The choice of the number of bins for the statistic

Abstract

Methods to monitor spatial patterns of disease in populations are of interest in public health practice. The M statistic uses interpoint distances between cases to detect abnormalities in the spatial patterns of diseases. This statistic compares the observed distribution of interpoint distances with that which is expected when no unusual spatial patterns exist. We show the relationship of M to Pearson's Chi Square statistic, . Both statistics require the discretization of continuous data into bins and then are formed by creating a quadratic form, scaled by an appropriate variance covariance matrix. We seek to choose the number and type of these bins for the M statistic so as to maximize the power to detect spatial anomalies. By showing the relationship between M to , we argue for the extension of the theory that has been developed for the selection of the number and type of bins for to M. We further show that spatial data provides a unique insight into the problem through examples with simulated data and spatial data from a health care provider. In the spatial setting, these indicate that the optimal number of bins depends on the size of the cluster. For large clusters, a smaller number of bins appears to be preferrable, however for small clusters having many bins increases the power. Further, results indicate that the number of bins does not appear to vary with m, the number of spatial locations. We discuss the implications of this result for further work.

Pedagogy of chi‐square goodness of fit test for continuous distributions

“…The binning discussed so far assumes that the bin widths are constant. One of the ways to mitigate the issue of the bin population with fixed bin width is to abandon constant bin widths and use the constant population bins (variable bin width) by fixing the population to be 5 or more . The chi‐square test is not impacted by the choice of the binning method and may be carried out as described above because we will be comparing the probabilities from the data and the hypothesized probabilities based on variable width bins giving the fixed bin population.…”

“…It has been suggested by a few researchers that the minimum count must be 5 while others have argued that it may even be 1. 9,[15][16][17][18] There are also suggestions that a rough estimate of the number of bins is N 2 2/5 . 19 Note that higher number of bins reduces the population in the bins.…”

Improving the power of chronic disease surveillance by incorporating residential history

“…Two, it has been shown to be powerful in detecting the presence of clusters[32, 33, 8]. And, three, it lends itself easily to the problem at hand when we actually do not know with certainty which address we wish to consider.…”

“…For a more complete discussion of the bin selection procedure, see [33]. When the interpoint distance distribution of the cases differs from the distribution of the controls, M R will have larger values.…”

“…This summary measure associated with the spatial distribution is not usually invertible (it is translation and rotation invariant) and thus does not uniquely identify the contemporaneous spatial distribution, but it does have advantages: One, a general advantage, is that it is univariate and thus much easier to manipulate and comprehend. Two, it has been shown to be powerful in detecting the presence of clusters [8,32,33]. And, three, it lends itself easily to the problem at hand when we actually do not know with certainty which address we wish to consider.…”