Gibbs posterior inference on the minimum clinically important difference

- Definition of a Gibbs posterior for inference on the minimum clinically important difference (MCID).
- Improved rates of convergence for M-estimator of MCID in a recent Biometrics paper on MCID.
- Proved a convergence rate theorem for the Gibbs posterior distribution and the corresponding posterior mean.
- Near-exact coverage probability for the Gibbs posterior credible intervals.

It is known that a statistically significant treatment may not be clinically significant. A quantity that can be used to assess clinical significance is called the minimum clinically important difference (MCID), and inference on the MCID is an important and challenging problem. Modeling for the purpose of inference on the MCID is non-trivial, and concerns about bias from a misspecified parametric model or inefficiency from a nonparametric model motivate an alternative approach to balance robustness and efficiency. In particular, a recently proposed representation of the MCID as the minimizer of a suitable risk function makes it possible to construct a Gibbs posterior distribution for the MCID without specifying a model. We establish the posterior convergence rate and show, numerically, that an appropriately scaled version of this Gibbs posterior yields interval estimates for the MCID which are both valid and efficient even for relatively small sample sizes.