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Comput Stat Data Anal. Author manuscript; available in PMC 2010 August 1.

Published in final edited form as:

Comput Stat Data Anal. 2009 August 1; 53(10): 3640–3649.

doi: 10.1016/j.csda.2009.03.005PMCID: PMC2703493

NIHMSID: NIHMS102768

See other articles in PMC that cite the published article.

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,
${x}_{n}^{2}$. 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
${x}_{n}^{2}$, we argue for the extension of the theory that has been developed for the selection of the number and type of bins for
${x}_{n}^{2}$ 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.

Syndromic surveillance and a growing interest in real time monitoring the patterns of diseases in populations has led to increased research in novel methods for detecting spatial anomalies. Among the methods being developed is the *M* statistic [1]. This statistic monitors the distribution of the interpoint distances between cases and is designed to detect deviations from expected behavior in this distribution. The *M* statistic has been shown to be effective at detecting exogenous clusters when compared with other statistics [1, 2, 3] designed for this same purpose. It claims some advantages, including the ability to detect a broad range of deviations, not just clusters. Additionally, it does not suffer from the curse of dimensionality and it does not make any parametric assumptions, or impose restrictions on the format of the spatial data (for example, exact locations, or aggregated over zip code, census tract, or some other region). A bivariate version of this statistic, incorporating temporal information with the spatial data, leads to greater power to detect aberrant behavior [4].

The *M* statistic is constructed by considering all the interpoint distances between *m* independent cases. Thus there are
$N=\left(\begin{array}{c}m\\ 2\end{array}\right)$ interpoint distances used in calculating the statistic. In order to check for goodness-of-fit, data is discretized into the *k* × 1 vector of the frequency distribution of the observed (dependent) distances and the vector is compared to the corresponding *k* × 1 frequency vector expected under the null hypothesis via the quadratic form,

$$M={({\text{o}}_{N}-\text{e})}^{\top}{S}^{-}({\text{o}}_{N}-\text{e})$$

where o* _{N}* is the observed vector of the frequencies of the distances, e is the vector of expected frequencies, and

It is also useful to see this statistic as a generalization of Pearson's Chi Square statistic,
${X}_{n}^{2}$ [5]. Pearson's statistic does exactly the same operation as *M*, but assumes that the underlying observations are independent. Thus, in that case, the observed *k* × 1 vector of the observed relative frequencies is distributed as a multinomial random variable. That statistic can be written as

$${X}_{n}^{2}={({\text{x}}_{n}-\text{e})}^{\top}{\sum}_{MN}^{-}\phantom{\rule{0.2em}{0ex}}({\text{x}}_{n}-\text{e})$$

where x* _{n}* is a multinomial random vector and
${\sum}_{MN}^{-}$ (here,

Both statistics have several appealing features. Among these are that they are intuitive and easy to interpret. It is also reasonable to assume that greater power is gained to detect aberrations from null behavior when the entire distribution is compared, as opposed to comparing a simple summary statistic, such as the mean or maximal value, as in the Kolmogorov-Smirnov test [6]. A challenge with
${X}_{n}^{2}$ and *M* arises when working with continuous, or nearly continuous data. In this case, one must aggregate the continuous data into *k* bins. The optimal method of doing this has been studied for
${X}_{n}^{2}$ (see references cited in the next section). This is an important consideration, as values for the power have been shown to vary dramatically, both for *M* [2] and
${X}_{n}^{2}$ [7] when *k* varies. Intuition might suggest that we select *k* to be large, so as to more closely represent the continuous distribution. However, there is a tradeoff for choosing large values for *k*, as this leads to an increase in the variance and the possibility of having bins with no observations. Further the particular anomaly to be detected in the data may dictate the optimal number of bins, as we show below.

