Background

Many tests for spatial randomness that adjust for a heterogeneous background population have been proposed. These test statistics are used to test whether the geographical distribution of disease is random or not. They are also used in many other areas such as geomorphology, ecology, genetics and geography (See, e.g., Fotheringham et al. [1], Gatrell et al. [2], Ruiz-Garcia [3], Aubry and Piegay [4], Clark and Richardson [5], Liebhold and Gurevitch [6], Gustine and Elwinger [7], Meirmans et al. [8]).

Among these test statistics, some are global clustering tests used to evaluate the presence of clustering throughout the study region. Others are used to detect and evaluate local clusters. In this paper, we are only concerned with the former. Examples of global clustering tests are Tango's maximized excess events test (MEET) [9], Cuzick and Edwards' k nearest neighbors (k-NN) [10] and Moran's I [11].

When we are using a global clustering test, it is important that it has good statistical power. We have previously [12, 13] evaluated the power of seven global clustering tests: Besag-Newell's R [14], Bonetti-Pagano's M statistic [15], Cuzick-Edwards' k-NN, Moran's I, Swartz' Entropy test [16], Tango's MEET and Whittemore's test [17]. The power varies greatly for different test statistics and Tango's MEET has the best power overall.

Tango's MEET depends on a weight function. Tango proposed a distance based exponential weight function for MEET', but other choices of weights are also possible. In this paper, we evaluate Tango's MEET using different weight functions. Nine weight functions are evaluated, and the power varies greatly with different choice of weight.

Methods

Notation

Denote c i as the number of cases in county i, n i as the population size of county i, C as the total number of cases, N as the total population size, H as the total number of counties, d ij as the distance between county i and j, uj(i) as the population size in county i and its j nearest neighbors. The maximum distance between county i and the other counties under study is denoted by dmax i = max1≤j≤nd ij .

Tango's MEET

Tango's [9] MEET is a maximized version of Tango's excess events test (EET) [18]. We first describe the latter.

For a given weight function w ij , Tango's EET is a weighted sum of excess events defined as

Tango proposed two distance based exponential weight functions

[9] and [18], where λ is a measure of the spatial scale of clustering. To avoid confusion with other weight functions, we denote the EET defined by these two distance based exponential weight functions as

and

DE1_EET and DE2_EET depends on the scale parameter λ. To be able to detect clustering irrespectively of its geographical scale, Tango [9] proposed the maximized excess events test (MEET). We use notation DE1_MEET and DE2_MEET to denote the maximized tests of DE1_EET and DE2_EET respectively, which are defined as

and

where de1_eet(λ) and de2_eet(λ) are the observed values of DE1_EET(λ) and DE2_EET(λ) conditioning on λ. U is an upper limit on λ. Basically, the maximized test is using the minimum of the profile p-values as the test statistics adjusting for the multiple testing resulting from the many parameter values considered.

Alternative weight functions

Since Tango's MEET performs very well, it is of interest to evaluate other potential weight functions, of which there are many. Nine weight functions including Tango's distance based exponential weights are evaluated in this paper. Five of them depend on a spatial scale parameter while four of them do not.

For all weights function, the weight decreases with increasing distance. The metric used for the decrease is different though. For example, the weight may be defined on Euclidean distance and depends only on distance. It may also be adjusted with population density, so that the weight declines faster in urban than in rural areas. We can also define the weight in terms of spatial contiguity of counties irrespective of the population density. Other choices of weight functions may take geographical or population size into consideration. We describe our weight functions next.

