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Re: st: Can we use the standard binary choice model?


From   Steven Samuels <sjsamuels@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Can we use the standard binary choice model?
Date   Tue, 25 Jan 2011 10:16:47 -0500

--

Quang-

I'm very curious: What event has a probability of 1 between the thresholds in your example? I'm having a hard time thinking one related child-bearing for which this model would be true. Possible exceptions: menstruation and ovulation. But these events would define the thresholds (one would not decide to bear children if one couldn't), so the model would be vacuous.


Steve
sjsamuels@gmail.com




On Jan 24, 2011, at 10:11 AM, Nick Cox wrote:

I understand the words, but not what they imply for your analyses.

There is a big difference between wanting to estimate these thresholds from the data and having numbers from somewhere else. Which do you want to do?

The example does not change my advice. Those thresholds are both highly stochastic, it seems to me, quite apart from any other details.

Nick
n.j.cox@durham.ac.uk

Quang Nguyen

The model is kinda comlicated to present in detail here. The main
point is that both x1 and x2 (the threshold levels) are derived from
other exgogenous parameters of the model. An example of such model is
women's childbearing decision in which x1=10 and x2=50.

On Mon, Jan 24, 2011 at 3:20 AM, Nick Cox <n.j.cox@durham.ac.uk> wrote:

Consider graphs like

. twoway function invlogit(5+ (x - 2) - 5*(x-2)^2), ra(-3 5)

These to me suggest that if this model is realistic it can be approximated using a quadratic in the predictor and a logit link. Models like this are common in ecology whenever there is some optimum for organisms (e.g. in moisture or temperature) and abundance is expected to be highest at that optimum.

In other words, the model is equivalent to one symmetric around (x1 + x2)/2.

There is no hint here why the theory suggests sharp thresholds at x1 and x2. In practice I'd expect those to be fuzzy, so I wouldn't feel guilty about not taking the model very literally.

On Behalf Of Maarten buis

--- On Mon, 24/1/11, Quang Nguyen wrote:
We have a theoretical model predicing the relationship between
a binary variable "y" and a continous variable in the following
pattern: y=1 if  y is in the range of [x1,x2] and y=0 if y is
smaller than x1 or greater than x2. Where x1 and x2 are some
threshold determined by the theoretical model's parameters.

Can you suggest an empirical model to verify the above
model's prediction?

Let me fix some terminology and notation:

y* is a latent continuous dependent/explained variable.
y  is the observed binary dependent/explained variable.
xb is the linear predictor: b0 + b1 x1 + b2 x2 + b3 x3, the xs
  are independent/explanatory variables not your theshholds.
a1 and a2 are your thresholds, (x1 x2 in your notation).

Your problem would be:
y* = xb

y  = 1 if a1 <= y* <= a2
y  = 0 if y* < a1 | y* > a2

This would be relatively easy to solve if a1 = - a2:

y*^2 = xb^2

y  = 1 if y*^2 <= a2^2
y  = 0 if y*^2 >  a2^2

y  = 1 if y*^2 - a2^2 <= 0
y  = 0 if y*^2 - a2^2 >  0

So now it is just a regular binary choice model whith a
lot of extra variables (squares and products of variables),
and a hard to interpret constant.


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