how to add confidence intervals (bounds) to Logistic function

Dear guys,

I want to add confidence intervals to logistic function.

Logistic function which is defined as : y = exp(z) / (1+exp(z)), also z = a+b*x.

For the convenience I provide the code:

a=-5;

b=38;

x=0.01:0.01:0.5;

z=a+b.*x;

y = exp(z)./(1+exp(z));

figure(1);plot(x,y);hold on;

Could anyone tell me how to calculate confidence intervals for y in the plot of fig1 (x&y plot).

Let's start with 95% two-tail.

Thank you in advance.

David

Answer 1

[Walter Roberson]

That's a tricky one: If you declare that x is to be uniformly randomly generated over [0,1/2] then the upper bound of the 95% two-tailed confidence interval is substantially above the upper limit of that range, near 1.105512795.

If x is held to be uniformly randomly generated over [0,T] then the upper bound of the 95% two-tailed confidence interval is larger than T until T reaches slightly more than 1.16 .

I proceeded with these calculations by finding the mean and standard deviation over U[0,T] of y with symbolic x. 95% confidence corresponds to 1.96 standard deviations, so for the upper bound calculate Mean + 1.96 * Std / 2, with the division by 2 being because the 95% range was being split evenly above and below the mean. Then evaluate for T=1/2 for U[0,1/2]

If you do the calculations with x in U(-T,T) then at T=1/2 the beginning of the upper tail is still above 1/2; in the U(-T,T) situation, the upper tail does not get smaller than the limit of the range until about T = 0.5808041659 . Note: for U(-T,T) the mean of y is exactly 1/2 .