  In various biomedical studies, the focus of analysis centers on the
magnitudes of data, particularly when algebraic signs are irrelevant or lost.
To analyze the magnitude outcomes in repeated measures studies, using models
with random effects is essential. This is because random effects can account
for individual heterogeneity, enhancing parameter estimation precision.
However, there are currently no established regression methods that incorporate
random effects and are specifically designed for magnitude outcomes. This
article bridges this gap by introducing Bayesian regression modeling approaches
for analyzing magnitude data, with a key focus on the incorporation of random
effects. Additionally, the proposed method is extended to address multiple
causes of informative dropout, commonly encountered in repeated measures
studies. To tackle the missing data challenge arising from dropout, a joint
modeling strategy is developed, building upon the previously introduced
regression techniques. Two numerical simulation studies are conducted to assess
the validity of our method. The chosen simulation scenarios aim to resemble the
conditions of our motivating study. The results demonstrate that the proposed
method for magnitude data exhibits good performance in terms of both estimation
accuracy and precision, and the joint models effectively mitigate bias due to
missing data. Finally, we apply proposed models to analyze the magnitude data
from the motivating study, investigating if sex impacts the magnitude change in
diaphragm thickness over time for ICU patients.
