Chapter 9 200 Scenario 2: Mixture In scenario 2 ’Mixture’, the differences in outcomes between the current data and external data are caused by a difference in covariate distributions between patient populations. For this situation, inclusion of covariates ought to improve the operating characteristics compared with excluding covariates (like in the MPP). Figure 4 presents the simulation results, and the RMSEs are depicted on the right-hand side. The RMSE of Ignore is a flat line at approximately 0.034 as there is no borrowing regardless of the outcome of the external data. Both Wang 10% (at 0.031) and Wang 20% (at 0.029) are also relative flat lines. Pooling reaches the lowest RMSE at 0.024 in non-zero drift, followed by the MPP and the ProPP at 0.0273 and 0.0274, respectively. Both the MPP and ProPP do show an increase in type I error rate, but remain more precise than Wang’s methodology across our simulation range. By accounting for covariate effects using propensity score (i.e., rightly only incorporating similar patients), all hybrid methods yield a relatively stable and well-controlled type I error rate. This result is most clearly seen in Figure 3A, where both naive methods suffer from a large increase in type I error rate compared with the hybrid methods in Figure 4. The results of this scenario show that the incorporation of covariates through propensity score methods provides an edge over the Pooling and MPP methods. The lower RMSE of these methods compared with ignoring external data is driven by the external patients that are similar to the current patients - and exactly these similar patients receive a higher weight. By including primarily similar patients, our estimate is improved. When there are more external patients to choose from (Setting 2), the chances of selecting the most similar patients increase, and the gain in precision becomes almost completely stable across settings. The increase in precision in Wang’s method is driven purely by the prespecified amount of borrowing, whereas our method seems not to be impeded by borrowing limits, generally leading to a lower RMSE. Only in the unlikely setting of smaller external than current data, the ProPP relatively underperforms compared with Wangs methods - but still outperforms the MPP.
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