BIOMETRICS 55, 544-552 June 1999 The Knox Method and Other Tests for Space-Time Interaction Martin Kulldorff Biometry Branch, DCP, National Cancer Institute, EPN 344, 6130 Executive Boulevard, Bethesda, Maryland 20892-7354, U.S.A. Current address: Division of Biostatistics, University of Connecticut School of Medicine, Farmington, Connecticut 06030-6205, U.S.A. email: [email protected] and Ulf Hjalmars Department of Pediatrics, Ostersund Hospital, 831 83 Ostersund, Sweden SUMMARY. The Knox method, as well as other tests for space-time interaction, are biased when there are geographical population shifts, i.e., when there are different percent population growths in different regions. In this paper, the size of the population shift bias is investigated for the Knox test, and it is shown that it can be a considerable problem. A Monte Carlo method for constructing unbiased space-time interaction tests is then presented and illustrated on the Knox test as well as for a combined Knox test. Practical implications are discussed in terms of the interpretation of past results and the design of future studies. KEY WORDS: Bias; Epidemiology; Jacquez’s test; Knox test; Mantel’s test; Population shifts; Power; Space- time clustering; Spatial statistics. 1. Introduction Space-time interaction tests are used to evaluate whether there is space-time clustering of events after adjusting for purely spatial and purely temporal clustering. These tests are fre- quently applied in epidemiological studies, where it is of in- terest to know whether cases of some disease are more clus- tered than what would be expected based on the underlying geographical population distribution and on any purely tem- poral trend. Two excellent surveys have been written by Man- tel (1967) and Williams (1984). Comparative evaluations and power studies have been done by Chen, Mantel, and Klingberg (1984) and by Jacquez (1996). The most widely used statistical technique for testing space-time interaction was proposed by Knox (1964). The time and geographical location of each case is noted, and for each possible pair of cases, the distances between them are calculated both in terms of time and space. If many of the cases that are ‘close’ in time are also ‘close’ in space or vice versa, then there is space-time interaction (‘close’ is defined by the user). This could be an indication that a disease is in- fectious or that it is caused by some other type of agent that appears locally at specific times, such as food poisoning. In a survey of epidemiological articles published between 1960 and 1990, Daniel Wartenberg and Michael Greenberg (personal communication) found 59 different studies utiliz- ing the Knox method. Many of these were concerned with leukemia, and the results from such studies have been used as evidence supporting a viral etiology of the disease (Alexander, 1992; Petridou et al., 1996). The Knox test is an elegant and in many ways attractive method. For example, it is simple and straightforward to cal- culate the test statistic, and it requires knowledge only of cases with no need for controls. At the same time, there is a well-known problem with the method. Mantel (1967) pointed out that the Knox test is biased if the rate of population growth is not constant for all gec- graphic subareas. We denote this as the population shift bias. Shifts in the population distribution create space-time inter- action among any random sample of individuals, including a set of cases generated under the null hypothesis of equal dis- ease risk. The Knox statistic is constructed so as to pick up any type of space-time interaction and does not distinguish whether it is due to shifting population distributions or to disease-related phenomena. This is not a flaw of the test per se if one is looking for any type of space-time interaction, but interest is typically focused on the latter so that the former should be adjusted for. In any such application, including epi- demiology, the population shift bias is a potential problem. Klauber and Mustacchi (1970) suggested that the popu- lation shift bias could be reduced by dividing the data into several parts corresponding to different time periods within which the population would be more stable. A test statistic is then calculated separately for each part and then summed to get an overall test. This reduces the bias but does not elimi- 544Methods for Space- Time Interactions 545 nate it. Unfortunately, it also decreases the power of the test. This is because pairs of cases falling in different data parts are no longer utilized, leading to loss of information. While the existence of the population shift bias has long been known, the magnitude of the bias has not been studied for any real data sets, and it has typically been ignored in practical applications. In Section 3, the bias of the ordinary Knox test is estimated for two different data sets, the child population in Sweden from 1976 to 1994, a fairly stable pop- ulation, and the total population in New Mexico from 1973 to 1991, where there have been large population shifts. There are situations for which the bias is considerable. A simple unbiased version of the Knox test is presented in Section 4. This test adjusts not only for the purely spatial and purely temporal variations but also for the space-time interaction inherent in the background population. It does so without the loss of power associated with the Klauber- Mustacchi approach. The one drawback is that it requires knowledge of the underlying populathn distribution. While this paper is focused on the Knox method, which is the most commonly used space-time interaction test, other space-time interaction tests suffer from the same population shift bias. This includes the methods proposed by David and Barton (1966), Mantel (1967), Pike and Smith (1968), Diggle et al. (1995), Jacquez (1996), and Baker (1996). The approach for constructing an unbiased Knox test, presented in this pa- per, can also be used to construct unbiased versions of these other methods. A second issue with the Knox method relates to the choice of critical distances to define which pairs of cases are close in space and time. Unless the investigator has a fairly clear idea at what scale potential