Tango's maximized excess events test with different weights
 Changhong Song^{1}Email author and
 Martin Kulldorff^{2}
DOI: 10.1186/1476072X432
© Song and Kulldorff; licensee BioMed Central Ltd. 2005
Received: 25 July 2005
Accepted: 15 December 2005
Published: 15 December 2005
Abstract
Background
Tango's maximized excess events test (MEET) has been shown to have very good statistical power in detecting global disease clustering. A nice feature of this test is that it considers a range of spatial scale parameters, adjusting for the multiple testing. This means that it has good power to detect a wide range of clustering processes. The test depends on the functional form of a weight function, and it is unknown how sensitive the test is to the choice of this weight function and what function provides optimal power for different clustering processes. In this study, we evaluate the performance of the test for a wide range of weight functions.
Results
The power varies greatly with different choice of weight. Tango's original choice for the weight function works very well. There are also other weight functions that provide good power.
Conclusion
We recommend the use of Tango's MEET to test global disease clustering, either with the original weight or one of the alternate weights that have good power.
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], RuizGarcia [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 (kNN) [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: BesagNewell's R [14], BonettiPagano's M statistic [15], CuzickEdwards' kNN, 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, u_{j(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 }= max_{1≤j≤n}d_{ 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
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 pvalues 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
Nearest neighbor adjusted weight
Distance adjusted weight
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
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
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 pvalue 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. H_{0} 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 cutoff point for the significance.
Results
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.
Fixed distance  Exponential distance  

0.00  0.5%  1%  2%  4%  8%  16%  0.5%  1%  2%  4%  8%  16%  average  
P_EET  0.16  0.15  0.14  0.12  0.10  0.07  0.04  0.15  0.14  0.13  0.11  0.09  0.07  0.11 
DA_EET  0.91  0.35  0.19  0.09  0.06  0.05  0.05  0.48  0.32  0.20  0.13  0.08  0.06  0.23 
DP_EET  0.64  0.42  0.27  0.12  0.07  0.06  0.04  0.46  0.34  0.23  0.15  0.10  0.07  0.23 
N_EET  0.71  0.57  0.46  0.30  0.13  0.07  0.05  0.61  0.51  0.38  0.25  0.15  0.09  0.33 
NN_MEET  0.99  0.68  0.45  0.27  0.16  0.10  0.07  0.79  0.62  0.42  0.27  0.17  0.10  0.39 
D_MEET  0.99  0.67  0.42  0.24  0.15  0.10  0.07  0.78  0.61  0.41  0.26  0.16  0.10  0.38 
PE_MEET  0.98  0.73  0.51  0.31  0.17  0.10  0.06  0.81  0.65  0.46  0.29  0.18  0.11  0.41 
DE1_MEET  0.99  0.64  0.41  0.26  0.18  0.12  0.07  0.75  0.57  0.39  0.26  0.17  0.11  0.38 
DE2_MEET  0.99  0.62  0.41  0.26  0.17  0.11  0.06  0.74  0.56  0.38  0.25  0.17  0.11  0.37 
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.
Fixed distance  Exponential distance  

