Scale and shape issues in focused cluster power for count data
- Robin C Puett^{1, 2}Email author,
- Andrew B Lawson^{1},
- Allan B Clark^{3},
- Tim E Aldrich^{1},
- Dwayne E Porter^{2},
- Charles E Feigley^{2} and
- James R Hebert^{1, 4}
https://doi.org/10.1186/1476-072X-4-8
© Puett et al; licensee BioMed Central Ltd. 2005
Received: 07 February 2005
Accepted: 31 March 2005
Published: 31 March 2005
Abstract
Background
Interest in the development of statistical methods for disease cluster detection has experienced rapid growth in recent years. Evaluations of statistical power provide important information for the selection of an appropriate statistical method in environmentally-related disease cluster investigations. Published power evaluations have not yet addressed the use of models for focused cluster detection and have not fully investigated the issues of disease cluster scale and shape. As meteorological and other factors can impact the dispersion of environmental toxicants, it follows that environmental exposures and associated diseases can be dispersed in a variety of spatial patterns. This study simulates disease clusters in a variety of shapes and scales around a centrally located single pollution source. We evaluate the power of a range of focused cluster tests and generalized linear models to detect these various cluster shapes and scales for count data.
Results
In general, the power of hypothesis tests and models to detect focused clusters improved when the test or model included parameters specific to the shape of cluster being examined (i.e. inclusion of a function for direction improved power of models to detect clustering with an angular effect). However, power to detect clusters where the risk peaked and then declined was limited.
Conclusion
Findings from this investigation show sizeable changes in power according to the scale and shape of the cluster and the test or model applied. These findings demonstrate the importance of selecting a test or model with functions appropriate to detect the spatial pattern of the disease cluster.
Background
Over the past several years, there has been an increased interest in the detection of focused clusters of disease, or disease clusters associated with a known pollution source [1]. Along with this interest, the development and use of various cluster detection methods have grown. Wartenberg and Greenberg [2] emphasized the importance of statistical power as a criterion in the selection of appropriate method for cluster investigations. A limited number of power evaluations in the literature have examined the issues of scale or shape with regard to focused disease cluster detection. As part of a more extensive evaluation, Sun [3] examined the power of Stone's [4] and Tango's Tests [5] to detect clusters of varying size. Waller [6] described two shapes of clusters, hot spot and clinal [2], and evaluated the power of Stone's and Besag and Newell's Tests to detect these types of clusters at varying levels of aggregation. Additional focused cluster shapes also have been proposed, such as a cluster of angular shape which could result from the effect of a dominant wind dispersing environmental contaminants [7]. However, this cluster shape has not been included in previous power evaluations. Additionally, information is lacking regarding the power of generalized linear models to detect focused clusters of varying scale and shape. This investigation evaluates the power of a number of focused cluster tests and generalized linear models to detect clusters of varying scale and shape. Count data are simulated for three total numbers of events (N = 200, 500 and 1000) to represent clustering around a single, centrally located pollution source in spatial patterns consistent with pollution dispersion principles.
Results
Focused cluster test results
Data simulated with distance decline (DD) in relative risk
Data simulated with peaked distance decline (PDD) in relative risk
Data simulated with a directional effect (DIR) in relative risk
Data simulated with a distance decline and directional effect (DDIR) in relative risk
Data simulated with a peaked distance decline and directional effect (PDDIR) in relative risk
Focused cluster model results
Comparison of the distance decline relative risk model to a model including a peaked distance decline function showed that no power was gained from adding the parameter for a peak. Power was less than 10% for all total event scenarios and variations of the α coefficients. However, better power curves resulted from other model comparisons. At values of α_{3} and α_{4} = 0.5 and 2, adding direction as a parameter to models with a function for distance decline vastly improved power for detecting clusters with distance decline and directional distributions. The increase in power was low for the addition of the directional parameter when α_{3} and α_{4} = 0.1. Similar trends were evident for the addition of parameters for a peak and for direction when compared to a distance decline only relative risk model. In general, power was high at all but the highest α_{1} value (α_{1} = 10) when α_{3} and α_{4} = 0.5 and 2. Yet, improvements in power were not very large for lower levels of α_{3} and α_{4} (α_{3} and α_{4} < 0.2).
