[R] how to get inflection point in binomial glm

[David Winsemius]

On Dec 1, 2011, at 8:24 AM, René 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,
> René
>
-- 

David Winsemius, MD
West Hartford, CT