  Power law scaling models have been used to understand the complexity of
systems as diverse as cities, neurological activity, and rainfall and
lightning. In the scaling framework, power laws and standard linear regression
methods are widely used to estimate model parameters with assumed normality and
fixed variance. Generalized linear models (GLM) can accommodate a wider range
of distributions where the chosen distribution must meet the assumptions of the
data to prevent model bias. We present a widely applicable Bayesian generalized
logistic regression (BGLR) framework to more flexibly model a continuous real
response addressing skew and heteroscedasticity. The Generalized Logistic
Distribution (GLD) was selected to flexibly model skewed continuous data. This
resulted in a nonlinear posterior distribution which may not have an analytical
solution which can be solved numerically with Markov Chain Monte Carlo (MCMC)
methods. We compared the BGLR model to standard and Bayesian normal models
having fixed and varying variance when fitting power laws to 759 days of
COVID-19 data. The BGLR yielded information beyond existing methods about the
evolution of skew and skedasticity while revealing parameter bias of widely
used methods. The BGLR flexibly modelled the complex characteristics necessary
for an improved understanding of the propagation and dynamics of this
infectious disease. The model is generally applicable and can be used as a
template for modeling complexity with other distributions.