Under certain conditions, power greatly increased when parameters were added to a directional only model. When comparing a directional only model to a model with parameters for distance decline and direction, power tended to increase with increasing α_{1}, α_{3} and α_{4} values. However a slight decrease in power was evident at α_{1} = 10 for most model comparisons. The power curves were very similar to those for the comparison between a directional-only model to models with parameters for peaked distance decline and direction at lower values of α_{2} (α_{2} < 0.5). At higher levels of α_{2} (α_{2} > 0.1), power trends differed greatly, with the highest power at lowest levels of α_{1} and greater decreases as α_{1} increased to 10.
Power curves from models with parameters for peaked distance decline were also compared to models with parameters for peaked distance decline and direction. Generally, power trends were similar for the two α_{2} levels with power inversely related to α_{1} value and directly related to α_{3} and α_{4} values. The improvement in power was very low for the addition of the directional component at the lowest α_{3} and α_{4} values (α_{3} and α_{4} < 0.2).
As with previous model comparisons, the addition of a parameter for peak generally resulted in low power when parameters for distance decline and direction were already in the model. The highest power achieved was modest (about 40%) and was evident only for 1000 events at the highest values for α_{2}, α_{3} and α_{4} values (α_{2} = 1, α_{3} and α_{4} = 2) at α_{1} = 1.
Conclusion
Focused cluster tests
Based on the results presented, we found that Tango's Test, the LRS Test for Distance Decline, Stone's Test and the Radial Score Test showed the most power for detecting a significant difference in relative risk simulated with DD from a centrally-located pollution source at a fixed location. As expected, power increased directly with number of total cases of disease involved in the cluster. For these tests, power also tended to increase as the slope of the DD became steeper or, in other words, as the relative risk changed more rapidly with increasing distance from the pollution source. The results of our power evaluations are not surprising as the best-performing cluster tests for detecting radial clusters with DD were generally developed for detecting these cluster shapes.
Power for detecting PDD clusters with N = 200 was generally low, with reasonable power demonstrated at N = 500 and N = 1000 for some combinations of peak distances from the pollution source and slopes of decline in risk. Evaluating the power of focused cluster tests to detect PDD clustering with peaks closest to the source revealed that the LRS Test for Distance Decline, Tango's Test, Stone's Test and the Radial Score Test showed the highest power among the tests evaluated. Power generally increased for these tests as the slope of the cluster became steeper. This trend reversed as peaks increased in distance from the pollution source so that power generally decreased as the slope became steeper. The Radial Score Test demonstrated the most power for detecting peaks furthest from the pollution source for N = 500 and N = 1000, while Besag and Newell's Test performed best for N = 200.
Supporting our hypothesis that focused cluster tests containing functions appropriate to the spatial pattern of pollution dispersion would be more powerful, the LRS test for direction and the Directional Score Tests revealed the most power for detecting focused clusters with DIR. Power increased for these tests with increased number of total events and as the angle of effect became more pronounced.
For data simulated with DDIR, the LRS Test for Direction performed best with wider angles of effect and the flattest declines in slope. But as the angle of effect became stronger, the LRS Test for Direction was powerful at all slopes. Overall, power to detect clusters with a DDIR distribution generally increased with narrower angles of effect and as the total number of events simulated increased. As the slope of the decline became steeper, Tango's Test, LRS Test for Distance Decline, Stone's Test and the Radial Score Test showed comparable power. Power for the Directional Score Test greatly improved with stronger angles of effect, becoming one of the best-performing tests for all slope levels. To summarize the power evaluation results for detecting DDIR clusters, directional tests demonstrated the most power with flatter declines in cluster slopes; whereas the radial distance decline tests were more powerful with steeper declines in slope. Both types of test improved with narrowing angles of effect. With those cluster patterns, directional tests became more powerful at all slope levels and distance decline tests remained powerful with steeper slopes.
