hello Bill

You hit the nail on the head I am having problems with language, as my parameter-variable faux-pax made clear. I have included a copy of the data I used to generate the plot. A glance will make it clear that it is a Binomial (n=12) data set and not Bernoulli. The column on the left is the predictor variable tested 12 times for each value of x and the column on the right is the response variable y. Or am I mistaken. { {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2.69897, 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {2., 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1.69897, 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {1., 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0.69897, 1.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 1.}, {0., 1.}, {0., 1.}, {0., 1.}, {0., 1.}, {-0.30103, 0.}, {-0.30103, 0.}, {-0.30103, 0.}, {-0.30103, 0.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-0.30103, 1.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1., 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 0.}, {-1.30103, 1.} }

I will try to express the question in a simpler way.

Starting with binomial odor responses to different concentrations of pairs of odorants we fit the data to a GLM model: y = GeneralizedLinearModelFit[p, x, x, ExponentialFamily -> "Binomial"]. The values of the model are the odor responses to the binomial choice A or B and y is expressed as A/( A+B). The arguments, a ratio of concentrations ([A]/[B]), is the independent variable. However we do not want to predict y from x but to determine x at a given value of y (0.5). If B is a blank X = x(at y = 0.50) is the threshold of A in the presence of B. We would like to determine a confidence interval (95%) in which X will fall. Does this mean that y is the independent variable (=0.5) and x is the dependent variable?

I think what you want are confidence intervals for $x$ given some probability $p$ (not prediction or predictional intervals - and I don't know what predictional intervals are).

Because the model you're fitting is

$$\log \left(\frac{\text{prob}}{1-\text{prob}}\right)=a+b x$$

you want confidence intervals for $x$

$$x = \frac{\log \left(\frac{\text{prob}}{1-\text{prob}}\right)-a}{b}$$

You can get the lower and upper limits for a 95% confidence interval for $x$ with the following:

prob = 0.5;
upper= tt /. Solve[prob == 1 - 1/(1 + Exp[parms.{1, tt} - 1.96 Sqrt[{1, tt}.cov.{1, tt}]]), tt]
lower= tt /. Solve[prob == 1 - 1/(1 + Exp[parms.{1, tt} + 1.96 Sqrt[{1, tt}.cov.{1, tt}]]), tt]