In the rest of this paper, we present two main results. First, the *M* statistic is indeed essentially equivalent to
${X}_{n}^{2}$. This also implies that any appropriately constructed quadratic form is equivalent to
${X}_{n}^{2}$, regardless of the nature of the underlying observations. Second, we explore the implication of this result for applying the theory that has been developed for
${X}_{n}^{2}$ with regard to the selection of *k* for spatial data using the *M*. Specifically, in the cluster detection setting with *M*, we show that the optimal number of bins depends on the size of the cluster to be detected. We discuss the implications of this result. First, however, we provide a summary of relevant work on selecting *k* for
${X}_{n}^{2}$.

Pearson's Chi Square statistic was introduced at the beginning of the 20th century and has been widely used since its introduction [5]. It is most suited for goodness-of-fit tests with discrete data, but is frequently extended for use with continuous data. We assume in what follows that the null hypothesis is of the form *H*_{0} : *X* ~ *f* (*x*; *θ*_{0}), with both *f* and *θ*_{0} fully specified. The null distribution of the
${X}_{n}^{2}$ statistic when testing for a specific distribution depends on the method of estimation of the parameters, if any are being estimated as part of the test, for example when testing the hypothesis that the observed data arises from a normal distribution without specifying the mean and variance [8]. When one uses the statistic with continuous data it then becomes necessary to categorize the data, and there are two main issues to consider in this process: how the bins should be partitioned (for instance equiprobable, equispaced or some other configuration) and how many bins, *k*, should be used. The focus of this paper addresses the latter question, however we begin by providing some comments on the former.

An analogous process to determining the type of bins to use is performed when creating a histogram of continuous data. In most of the popular statistical computing packages (for instance Splus/R, Stata and SAS), the default setting for this is to use equispaced bins when creating the histogram. One might consider this same process when using ${X}_{n}^{2}$ for continuous data. The benefit of this method is that when the aberration we wish to detect occurs in one of the cells with a small expected count, we are more likely to detect even a small absolute change. For instance, if only one observation is expected in each cell, then it would not take a large increase for a strong signal to be generated. However, a major problem of this method is that inevitable disparities in the cell counts will be created. If one cell contains a large portion of the data, while the remaining cells are sparse, then subtle changes between the null and alternative distributions could be masked within the large cell, and power may be compromised. Additionally, having cells with no observations is not only inefficient, but also leads to instability in the statistic, since ${X}_{n}^{2}$ is calculated by dividing by the expected number in each cell. If that number is approaching zero for at least one cell, then we can expect that the statistic will behave aberrantly. For instance, [9] cites research showing that the asymptotic behavior of ${X}_{n}^{2}$ is not as expected, giving some explanation for the well-known rule of thumb of having at least five as the expected count in each cell (though subsequent work has shown this particular cutoff to be conservative [10]).

As an alternative to equispaced cells, [7] suggest that equiprobable cells are probably better, unless there is a particular alternative that one is interested in detecting. In that case, it is optimal to use equiprobable cells, except in the area closest to the anticipated aberration; in that vicinity one should use more cells. For instance, if the type of alternative we are interested in detecting would lead to deviations in values on the extremes of the distribution, we should consider increasing the number of bins in the area at the extremes of the data. The use of equiprobable binning is also recommended when using the *M* statistic, as work with that statistic suggests that this is more powerful than the equispaced approach [2, 3].

We now focus our attention on the selection of *k*, the number of bins. The goal is to maintain a balance between accurately portraying the distribution by having a sufficient number of bins, but without allowing that number to be so high as to create excessive variability. Table 1 provides a brief summary of some of the more conclusive papers addressing this topic for the
${X}_{n}^{2}$ test. Mann and Wald provide a rule that is easy to implement in this problem. They prescribe a formula that has *k* increase with *n*^{2/5}. However much of the subsequent work showed that their approach often prescribes too many bins [11]. We give particular attention to the work done by [7] and [12]. This work describes the asymptotic behavior of the power of
${X}_{n}^{2}$ as both *k* and *n* tend to infinity. We find this particularly useful for understanding how to select *k* with respect to *n* for scenarios that have not been previously characterized, as the results are general and can be applied to detect any alternative against a given null distribution.

