André Luiz Fernandes Cançado

Nonparametric intensity bounds for the delineation of spatial clusters

BackgroundThere is considerable uncertainty in the disease rate estimation for aggregated area maps, especially for small population areas. As a consequence the delineation of local clustering is subject to substantial variation. Consider the most likely disease cluster produced by any given method, like SaTScan, for the detection and inference of spatial clusters in a map divided into areas; if this cluster is found to be statistically significant, what could be said of the external areas adjacent to the cluster? Do we have enough information to exclude them from a health program of prevention? Do all the areas inside the cluster have the same importance from a practitioner perspective?ResultsWe propose a method to measure the plausibility of each area being part of a possible localized anomaly in the map. In this work we assess the problem of finding error bounds for the delineation of spatial clusters in maps of areas with known populations and observed number of cases. A given map with the vector of real data (the number of observed cases for each area) shall be considered as just one of the possible realizations of the random variable vector with an unknown expected number of cases. The method is tested in numerical simulations and applied for three different real data maps for sharply and diffusely delineated clusters. The intensity bounds found by the method reflect the degree of geographic focus of the detected clusters.ConclusionsOur technique is able to delineate irregularly shaped and multiple clusters, making use of simple tools like the circular scan. Intensity bounds for the delineation of spatial clusters are obtained and indicate the plausibility of each area belonging to the real cluster. This tool employs simple mathematical concepts and interpreting the intensity function is very intuitive in terms of the importance of each area in delineating the possible anomalies of the map of rates. The Monte Carlo simulation requires an effort similar to the circular scan algorithm, and therefore it is quite fast. We hope that this tool should be useful in public health decision making of which areas should be prioritized.

Delineation of Irregularly Shaped Disease Clusters Through Multiobjective Optimization

Irregularly shaped spatial disease clusters occur commonly in epidemiological studies, but their geographic delineation is poorly defined. Most current spatial scan software usually displays only one of the many possible cluster solutions with different shapes, from the most compact round cluster to the most irregularly shaped one, corresponding to varying degrees of penalization parameters imposed on the freedom of shape. Even when a fairly complete set of solutions is available, the choice of the most appropriate parameter setting is left to the practitioner, whose decision is often subjective. We propose quantitative criteria for choosing the best cluster solution, through multiobjective optimization, by finding the Pareto-set in the solution space. Two competing objectives are involved in the search: regularity of shape and scan statistic value. Instead of running sequentially a cluster-finding algorithm with varying degrees of penalization, the complete set of solutions is found in parallel, employing a genetic algorithm. The cluster significance concept is extended for this set in a natural and unbiased way, being employed as a decision criterion for choosing the optimal solution. The Gumbel distribution is used to approximate the empirical scan statistic distribution, speeding up the significance estimation. The multiobjective methodology is compared with the genetic mono-objective algorithm. The method is fast, with good power of detection. We discuss an application to breast cancer cluster detection. The introduction of the concept of Pareto-set in this problem, followed by the choice of the most significant solution, is shown to allow a rigorous statement about what is a “best solution,” without the need of any arbitrary parameter.

Internal cohesion and geometric shape of spatial clusters

The geographic delineation of irregularly shaped spatial clusters is an ill defined problem. Whenever the spatial scan statistic is used, some kind of penalty correction needs to be used to avoid clusters’ excessive irregularity and consequent reduction of power of detection. Geometric compactness and non-connectivity regularity functions have been recently proposed as corrections. We present a novel internal cohesion regularity function based on the graph topology to penalize the presence of weak links in candidate clusters. Weak links are defined as relatively unpopulated regions within a cluster, such that their removal disconnects it. By applying this weak link cohesion function, the most geographically meaningful clusters are sifted through the immense set of possible irregularly shaped candidate cluster solutions. A multi-objective genetic algorithm (MGA) has been proposed recently to compute the Pareto-sets of clusters solutions, employing Kulldorff’s spatial scan statistic and the geometric correction as objective functions. We propose novel MGAs to maximize the spatial scan, the cohesion function and the geometric function, or combinations of these functions. Numerical tests show that our proposed MGAs has high power to detect elongated clusters, and present good sensitivity and positive predictive value. The statistical significance of the clusters in the Pareto-set are estimated through Monte Carlo simulations. Our method distinguishes clearly those geographically inadequate clusters which are worse from both geometric and internal cohesion viewpoints. Besides, a certain degree of irregularity of shape is allowed provided that it does not impact internal cohesion. Our method has better power of detection for clusters satisfying those requirements. We propose a more robust definition of spatial cluster using these concepts.

