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

From   "Joseph Coveney" <>
To   "Statalist" <>
Subject   Re: st: reverse prediction - confidence interval for x at given y in nonlinear model
Date   Fri, 26 Oct 2007 01:15:05 -0700

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.

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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