In many of the cases for
${X}_{n}^{2}$, research indicates that *k* should increase with *n*, though *k* = *n* is problematic as sparse cell counts will occur. Exceptions occur when testing for normality as shown in [13] and when the alternative has light tails compared to the hypothesized distribution [7, 12]. In these cases, small *k* is optimal for all values of *n*. The work presented here indicates that when testing for spatial clusters with the *M* statistic the value of *k* does not appear to depend on *m*, the number of independent spatial locations, as much as it depends on the size of the cluster to be detected. In the following sections we give the theoretical underpinning for the extension of the
${X}_{n}^{2}$ results to the *M* statistic and then illustrate it through examples.

The above discussion on bin selection deals with the case where the continuous observations on which the frequencies are calculated are independent and identically distributed. We now seek to show their applicability to the case of observations that are not independent. This is motivated by work done in spatial statistics. [1, 14] have studied a statistic designed to detect spatial anomalies. This statistic is being used in syndromic surveillance [4], as well as genetics research [15, 16]. The statistic considers the distribution of all of the interpoint distances between observations. For the case of syndromic surveillance, we consider the Euclidean distances between cases with a syndrome of interest. In the genetics setting, one uses genetic distances between, for example, viruses. In what follows,
$N=\left(\begin{array}{c}m\\ 2\end{array}\right)$ is the number of interpoint distances between the *m* independent spatial locations. The matrix *S* has been shown empirically to be of rank *k* − 1 in most cases. We use the generalized inverse described in [17, p. 27]. This generalized inverse is obtained by partitioning the matrix as

$$S=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),$$

where the rank of *A* is equal to that of *S*. The generalized inverse is computed as

$${S}^{-}=\left(\begin{array}{cc}{A}^{-1}& 0\\ 0& 0\end{array}\right).$$

As we pointed out, the *M* statistic bears strong resemblance to
${X}_{n}^{2}$. In what follows we show that that these two statistics are essentially equivalent and therefore argue that the theory that has been developed for
${X}_{n}^{2}$ can be applied to *M*. These results can be generalized to any quadratic form with an appropriate variance covariance matrix. Some consideration, however, must be given to the rate at which the asymptotic results apply. This is because with dependent data generated from *m* independent observations, we observe a much larger number of dependent observations (of order *m*^{2}). We can rewrite the statistics *M* and
${X}_{n}^{2}$ as

$$\begin{array}{ll}M\hfill & ={({\text{o}}_{N}-\text{e})}^{\top}{\left(\begin{array}{cc}{S}_{1}& {\text{s}}_{2}\\ {\text{s}}_{2}& {s}_{3}\end{array}\right)}^{-}({\text{o}}_{N}-\text{e})={({\stackrel{\sim}{\text{o}}}_{N}-\stackrel{\sim}{\text{e}})}^{\top}{S}_{1}^{-1}({\stackrel{\sim}{\text{o}}}_{N}-\stackrel{\sim}{\text{e}})\hfill \\ \hfill & ={\stackrel{\sim}{\text{a}}}_{N}^{\top}{S}_{1}^{-1}{\stackrel{\sim}{\text{a}}}_{N}\hfill \end{array}$$

$$\begin{array}{ll}{X}_{n}^{2}\hfill & ={({\text{x}}_{n}-\text{e})}^{\top}{\left(\begin{array}{cc}{\sum}_{1}& {\sigma}_{2}\\ {\sigma}_{2}& {\sigma}_{3}\end{array}\right)}^{-}({\text{x}}_{n}-\text{e})={({\stackrel{\sim}{\text{x}}}_{n}-\stackrel{\sim}{\text{e}})}^{\top}{\sum}_{1}^{-1}({\stackrel{\sim}{\text{x}}}_{n}-\stackrel{\sim}{\text{e}})\hfill \\ \hfill & ={\stackrel{\sim}{\text{z}}}_{n}^{\top}{\sum}_{1}^{-1}{\stackrel{\sim}{\text{z}}}_{n},\hfill \end{array}$$

where a tilde represents a *k* − 1 dimensional vector.

