As a useful tool for geographical cluster detection of events, the spatial scan statistic is widely applied in many fields and plays an increasingly important role. The classic version of the spatial scan statistic for the binary outcome is developed by Kulldorff, based on the Bernoulli or the Poisson probability model. In this paper, we apply the Hypergeometric probability model to construct the likelihood function under the null hypothesis. Compared with existing methods, the likelihood function under the null hypothesis is an alternative and indirect method to identify the potential cluster, and the test statistic is the extreme value of the likelihood function. Similar with Kulldorff’s methods, we adopt Monte Carlo test for the test of significance. Both methods are applied for detecting spatial clusters of Japanese encephalitis in Sichuan province, China, in 2009, and the detected clusters are identical. Through a simulation to independent benchmark data, it is indicated that the test statistic based on the Hypergeometric model outweighs Kulldorff’s statistics for clusters of high population density or large size; otherwise Kulldorff’s statistics are superior.
Citation: Zhao X, Zhou X-H, Feng Z, Guo P, He H, Zhang T, et al. (2013) A Scan Statistic for Binary Outcome Based on Hypergeometric Probability Model, with an Application to Detecting Spatial Clusters of Japanese Encephalitis. PLoS ONE 8(6): e65419. doi:10.1371/journal.pone.0065419
Editor: Jaymie Meliker, Stony Brook University, Graduate Program in Public Health, United States of America
Received: April 2, 2013; Accepted: April 24, 2013; Published: June 13, 2013
Copyright: © 2013 Zhao et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by Natural Science Foundation of China (No. 30571618 and No. 61103042), Grants of the Ministry of Health, China (No. 200802133), Research Fund for the Doctoral Program of Higher Education of China (No. 20100181120029), State Key Laboratory of Software Engineering of China (No. SKLSE2012-09-32), and China Scholarship Council (http://en.csc.edu.cn/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
In epidemiological studies, it is often important to evaluate whether the occurrence of a disease is randomly distributed or tends to occur as clusters over time and/or space after adjusting for known confounding factors, which may provide clues to the etiology of the disease . In addition, outbreak detection becomes possible thanks to growing geographically referenced health-related data, such as data from sales of Over-the-Counter (OTC) Healthcare Products.
Likelihood ratio based spatial scan statistic is a cluster detection test. Because of its ability of both identifying localized clusters and evaluating their significance, the spatial scan statistic becomes more popular relative to other statistical methods for disease clustering. In order to investigate excessive risk of disease after adjusting for the unevenly distributed population, Kulldorff proposed the spatial scan statistic for the binary outcome in his seminal papers , . Those included the Bernoulli and the Poisson models. The spatial scan statistic quickly has become a popular research field and various new methods have been developed, which could roughly be divided into two classes: those extending the shape of the scanning window to detect irregularly shaped clusters –, and those modifying the test statistic to exploit more information – or to accommodate more complex data structures, such as multivariate data –, ordinal data , survival data , , normally distributed data ,  and multinomial data . For all methods, the significance of the test statistic is evaluated using the Monte Carlo test .
While a great variety of methods have been developed for diverse purposes, the two classic models for the binary outcome play a central role in the spatial scan statistic due to their wide application. The Bernoulli model is more appropriate for case-control data, while the Poisson model is more appropriate for case-population data. When the number of cases is small compared to the target population, the two models approximate each other. Both of them are based on the likelihood ratio test theory and use the likelihood ratio (LR), a generalized measuring index of clustering for each window, thereby making windows of different size comparable. In a word, the larger the LR of a window is, the more likely it is a true cluster. The window with the largest LR is called the most likely cluster (MLC) and the test statistic is the largest LR. In general, as summarized by Kulldorff , a test for spatial randomness adjusted for an inhomogeneity comprises 7 steps, of which the 3rd step is to construct a measuring index for each defined area and the 5th step is to define a summary quantification for all defined areas. The LR and the largest LR correspond with the 3rd and 5th steps, repsectively. Neill  presents a heuristic interpretation that the LR is a sort of distance away from the null hypothesis of no clustering. To the best of our knowledge, quite on the contrary, the LR is an index of measuring the closeness to the alternative hypothesis of an existing window of clustering. However, it is interesting to study whether there exists an index of measuring the departure from the null hypothesis of no clustering and, if so, how the new measuring index performs compared with the existing methods.
