Logistic function between arbitrary parameter values

Phil,

One reasonable solution is add two additional parameters for the bottom and top (or bottom and span) of your logistic.:

Your function: K / (1 + ((K - x.0)/x.0) * exp(-r * x))  
becomes        alpha + beta*K / (1 + ((K - x.0)/x.0) * exp(-r * x)),

This logistic spans Y values from -alpha to -alpha + beta.
You may have trouble fitting it, because Yhat will be undefined if alpha is not large enough and I would be surprised if there were high collinearity between \hat\alpha and \hat\x.0.  However, the choices of alpha and beta are no longer arbitrary.

Best wishes,
Philip Dixon