A spatial scan statistic for zero-inflated Poisson process

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. However, real data may present substantial departure from the underlying Poisson process. One of the possible departures has to do with zero excess. Some studies point out that when applied to data with excess zeros, the spatial scan statistic may produce biased inferences. In this work, we develop a closed-form scan statistic for cluster detection of spatial zero-inflated count data. We apply our methodology to simulated and real data. Our simulations revealed that the Scan-Poisson statistic steadily deteriorates as the number of zeros increases, producing biased inferences. On the other hand, our proposed Scan-ZIP and Scan-ZIP+EM statistics are, most of the time, either superior or comparable to the Scan-Poisson statistic.

An Item Response Theory approach to spatial cluster estimation and visualization

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. The most popular variant of the spatial scan is the circular scan. However, such approach has several limitations, in particular, the circular window is not suitable to make the correct description of irregularly shaped and/or unconnected clusters. Additionally, such methodology does not incorporate the tools needed for quantifying the uncertainty in the description of the most likely cluster in the analysis. In the present work we build upon the previously proposed methodology called intensity function a more efficient and accurate way of defining the uncertainty in the identification of spatial clusters using Item Response Theory ideas. Using simulated data we show that the proposed method can correctly identify primary, secondary and irregular clusters.

Multiple source spatial cluster detection via multi-criteria analysis

Multiple data sources are essential to provide reliable information regarding the emergence of potential health threats, compared to single source methods. Spatial Scan Statistics have been adapted to analyze multivariate data sources, but only ad hoc procedures have been devised to address the problem of selecting the most likely cluster and computing its significance. In this work, information from multiple data sources of disease surveillance is incorporated to achieve more coherent spatial cluster detection using tools from multi-criteria analysis. The best cluster solutions are found by maximizing two objective functions simultaneously, based on the concept of dominance. To evaluate the statistical significance of solutions, a statistical approach based on the concept of attainment function is used. The multi-criteria approach has several advantages: the representation of the evaluation function for each data source is clear, and does not suffer from an artificial, and possibly confusing mixture with the other data source evaluations; it is possible to attribute, in a rigorous way, the statistical significance of each candidate cluster; and it is possible to analyze and pick-up the best cluster solutions, as given naturally by the non-dominated set. The methodology is illustrated with real datasets.

The Touchard distribution

ABSTRACT We present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.

Spatial Cluster Estimation and Visualization using Item Response Theory

In recent years Kulldorff’s circular scan statistic has become the most popular tool for detecting spatial clusters. However, window-imposed limitation may not be appropriate to detect the true cluster. To work around this problem we usually use complex tools that allow the detection of clusters with arbitrary format, but at the expense of an increase in computational effort. In this chapter we describe a methodology that assists the detection of unconnected and arbitrarily shaped clusters and that provides a measure of uncertainty in the design of such clusters.

A Comparison of Simulated Annealing, Elliptic and Genetic Algorithms for Finding Irregularly Shaped Spatial Clusters

Methods for the detection and evaluation of the statistical significance of spatial clusters are important geographic tools in epidemiology, disease surveillance and crime analysis. Their fundamental role in the elucidation of the etiology of diseases (Lawson, 1999; Heffernan et al., 2004; Andrade et al., 2004), the availability of reliable alarms for the detection of intentional and non-intentional infectious diseases outbreaks (Duczmal and Buckeridge, 2005, 2006a; Kulldorff et al., 2005, 2006) and the analysis of spatial patterns of criminal activities (Ceccato, 2005) are current topics of intense research. The spatial scan statistic (Kulldorff, 1997) and the program SatScan (Kulldorff, 1999) are now widely used by health services to detect disease clusters with circular geometric shape. Contrasting to the naive statistic of the relative count of cases, the scan statistic is less prone to the random variations of cases in small populations. Although the circular scan approach sweeps completely the configuration space of circularly shaped clusters, in many situations we would like to recognize spatial clusters in a much more general geometric setting. Kulldorff et al. (2006) extended the SatScan approach to detect elliptic shaped clusters. It is important to note that for both circular and elliptic scans there is a need to impose size limits for the clusters; this requisite is even more demanding for the other irregularly shaped cluster detectors. Other methods, also using the scan statistic, were proposed recently to detect connected clusters of irregular shape (Duczmal et al., 2004, 2006b, 2007, Iyengar, 2004, Tango & Takahashi, 2005, Assuncao et al., 2006, Neill et al., 2005). Patil & Tallie (2004) used the relative incidence cases count for the objective function. Conley et al. (2005) proposed a genetic algorithm to explore a configuration space of multiple agglomerations of ellipses; Sahajpal et al. (2004) also used a genetic algorithm to find clusters shaped as intersections of circles of different sizes and centers. Two kinds of maps could be employed. The point data set approach assigns one point in the map for each case and for each non-case individual. This approach is interested in finding, among all the allowed geometric shape candidates defined within a specific strategy, the one that encloses the highest ratio of cases vs. non-cases, thus defining the most likely cluster. The second approach assumes that a map is divided into M regions, with total population N and C total cases. Defining the zone z as any set of connected regions, the