🚀 go-pugleaf

I'm (trying to) design an experiment. The input, x, is a real number.
The output, y, is random, but either 0 or 1. I believe that either x
or x^2 has linear relationship with the expected value of y. How can I
test this and find the best model?

(Can anyone suggest a reference that covers this?)

- Carl


p.s. I've considered three designs, but each has problems or questions.
  
Design 1:
  Run m trials, producing data such as
    { (.34,0), (.53,1), (.13,0), ...}
  Do linear regression on the data.

  Problem:
     Linear regression assumes that y values are distributed
     normally, these y values actually have the Bernoulli distribution,
     which is quite different.

     Lack of fit not tested.

Design 2 -- (Repeated experiments to allow y's to be replaced with sample mean)
   Get m1 values of x
      For each value of x, get m2 measure of y, find the sample mean.
      Producing data such as
      { (.34,.80), (.53,.92), (.13,.76), ...}

   Do linear regression on the data.

   Problem:
       What relationship between m1 and m2 minimizes the number of
       experiments needed?

       Linear regression assumes that y values are distributed
       normally, these y values actually have the Binomial-related
       distribution. When the expect value of y is near 0 or 1,
       a large m2 will be needed to make the Binomial-related
       distribution approximate a normal distribution.
          
       Lack of fit not tested.

Design 3 -- (More repeated experiments to allow lack of fit tests)
   Get m1 values of x
      For each value of x, get m2 measure of y, find the sample mean.
        Do this m3 times.

         Producing data such as
          { (.34,.80), (.34,.82), (.34,.73),  ...
            (.53,.92), (.53,.90), (.53,.98),  ...
            (.13,.76), (.13,.82), (.13,.77),  ...
             ...}   
 Do linear regression (and lack of fit testing) on the data.

   Problem:
       What relationship between m1, m2, and m3 minimizes the number of
       experiments needed?

       Linear regression assumes that y values are distributed
       normally, these y values actually have the Binomial-related
       distribution. When the expect value of y is near 0 or 1,
       a large m2 will be needed to make the Binomial-related
       distribution approximate a normal distribution.

-- 
Carl Kadie -- kadie@cs.uiuc.edu -- University of Illinois at Urbana-Champaign