Roche
clarify role of reference dose
Summary
The documentation several times mentions reference dose but does not explain it any further.
Particularly the function MinimalInformative() requires the reference dose as parameter.
On page 2423 of Statist. Med. 2008; 27:2420–2439 the reference dose d* is introduced as:
"reference dose allowing for the interpretation of alpha as the odds of a DLT at d*"
for the logistic transform log(alpha) + beta*log(d/d*).
If the reference dose d* is just for convenience for interpretation of alpha - could it not instead be set to 1 for convenience of programming?
As discussed with @reikoch, the reference dose has a geometric interpretation: it is the dose that defines the inflexion point of the dose toxicity curve: the point at which the second derivative of the curve is zero. In a sense, the dose-toxicity curve "pivots" about d*. With this insight, I believe the benefit of having the reference dose within the dose range being studied is clear: if it is not, the model will try to pivot the toxicity cure around a point that is outside the range of observed doses. This will lead to instability in estimates of the model's parameters.
This geometric interpretation shows why a small amount of negative correlation between alpha and beta can be useful. Randomly "low" sampled values of alpha are more likely to be offset by randomly "high" values of beta, and vice versa.
Moreover, as @reikoch points out, it also provides a similar geometric interpretation of log(alpha): d/d* is 1 when d = d* meaning that beta*log(d/d*) is 0 regardless of the value of beta. Hence (log) alpha is the (log) odds of toxicity at d*.
In my opinion, fixing the reference dose may be programmatically useful, but it is practically detrimental.
As discussed with @reikoch, the reference dose has a geometric interpretation: it is the dose that defines the inflexion point of the dose toxicity curve: the point at which the second derivative of the curve is zero...
Hi @PuzzledFace - Are you sure about this?
p(d) = log(alpha) + beta*log(d/d*) function (which is just a shifted and scaled log function) does not have any inflection (given any valid triple (alpha, beta, d*)). This is also confirmed by the 2nd derivative test. The 2nd derivative p'' of p with respect to d is
p''(d) = - beta * 1/d^2,
Then, p''(d) !=0 for all d (of course, given that beta != 0). So the necessary (2nd derivative) condition for p to have an inflection at some d (i.e. p''(d) = 0) is not met for any d.
