Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2, λ) penalty with λ → 0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case-pure penalization or prior information-our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Statisticians have made great progress in creating methods that reduce our reliance on parametric assumptions. However this explosion in research has resulted in a breadth of inferential strategies that both create opportunities for more reliable inference as well as complicate the choices that an applied researcher has to make and defend. Relatedly, researchers advocating for new methods typically compare their method to at best 2 or 3 other causal inference strategies and test using simulations that may or may not be designed to equally tease out flaws in all the competing methods. The causal inference data analysis challenge, "Is Your SATT Where It's At?", launched as part of the 2016 Atlantic Causal Inference Conference, sought to make progress with respect to both of these issues. The researchers creating the data testing grounds were distinct from the researchers submitting methods whose efficacy would be evaluated. Results from 30 competitors across the two versions of the competition (black box algorithms and do-it-yourself analyses) are presented along with post-hoc analyses that reveal information about the characteristics of causal inference strategies and settings that affect performance. The most consistent conclusion was that methods that flexibly model the response surface perform better overall than methods that fail to do so. Finally new methods are proposed that combine features of several of the top-performing submitted methods.

When fitting hierarchical regression models, maximum likelihood (ML) estimation has computational (and, for some users, philosophical) advantages compared to full Bayesian inference, but when the number of groups is small, estimates of the covariance matrix (S) of group-level varying coefficients are often degenerate. One can do better, even from a purely point estimation perspective, by using a prior distribution or penalty function. In this article, we use Bayes modal estimation to obtain positive definite covariance matrix estimates. We recommend a class of Wishart (not inverse-Wishart) priors for S with a default choice of hyperparameters, that is, the degrees of freedom are set equal to the number of varying coefficients plus 2, and the scale matrix is the identity matrix multiplied by a value that is large relative to the scale of the problem. This prior is equivalent to independent gamma priors for the eigenvalues of S with shape parameter 1.5 and rate parameter close to 0. It is also equivalent to independent gamma priors for the variances with the same hyperparameters multiplied by a function of the correlation coefficients. With this default prior, the posterior mode for S is always strictly positive definite. Furthermore, the resulting uncertainty for the fixed coefficients is less underestimated than under classical ML or restricted maximum likelihood estimation. We also suggest an extension of our method that can be used when stronger prior information is available for some of the variances or correlations.

When estimating causal effects, unmeasured confounding and model misspecification are both potential sources of bias. We propose a method to simultaneously address both issues in the form of a semi‐parametric sensitivity analysis. In particular, our approach incorporates Bayesian Additive Regression Trees into a two‐parameter sensitivity analysis strategy that assesses sensitivity of posterior distributions of treatment effects to choices of sensitivity parameters. This results in an easily interpretable framework for testing for the impact of an unmeasured confounder that also limits the number of modeling assumptions. We evaluate our approach in a large‐scale simulation setting and with high blood pressure data taken from the Third National Health and Nutrition Examination Survey. The model is implemented as open‐source software, integrated into the treatSens package for the R statistical programming language. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

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