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The CRASH computer model simulates the effect of a vehicle colliding against different barrier types. If it accurately represents real vehicle crashworthiness, the computer model can be of great value in various aspects of vehicle design, such as the setting of timing of air bag releases. The goal of this study is to address the problem of validating the computer model for such design goals, based on utilizing computer model runs and experimental data from real crashes. This task is complicated by the fact that (i) the output of this model consists of smooth functional data, and (ii) certain types of collision have very limited data. We address problem (i) by extending existing Gaussian process-based methodology developed for models that produce real-valued output, and resort to Bayesian hierarchical modeling to attack problem (ii). Additionally, we show how to formally test if the computer model reproduces reality. Supplemental materials for the article are available online. [ABSTRACT FROM AUTHOR]

Hierarchical models are increasingly used in many applications. Along with this increased use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate on objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated and critically compared. [ABSTRACT FROM AUTHOR]

P values are the most commonly used tool to measure evidence against a hypothesis or hypothesized model. Unfortunately, they are often incorrectly viewed as an error probability for rejection of the hypothesis or, even worse, as the posterior probability that the hypothesis is true. The fact that these interpretations can be completely misleading when testing precise hypotheses is first reviewed, through consideration of two revealing simulations. Then two calibrations of a p value are developed, the first being interpretable as odds and the second as either a (conditional) frequentist error probability or as the posterior probability of the hypothesis. [ABSTRACT FROM AUTHOR]

The problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters. The most frequently used measures of compatibility are p values, based on statistics T for which large values are deemed to indicate incompatibility of the data and the model. When the null model has unknown parameters, p values are not uniquely defined. The proposals for computing a p value in such a situation include the plug-in and similar p values on the frequentist side, and the predictive and posterior predictive p values on the Bayesian side. We propose two alternatives, the conditional predictive p value and the partial posterior predictive p value, and indicate their advantages from both Bayesian and frequentist perspectives. [ABSTRACT FROM AUTHOR]

This article presents a response to several commentaries regarding a study which proposed computations for Bayesian p values in composite null models. In the double use of data and conditioning, Michael Evans and Hal S. Stern both discussed what constitutes double use of the data in computing a p value. They take opposite perspectives on this issue. Evans argues that the only way to avoid double use of the data is to separate the data into two independent subsets, using one subset to fit the model and the other to assess the validity of the model. Because the procedures that we recommend do not formally split the data into two parts, Evans suggests that they can be guilty double use of the data. In contrast, Stern does not feel that it is necessary to worry about double use of the data the extent that we worry, he does not feel that even the posterior predictive p value involves double use of the data. These contrasting positions perfectly highlight what we view to be the main motivation for our paper. We feel that it is very important to avoid a double use of the data, but that separating the data into two independent subsets is a too drastic solution that can lose considerable power. We consider each of these issues in turn.