We use the Cholesky decomposition and let

$$\begin{array}{l}{S}_{1}={L}^{\top}L,\text{so that}\phantom{\rule{0.2em}{0ex}}{S}_{1}^{-1}=({L}^{-1})\phantom{\rule{0.2em}{0ex}}{({L}^{-1})}^{\top}\\ {\sum}_{1}={T}^{\top}T,\phantom{\rule{0.2em}{0ex}}\text{so that}\phantom{\rule{0.2em}{0ex}}{\sum}_{1}^{-1}=({T}^{-1})\phantom{\rule{0.2em}{0ex}}{({T}^{-1})}^{\top}.\end{array}$$

Then we can rewrite *M* as

$$\begin{array}{ll}M\hfill & ={\stackrel{\sim}{\text{a}}}_{N}^{\top}\phantom{\rule{0.2em}{0ex}}{L}^{-1}{({L}^{-1})}^{\top}{\stackrel{\sim}{\text{a}}}_{N}={\stackrel{\sim}{\text{a}}}_{N}^{\top}\phantom{\rule{0.2em}{0ex}}{L}^{-1}T{T}^{-1}{({T}^{-1})}^{\top}{T}^{\top}{({L}^{-1})}^{-\top}{\stackrel{\sim}{\text{a}}}_{N}\hfill \\ \hfill & ={\stackrel{\sim}{\text{a}}}_{N}^{\top}\phantom{\rule{0.2em}{0ex}}{L}^{-1}T{\sum}_{1}^{-1}{T}^{\top}{({L}^{-1})}^{-\top}{\stackrel{\sim}{\text{a}}}_{N}={\stackrel{\sim}{\text{b}}}_{N}^{\top}{\sum}_{1}^{-1}{\stackrel{\sim}{\text{b}}}_{N}.\hfill \end{array}$$

Given the transformation
${\stackrel{\sim}{\text{b}}}_{N}={T}^{-1}{({L}^{-1})}^{\top}{\stackrel{\sim}{\text{a}}}_{N}$, Σ_{1} is the variance covariance matrix of * _{N}*:

$$\begin{array}{ll}\text{var}({\stackrel{\sim}{\text{b}}}_{N})\hfill & =\text{var}({T}^{\top}{({L}^{-1})}^{-\top}{\stackrel{\sim}{\text{a}}}_{N})={T}^{\top}{({L}^{-1})}^{-\top}\text{var}({\stackrel{\sim}{\text{a}}}_{N}){L}^{-1}T\hfill \\ \hfill & ={T}^{\top}{({L}^{-1})}^{-\top}{L}^{\top}L{L}^{-1}T={T}^{\top}T={\sum}_{1}.\hfill \end{array}$$

Therefore we see that *M* and
${X}_{n}^{2}$ are essentially equivalent. Both are quadratic forms constructed from vector random variables that (when normalized) converge in distribution to multivariate normal distributions.

This equivalence motivates our use of the theory for the behavior of
${X}_{n}^{2}$ to *M _{N}* as

The results with
${X}_{n}^{2}$ provided by [12] and [7] suggest that *k* should increase with *m* in many cases, however there are exceptions, as we show. Simulations indicate that in fact the optimal number of bins depends on the size of the cluster. We illustrate this point by example in the following section. We also observe that the optimal value for *k* does not appear to change as *m* increases.

We provide empirical examples of the effect of varying *k* on the power to detect a spatial cluster. The first examples arise from data of simulated locations based on a parametric model. A final example comes from a real dataset composed of the locations, aggregated by census tract, of patients with upper respiratory tract infections seeking care at an eastern Massachusetts healthcare provider. In both examples, spatial clusters are superimposed on the data and the power to detect these clusters with the *M* statistic is estimated empirically. The question that we address is how to determine the value of *k* for a given value of *m* such that the power to detect the cluster will be maximized.