In this paper, we apply the Hypergeometric probability model to construct a likelihood function under the null hypothesis, which sometimes is complementary to the alternative hypothesis. The idea originates from the email exchange with Dr. Kulldorff and another method from Dr.Wong et al for syndrome surveillance, which is named WSARE . By analogy with our situation, WSARE employs the P value of testing heterogeneity of proportions inside and outside each window as a measure of the departure from the null hypothesis. That is, the window with the minimum P value is the ‘most strange window’ (in Dr.Wong’s words) and the minimum P value is the test statistic. The proposed and existing methods are both applied to the real data of Japanese Encephalitis (JE) in Sichuan province, China. A simulation between the proposed test statistic and the Kulldorff’s statistics is carried out using a set of independent benchmark data.
Materials and Methods
In 2003, there was a outbreak of SARS in China, and this exposed the underdevelopment of the public health system in handling public health emergencies in China. The China Central government requested to strengthen the construction of an infectious disease and public health emergency system, with focus on promoting the timeliness, sensitivity and accuracy of reporting. The Chinese Center for Disease Control and Prevention (CCDC) made the construction of a new operation model, Chinese Information System for Infectious Diseases Control and Prevention (CISIDCP). CISIDCP was established on the basis of individual cases and public health emergencies. A Virtual Private Network (VPN) has been constructed using the information safety technology, and information of individual cases is directly reported to the national database through the internet. This system will report 39 notifiable infectious diseases to CCDC within 24 hours. However, the management is classified into national, provincial, prefecture and county levels. CISIDCP makes feedback with health authority departments at every level. In 2005, CISIDCP had covered at least 93.3% of medical units at the county and above.
JE is among the 39 notifiable infectious diseases and, therefore JE case will be reported routinely by CISIDCP. JE is a vector-borne viral disease with a high mortality rate and a high percentage of neuropsychiatric sequelae. The JE virus is spread by marsh birds and intensified by pigs, mainly transmitted via the bite of infected Culex mosquito. Humans are dead-end hosts . Many of the ecological, environmental, climatic and human behavioral factors are involved in the spread of the JE virus . Contextual determinants of JE include irrigated rice farming, pig rearing and the rural population. Sichuan is a province in Southwest China. It is one one of the major agricultural production bases of China, including rice and pork production. Hence, Sichuan province is a high-incidence region for JE with the incidence of JE ranked the 5th in 2009 among 31 provinces in Mainland China. As a subordinate unit of CCDC, Sichuan Center for Disease Control and Prevention (SCDC) has the permission to access the data of Sichuan from CISIDCP. It is interesting for SCDC to investigate the geographic distribution pattern of JE. This analysis may help further learn the disease cluster areas and influencing factors of JE, finally assisting health officials in allocating the health resources.
Notation and Kulldorff’s Test Statistics for the Binary Outcome
- : the whole study region divided into many counties
- : a circular window in the study region
- : all overlapped windows formed by circles of arbitrary radius centered in each one of counties,
- : the number of population within window
- : the observed number of cases within window
- : the total population in the study region
- : total number of cases in the study region
- : one cluster such that all individuals within have probability be a case and is the same probability for individuals outside
When scanning for the high rates only, such as identifying areas with high rate of leukemia  or Breast cancer , Kulldorff’s statement on the hypotheses is as following: , for . Under the null hypothesis, the expected number of cases within , , is calculated as follows:(1)
(3)The search space for the candidate cluster is ristricted by equation 3, where the first inequality specifies the maximum spatial cluster size, and the second inequality determines the aim of scanning for high risk area. Mathematical details have been provided by Kulldorff . Sometimes it is interesting to identify areas with low rates only, such as detecting low clusters of sex ratio . We just need to change the direction of the second inequality. The window attaining the maximum is defined as the Most Likely Cluster (MLC). The MLC is least likely to be a chance occurrence under the null hypothesis. And the test statistic is the in equation 2. The significance of the test statistic is evaluated by Monte Carlo test.
A Test Statistic based on the Hypergeometric Probability Model
As with the Poisson- and Bernoulli-based statistics, the Hypergeometric-based statistic has two aims: detecting the potential clusters; and evaluating the significance of the detected clusters.