In order to simulate data with clusters of PDDIR, only one component was added to the simulation equation for clusters of DDIR: α_{2} * log(d). Therefore, as one would expect, at very low values of α_{2} (e.g. 0.005), the results of power evaluations for detecting PDDIR clusters were very similar to those for detecting DDIR clusters. However, as α_{2} values increased, or as the peak of the cluster increased in distance from the pollution source, findings differed. With the widest angles, overall power for detecting PDDIR with N = 200 was low. At these same angles, the Radial Score Test proved best for detecting clustering with N = 500 and N = 1000. This test decreased in power with increasing steepness of slope. As the angle of the directional effect narrowed, the LRS Test for Direction and the Directional Score Test also showed comparable power, particularly with a greater number of events. The power for these tests decreased as the cluster slope flattened. Interestingly, the Radial Score Test also followed this trend and demonstrated power second only to the directional tests. These results show that although the Radial Score Test could detect DIR effects combined with PDD effects, it was generally less powerful than directional tests at the narrowest angles of effect.
Focused cluster models
Overall, the addition of model parameters for peaks did not appear to contribute to improved power, particularly when a distance decline parameter was already included in the model. Only at α_{1} < 1, or the flattest slope, did the null versus peaked distance decline model comparison show higher power than the null versus distance decline model comparison. Also, very low power resulted when comparing the distance decline only model to the peaked distance decline model and when adding a parameter for peak to models already containing distance decline and directional parameters. Lastly, the power curves in which only a directional component was added are very similar to those resulting from the addition of peaked and directional components. Low power to detect a peaked distance decline cluster of elevated risk may be related to the variation in relative risk. For example, Figure 3 shows higher power for lower values of α_{1} when α_{2} = 1. For α_{2} = 1, the range of relative risk is much greater at lower values of α_{1} and decreased directly with α_{1}.
The addition of a directional parameter improved the power of tests from models, particularly for detecting disease clusters distributed over narrower angles of effect. Evidence of this outcome was provided by the power curves resulting from the comparison between the model of null relative risk and the model with a directional component. When a directional parameter was added to distance decline models and to peaked distance decline models, power was very high for clustering with gentler declines (α_{1} < 2) and narrower angles (α_{3} and α_{4} > 0.1). However, power tended to decrease as α_{1} increased from 2 to 10 and as the angle of effect widened, or as α_{3} and α_{4} increased.
Similarly, distance decline also appears to be an important parameter to include in focused cluster models, with respect to power. In opposition to the directional component, parameters for distance decline appeared most beneficial with clusters of steeper declines (larger α_{1} values). However, this observation may be due to the main effect extending outside the window of the simulated region for clusters with gentler declines. Comparison of the null model to the distance decline model showed power generally increasing as the steepness of the cluster slope increased. Also, the addition of a parameter for distance decline to a directional model showed similar power curves. One interesting effect in many of the model comparisons involving a distance decline parameter was the decrease in power from α_{1} = 2 to α_{1} = 10. This may be caused by the steepness of the slope resulting in fewer data points demarcating the decline in slope. As the number of observations decreases, power will also decrease.
