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GLM fit or Cubic smoothing spline for categorical boundary data??
The simpler model y(t)=1/(1+exp(-r(t-t0))) is a special case of the more complex model
k1/(1+exp(-r(t-t0)))+k2 (with k1 = 1 and k2 = 0). With maximum likelihood estimation, the more
complex model will fit at least as well as the simpler model. Also, MLE will enable you to test (e.g.,
with a likelihood ratio) whether or not the additional parameters are justified. You should also just
look at the data and model predictions/fit to get a sense of which of the k parameters, if either, are
likely to help improve the fit at all (e.g., if the maximum categorization probabilities in your data are
less than one, the k1 parameter can scale the curve to account for that, and if the minimum
categorization probabilities in your data are above zero, k2 can raise the curve appropriately, with k1
scaling the curve to keep it from exceeding one, etc...).