The null hypothesis signifies complete spatial randomness with each individual in the study region, implying that each person in the study region are equally likely to become the case. There are individuals of which are cases in total. Under null hypothesis, we can think of that all individuals are equally likely to be ‘labeled’. For a window with people, the probability of ‘labeled’ individuals has probability as:(4)
This is the classic application of Hypergeometric distribution, sampling from a finite population without replacement. The window with the minimum probability is least likely occur under the null hypothesis. This can be written:(5)
Where the is the same as Kulldorff's method, which is defined by equation 3. The window attaining the minimum probability is least likely occur under the null hypothesis. We may call this window the most strange window, as it have the minimum probability of satisfying the null hypothesis that .
Test of significance.
The test statistic is the in equation 5. To evaluate whether the identified cluster is statistically significant or just can be explained by random noise, the value is obtained through Monte Carlo test, by comparing the rank of the test statistic from the real data set with those from the random replications. The test statistic is calculated for each random replication as well as for the real data set, and if the latter is among the 5 percent lowest, then the test is significant at the 0.05 level. When there are multiple clusters in the data set, many methods can deal with this problem , , .
Detection of Clusters of JE in Sichuan Province
We applied both the proposed and existing methods to analyze data on JE from 2009 in Sichuan province, China. Our analysis used both methods to investigate whether the high JE incidence is evenly spread over Sichuan province. This would examine whether any observed clusters of JE cases could be explained by chance alone, or whether there were clusters of statistical significance.
In 2009, Sichuan province had a population of 81,379,919 and consisted of 181 counties. It had a total number of 598 cases, in which 2 cases lost the geographical information. The JE cases data in Sichuan province came from CISIDCP and this data was not publicly available. The population data came from the Nation Bureau of Statistics of China (http://www.stats.gov.cn/tjsj/ndsj/). To eliminate the random noise in the incidence map, we utilized a method of empirical Bayes estimate to visualize the spatial distribution of JE in Sichuan (Figure 1). It seems that there is a high risk in the east borderline region, especially in the northeast and southeast areas.
Because of the case-population data structure, we used the Poisson model representing Kulldorff’s method. The 9,999 random data were generated under the null hypothesis to evaluate the significance of the detected clusters. On a significance level of 0.05, the two methods obtained almost the same results (Figure 2). They detected two significant clusters, the most likely cluster in the southeast with 18 counties and one secondary cluster in the northeast with 12 counties. Both clusters had a P value of 0.0001. In addtion, the two methods detected identical areas as a third likely cluster, but with different P values: 0.0909 for the existing method and 0.1334 for the new method. This area is not colored in the figure. Overall, the detected clusters account for 53.2% (317/596) of the JE cases in Sichuan province. Cao  had analyzed the influencing factors of JE in the southwest of China on the Prefecture-city level, an administrative division below a province and above a county in China’s administrative structure, but found no statistically significant factors. As Cao presented, this is probably due to the little spatial variation of the influencing factors in the southwest of China. On the other hand, the financial support may, on a large extent, determine the spatial variation of JE in Sichuan. We ranked the counties by GDP per capita in a descending order. It turned out that 86.7% (26/30) of the counties in the significant clusters ranked in the last two thirds of the 181 counties, and 33.3% (10/30) ranked in the last one third parts. As pointed by Zhen , the less-developed region of Sichuan province was often short of funds for JE control, especially the remote rural areas. These constitute the high risk areas in Sichuan, which indicates that more financial and policy support is required to control JE in these areas.
On a significance level of 0.05, the test statistics based on Poisson and Hypergeometric models obtained almost the same results. They detected two significant clusters, the most likely cluster in the southeast with 18 counties and one secondary cluster in the northeast with 12 counties, both with P value of 0.0001. They differs slightly on statistically insignificant clusters.
To carry out the comparison study between the two kinds of methods, we use public domain benchmark data sets. The benchmark data sets are 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. Each county is represented by a centroid coordinate. The benchmark data was created by Kulldorff, M., T. Tango and P.J. Park (2003) to investigate the performance of different statistical methods for clustering. They have been used in many research studies –. It is available at ‘http://www.satscan.org/datasets.html’. The benchmark data and how it was generated has been described in detail elsewhere . We provide a brief summary here.
Under the null hypothesis of no clustering, 99,999 random data sets were generated by randomly allocating 600 cases to the various counties, with probabilities proportional to the county population. The null data is used to estimate the critical values, which is the cut-off point for the significance. Hot-spot clusters were generated by setting the relative risk in some counties to be larger than 1. Different real hot-spot clusters corresponds to different combinations of population density, number of counties. For each kind of hot-spot clusters, 10,000 random data sets were generated using a multinomial probability distribution with the relative risks such that if the exact location of the real cluster was known in advance, the power to detect it should be 0.999. In order to clearly examine the performance of these methods when applied in high and low population density, we focused on the relatively extreme combinations. urban area, rural area, rural & mixed area and urban & mixed area.
