[R] Lognormal with 0/1 "tag" [theoretical]

(Ted Harding) Ted.Harding at manchester.ac.uk
Mon Dec 29 16:12:34 CET 2008

Hi Folks,
This is not (well, not yet) an R question as such, though
it is preparatory to embarking on an R-based study.

Scenario: X has a lognormal distribution. Jointly distributed
with X is a "tag" Y: Y = 0 or 1, with

   Prob(Y=1|X) = exp(L)/(1+exp(L)), L = a + b*X

I am interested in the distribution of X conditional on Y,
which is proportional to

  f(x,Y) = dlnorm(x,mu,sigma)*exp(Y*L)/(1+exp(L))
  (Y = 0 or 1, L = a + b*x)

In particular, before gettiong down to the computational work,
I am interested in studying its moments, and its "incomplete

  Mj(X,Y) = Integral[x=0:X] x^j * f(x,Y) dx

What I'd like to find out (and the chances are that some people
on this list should know!) is what analytical forms may be
available for such things. They are certainly known for the
lognormal on its own, but the additional logistic factor has
taken it beyond my immediate analytical capabilities.

This has an obvious practical illustration: X may be a predictor
for an event which, given X, has the logistic probability of
occurring. The question relates therefore to the distrbution
of X in cases where the event occurred, and in cases where the
event did not occur. As such, I would expect that it has often
turned up in the epidemiological world, and the resulting
distribution may well have a name and well-known analytical
properties. It just happens that I'm not acquainted with these!

Any help will be much appreciated.
With thanks,

E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 29-Dec-08                                       Time: 15:12:31
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