The first examples that we present arise from simulated data. There are three scenarios considered here. First, locations are simulated on the unit line and distances are calculated as the difference between these values. Second, data is simulated from the uniform distribution on the unit square. The last model simulates cases according to a bivariate normal distribution, centered at the origin with variance covariance matrix given by the identity, **I**_{2}. For each of these models, *m* assumes four fixed values, 50, 100, 150 and 200 and we vary *k* over 31 values between two and 3000.

The bin breaks, *S* matrix and null distribution of the *M* statistic are calculated using resampling methods. The datasets used to calculate power contain a cluster placed in a region that composes 0.1, 0.01, 0.001, and 0.0001 of the probability space (these are heretofore denoted as *p*_{0}). The number of points placed in the cluster region is *p _{A}m*, where

Configuration for the simulations for the Unit Square and Bivariate Normal scenarios when *p*_{A} = 0.1 or *p*_{A} = 0.001. The x's denote the points in the cluster and the o's denote non cluster points.

Power for the dense cluster model simulations. Within each plot the powers for *m* = 50, 100, 150, and 200 are shown. Each column represents a different value for *p*_{0}, indicated by the title. Power decreases with *m*. Each row of plots corresponds to a different **...**

Power for the diffuse cluster model simulations. Within each plot the power for *m* = 50, 100, 150, and 200 are shown. Power is monotonically decreasing with *m*. The x axis denotes values of *k*, where *k* ranges between 2 and 3000.

In the first example, data are generated from a uniform distribution on the interval [0, 1]. The cluster comes from a uniform random variable with positive probability centered at 0.50. The probabilities of points being in the cluster region under the cluster model, *p _{A}*, were 0.2253, 0.0700, 0.0600, and 0.0500 to correspond with the four values of

The third and fourth rows of plots in Figure 2 illustrate the power to detect a dense cluster superimposed on spatial data uniformly distributed on a unit square. Four single cluster models were simulated with clusters centered at (0.5, 0.5), (0.25, 0.25), (0.16, 0.16) (on the corner of the region), and (0.5, 0.25). We report only those results from the cluster being centered at (0.5, 0.5) since the results were consistent, regardless of cluster location. The values of *p _{A}* considered were 0.2253, 0.1000, 0.0600, and 0.0500. Additionally we consider a double cluster model with clusters centered at (0.25, 0.25) and (0.75, 0.50) and the values of

The final simulated example arises from data that are generated from a bivariate normal distribution centered at the origin and with variance covariance matrix of **I**_{2}. The values for *p _{A}* used in this scenario are 0.2253, 0.1000, 0.0600, and 0.0500. The points of the clusters are obtained by identifying a square region centered at the origin that contains approximately

From these examples the optimal value of *k* does not appear to be increasing with *m*. Also, the best performing value of *k* depends on the cluster size. Figure 4 illustrates the reason for this, which is that a key disturbance created in the distance distribution comes from the interpoint distances within the cluster. So, clusters that are very small create a bump in the distribution around very small distances. As the cluster size increases, the area of the distribution that is affected increases. Power is likely to be optimized when the disturbance is contained within a single bin. Figure 4 illustrates this phenomenon and Table 2 shows the length of the first bin compared to the size of the cluster. The dependence of the optimal value of *k* on the size of the cluster poses challenges for exploratory work and generally in surveillance settings where the size of the cluster that one is interested in is likely not known a priori. Additionally it is not clear how to best choose *k* if the spatial abnormality is not a cluster, but some other modification in the distribution.

The null (black) and alternative densities (dotted) of the distances for the unit square simulations. The x axis represents distance. The width of the first bin for *k* = 5, 50, 100, and 200 are shown as vertical dotted lines with the optimal width being **...**

This study is based on data collected over a period of 1399 days from a large healthcare provider in eastern Massachusetts. The data contains the locations, aggregated by census tract, of patients with upper respiratory tract infections (URI). This information is downloaded from a central database daily. The data has been previously analyzed in [4]. Days with case counts less than six were excluded from the analysis. The average daily case count for the days included in the analysis was 40.12 (sd = 22.30). We use this data to calculate the bin breaks, *S* matrix and null distribution of *M*.

