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Re: st: reverse prediction - confidence interval for x at given y in nonlinear model


From   "Rosy Reynolds" <rr@dandr.demon.co.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   Re: st: reverse prediction - confidence interval for x at given y in nonlinear model
Date   Fri, 26 Oct 2007 10:38:54 +0100

Joseph,
That's a very helpful contribution, thank you. I realised that I wasn't quite clear in my mind on the naming of these models, and you have helped to clear that up.

I had previously followed through the algebra from
E = Emin + Emax * Dose^Hill / (Dose^Hill + ED50^Hill)
(which you have very nicely clarified is the generally accepted so-called "sigmoid Emax" model)
and found that it is the same model as the 4-parameter logistic fitted to ln(dose).

That explains a lot of my confusion - when the models are discussed afterwards, they are discussed as sigmoid Emax models but, at least sometimes, they have actually been fitted via 4-parameter logistic on log(dose).

What I hadn't thought through was what you have pointed out about zero dose. By fitting to log(dose), strictly speaking, we cannot include zero doses in the analysis.
I know that it offends against mathematical purity, but how bad would it be really to allocate an arbitrarily small value to the zeros, so that their logs are suitably large negative numbers? So long as you place them way out on the tail of the S-shaped curve, where the response hardly alters with changes in ln(dose), how much would it matter that they are not actually at -infinity?.

best wishes
Rosy

----- Original Message ----- From: "Joseph Coveney" <jcoveney@bigplanet.com>
To: "Statalist" <statalist@hsphsun2.harvard.edu>
Sent: Friday, October 26, 2007 9:15 AM
Subject: Re: st: reverse prediction - confidence interval for x at given y in nonlinear model



Rosy Reynolds wrote:

Sigmoid models are customary in pharmacodynamics (dose-response studies).
According to custom, I am using a 4-parameter logistic (sigmoid Emax) model.
This is very easily done with -nl- as Stata has this model already built in.

The model is y= b0 + b1/(1 + exp(-b2*(x-b3))) + error

and the coefficients can be interpreted as
b0 = baseline outcome
b1 = Emax i.e. largest change from baseline
b2 = Hill or slope coefficient
b3 = ED50 i.e. value of x (dose) required to produce half-maximal effect,
that is x required for y=b0 + b1 / 2

As the ED50 is a parameter of the model, -nl- reports it with a standard
error and confidence interval.
What I would like to do is obtain estimates with standard errors and
confidence intervals for other similar measures e.g. the ED90, the dose
required for 90% of maximal effect.
[redacted]
--------------------------------------------------------------------------------

Maarten gave a solid, well documented answer, as usual.

Just a comment and a follow-on question. First the comment. If I'm not
mistaken, the four-parameter logistic model Rosy used is for the *logarithm*
of dose and *logarithm* of ED50, and not the dose and ED50, per se (cf.
Maarten's y-axis values). So, Rosy will need to remember to logarithmically
transform drug doses before fitting the model, and to back-transform the
log-ED50 (log-ED90) values and their confidence limits afterward. I believe
that this parameterization is sometimes advocated in order to guarantee
strictly positive values when lower confidence limits for ED50 (ED10, etc.)
are back-transformed, that is, in order to assure that you won't end up with
a value of, say, -10 mg of drug as a lower confidence limit for a poorly
estimated ED10. Perhaps the parameterization is numerically stabler, too,
in some sense, but you cannot use it if you wanted to include results from
a placebo treatment group in a clinical study or a vehicle-control condition
in a laboratory setting.

As to the question: given that the generally accepted four-parameter
(so-called, sigmoid Emax) model for a dose-response curve is

E = Emin + Emax * Dose^Hill / (Dose^Hill + ED50^Hill)

(E is response, Emin is response at zero dose, Emax is asymptotic response
at infinite dose, Dose is untransformed dose, Hill is the coefficient of
receptor cooperativity, ED50 is dose yielding a response that is Emin + half
of Emax),

does the -nl log4:- four-parameter logistic model give rise to biased
estimates of ED50 (ED10, ED90, etc.) and confidence intervals in the
original measurement scale with nonasymptotic sample sizes? That is, should
a pharmacologist ever use -nl log4:- in lieu of the model shown just above?

Joseph Coveney


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