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The Stata Journal
Volume 11 Number 2: pp. 271-289

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M statistic commands: Interpoint distance distribution analysis

Pietro Tebaldi
Bocconi University
Milan, Italy
Marco Bonetti
Bocconi University
Milan, Italy
Marcello Pagano
Harvard School of Public Health
Boston, MA
Abstract.  We implement the commands mstat and mtest to perform inference based on the M statistic, a statistic that can be used to compare the interpoint distance distribution across groups of observations.

The analyses are based on the study of the interpoint distances between n points in a k-dimensional setting to produce a one-dimensional real-valued test statistic. The locations are distributed in a region of the plane. When we consider all interpoint distances, the dependencies among them are difficult to express analytically, but their distribution is informative, and the M statistic can be built to summarize one aspect of this information.

The two commands can be used on a wide class of datasets to test the null hypothesis that two groups have the same (spatial) distribution. mstat and mtest return the exact M test statistic. Moreover, mtest executes a Monte Carlo–type permutation test, which returns the empirical p-value together with its confidence interval. This is the command to use in most situations, because the convergence of M to its asymptotic chi-squared distribution is slow.

Both commands can be used to obtain graphical output of the empirical density function of the interpoint distance distributions in the two groups and the twodimensional map of the n observations in the plane.

The descriptions of the commands are accompanied by examples of applications with real and simulated data. We run the test on the Alt and Vach grave site dataset (Manjourides and Pagano, forthcoming, Statistics in Medicine) and reject the null hypothesis, in contradiction to other published analyses. We also show how to adapt the techniques to discrete datasets with more than one unit in each location. Finally, we report an extensive application on breast cancer data in Massachusetts; in the application, we show the compatibility of the M commands with Pisati’s spmap package.
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