Overall conclusions
Though a variety of spatial scales and shapes of clusters were examined, further power evaluations are needed in order to explore fully the range of disease clustering that could result from various pollution dispersion spatial patterns. Given the overall finding that more complex spatial cluster patterns can be more difficult to detect, the development and use of additional sampling schemes for power evaluations of these clusters would be beneficial. We chose to simulate three total numbers of events because count data are typically most accessible in cluster investigations. Work is currently underway to examine power for case-event data, however further investigations are needed to examine a greater range of total numbers of events for count data. Smaller numbers of total events are of particular importance in cluster investigations of rare diseases or sparsely populated areas. A number of variations were examined for the four α coefficients, yet a more comprehensive range of coefficients, representing additional changes in shape and scale, would provide important information. Other spatial components also should be examined, such as azimuth, which could be of primary concern for air pollution sources located in valleys. Additionally, these power evaluations were performed simulating dispersion from a single, centrally-located pollution source. Further power evaluations are needed to address cluster detection in situations where pollutants are dispersed from multiple sources. Williams and Ogston [8] compared observed and simulated spatial distributions of environmental exposures. Additional comparisons between measured levels of environmental pollutant spatial dispersion with those simulated here would also be useful in evaluating the accuracy of spatial functions in a variety of situations. Similar power evaluations of focused cluster tests and models using individual-level data are also needed to improve cluster investigation techniques, and work in this area is proceeding.
Sample Monte Carlo standard errors for focused cluster tests with N = 1000
Radial Score Test | DIR Score Test | Besag and Newell's Test (k= 7) | Cuzick and Edwards' Test (k= 7) | Tango's Test (τ= 5) | LRS DIR Test | LRS DD Test | Stone's Test | |
---|---|---|---|---|---|---|---|---|
DD | ||||||||
α_{1} = 0.005 | 0.02 | 0.02 | 0.01 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 |
α_{1} = 0.05 | 0.02 | 0.02 | 0.01 | 0.02 | 0.03 | 0.02 | 0.03 | 0.02 |
α_{1} = 0.1 | 0.02 | 0.02 | 0.01 | 0.02 | 0.03 | 0.02 | 0.03 | 0.03 |
α_{1} = 1 | 0.05 | 0.02 | 0.01 | 0.03 | 0.04 | 0.02 | 0.04 | 0.05 |
α_{1} = 2 | 0.03 | 0.02 | 0.01 | 0.04 | 0.02 | 0.03 | 0.01 | 0.03 |
α_{1} = 10 | 0.04 | 0.02 | 0.01 | 0.05 | 0.03 | 0.02 | 0.03 | 0.03 |
DIR | ||||||||
α_{3} and α_{4} = 0.005 | 0.03 | 0.04 | 0.01 | 0.02 | 0.03 | 0.04 | 0.03 | 0.02 |
α_{3} and α_{4} = 0.1 | 0.03 | 0.04 | 0.02 | 0.02 | 0.02 | 0.05 | 0.02 | 0.02 |
α_{3} and α_{4} = 0.2 | 0.03 | 0.03 | 0.01 | 0.01 | 0.03 | 0.01 | 0.03 | 0.02 |
α_{3} and α_{4} = 0.5 | 0.03 | 0.00 | 0.02 | 0.01 | 0.02 | 0.00 | 0.02 | 0.02 |
α_{3} and α_{4} = 1 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.00 | 0.03 | 0.02 |
α_{3} and α_{4} = 2 | 0.03 | 0.00 | 0.04 | 0.00 | 0.04 | 0.00 | 0.04 | 0.03 |
PKDD α_{1} = 0.005 | ||||||||
α_{2} = 0.05 | 0.03 | 0.02 | 0.02 | 0.02 | 0.01 | 0.02 | 0.02 | 0.01 |
α_{2} = 0.1 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.02 | 0.01 | 0.00 |
α_{2} = 0.5 | 0.03 | 0.02 | 0.04 | 0.01 | 0.00 | 0.02 | 0.00 | 0.00 |
α_{2} = 1 | 0.01 | 0.02 | 0.04 | 0.01 | 0.00 | 0.02 | 0.00 | 0.00 |
Sample Monte Carlo standard errors for fitted models with N = 1000
Null vs. DD | α_{1} = 0.005 | α_{1} = 0.05 | α_{1} = 0.1 | α_{1} = 1 | α_{1} = 2 | α_{1} = 10 |
---|---|---|---|---|---|---|
Standard Error | 0.02 | 0.02 | 0.03 | 0.05 | 0.02 | 0.04 |
Null vs. DIR | α _{3} and α_{4} = 0.005 | α_{3} and α_{4} = 0.1 | α_{3} and α_{4} = 0.2 | α_{3} and α_{4} = 0.5 | α_{3} and α_{4} = 1 | α_{3} and α_{4} = 2 |
Standard Error | 0.04 | 0.05 | 0.03 | 0.00 | 0.00 | 0.00 |
Null vs. PKDD α_{1} = 0.005 | α_{2} = 0.05 | α_{2} = 0.1 | α_{2} = 0.5 | α_{2} = 1 | ||
Standard Error | 0.03 | 0.03 | 0.04 | 0.00 |
This study examined the power of a number of focused cluster tests and generalized linear models to detect a wide range of simulated focused cluster shapes and scales. The results of this study provide information that can improve the choice of statistical method in focused cluster investigations. To summarize the overall findings from this investigation:
1) Focused cluster tests and tests from models containing functions appropriate to the spatial pattern of pollution dispersion are more powerful. DIR tests were more powerful detecting clusters with narrower angles of effect and DD tests were more powerful detecting clusters with steeper declines in slope.
