^{1}

^{2}

The authors have declared that no competing interests exist.

The spatial scan statistic is commonly used to detect spatial and/or temporal disease clusters in epidemiological studies. Although multiple clusters in the study space can be thus identified, current theoretical developments are mainly based on detecting a ‘single’ cluster. The standard scan statistic procedure enables the detection of multiple clusters, recursively identifying additional ‘secondary’ clusters. However, their

Whether the distribution of disease spreads randomly or clusters around particular epicenters, this has been a crucial concern in epidemiological studies. Any indication of disease clustering at an early phase of outbreak offers valuable insights into preventing us from a worse pandemic scenario or may provide us with clues to the etiology of the disease [

Although the extent to which disease distributes is often discussed in a spatial context, like Snow’s map, there is parallel to purely temporal events which also attract attention from a broad public today. For example, identifying the threat of emerging infections or the risk of bioterrorist attacks is a critical role of surveillance systems, the main focus of which is to monitor the incidence or prevalence of specific health problems over time within a well-defined population [

To identify meaningful clusters, in other words, to investigate a regional or temporal tendency in the presence of certain diseases, whether the disease risk is relatively high to other surrounding regions or subsequent time periods, a number of statistical tests have been proposed and are widely used [

The scan statistic is one of the most powerful elements of the CDT since it is based on a concrete statistical framework—i.e., the maximum likelihood ratio; examples include the circular scan statistic [

A shortcoming of these approaches is, however, the fact that most of them focus on ‘single’ cluster detection while investigating the extended study space or period within which more than one cluster is expected. To detect more than one cluster, the ordinary scan statistic procedure, including the circular and flexibly shaped ones, is iteratively applied after the identification of the first (primary) cluster; additional, mutually exclusive ‘secondary’ clusters are then sequentially detected by the likelihood ratio statistic—we hereafter refer to this conventional procedure as the secondary-cluster procedure. The procedure can only evaluate these clusters one by one, and each corresponding

In the present work, we construct a general test procedure that enables the simultaneous evaluation of multiple clusters, focusing on a purely temporal Poisson model. Combining generalized linear models (GLMs) and the ordinary scan statistic procedure, the new testing framework stands directly on the full-likelihood principle that can easily be amalgamated with an information criterion approach to select clusters via GLMs. This procedure becomes, as will be described in a later section, a natural extension of scan statistic—i.e., the conventional secondary-cluster procedure, and can accurately evaluate multiple clusters as a whole. An application study adopting the proposed procedure is then discussed in the context of a real-world example—i.e., temporal data on the daily incidence of out-of-hospital cardiac arrest cases in Japan [

In this section, we first present an overview of the ordinary scan statistic framework implemented in SaTScan and FleXScan for a single-cluster detection and then describe the conventional secondary-cluster procedure for the multiple-cluster detection proposed by Kulldorff [

A study space (area or time period) _{i}—is presumed to follow an independent Poisson distribution, with an expected value _{i}—i.e., _{i}|_{i} ∼ Poisson(_{i}), which is henceforth denoted in lowercase as _{i}, _{i}, is higher than in other parts of the study space, the expected number of cases can be modelled as
_{i} = 1 if _{i} = 0 otherwise (_{i} = _{w} = exp(_{i}

The likelihood function of model (_{i} = _{i} given the location of hot-spot window, _{1}, _{2}, …, _{m}), and the parameters _{0} states that there is no cluster—i.e., _{1} asserts that there is a hot-spot window _{0}: _{1}:

The maximum log likelihood ratio (LLR) for a given

The described procedure above was intended to identify only the primary cluster,

To construct a new procedure, we first show here that the multi-cluster model can be formulated in the form of mixture Poisson GLMs and then derive the likelihood function. This new formulation recognizes the cluster selection problem as a model selection problem for which we can embrace the information criterion paradigm. To choose an appropriate number of clusters, we then propose a new criterion delivered from the likelihood function in the same manner as BIC. The computational aspect of the proposed methods is reasonably straightforward utilizing existing statistical software although, some technical details are explained at the end of this section.

Assuming that there are _{1}, _{2}, …, _{K}_{k} contains a set of adjacent segments as a cluster—i.e., _{i} of segment _{ki} = 1 if _{k} and _{ki} = 0 otherwise. Note that _{k} > 0.

