Whenever parameter estimates are uncertain or observations are contaminated by measurement error, the Pearson correlation coefficient can severely underestimate the true strength of an association.... Show moreWhenever parameter estimates are uncertain or observations are contaminated by measurement error, the Pearson correlation coefficient can severely underestimate the true strength of an association. Various approaches exist for inferring the correlation in the presence of estimation uncertainty and measurement error, but none are routinely applied in psychological research. Here we focus on a Bayesian hierarchical model proposed by Behseta, Berdyyeva, Olson, and Kass (2009) that allows researchers to infer the underlying correlation between error-contaminated observations. We show that this approach may be also applied to obtain the underlying correlation between uncertain parameter estimates as well as the correlation between uncertain parameter estimates and noisy observations. We illustrate the Bayesian modeling of correlations with two empirical data sets; in each data set, we first infer the posterior distribution of the underlying correlation and then compute Bayes factors to quantify the evidence that the data provide for the presence of an association. Show less
A common problem in Bayesian statistics is to determine whether a quantity obtained from a Bayesian posterior distribution is also meaningful in a frequentist context. In this thesis, we try to... Show moreA common problem in Bayesian statistics is to determine whether a quantity obtained from a Bayesian posterior distribution is also meaningful in a frequentist context. In this thesis, we try to answer this question for credible sets in the so-called fixed design model. Taking a specific prior distribution, we study whether credible sets based on this prior can also be used as confidence intervals. In particular, our aim is to construct a credible set for a parameter function. Under certain assumptions on the smoothness of the function, it turns out that we can obtain meaningful results about both the frequentist coverage and the width of the credible set. We consider several different classes of functions in this thesis, each with a different set of assumptions. In the first chapter, we assume that we know rather a lot about the structure of the function, and use this to obtain a useful method. In chapter 2, we extend this to a method that adapts to the structure of the function. Finally, in the last chapter, we extend this to so-called credible bands, that describe the behaviour of the entire function, rather than at a specific point. Show less
