  Calibration data are often obtained by observing several well-understood
objects simultaneously with multiple instruments, such as satellites for
measuring astronomical sources. Analyzing such data and obtaining proper
concordance among the instruments is challenging when the physical source
models are not well understood, when there are uncertainties in "known"
physical quantities, or when data quality varies in ways that cannot be fully
quantified. Furthermore, the number of model parameters increases with both the
number of instruments and the number of sources. Thus, concordance of the
instruments requires careful modeling of the mean signals, the intrinsic source
differences, and measurement errors. In this paper, we propose a log-Normal
hierarchical model and a more general log-t model that respect the
multiplicative nature of the mean signals via a half-variance adjustment, yet
permit imperfections in the mean modeling to be absorbed by residual variances.
We present analytical solutions in the form of power shrinkage in special cases
and develop reliable MCMC algorithms for general cases. We apply our method to
several data sets obtained with a variety of X-ray telescopes such as Chandra.
We demonstrate that our method provides helpful and practical guidance for
astrophysicists when adjusting for disagreements among instruments.