0.00  0.5%  1%  2%  4%  8%  16%  0.5%  1%  2%  4%  8%  16%  average  
DE1_EET(λ) with  
λ = 66, 000  0.37  0.32  0.29  0.26  0.20  0.14  0.08  0.32  0.30  0.26  0.22  0.17  0.11  0.23 
λ = 32, 000  0.44  0.36  0.32  0.27  0.21  0.14  0.08  0.37  0.33  0.29  0.23  0.17  0.12  0.26 
λ = 15, 000  0.58  0.43  0.37  0.29  0.22  0.14  0.07  0.46  0.40  0.33  0.25  0.18  0.12  0.29 
λ = 4, 000  0.95  0.59  0.41  0.26  0.16  0.10  0.06  0.68  0.54  0.38  0.25  0.16  0.10  0.36 
λ = 500  1.00  0.67  0.34  0.13  0.06  0.06  0.06  0.80  0.60  0.37  0.20  0.12  0.07  0.34 
DE1_MEET  0.99  0.64  0.41  0.26  0.18  0.12  0.07  0.75  0.57  0.39  0.26  0.17  0.11  0.38 
DE2_EET(λ) with  
λ = 66, 000  0.30  0.28  0.27  0.25  0.20  0.14  0.08  0.28  0.27  0.25  0.21  0.16  0.11  0.22 
λ = 32, 000  0.42  0.37  0.33  0.28  0.21  0.13  0.06  0.38  0.34  0.29  0.23  0.17  0.11  0.25 
λ = 15,000  0.68  0.47  0.38  0.28  0.18  0.11  0.06  0.51  0.43  0.33  0.24  0.16  0.10  0.30 
λ = 4, 000  0.99  0.61  0.34  0.17  0.10  0.07  0.05  0.74  0.55  0.35  0.21  0.13  0.08  0.34 
λ = 500  1.00  0.66  0.33  0.13  0.06  0.06  0.06  0.80  0.59  0.36  0.19  0.11  0.07  0.34 
DE2_MEET  0.99  0.62  0.41  0.26  0.17  0.11  0.06  0.74  0.56  0.38  0.25  0.17  0.11  0.37 
PE_EET(k) with  
k = 50%population  0.48  0.42  0.37  0.31  0.22  0.13  0.07  0.43  0.38  0.33  0.25  0.18  0.11  0.28 
k = 25%population  0.72  0.57  0.47  0.33  0.20  0.10  0.05  0.60  0.52  0.41  0.28  0.18  0.11  0.35 
k = 10%population  0.96  0.75  0.53  0.27  0.11  0.06  0.05  0.81  0.66  0.47  0.28  0.15  0.09  0.40 
k = 5%population  0.99  0.75  0.45  0.17  0.06  0.06  0.06  0.84  0.67  0.43  0.23  0.13  0.07  0.38 
PE_MEET  0.98  0.73  0.51  0.31  0.17  0.10  0.06  0.81  0.65  0.46  0.29  0.18  0.11  0.41 
NN_EET(s) with  
s = 0.1  0.72  0.52  0.42  0.32  0.22  0.13  0.07  0.56  0.48  0.38  0.28  0.19  0.12  0.34 
s = 0.5  0.93  0.64  0.47  0.31  0.18  0.11  0.06  0.71  0.59  0.43  0.29  0.18  0.11  0.39 
s = 1  0.99  0.70  0.46  0.25  0.13  0.08  0.06  0.79  0.64  0.44  0.27  0.16  0.09  0.39 
s = 2  1.00  0.70  0.40  0.17  0.08  0.06  0.06  0.82  0.63  0.41  0.23  0.13  0.08  0.36 
s = 8  1.00  0.66  0.33  0.13  0.06  0.06  0.06  0.80  0.59  0.36  0.19  0.11  0.07  0.34 
NN_MEET  0.99  0.68  0.45  0.27  0.16  0.10  0.07  0.79  0.62  0.42  0.27  0.17  0.10  0.39 
D_EET(s) with  
s = 0.1  0.91  0.57  0.41  0.28  0.19  0.13  0.07  0.65  0.52  0.39  0.27  0.19  0.11  0.36 
s = 0.5  0.99  0.68  0.44  0.25  0.15  0.10  0.06  0.78  0.62  0.43  0.28  0.17  0.10  0.37 
s = 1  1.00  0.69  0.39  0.18  0.09  0.07  0.06  0.81  0.62  0.41  0.23  0.14  0.08  0.38 
s = 2  1.00  0.67  0.34  0.13  0.06  0.06  0.06  0.80  0.60  0.37  0.20  0.12  0.07  0.34 
s = 8  1.00  0.66  0.33  0.13  0.06  0.06  0.06  0.80  0.59  0.36  0.19  0.11  0.07  0.34 
D_MEET  0.99  0.67  0.42  0.24  0.15  0.10  0.07  0.78  0.61  0.41  0.26  0.16  0.10  0.38 
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.
List of abbreviations
 EET:

Excess Events Test.
 MEET:

Maximized Excess Events Test.
Declarations
Acknowledgements
This study was funded by grant RO1CA09597901 from the National Cancer Institute.
Authors’ Affiliations
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