Recalling the notation in the previous section, _{i} = _{i} given _{ki})—which is now a _{1}, _{2}, …, _{K}_{0i} = 1 if _{0i} = 0 otherwise. We assume _{1}.

The multiple-cluster model (_{1} in _{max} (≥ 1)—(_{max} − 1) secondary clusters, _{max} = 10, 20, …) or a P-value threshold, _{s}, (e.g., _{s} < 0.5, 0.8, 1.0). It should be noted that _{max} = 1 corresponds to detection of a single cluster using the conventional scan statistic procedure. Candidate selection may differ depending on the scanning method that is adopted (e.g., circular, flexible, etc.).

To select an appropriate number of clusters, _{max}), we propose a new information criterion approach that chooses _{0} = _{K} _{w} _{max} = 1.

To calculate the proposed criterion (^{K} as an approximation of the probability of selecting locations

The statistical significance of appropriate models is evaluated by the Monte Carlo hypothesis testing procedure in the same manner as with the standard scan statistic. Under the null hypothesis, a large number of random datasets are generated; however, for each of these, max_{K}

All the numerical computation can easily be carried out by R [_{rep} of data sets which are simulated independently under the null hypothesis of no clustering. The simulated _{rep} + 1 values of the statistic. If this rank is _{rep} + 1). The Monte Carlo simulated _{rep} realisations will result in a slightly different _{rep} consequently provides a more stable _{rep} is usually set as 999 or 9,999.

To illustrate how the proposed procedure performs in detecting temporal multiple-clusters, we apply it to real-world data, the daily out-of-hospital cardiac arrest (OHCA) cases in Japan and compare the results with those obtained from the conventional secondary-cluster procedure. Further, the consistency of the proposed procedure is also investigated via a simulation study. The aspect of consistency, whether the proposed procedure tends to select the correct number of clusters when the truth is known, is an essential part in evaluating the performance of the proposed framework and the desired property as a reliable procedure.

As an example, the Japanese OHCA data for the period of 2005–2011 (2,260 days) were studied and a total of 701,651 cases (MNC; male con-cardiac cases) were analyzed following a previous work [_{i}, while accommodating the following five factors into the model: year, month, day of the week, holidays, and temperature, in two-by-two stratification by sex (male/female) and the etiology of arrest (cardiac/non-cardiac). For more details see Takahashi and Shimadzu [

To determine whether OHCA incidence showed specific temporal clustering patterns, we calculated the scan statistic implemented with the restricted likelihood ratio [_{1} = 0.2. The significance level of the test was set as 0.05, and the

Daily ambulance records of OHCA cases were obtained from the All-Japan Utstein registry data of cardiopulmonary arrest patients provided by the Fire and Disaster Management Agency (FDMA). This was a nationwide and population-based registry system of OHCA cases available since 2005, in accordance with the Utstein guidelines. Since all the records were made anonymous by FDMA, according to the informed consent guidelines in Japan, we were exempt from obtaining informed consent from each patient to use this dataset. All the data set used in the present study is public data and available from the Ministry of Internal Affairs and Communications, Japan upon request at

We carried out simulation studies to assess the performance of the proposed framework in detecting multiple clusters from time-series data. The purpose of it is to check if the proposed framework possesses a consistent property that tends to select the correct number of clusters when the truth is known. Here, the expected count of daily OHCA MNC cases from the null hypothesis served as an example time series. We assumed six periods [A–F;

period | # days | expected counts | |
---|---|---|---|

A | 2006/01/01–03 | 3 | (115.10, 131.43, 122.34) |

B | 2005/01/01–03 | 3 | (103.08, 108.53, 124.18) |

C | 2005/04/01–03 | 3 | (76.44, 77.10, 84.35) |

D | 2005/02/01 | 1 | (87.27) |

E | 2007/01/01–05 | 5 | (127.77, 117.05, 114.54, 99.36, 100.34) |

F | 2007/04/01–05 | 5 | (84.83, 81.85, 78.47, 79.52, 82.61) |

scenario | # clusters | # days | Relative Risk (RR) |
|||||
---|---|---|---|---|---|---|---|---|

A | B | C | D | E | F | |||

S0 | 0 | 0 | ||||||

S1 | 1 | 3 | 1.5 | |||||

S2.1 | 3 | 9 | 1.2 | 1.2 | 1.2 | |||

S2.2 | 3 | 9 | 1.3 | 1.3 | 1.3 | |||

S2.3 | 3 | 9 | 1.5 | 1.5 | 1.5 | |||

S2.4 | 3 | 9 | 2.0 | 2.0 | 2.0 | |||

S3.1 | 6 | 20 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 |

S3.2 | 6 | 20 | 1.3 | 1.5 | 2.0 | 2.0 | 1.3 | 2.0 |

* All the blanks should read as Relative Risk (RR) is 1.