In this example, *m* is varying from day to day. Additionally, the location data is aggregated. The impact of these two features on the choice of *k* is not clear. Therefore, these simulations are useful for observing how *k* behaves in this more realistic scenario.

We consider three alternative models. The first two contain single clusters of size six. The final model is a combination of the previous two, containing both clusters used in the previous example simultaneously, in other words, there are 12 points that are in clusters. For each of these three examples, *k* ranged over eight values between 2 and 50. The results of these simulations are given in Figure 5. The power curves demonstrate the same basic pattern observed in the previous simulated examples. That is, we see a sharp initial increase in power as *k* increases and then a very gradual decline as *k* approaches and extends beyond *m*.

These results are consistent with those observed for the simulated data suggesting that the clusters comprise a relatively large part of the probability space. In fact, based on our observations here, it is reasonable to suggest a test that is performed for varying *k* and the final result chosen to be for the value of *k* that optimizes some criteria, analogous to the AIC criterion. For cluster models, this appears to be a reasonable approach. Additionally, we note that the differences in power from choosing *k* within a reasonably broad range of values are not dramatic.

We noted the equivalence of the *M* and
${X}_{n}^{2}$ statistics, implying that the underlying distribution of the observations is unimportant, as long as appropriate variance covariance matrix is used when constructing the quadratic form. In other words, we can say that
${X}_{n}^{2}$ is a special case of a broader class of statistics composed of quadratic forms that can be constructed as

$$Q={\text{a}}_{n}^{\top}{\sum}^{-}{\text{a}}_{n}$$

where Σ is the *k* × *k* variance covariance matrix, with rank *k* − 1 of the *k* × 1 vector **a**. In the case of
${X}_{n}^{2}$, the vector of data follows a multinomial distribution and the variance covariance matrix takes on a known form. On the other hand, for the M statistic, in general, it difficult to specify the variance covariance matrix. In practice we use resampling methods to estimate the variance covariance matrix. In simple cases, such as the unit line, we can obtain an exact form of the variance covariance matrix.

Pearson's Goodness-of-Fit test and the *M* statistic compare an observed distribution with a hypothesized distribution, the null distribution, and quantify the differences between the two. While these statistics are intuitive and very useful, there can be some difficulty in implementing them with continuous data. As we have seen, the power of the test is impacted substantially by the choice of the type and number of classes.

We confirm the recommendation of using of equiprobable binning also with *M*, unless there is prior interest in detecting a particular alternative. In that case one should use a finer probability grid in the vicinity of the expected deviation and a coarser equiprobable grid elsewhere.

The power of the test is affected by the selection of the number of discrete categories, *k*. Our simulations indicate that in the case of spatial cluster models, the best performing number of bins depends on the size of the cluster to be detected. This would argue for a modification to the *M* statistic that would allow one to search over a range of values of *k*. The optimal choice of *k* will also be informative in indicating the size of the cluster that has been detected. Additionally, our simulations affirm that overall power increases as *m* increases, regardless of the choice of *k*.

We recognize that these guidelines have limitations. First, the results are based on asymptotic theory of the behavior of the power as *k* increases with respect to *n* or *m*. In application, the point where the asymptotic results provide an accurate guide may not be attained. Therefore, it is possible that the rule of thumb given may not provide the best value of *k* for a practical dataset. It is hoped that by examining simulations, we can shed some insight on finite sample behaviors. It should also be noted that the results obtained for
${X}_{n}^{2}$ often contradict each other. For instance, [7] and [12] approach the same problem of describing the behavior of *k* as *n* goes to infinity and find results that mostly coincide, but do differ significantly in some cases. This, and the large body of literature that has attempted to address this problem, lead to the conclusion that this is clearly not a simple issue, and that to this point it has not yet been addressed in a definitive way. We contend, however, that the literature that does exist and the results shown in this paper are sufficient to provide guidelines that appear to be reasonable for this problem.

^{*}Research supported in part by grants from the National Institutes of Health AI28076 and T32 AI007358, and NLM ILM007677.

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