st: Zeros and measures of inequality or concentration

[<S.Jenkins@lse.ac.uk>]

------------------------------

Date: Wed, 8 Feb 2012 20:17:24 -0900
From: Troy Payne <paynetc@gmail.com>
Subject: st: Zeros and measures of inequality or concentration

I have a more statistical question than a Stata-related question:  Which
measure of inequality or concentration is best for data with a large
number of observations with a value of zero? 

While I haven't used them before, it seems that Lorenz curves, Gini
coefficients, and other related measures of inequality would be a good
way to examine concentrations of crime at addresses.  Like income, crime
tends to be highly concentrated, with a relative handful of places
contributing large proportions to the total crime count.  In fact, at
the place-level (address or street segment) the most common crime count
is often zero.

I have crime data at apartment buildings in a midwestern city.  In my
data, 45% of apartments had zero crimes in any given year.  If I include
only violent crimes, then 74% of apartments have zero crimes in any
given year.  

Posts here on Statalist lead me to -inequal-, -sgini-, -lorenz-, and
-glcurve- (all installed in Stata 12.1, all available via SSC).  Judging
from the r(N) returned, -inequal- seems to explicitly exclude
observations with values of zero, while -sgini- does not.  It's
difficult for me to tell if -lorenz- and -glcurve- include observations
with values of zero, even after reading the help files and other
documentation provided.  

Nearly all of what I've read about these various inequality measures so
far seems to assume non-zero values, or at least that zero values are
rare.  I'm unsure what the practical impact of a large proportion of
zeros would have, even for user-written commands that appear to allow
them. 

Until two days ago, I had never dug into the details of Gini
coefficients.  It's possible that the documentation has the answer and
I've just missed it.  I'd very much appreciate any guidance list members
could give.

-
Troy Payne
paynetc@gmail.com

+++++++++++++++++++++++++++

The Lorenz curve is defined for zero values, and indeed negative ones.
-glcurve- will draw a curve if the data include such values. (It reports
the number of negative ones.)

Many "standard" indices of inequality are defined only for positive
values. As Troy says, the Gini is well-defined in the case in which
there are zeros; so too is the coefficient of variation (CV) and
transformations of it, such as .5*CV^2 (generalised entropy index with
coeff = 2).

-ineqdec0- on SSC will calculate these 2 indices, allowing zero values.
(Its sibling, -ineqdeco-, does calculations using positive values only,
and for a wider range of indices.)


Stephen
------------------
Professor Stephen P. Jenkins <s.jenkins@lse.ac.uk>
Department of Social Policy and STICERD
London School of Economics and Political Science
Houghton Street, London WC2A 2AE, UK
Tel: +44(0)20 7955 6527
Changing Fortunes: Income Mobility and Poverty Dynamics in Britain, OUP
2011, http://ukcatalogue.oup.com/product/9780199226436.do
Survival Analysis Using Stata:
http://www.iser.essex.ac.uk/survival-analysis
Downloadable papers and software: http://ideas.repec.org/e/pje7.html



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