Pedagogical example: search for a "good" non-trivial unidentifiable model

Modeling
21

Not sure if this is considered trivial, but fitting a 4-parameter sigmoid:

y(x) = y0 + A / (1 + exp(-b*(x-x0)))

has lots of cases where there the data do not uniquely identify a set of parameters. The “boring” cases are when the y is not a function of x in the data. The more interesting case is when there isn’t enough data near the intercept, so there are many equally good fits for b and x0. This is particularly nice as an example in stan because a prior that favors small values of b (i.e. shallower curves) can make it so that the MAP estimate is easy to find even though the MLE estimate is not.

22

I was also thinking of a sigmoid Emax dose response model (as e.g. used for clinical dose finding trials). I’d normally specify that as f(d) = E_0 + E_\text{max} d^h/(ED50^h+d^h). You tend to have trouble, if you have a placebo (d=0) and just 2 or 3 doses, or if all doses have near maximal effect (>>ED50) or if all doses are at most a bit above the ED50 (i.e. we don’t see the plateau).