The sensitivity (SEN) and the positive predictive value (PPV) were estimated to examine the performance of different methods. The SEN and PPV of the spatial scan statistic were introduced by Huang et al , and can be defined in terms of either the number of regions or the population. First, we define the SEN as the probability of detecting the regions that actually constitute the cluster, i.e, proportion of the number of regions correctly detected from the true clusters.(6)
For the SEN and PPV, the larger the better they are, with 100% being the optimal. Here, the detected clusters are defined as follows: a critical value corresponding to a 0.05 significance level was computed by identifying the 5000th highest maximum index from among the 99,999 random data sets generated under the null model. For each kind of hot spot cluster, all windows with index exceeding the critical value are candidate clusters. The window with the maximum index is reported as the most likely cluster. Then, we eliminate the remaining candidate clusters that overlap with most likely cluster, and report the one having the largest index as the second cluster. We then repeated this procedure for the third and the fourth clusters and so on. All reported clusters are detected clusters.
We obtain the estimate of SEN and PPV for the test statistics based on the Bernoulli, the Poisson and the Hypergeometric models, respectively, and find that the results of the two test statistics proposed by Kulldorff are exactly the same except that a few values differ slightly. Thus, we take the Poisson model as an example for Kulldorff’s methods (Table 1). There are several common features between Hypergeometric and Poisson models: 1) In general, the two test statistics perform very similarly; 2) The SEN decreases as the number/size of hot spot clusters increases; 3) The PPV does not show the 2nd feature and always maintains a high level; 4) In general, the SEN based on population is greater than that based on counties, likewise for the PPV.
The SEN and PPV represent two sides of a test statistic for cluster detection. If the SEN and PPV based on one probability model both are greater than the corresponding values based on the other model, the test statistic based on the former model outweighs the latter under that scenario. From Table 1 it appears that: 1) the Hypergeometric model outweighs the Poisson model when the hot-spot cluster is in urban area, whereas the Poisson model outweighs the Hypergeometric model when the hot spot cluster is in the rural area. In other words, with the densely populated hot-spot cluster, the Hypergeometric model performs better, otherwise the Poisson model performs better; 2) the Hypergeometric model outweighs the Poisson model when the size of hot-spot cluster is large, and otherwise the Poisson model performs better. In summary, with densely populated or largely sized hot-spot clusters, the Hypergeometric model performs better, otherwise the Poisson model performs better.
First, we would like to review the motivation of this study. Our initial test statistic was the P value, but it failed in the simulation study. There is a lot of literature relating to the role of the P value and its origin, in which a great part are concerned with what is evidence in statistics and the debate over Fisher’s test of significance and Neyman-Pearson’s hypothesis testing. That is beyond the scope of this paper, and plenty of literature has interestingly discussed those questions –.
The key assumption to the two kinds of test statistics is that there is no positive spatial autocorrelation, which implies that pairs of observations taken nearby are more similar than those taken far apart. As summarized by Tango , the positive spatial autocorrelation will be a key issue in statistical modeling of spatial epidemiology. For instance, when spatial regression is performed to determine what covariates contribute to a higher risk for a disease under study, it is critical to adjust for the spatial autocorrelation in the data. Otherwise, the risk will be overestimated, with biased p-values that are too small, providing “statistically significant” results when none exist. However, for detecting disease clustering or disease clusters, we should not adjust for the spatial autocorrelation since we are interested in detecting clusters due to such autocorrelation and, if they are adjusted away, important clusters might go undetected. The test statistic based on the Hypergeometric model is more similar with the one based on Poisson model in terms of the data structure, both for case-population data.
As the simulation study shows, with the identical assumption, the two kinds of methods perform similarly. The SEN and PPV based on population usually are greater than that based on counties, indicating that the cluster detection methods often “capture” the hot spot areas of high population, which would benefit more overall. Furthermore, the test statistic based on the Hypergeometric model performs better when used with densely populated or largely sized hot spot clusters; otherwise Kulldorff’s test statistics perform better. Although it appears relatively small, given the scarce resources available to most local health departments, a greater improvement would reduce the cost to investigate potential disease outbreaks. In the application of JE in Sichuan province, the two kinds of methods identified the same cluster, which may greatly help the health department allocate relevant resources to these areas for JE prevention.