_{i} and the expected counts

Grey line overlays the null expected counts (Takahashi and Shimadzu [

The conventional secondary-cluster procedure first detected the most likely cluster of 2-day length and then seven other secondary clusters (

rank | cluster | clustered period | cases | expects | RR | _{s} |
---|---|---|---|---|---|---|

1 | _{1} |
2010/01/01—2010/01/02 | 423 | 232.43 | 1.82 | 0.001 |

2 | _{2} |
2005/01/01—2005/01/02 | 381 | 211.61 | 1.80 | 0.001 |

3 | _{3} |
2011/01/01 | 228 | 111.40 | 2.05 | 0.001 |

4 | _{4} |
2008/12/31—2009/01/03 | 630 | 444.54 | 1.42 | 0.001 |

5 | _{5} |
2006/01/01 | 207 | 115.10 | 1.80 | 0.001 |

6 | _{6} |
2008/01/01—2008/01/04 | 614 | 446.72 | 1.37 | 0.001 |

7 | _{7} |
2007/01/01—2007/01/02 | 344 | 244.83 | 1.41 | 0.001 |

8 | _{8} |
2011/01/02—2011/01/06 | 711 | 589.98 | 1.21 | 0.011 |

RR: relative risk

We then applied our proposed framework to the same data. Candidate clusters were selected with a threshold _{s} < 1.0, which was equivalent to setting _{max} = 25 for the observed data. The suggested multiple cluster model had eight clusters, _{0} = 17433.89,

_{5%} = −0.00025. The selected multiple-cluster model by the proposed procedure is shown in

clustered period | coef. | OR | 95%CI | ||
---|---|---|---|---|---|

intercept | −0.006 | 0.0077 | |||

_{1} |
2010/01/01—2010/01/02 | 0.605 | 1.831 | (1.662, 2.012) | < 0.0001 |

_{2} |
2005/01/01—2005/01/02 | 0.594 | 1.812 | (1.636, 2.000) | < 0.0001 |

_{3} |
2011/01/01 | 0.722 | 2.059 | (1.803, 2.339) | < 0.0001 |

_{4} |
2008/12/31—2009/01/03 | 0.355 | 1.426 | (1.317, 1.541) | < 0.0001 |

_{5} |
2006/01/01 | 0.593 | 1.810 | (1.574, 2.068) | < 0.0001 |

_{6} |
2008/01/01—2008/01/04 | 0.324 | 1.383 | (1.276, 1.500) | < 0.0001 |

_{7} |
2007/01/01—2007/01/02 | 0.346 | 1.414 | (1.270, 1.569) | < 0.0001 |

_{8} |
2011/01/02—2011/01/06 | 0.193 | 1.213 | (1.126, 1.304) | < 0.0001 |

coef.: estimated coefficients; OR: odds ratio; 95%CI: its 95% confidence interval; and

_{K}

Takahashi and Shimadzu [_{s} < 0.05 based on max_{s} > 0.01.

We generated 1,000 datasets for each scenario and compared the estimated power calculated from the two cluster detection tests with the significance level 0.05.

