Table 2.

Common classes of borrowing methods.

Statistical method	Description	Tuning parameter	Pros/cons
Static
Power prior with fixed power parameter (Chen et al., 2000; Ibrahim et al., 2000)	The contribution of each external patient to the likelihood is weighted by a common “power parameter” between 0 and 1. Typically implemented as a Bayesian model.	Power parameter: Setting it to 1 is equivalent to pooling, and setting it to 0 is equivalent to ignoring external data	Pro: Simple and interpretable downweighting factor Con: Does not cap type I error inflation or decreases in power
Dynamic
Test-then-pool (Viele et al., 2014)	A hypothesis test is done to compare the outcomes of external and trial controls after steps 1–3. ●For point null hypotheses, the data are pooleda if the null hypothesis of no difference is not rejected, and is ignored otherwise. ●For non-equivalency null hypotheses, the external data are pooled if the null is rejected, and is ignored otherwise	For point null hypotheses: ●The significance level of the test (smaller alpha makes it more difficult to reject the null, and thus more likely to pool) For non-equivalency null hypotheses: ●The significance level of the test (smaller alpha results in wider confidence intervals, making it harder to reject the null and thus less likely to pool) ●The equivalency bounds (larger bounds are more likely to contain the confidence interval, thus making it more likely to reject the null and pool)	Pro: ●Simple ●Does not require outcome data for experimental group to determine downweighting factor Con: All or nothing approach, resulting in greater variability and uncertainty about how much information will be borrowed
Adaptive/modified power prior model (Duan et al., 2006; Neuenschwander et al., 2009)	Similar to the (static) power prior, but the power parameter is given a prior distribution and allowed to be selected based on the data. The power parameter is estimated simultaneously with all other parameters in the model, including the treatment effect.	Hyperpriors on the power parameter	Pro: Retains some of the interpretability of the fixed power prior method Con: ●Can be difficult to implement in standard software and can be computationally intensive ●Requires outcome data on experimental group to estimate the downweighting factor
Frequentist version of modified power prior (See two-step approach in Web Appendix A)	Step 1: A regression model is fit to the external and trial controls to estimate the HR between these two arms. The estimated HR is mapped to a downweighting factor, such that HRs near 1 give a downweighting factor close to 1 and HRs far from 1 give a downweighting factor close to 0. Step 2: A second regression model is fit to the pooled external and trial data, giving all external patients the common downweighting factor determined in step 1 and giving all trial patients a weight of 1.	The rate at which the common weights decay to 0 as the HR moves away from 1. For example, the downweighting factor could be defined by the function $w=exp(c\ast |log(HR)|)$ for a tuning parameter $c>0$. Larger values of $c$ result in a faster decay to 0 as the HR moves away from 1.	Pro: ●Simple and interpretable downweighting factor that is chosen dynamically ●Does not require outcome data from experimental group to determine downweighting factor, as the downweighting factor is determined in step 1 and outcome data for the experimental group is not required until step 2 Con: Still pending a full evaluation of performance in different settings
Commensurate prior model (Hobbs et al., 2011, 2012)	The outcomes in the randomized controls are centered around the outcomes in the external controls. For example, the log hazard rate of the trial controls might be given a normal prior, centered around the log hazard in the external controls and with hyperprior on the precision of the normal prior.	The hyperpriors on the precision of the normal distribution that shrinks the hazard rate in the randomized controls toward the hazard rate in the external controls. The more this precision is pushed toward zero, the less the hazard in the trial controls is shrunk toward the hazard in the external controls and the more the external controls are effectively downweighted.	Pro: Dynamic Bayesian borrowing method that is straightforward to implement in standard software Con: ●Downweighting is implicit, so can be more difficult to interpret the amount of borrowed information. ●Requires outcome data on experimental group to estimate the downweighting factor

^aIn this context, pooling refers to combining RWD and trial control data into a single dataset that is then analyzed as though the data were collected together.