Population density adjusted exponential weight

The scale of the spatial clustering usually depends on the underlying population. It will be reasonable to adjust the weight function with the underlying population density. We define this weight function as

, where and m i = max{j : uj(i) ≤ k}. The parameter k is set by the user and can be viewed as a population measure for the clustering. Note that for a given k, λ i in the rural area, which has a small population density, is larger. This means the hazard rate in the rural area decreases slower than that of urban area as the Euclidean distance becomes large. Usually, large k is more sensitive to large clusters and small k is more sensitive to small clusters. To study the strength of the parameter, we take the value of k equal to 50%, 25%, 10% and 5% of the overall population. We denote the test statistic with this weight function as PE_EET and

Nearest neighbor adjusted weight

Another potential weight function, based on the nearest neighbors property, is defined by

, where l indicates that county j is the lth closest county to county i. So the weight for county i itself is 1, the weight for its closest neighbor is , the weight for its second closest neighbor is , and so on. This weight function is based on spatial contiguity of counties adjusted with distance. It may be desirable when the hazard risk does not decrease proportionally with distance. A small value of the parameter s will give more weight to the counties far from county i, while a large s will give more weight to county i and its closest neighbors. We set s = 0.1, 0.5, 1, 2, 8 to study the property of the parameter. We denote the EET test statistic with this weight as

Distance adjusted weight

The next weight function is defined by

. This weight function gives more weight to the counties that are geographically close to each other. The 1 in the denominator is used to adjust the weight so that when the distance is very small, the weight will not be too large. H is used to adjust the weight with the total number of counties. The EET test with this weight is written as

Distance and area adjusted weight

For a different spatial statistical method, Gangnon and Clayton [19] used the weight function , where a i denotes the area of county i, A denotes the total area of all counties. The test statistic is denoted as

Distance and population adjusted weight

By replacing the area size in the above weight function with population size, we get another weight function

. The test statistic is

Adjacent neighbor weight

If we define two individual persons to be neighbors if they are in the same county or neighboring county, then we can get the stepwise weight function

Test statistic with this weight is

Population based weight

Another possible weight function is to use the product of the proportion of the corresponding counties to the total population, so that

. The test statistic is then

Note that this population based weight does not take into account any distance information between counties.

Maximized tests over spatial scale parameters

Three EET tests, PE_EET, NN_EET and D_EET, depend on a parameter. By using Tango's maximization technique, which uses the minimum profile p-value of EET for the parameter, we get the maximized tests for these three tests. For PE_EET(k), the MEET is defined as

where pe_eet(k) is the observed value of the excess events test statistic conditioning on k, and V is an upper limit on k. H0 denotes the null hypothesis of no spatial correlation for the data. Our implementation of the test is carried out by choosing k as 5%, 10%, 15%, ..., 50% of the population.

Similarly, for NN_EET(s), we define the MEET as

where nn_eet(s) is the observed value of the excess events test statistic conditioning on s. The implementation of this test is carried out by choosing s as 0.1, 0.25, 0.5, 1, 1.5, 2, 4, 8, 10. The MEET for D_EET(s) is similar to NN_EET(s), and it is defined as

The implementation of this test is carried out by the same collection of s as for NN_MEET.

Benchmark data

To evaluate statistical power, we used a collection of benchmark data sets based on the 1990 female population in the 245 counties and county equivalents in the northeastern United States, consisting of the states of Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, Connecticut, New York, New Jersey, Pennsylvania, Delaware, Maryland and the District of Columbia. The benchmark data has been described in detail elsewhere [13]. It can be downloaded at 'http://www.satscan.org/datasets'.

Under the null hypothesis of no clustering, 99,999 random data sets were generated by randomly allocating 600 cases to various counties, with the probabilities proportional to the county population. The null data is used to estimate the critical values, which is the cut-off point for the significance.

For each clustering model, 10,000 random data sets were used to estimate the power. The counties are tied together sequentially on a chain that passes through each county exactly once, after which it reconnects with the first county on the chain, forming a Hamiltonian cycle. A map of the Hamiltonian cycle used has been illustrated in figure 1. The clusters are generated by first locating 300 cases randomly on the map under the null hypothesis. Then each of these original cases generates one new case for a total of 600. There are three types of clustering models with the distance between the twins along the chain being either constant or exponentially distributed with different means. For the first type of clustering models, the distance between twins is zero, which means the twins are always in the same county. For the second type of clustering, six clustering models were constructed by setting the distance between twins to be fixed with the mean corresponding to 0.5%, 1%, 2%, 4%, 8% and 16% of the overall population along the chain. For the third type of clustering models, the distance was set to be exponentially distributed and span over 0.5%, 1%, 2%, 4%, 8% and 16% of the overall population size. The Hamiltonian cycle does not imply that the disease itself spreads around the chain, just that twin cases are located in either of the two directions, as defined by the chain.

