Search for: All records

Summary Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g., the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function. Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies. 

Two popular approaches for relating correlated measurements of a non‐Gaussian response variable to a set of predictors are to fit amarginal modelusing generalized estimating equations and to fit ageneralized linear mixed model(GLMM) by introducing latent random variables. The first approach is effective for parameter estimation, but leaves one without a formal model for the data with which to assess quality of fit or make individual‐level predictions for future observations. The second approach overcomes these deficiencies, but leads to parameter estimates that must be interpreted conditional on the latent variables. To obtain marginal summaries, one needs to evaluate an analytically intractable integral or use attenuation factors as an approximation. Further, we note an unpalatable implication of the standard GLMM. To resolve these issues, we turn to a class of marginally interpretable GLMMs that lead to parameter estimates with a marginal interpretation while maintaining the desirable statistical properties of a conditionally specified model and avoiding problematic implications. We establish the form of these models under the most commonly used link functions and address computational issues. For logistic mixed effects models, we introduce an accurate and efficient method for evaluating the logistic‐normal integral.