Re: Optimal seperability of distribution


newsgroupie2003@xxxxxxxx wrote:
> Hi,
> I have the following problem that I am not sure how to best go about.
> I have 2 populations (healthy and ill) and I then have a choice of
> different measurement methods to try and seperate them. The measurement
> is univariate.
> What is the best way of characterizing the separability between the
> healthy and sick populations?

There is no, apriori, best way. You have to decide, given prior
probabilities of the two distributions and misclassification costs
what you want to optimize.

A classical measure of performance is misclassification risk. One
form for c mutually exclusive classes is

R = sum(i = 1,2,..c){ R(i|x) },

R(i|x) = sum(j=1,c){ Cij * P(j|x) },

where

R(i|x) The conditional risk of choosing class i given the
measurement x
Cij The cost of choosing class i when the true class is j
P(j|x) The conditional probability that the true class is j
given the measurement x (which depends on the
prior probability, Pj).

Often, the true values of Pj and Cij are not known and have
to be guesstimated.

When the classes are unbalanced in number and in importance
(e.g., P1 >> P2 but C12 >> C21) it can be dangerous to ignore
the priors and costs.

> I was thinking of ROC curves summarized by the integral of the ROC
> curve, but i guess the problem is quite general in the sense that what
> is needed is measure of how well the 2 distributions are seperated.
> Any input appreciated!

Measuring how much the probability density functions p(x|1) and
p(x|2) are separated doesn't mean much when the products
P1*p(x|1) and P2*(p(x|2) determine the optimal decisions.

Hope this helps.

Greg

.

Relevant Pages

  • Re: Optimal seperability of distribution
    ... >> healthy and sick populations? ... >> I was thinking of ROC curves summarized by the integral of the ROC ... but that addresses only one aspect of separation. ...
    (sci.stat.math)
  • Optimal seperability of distribution
    ... I have 2 populations and I then have a choice of ... different measurement methods to try and seperate them. ... healthy and sick populations? ...
    (sci.stat.math)
  • Re: Optimal seperability of distribution
    ... > healthy and sick populations? ... > I was thinking of ROC curves summarized by the integral of the ROC ... there is no measure of separation that is best for all ...
    (sci.stat.math)