Further refinements of the new test statistic may include clusters that are not circular, but instead irregularly shaped ones. Space-time clusters extensions to the proposed method are also straightforward.
The authors wish to thank the Odegaard Writing and Research Center at the University of Washington, especially Henlen Olsen for their help on the language.
Conceived and designed the experiments: XZ PFG HYH TZ. Performed the experiments: XZ. Analyzed the data: XZ. Contributed reagents/materials/analysis tools: ZJF LD. Wrote the paper: XZ XHZ XSL LD.
- 1. Tango T (2010) Statistical methods for disease clustering. Springer.
- 2. Kulldorff M, Nagarwalla N (1995) Spatial disease clusters: detection and inference. Statistics in medicine 14: 799–810.
- 3. Kulldorff M (1997) A spatial scan statistic. Communications in Statistics-Theory and methods 26: 1481–1496.
- 4. Patil G, Taillie C (2004) Upper level set scan statistic for detecting arbitrarily shaped hotspots. Environmental and Ecological Statistics 11: 183–197.
- 5. Duczmal L, Assuncao R (2004) A simulated annealing strategy for the detection of arbitrarily shaped spatial clusters. Computational Statistics & Data Analysis 45: 269–286.
- 6. Tango T, Takahashi K (2005) A flexibly shaped spatial scan statistic for detecting clusters. International Journal of Health Geographics 4: 11.
- 7. Assuncao R, Costa M, Tavares A, Ferreira S (2006) Fast detection of arbitrarily shaped disease clusters. Statistics in Medicine 25: 723–742.
- 8. Kulldorff M, Huang L, Pickle L, Duczmal L (2006) An elliptic spatial scan statistic. Statistics in medicine 25: 3929–3943.
- 9. Christiansen LE, Andersen JS, Wegener HC, Madsen H (2006) Spatial scan statistics using elliptic windows. Journal of agricultural, biological, and environmental statistics 11: 411–424.
- 10. Duczmal L, Cancado AL, Takahashi RH, Bessegato LF (2007) A genetic algorithm for irregularly shaped spatial scan statistics. Computational Statistics & Data Analysis 52: 43–52.
- 11. Duarte AR, Duczmal L, Ferreira SJ, Cançado ALF (2010) Internal cohesion and geometric shape of spatial clusters. Environmental and Ecological Statistics 17: 203–229.
- 12. Gangnon RE, Clayton MK (2001) A weighted average likelihood ratio test for spatial clustering of disease. Statistics in Medicine 20: 2977–2987.
- 13. Kulldorff M (2001) Prospective time periodic geographical disease surveillance using a scan statistic. Journal of the Royal Statistical Society: Series A (Statistics in Society) 164: 61–72.
- 14. Duczmal L, Buckeridge DL (2006) A workflow spatial scan statistic. Statistics in Medicine 25: 743–754.
- 15. Loh JM, Zhu Z (2007) Accounting for spatial correlation in the scan statistic. The Annals of Applied Statistics : 560–584.
- 16. Wen S, Kedem B (2009) A semiparametric cluster detection methoda comprehensive power comparison with kulldorff’s method. International Journal of Health Geographics 8.
- 17. Tango T, Takahashi K, Kohriyama K (2011) A space–time scan statistic for detecting emerging outbreaks. Biometrics 67: 106–115.
- 18. Kulldorff M, Mostashari F, Duczmal L, Katherine Yih W, Kleinman K, et al. (2007) Multivariate scan statistics for disease surveillance. Statistics in Medicine 26: 1824–1833.
- 19. Neill DB, Cooper GF (2010) A multivariate bayesian scan statistic for early event detection and characterization. Machine learning 79: 261–282.
- 20. Neill DB (2011) Fast bayesian scan statistics for multivariate event detection and visualization. Statistics in Medicine 30: 455–469.
- 21. Jung I, Kulldorff M, Klassen AC (2007) A spatial scan statistic for ordinal data. Statistics in Medicine 26: 1594–1607.
- 22. Huang L, Kulldorff M, Gregorio D (2006) A spatial scan statistic for survival data. Biometrics 63: 109–118.
- 23. Cook AJ, Gold DR, Li Y (2007) Spatial cluster detection for censored outcome data. Biometrics 63: 540–549.
- 24. Kulldorff M, Huang L, Konty K (2009) A scan statistic for continuous data based on the normal probability model. International journal of health geographics 8: 58.
