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Hi everybody,

We have a nonparametric density estimate for a population and want to test whether or not it is composed of 2 subpopulations (i.e. males and females)

My questions are:

1. Is it possible to test similarity between the whole population and each sub population? 

2. If yes, is the test same as the test between sub populations? 

3. Which tests can be performed for 

(1) a nonparametric (univariate) density for directional data (estimated by Von Mises kernel) and
(2) a nonparametric bivariate (Gaussian) density estimates?

4. If there are more than 2 sub populations, for instant 10 subpopulations, how does multiple testing correction for multiple density similarity test?  

 

I would really appreciate any comments or suggestions from the group.

Best Regards,

Mina

I would not look directly at the non-parametric density estimates.  I would consider one or both of

1) Testing whether some aspect(s) of the populations are different.  For example a t-test to see if the means are different or a non-parametric comparison of the populations or a comparison of the variabilities.

2) A Kolmogorov-Smirnov comparison of the distributions if the concern really is the density.  Though I would be concerned with the power of the K-S test.

If different (or different enough), then I would fit the two populations separately.

Other than a Bonferroni correction, I'm not sure how to allow for multiple comparisons.  And with such a large number of comparisons that feels very conservative.

Best regards,
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Michael Morton
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Raymond Hoffmann
Professor
Medical College of Wisconsin
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See below for comments.

I have a question.  If it consists of two groups (i.e. males and females), then isn't it composed of two subpopulations by definition?  And then doesn't the question become, do the two subpopulations have different densities?

With regard to questions 1 and 2, the answer should be Yes.  Moreover, with respect to question 2, not only should the test be the same, but also the test statistic should be the same.  If the density of your whole population is the weighted average of the densities of its two subpopulations, then the difference between one subpopulation and the weighted average is equal to a scalar multiple times the difference between the two subpopulations. 

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Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences
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