that involve stratified randomization, which is commonly used. It is

important to understand the large sample properties of estimators that

adjust for stratum variables (those used in the randomization

procedure) and additional baseline variables, since this can lead to

substantial gains in precision and power. Surprisingly, to the best

of our knowledge, this is an open problem. It was only recently that a

simpler problem was solved by Bugni et al. (2018) for the case with no

additional baseline variables, continuous outcomes, the analysis of

covariance (ANCOVA) estimator, and no missing data. We generalize

their results in three directions. First, in addition to continuous

outcomes, we handle binary and time-to-event outcomes; this broadens

the applicability of the results. Second, we allow adjustment for an

additional, preplanned set of baseline variables, which can improve

precision. Third, we handle missing outcomes under the missing at

random assumption. We prove that a wide class of estimators is

asymptotically normally distributed under stratified randomization and

has equal or smaller asymptotic variance than under simple

randomization. For each estimator in this class, we give a consistent

variance estimator. This is important in order to fully capitalize on

the combined precision gains from stratified randomization and

adjustment for additional baseline variables. The above results also

hold for the biased-coin covariate-adaptive design. We demonstrate our

results using completed trial data sets of treatments for substance

use disorder, where adjustment for additional baseline variables

brings substantial variance reduction.

]]>DISCUSSION: Our findings suggest that national prediction models can be built on only a small number (30 or fewer) of important variables and provide robust concentration estimates. Model estimates are freely available online.

]]>We compare TMLE versus MMRM by analyzing data from a completed Alzheimer's disease trial data set and by simulation studies. The simulations involved different missing data distributions, where loss to followup at a given visit could depend on baseline variables, treatment assignment, and the outcome measured at previous visits. The TMLE generally has improved robustness in our simulated settings, i.e., less bias and mean squared error, and better confidence interval coverage probability. The robustness is due to the TMLE correctly modeling the dropout distribution. We illustrate the tradeoffs between these estimators and give recommendations for how to use these estimators in practice.

]]>**Methods:** A common method used to capture the correlated endpoints across baskets is Bayesian hierarchical modeling. We evaluate a Bayesian adaptive design in the context of a basket trial and investigate two popular prior specifications: an inverse-gamma prior on the basket-level variance and a uniform prior on the basket-level standard deviation.

**Results:** From our simulation study, we see the inverse-gamma prior is highly sensitive to the input hyperparameters. When the prior mean value of the variance parameter is set to be near zero (<0.5), this can lead to unacceptably high false positive rates (>40%) in some scenarios. Thus, use of this prior requires a fully comprehensive sensitivity analysis before implementation. Alternatively, we see that a prior that moves the mass of the variance parameter away from zero, such as the uniform prior, displays desirable and robust operating characteristics over a wide range of prior specifications, with the caveat that the upper bound of the uniform prior must be larger than 1.

**Conclusion:** Based on our results, we recommend that those involved in designing basket trials that implement hierarchical modeling avoid using a prior distribution that places a large density mass near zero for the variance parameter. Priors with this property force the model to share information regardless of the true efficacy configuration of the baskets. Many commonly used inverse-gamma prior specifications have this undesirable property. We recommend to instead consider the more robust uniform prior on the standard deviation.