- 25. Huang L, Tiwari RC, Zou Z, Kulldorff M, Feuer EJ (2009) Weighted normal spatial scan statistic for heterogeneous population data. Journal of the American Statistical Association 104: 886–898.
- 26. Jung I, Kulldorff M, Richard OJ (2010) A spatial scan statistic for multinomial data. Statistics in medicine 29: 1910–1918.
- 27. Dwass M (1957) Modified randomization tests for nonparametric hypotheses. The Annals of Mathematical Statistics 28: 181–187.
- 28. Kulldorff M (2006) Tests of spatial randomness adjusted for an inhomogeneity. Journal of the American Statistical Association 101: 1289–1305.
- 29. Neill DB (2006) Detection of spatial and spatio-temporal clusters. Ph.D. thesis, Carnegie Mellon University.
- 30. Wong WK, Moore A, Cooper G, Wagner M (2005) What’s strange about recent events (wsare): an algorithm for the early detection of disease outbreaks. The Journal of Machine Learning Research 6: 1961–1998.
- 31. Solomon T (2006) Control of japanese encephalitiswithin our grasp? New England Journal of Medicine 355: 869–871.
- 32. Diagana M, Preux PM (2007) DumasM (2007) Japanese encephalitis revisited. Journal of the neurological sciences 262: 165–170.
- 33. Hjalmars U, Kulldorff M, Gustafsson G, Nagarwalla N (1998) Childhood leukaemia in sweden: using gis and a spatial scan statistic for cluster detection. Statistics in medicine 15: 707–715.
- 34. Kulldorff M, Feuer EJ, Miller BA, Freedma LS (1997) Breast cancer clusters in the northeast united states: a geographic analysis. American Journal of Epidemiology 146: 161–170.
- 35. Viel JF, Floret N, Mauny F (2005) Spatial and space-time scan statistics to detect low rate clusters of sex ratio. Environmental and Ecological Statistics 12: 289–299.
- 36. Zhang Z, Assunção R, Kulldorff M (2010) Spatial scan statistics adjusted for multiple clusters. Journal of Probability and Statistics 2010.
- 37. Li XZ, Wang JF, Yang WZ, Li ZJ, Lai SJ (2011) A spatial scan statistic for multiple clusters. Mathematical biosciences 233: 135–142.
- 38. Cao M (2009) A Study on the multiple membership multiple classification models and disease mapping for analyzing spatial dependence and heterogeneity of regional distribution of Japanese encephalitis in southwestern China. Ph.D. thesis, Sichuan University.
- 39. Yang Z, Jia-qi M, Zi-jian F, Xiaosong L (2008) Analysis of epidemiological characteristics of encephalitis b in sichuan in 2004. Modern Preventive Medicine 10: 003.
- 40. Kulldorff M, Tango T, Park PJ (2003) Power comparisons for disease clustering tests. Computational Statistics & Data Analysis 42: 665–684.
- 41. Song C, Kulldorff M (2003) Power evaluation of disease clustering tests. International Journal of Health Geographics 2: 9.
- 42. Costa MA, Assunção RM (2005) A fair comparison between the spatial scan and the besag–newell disease clustering tests. Environmental and Ecological Statistics 12: 301–319.
- 43. Song C, Kulldorff M (2006) Likelihood based tests for spatial randomness. Statistics in medicine 25: 825–839.
- 44. Berkson J (1942) Tests of significance considered as evidence. Journal of the American Statistical Association 37: 325–335.
- 45. Gibbons JD, Pratt JW (1975) P-values: interpretation and methodology. The American Statistician 29: 20–25.
- 46. Berger JO, Sellke T (1987) Testing a point null hypothesis: the irreconcilability of p values and evidence. Journal of the American Statistical Association 82: 112–122.
- 47. Goodman SN (1993) P values, hypothesis tests, and likelihood: implications for epidemiology of a neglected historical debate. American Journal of Epidemiology 137: 485–496.
- 48. Lehmann EL (1993) The fisher, neyman-pearson theories of testing hypotheses: One theory or two? Journal of the American Statistical Association 88: 1242–1249.
- 49. Schervish MJ (1996) P values: what they are and what they are not. The American Statistician 50: 203–206.
- 50. Goodman SN (1999) Toward evidence-based medical statistics. 1: The p value fallacy. Annals of internal medicine 130: 995–1004.