Figure 1
figure 1

Hamiltonian chain of counties used for the global chain clustering.

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Results

Table 1 shows the estimated power of Tango's EET and MEET with different choice of weight. The highest power for each clustering model is highlighted. The power varies greatly with different choice of weight. PE_MEET, DE1_MEET, DE2_MEET, NN_MEET and D_MEET all have very good power. DA_EET, DP_EET and N_EET have good power for some clustering models, but not as good as the MEET tests. P_EET does not perform well. All tests have lower power as the distance between twins increases since there is less clustering in the data.

Table 1 Power of the test statistics for the global twin clustering. The row variable denotes the test statistics. The column variable denotes the clustering models. The last column is the average power for each test statistic.
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In Table 2, we present the estimated power of the four weight functions that depend on a parameter. For each weight function, we choose different parameter values that can represent its overall strength and compare them with the maximized test. For DE1_EET(λ), DE2_EET(λ) and PE_EET(k), large parameter value is more sensitive to clustering with large scale, while small parameter value is more sensitive to clustering with small scale. For NN_EET(s) and D_EET(s), large parameter value is more sensitive to clustering with small distance, while small parameter value is more sensitive to clustering with large distance. DE1_EET(λ) and DE2_EET(λ) have very similar performance, but DE2_EET(λ) is more sensitive to the choice of parameter. For all the five tests, the maximized tests have less power compared to the maximum power that can be obtained by the EET test with an appropriately chosen parameter, but the maximized test has reasonably good power overall. For example, for the fixed 1% distance clustering, the maximized test PE_MEET has the power of 0.73. The original test PE_EET(k) has the power of 0.75 with k equal to 5% or 10% of the population, but it has a low power of 0.42 when k is 50% of the population.

Table 2 Power of the test statistics for the global twin clustering using different spatial scale parameters. The row variable denotes the test statistics. The column variable denotes the clustering models. The last column is the average power for each test statistic.
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Discussion

In this paper, we evaluated Tango's EET and MEET using both used and unused weight functions. The power can vary greatly with the choice of weight. This indicates that for global clustering test, consideration of weight is important. For the weight functions that incorporate good distance information, the power of the test is much better than the weight functions that do not incorporate the spatial relationship between counties.

With reasonable parametric distance based weights, the power of Tango's MEET is rather robust. For this study, PE_MEET, DE1_MEET, DE2_MEET, NN_MEET and D_MEET all have good power, and their average power for all clustering models considered are very similar.

Tango's DE1_MEET and DE2_MEET scan over the study area by distance. These two tests have similar performance. Both tests are based on the summation of the weighted excess events and they collect clustering information throughout the map, which makes them good global tests. Previous studies [12, 13] indicated that DE2_MEET perform well when the cluster is large in population size.

For the clustering models considered in this paper, PE_MEET performs a little better than the tests with other weights. The reason for this may be due to the way that the data were generated based on the population density. We believe some of the test statistics may have better strength under other alternate models. We use the female population in the 245 counties and county equivalent in Northeastern United States as the underlying population. It is possible that the relative strength of the various test statistics may be different for other underlying population or different alternative clustering models. The type of power evaluations done in this paper are, in spite of these limitations, very important. For practical applications, the power estimates presented in this paper provides some help when we choose a test.

Conclusion

The power of Tango's MEET varies greatly with different choice of weight. In general, with reasonable parametric distance based weights, the power of Tango's MEET is robust. Tango's original choice for the weight function works well. At the same time, there are also other weight functions for which the test has good power.