The simulations were carried out in the context of binary treatment and binary outcome with biases resulting from unmeasured confounder, model misspecification, measurement error, and lack of positivity. In this paper, we systematically examine the effect of empirical calibration of the confidence interval by simulating different types of residual confounding. While a prior study examined the limitations of empirical calibration of p-values, no study to date has assessed the effectiveness of empirical calibration of confidence intervals under different bias scenarios. Empirical calibration of confidence intervals has been applied in several observational studies, where the calibration increased the coverage of the confidence intervals-to the “nominal” 95% coverage for 95% confidence intervals. This idea was extended to the empirical calibration of the confidence intervals of treatment effects by using negative controls and positive controls-synthetically generated outcomes with known treatment effects. The empirical calibration of p-values uses a Gaussian model of the negative controls to shift and scale the test statistics used to calculate the p-values. Initially, empirical calibration was proposed to calibrate the p-values of treatment effects through an empirical null distribution derived from the negative controls. Estimates of treatment effect from negative control outcomes can be used to adjust for the biases in the estimate of treatment effect on the outcome of interest, with the assumption that the negative controls and outcome of interest share the same casual structure. One technique to account for residual confounding is through the use of negative control outcomes, which are outcomes not believed to be affected by the treatment of interest. Residual confounding occurs when confounding variables are not measured, are measured incorrectly, or when the relationships between the confounders and the outcome are incorrectly modelled. Not all confounding can be accounted for. Thus, treatment effect estimation in observational studies should include adjustments for confounders, for example, using inverse probability score weighting. This lack of randomisation can introduce confounding, which can lead to biases in the estimate of the treatment effect. The trade-off is the loss of randomisation of treatment assignment, which is not guaranteed in observational studies. Observational studies are often used when a randomised controlled trial design is unethical, costly, or time-consuming. Further research is needed on the selection of suitable negative controls. ![]() ![]() Calibration of confidence intervals is most effective where there are biases due to unmeasured confounding. This work adds evidence to the efficacy of empirical calibration of the confidence intervals in observational studies. Suitable negative controls had a large impact on the adjustment made by empirical calibration, but small improvements in the coverage of the outcome of interest were also observable when using unsuitable negative controls. ![]() Empirical calibration of confidence intervals was most effective when adjusting for the unmeasured confounding bias. ResultsĮmpirical calibration increased coverage of the 95% confidence interval of the treatment effect estimate under most bias scenarios but was inconsistent in adjusting the bias in the treatment effect estimate. The performance of the empirical calibration was evaluated by determining the change in the coverage of the confidence interval and the bias in the treatment effect estimate. The simulations consisted of binary treatment and binary outcome, with biases resulting from unmeasured confounder, model misspecification, measurement error, and lack of positivity. The effect of empirical calibration of confidence intervals was analyzed using simulated datasets with known treatment effects. Although empirical calibration has been used in several large observational studies, there is no systematic examination of its effect under different bias scenarios. An extension of this technique calibrates the coverage of the 95% confidence interval of a treatment effect estimate by using negative control outcomes as well as positive control outcomes, which are outcomes for which the treatment of interest has known effects. The empirical calibration procedure is a technique that uses negative control outcomes to calibrate p-values. One method for adjusting for the residual biases in the estimation of treatment effects is through the use of negative control outcomes, which are outcomes not believed to be affected by the treatment of interest. Estimations of causal effects from observational data are subject to various sources of bias.
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