2) Power increased with stronger DD (steeper slopes) and DIR (narrower angles) effects.
3) Power for detecting clusters with peaked effect patterns was generally low.
Methods
Data simulation
As count data are generally more widely available in disease mapping studies than individual-level data, count data were simulated for this study. A basic Poisson model was assumed for the counts: y_{ i }~ Poisson(e_{ i }θ_{ i }), where an expected count e_{ i }is modified by a relative risk θ_{ i }and y_{ i }, i = 1,...,M, is the count of disease in the ith region. Clusters of three sizes (N = 200, 500 and 1000 events) were simulated from the multinomial distribution, where the probability of a case in the ith region is .
Five shapes of clusters were simulated, including: 1) distance decline, where risk declines with increasing radial distance from the pollution source (DD, Model 1, Additional File 1: Simulated models of the five focused cluster shapes); 2) peaked distance decline, where risk peaks and then declines with increasing radial distance from the source (PDD, Model 2, Additional File 1); 3) direction, increasing disease risk in a particular angular direction from the pollution source (DIR, Model 3, Additional File 1); 4) distance decline combined with a directional effect (DDIR, Model 4, Additional File 1); and 5) peaked distance decline combined with a directional effect (PDDIR, Model 5, Additional File 1). These shapes correspond to frequently encountered air pollution dispersion patterns from point sources. For instance, shape 1 typifies dispersion from a ground level source with a relatively uniform distribution of wind directions, shape 3 represents ground level dispersion with a dominant wind direction and shape 5 represents dispersion from an elevated source with a dominant wind direction.
The count data simulated for these five cluster shapes under the alternative hypothesis as well as data simulated under the null hypothesis of randomly distributed counts of disease were assigned to regional centroids of a 16*16 unit square grid. The grid of 256 regions of uniform size and shape, unitless in geographic terms, also contained a centrally located pollution source. Expected disease rates were considered to be uniform throughout the regions composing the simulated study area. The model for the relative risk at location x, θ(x, β) represents the relationship between the pollution source and spatial distribution of associated disease, for some choice of parameters β. As shown in Additional File 1: Simulated models of the five focused cluster shapes, coefficients in the model equations were varied in order to represent further variations of scale for the five main cluster shapes (DD, PDD, DIR, DDIR, and PDDIR).
Power evaluation methods for focused cluster tests
In this study, we evaluated the power to detect the five general cluster shapes for eight widely known focused cluster tests, including: Stone's Maximum Likelihood Test [4], the focused adaptation of Besag and Newell's Test [1], Cuzick and Edwards' Test [9], Tango's Focused Test [5], variations of the Lawson-Waller Score Test [7, 10], and variations of Bithell's Linear Risk Score (LRS) Test [11]. Power was evaluated through the use of Monte Carlo significance testing with 100 datasets simulated under the null hypothesis and 100 datasets simulated under each alternative hypothesis (i.e., each variation of cluster shape, size and scale). The formulations used for each test are briefly described.