N.S. |
total |
Sen |
Sen |
PPV |
PPV |
||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

S0: null RR = 1.0 | |||||||||||||

s-c proc. | 0.951 | 0.049 | 0.049 | — | — | — | — | ||||||

proposed proc. | 0.951 | 0.049 | 0.049 | — | — | — | — | ||||||

S1: one cluster (three days) RR = 1.5 | |||||||||||||

s-c proc. | 0.959 | 0.040 | 0.001 | 1.000 | 0.996 | 0.988 | 0.971 | 0.919 | |||||

proposed proc. | 0.994 | 0.006 | 1.000 | 0.996 | 0.988 | 0.986 | 0.954 | ||||||

S2.1: three clusters (nine days) RR = 1.2 | |||||||||||||

s-c proc. | 0.430 | 0.436 | 0.123 | 0.011 | 0.570 | 0.361 | 0.003 | 0.870 | 0.376 | ||||

proposed proc. | 0.426 | 0.531 | 0.043 | 0.574 | 0.313 | 0.000 | 0.878 | 0.405 | |||||

S2.2: three clusters (nine days) RR = 1.3 | |||||||||||||

s-c proc. | 0.009 | 0.118 | 0.455 | 0.406 | 0.012 | 0.991 | 0.728 | 0.266 | 0.935 | 0.658 | |||

proposed proc. | 0.009 | 0.315 | 0.469 | 0.206 | 0.001 | 0.991 | 0.605 | 0.140 | 0.946 | 0.734 | |||

S2.3: three clusters (nine days) RR = 1.5 | |||||||||||||

s-c proc. | 0.004 | 0.962 | 0.034 | 1.000 | 0.992 | 0.940 | 0.978 | 0.847 | |||||

proposed proc. | 0.011 | 0.984 | 0.005 | 1.000 | 0.990 | 0.934 | 0.984 | 0.872 | |||||

S2.4: three clusters (nine days) RR = 2.0 | |||||||||||||

s-c proc. | 0.960 | 0.039 | 0.001 | 1.000 | 1.000 | 1.000 | 0.991 | 0.957 | |||||

proposed proc. | 0.990 | 0.010 | 1.000 | 1.000 | 1.000 | 0.997 | 0.987 | ||||||

S3.1: six clusters (20 days) RR = 2.0 | |||||||||||||

s-c proc. | 0.976 | 0.024 | 1.000 | 0.999 | 0.999 | 0.997 | 0.972 | ||||||

proposed proc. | 0.002 | 0.989 | 0.009 | 1.000 | 0.999 | 0.997 | 0.998 | 0.987 | |||||

S3.2: six clusters (20 days) RR = 1.3, 1.5, 2.0 | |||||||||||||

s-c proc. | 0.003 | 0.170 | 0.795 | 0.032 | 1.000 | 0.952 | 0.597 | 0.976 | 0.669 | ||||

proposed proc. | 0.012 | 0.276 | 0.704 | 0.008 | 1.000 | 0.934 | 0.526 | 0.979 | 0.699 |

* averages among the custers detected as

For the null scenario S0, the total power—i.e. probability of type I error—was 0.049 for both procedures and was very close to the significance level 0.05. In the single-cluster scenario [S1 (RR = 1.5)], both procedures had a total power equal to 1.0 but ours performed slightly better in the detection of a single cluster (

We have proposed a general test procedure that enables the simultaneous evaluation of multiple clusters as an extension of the conventional secondary-cluster procedure, focusing on a purely temporal Poisson model. The Japanese OHCA data analysis has highlighted the most advantageous aspect of the new procedure, that is, it can evaluate the _{s}, of the clustered periods commonly identified by both of the procedures state high significance with _{s} = 0.001. On the other hand, for those clusters being excluded by the proposed procedure indicate relatively large

Several studies have reported the detection of multiple clusters using scan statistics. In one study proposing an adjusted

In the spatial context, a multiple-cluster detection procedure using spatial scan statistics has also been proposed [

We have proposed an information criterion procedure to select an appropriate number of clusters for detection. This approach has been used for model selection in more general statistical modelling contexts—for instance, to estimate the number of multiple clusters [

A more conservative

Here, it is also worth noting about the account for dependence structures in time series data. Although our likelihood approach assumes a (conditional) independent structure among observations, the approach also shares the notion of partial likelihood [

In the spatial context particular, it has been pointed out that the original scan statistic tends to produce more false-positives, when the data contain underlying overdispersion and/or spatial correlation, for which some modification has been proposed in the single-cluster detection test framework [

We have proposed a new testing framework for the simultaneous evaluation of purely temporal multiple-clusters, combining GLM and information criterion approaches that directly stand on the likelihood principle. The framework can, thus, treat the cluster selection problem as the model selection problem and provide a single

The authors are grateful to the editor and the anonymous reviewers for their helpful and constructive comments that have improved the manuscript greatly.