of as clusters (many events bunched together), trends (a gradual increase or decrease in the rate of events over time), or cycles (repeating patterns, such as high rates in certain seasons). For example, if one is concerned about a sudden increase in the incidence of asthma following the opening of a new industrial facility, one might want to look for a trend in asthma incidence over time, both before and after the opening. If one is concerned that events such as copycat suicides come in bunches, one might look for clusters of reported suicides using death certificates. If one believes that events come in cycles, such as asthma attacks during periods with high ozone levels in the summer, one might use a method that detects annual cycles.
Assessment over space is two—dimensional and has increased complexity compared with the assessment of patterns over time (Cuzick and Edwards, 1990; Diggle, 1991; Geary, 1954; Grimson et al., 1981; Moran, 1948, 1950; Ohno et al., 1979; Openshaw et al., 1988; Schulman et al., 1988; Tango, 1984; Whittemore et al., 1987). Spatial patterns may be clusters (regions with more events than other regions) or trends (gradual increases in the number of events across the study area). For example, one may suspect that there was a large spill of toxic material somewhere nearby but may not be sure where. Then, one might want to look for clusters of adverse health events that might signal the location (and effect) of the spill. On the other hand, one might be concerned about the confluence of emissions for a variety of pollution emitters, although one might not know the specific dispersion patterns of the mixed releases. For that situation, one might postulate both clusters of adverse health outcomes near the facilities, signifying hot spots, and a decreasing trend of adverse health effects as one moves away from the general area of the facilities, indicating the atmospheric processes of dilution, dispersion, and transport. In such cases, one would use spatial methods.
A specialized group of spatial methods assesses clustering in proximity to a particular source of hazard. They are called focused methods (Besag and Newell, 1991; Bithell and Stone, 1989; Lawson and Williams, 1993; Stone, 1988; Waller et al., 1994). Most often, these methods use distance as a surrogate for exposure and assess whether cases are closer to the source than expected. The advantage of these methods over other spatial methods is that they address a specific hypothesis of concern and, because of their specificity, have increased sensitivity. The disadvantage of these methods is that their specificity limits their ability to detect more general patterns of clustering.
Space-time assessments are three—dimensional (Abe, 1973; Barton et al., 1965; David and Barton, 1966; Klauber, 1975; Knox, 1964 a,b; Mantel, 1967; Pike and Smith, 1968, 1974; Pinkel and Nefzger, 1959; Pinkel et al., 1963; Symons et al., 1983; Williams, 1984). However, distances in space are not commensurate with distances in time, which further increases the complexity of the assessment problem. In general, space-time methods look for corresponding patterns of events that occur in both space and time. For example, finding that asthma cases occur when there are releases from an industrial facility and that those asthma cases tend to occur in specific geographic regions (say, mainly