R help - Dec 2011 - how to get inflection point in binomial glm

Dear All,

I have a binomial response with one continuous predictor (d) and one  
factor (g) (8 levels dummy-coded).

glm(resp~d*g, data, family=binomial)

Y=b0+b1*X1+b2*X2 ... b7*X7

how can I get the inflection point per group, e.g., P(d)=.5

I would be grateful for any help.

Thanks in advance,
Ren?

Assuming a logistic model, for each group solve for d at Y=0, or solve for d at
p=0.5, where d is your continuous predictor, Y is the logit score, and p is the
probability of success in the binomial model. After that you can get the
standard error of the inflection point by Taylor series (aka delta method).

Another idea is to re-parameterize the logistic model to include explicitly the
inflection point, thus you'll get its estimate and standard error directly
as a result of the fit.
For example, disregarding the g factor predictor (or for each group), a logistic
model such as
P(d) = 1/(1+exp(log(1/19)(d-d50)/(d95-d50))
includes the inflection point directly (d50) and is a re-parameterized version
of the usual logistic model
P(d) =1/(1+exp(b0+b1*d))
where parameters b0 and b1 have been replaced by d95 and d50, the predictor at
50% probability (inflection point), and the predictor at 95% probability,
respectively.

HTH

Rub?n

-----Mensaje original-----
De: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] En
nombre de Ren? Mayer
Enviado el: jueves, 01 de diciembre de 2011 14:25
Para: r-help at r-project.org
Asunto: [R] how to get inflection point in binomial glm

Dear All,

I have a binomial response with one continuous predictor (d) and one factor (g)
(8 levels dummy-coded).

glm(resp~d*g, data, family=binomial)

Y=b0+b1*X1+b2*X2 ... b7*X7

how can I get the inflection point per group, e.g., P(d)=.5

I would be grateful for any help.

Thanks in advance,
Ren?

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On Dec 1, 2011, at 8:24 AM, Ren? Mayer wrote:
> Dear All,
>
> I have a binomial response with one continuous predictor (d) and one  
> factor (g) (8 levels dummy-coded).
>
> glm(resp~d*g, data, family=binomial)
>
> Y=b0+b1*X1+b2*X2 ... b7*X7
Dear Dr Mayer;

I think it might be a bit more complex than that. I think you should  
get 15 betas rather than 8. Have you done it?
>
> how can I get the inflection point per group, e.g., P(d)=.5
Wouldn't that just be at d=1/beta in each group? (Thinking, perhaps  
naively, in the case of X=X1 that

(Pr[y==1])/(1-Pr[y==1])) = 1 = exp( beta *d*(X==X1) )  # all other  
terms = 0

And taking the log of both sides, and then use "middle school" math to
solve.

Oh, wait. Muffed my first try on that for sure.  Need to add back both  
the constant intercept and the baseline "d" coefficient for the non-b0
levels.

(Pr[y==1])/(1-Pr[y==1])) = 1 = exp( beta_0 + beta_d_0*d +
                                     beta_n + beta_d_n *d*(X==Xn) )

And just

(Pr[y==1])/(1-Pr[y==1])) = 1 = exp( beta_0 + beta_d_0*d ) # for the  
reference level.

This felt like an exam question in my categorical analysis course 25  
years ago. (Might have gotten partial credit for my first stab,  
depending on how forgiving the TA was that night.)
>
> I would be grateful for any help.
>
> Thanks in advance,
> Ren?
>
-- 

David Winsemius, MD
West Hartford, CT