A better (Bayesian) interval estimate for within-subject designs

We develop a Bayesian highest-density interval (HDI) for use in within-subject designs. This credible interval is based on a standard noninformative Jeffreys prior and a modified posterior distribution that conditions on both the data and point estimates of the subject-specific random effects. Conditioning on the estimated random effects removes between-subject variance and produces intervals that are the Bayesian analogue of the within-subject confidence interval proposed in Loftus and Masson (1994). We show that the latter interval can also be derived as a Bayesian within-subject HDI under a certain improper prior. We argue that the proposed new interval is superior to the original within-subject confidence interval, on the grounds of (a) it being based on a more sensible prior, (b) it having a clear and intuitively appealing interpretation, and (c) because its length is always smaller. A generalization of the new interval that can be applied to heteroscedastic data is also derived, and we show that the resulting interval is numerically equivalent to the standardization method discussed in Franz and Loftus (2012); however, our work provides a Bayesian formulation for the standardization method, and in doing so we identify the associated prior distribution.