  Intuitively, a scientist might assume that a more complex regression model
will necessarily yield a better predictive model of experimental data. Herein,
we disprove this notion in the context of extracting the proton charge radius
from charge form factor data. Using a Monte Carlo study, we show that a simpler
regression model can in certain cases be the better predictive model. This is
especially true with noisy data where the complex model will fit the noise
instead of the physical signal. Thus, in order to select the appropriate
regression model to employ, a clear technique should be used such as the Akaike
information criterion or Bayesian information criterion, and ideally selected
previous to seeing the results. Also, to ensure a reasonable fit, the scientist
should also make regression quality plots, such as residual plots, and not just
rely on a single criterion such as reduced chi2. When we apply these techniques
to low four-momentum transfer cross section data, we find a proton radius that
is consistent with the muonic Lamb shift results. While presented for the case
of proton radius extraction, these concepts are applicable in general and can
be used to illustrate the necessity of balancing bias and variance when
building a regression model and validating results, ideas that are at the heart
of modern machine learning algorithms.