For Stone's Maximum Likelihood Test [4] (hereafter referred to as Stone's Test), we selected a number of distances (d_{1},...,d_{ k }) as bins and placed regions falling between these distances in the appropriate bin. Stone's Test was then defined as follows:
For the purposes of this investigation, the focused cluster adaptation of Besag and Newell's Test [1], Waller and Lawson reference} was defined as: M = min(i : D_{ i }≥ k), where D_{ i }is the number of cases accumulated among i regions and k is defined as the number of cases specified to define a cluster. Four variations of k (2, 4, 7 and 10) were evaluated.
In the one sample approach of Cuzick and Edwards' Test [9], data are ordered by distance to cluster center; and the test statistic is defined as:
where n_{0} is the number of regions required until we have k events (k = the number of cases designating a cluster). We examined four values of k: 2, 4, 7 and 10. The application of this test involves the construction of increasingly larger circles around the point source of interest until the number of cases in the regions contained by the circle equals k cases.
We applied the following formulation of Tango's Focused Test [5]:
C_{ F }= A(r - p),
where, A is a vector with ith element given by a_{ i }= exp(-d_{ i }/τ), d_{ i }= distance of the i th region centroid from the pollution source, and r and p are vectors with ith element y_{ i }/ N and e_{ i }/ N respectively. For the purposes of this study, τ was defined as 1 and 5.
Bithell [11] indicates that LRS Tests can incorporate various functions of distance and rank to describe exposure. We applied functions of distance decline and direction, as described by Lawson [7], to represent exposure to environmental contaminants from a centrally located pollution source. Bithell's LRS Test statistic formulation [11] is described as:
where θ_{1i}is the area-specific relative risk based on the alternative hypothesis. We therefore defined:
Two formulations of the Lawson-Waller Score Test [7, 10] also were evaluated. We applied the Radial Distance Decline Score Test as defined by Lawson [7]:
Lawson's [7] formulation of the Directional Score test also was used:
where μ is the mean angle estimated under the null hypothesis during model fitting; however, during data simulation, was selected.
Power evaluation methods for focused cluster tests from models
Model comparisons
Base Model | Spatial Distribution Function of Alternative Model | Base Model Observed Dataset | More Explicit Model Observed Dataset |
---|---|---|---|
Null | Distance Decline | Relative Risk = 1 | Distance Decline Relative Risk |
Null | Direction | Relative Risk = 1 | Directional Relative Risk |
Null | Peaked Distance Decline | Relative Risk = 1 | Peaked Distance Decline Relative Risk |
Null | Distance Decline and Direction | Relative Risk = 1 | Distance Decline and Direction Relative Risk |
Null | Peaked Distance Decline and Direction | Relative Risk = 1 | Peaked Distance Decline and Direction Relative Risk |
Distance Decline | Distance Decline and Direction | Distance Decline Relative Risk | Distance Decline and Direction Relative Risk |
Distance Decline | Peaked Distance Decline | Distance Decline Relative Risk | Distance Decline and Direction Relative Risk |
Distance Decline | Peaked Distance Decline and Direction | Distance Decline Relative Risk | Peaked Distance Decline and Direction Relative Risk |
Direction | Distance Decline and Direction | Directional Relative Risk | Distance Decline and Direction Relative Risk |
Direction | Peaked Distance Decline and Direction | Directional Relative Risk | Peaked Distance Decline and Direction Relative Risk |
Peaked Distance Decline | Peaked Distance Decline and Direction | Peaked Distance Decline Relative Risk | Peaked Distance Decline and Direction Relative Risk |
Distance Decline and Direction | Peaked Distance Decline and Direction | Distance Decline and Direction Relative Risk | Peaked Distance Decline and Direction Relative Risk |
Declarations
Acknowledgements
We would like to acknowledge the support of NIH grant 5R01CA092693-2 which has supported the authors during the development and completion of this work.
Authors’ Affiliations
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