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

["Rosy Reynolds" <rr@dandr.demon.co.uk>]

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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