I think a third answer that belongs here is this: You need to be clear what you want in a "robust" estimator.
Robust is used with a lot of ambiguity in the statistical literature. It can mean:
- that error estimate accounts for model misspecification
- that the method is a generalization of a known or accepted technique that downweights the influence of extreme, outlying observations.
- that the model is capable of fitting a large number of scenarios
My contention with the second sense of robust is that it exonerates statisticians from doing any thinking. If the "value" of 120 is wrong (clinical error), then delete it. But if it's not an error you need to figure out why it's there. Is this a longitudinal design? Was second dose applied before first was metabolized/cleared? Did the subject have liver or kidney damage and drug metabolism was slowed and response shows the cumulative effect? Could there be a U-shaped (logistic) curve relating dose and response? This is the power of 3+3 designs for evaluating dose-response.
But mainly it is the third sense on which there is some disagreement. The 4 parameter logistic model is indeed quite robust, having intercept, shift, slope, and scale terms. However, it cannot fit a "non-linear" sigmoid. So if 120 is "right" you need a different model, or a clearer understanding of the drug and its properties.
If you just want to fit an x-y curve, consider using a smoothing spline as an input to a least squares model. This can be quite challenging both in terms of fitting, interpreting, and generalizing to predict response in other scenarios.
You might consider a more sophisticated time series if the 120 was just a result of applying the second dose too soon. Consider modeling the area under the (dose concentration) curve as an input or "x" variable in the analysis instead.