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The MannKendallSneyers test to identify the change points of COVID19 time series in the United States
BMC Medical Research Methodology volumeÂ 22, ArticleÂ number:Â 233 (2022)
Abstract
Background
One critical variable in the time series analysis is the change point, which is the point where an abrupt change occurs in chronologically ordered observations. Existing parametric models for change point detection, such as the linear regression model and the Bayesian model, require that observations are normally distributed and that the trend line cannot have extreme variability. To overcome the limitations of the parametric model, we apply a nonparametric method, the MannKendallSneyers (MKS) test, to change point detection for the statelevel COVID19 case time series data of the United States in the early outbreak of the pandemic.
Methods
The MKS test is implemented for change point detection. The forward sequence and the backward sequence are calculated based on the new weekly cases between March 22, 2020 and January 31, 2021 for each of the 50 states. Points of intersection between the two sequences falling within the 95% confidence intervals are identified as the change points. The results are compared with two other change point detection methods, the pruned exact linear time (PELT) method and the regressionbased method. Also, an openaccess tool by Microsoft Excel is developed to facilitate the model implementation.
Results
By applying the MKS test to COVID19 cases in the United States, we have identified that 30 states (60.0%) have at least one change point within the 95% confidence intervals. Of these states, 26 states have one change point, 4 states (i.e., LA, OH, VA, and WA) have two change points, and one state (GA) has three change points. Additionally, most downward changes appear in the Northeastern states (e.g., CT, MA, NJ, NY) at the first development stage (March 23 through May 31, 2020); most upward changes appear in the Western states (e.g., AZ, CA, CO, NM, WA, WY) and the Midwestern states (e.g., IL, IN, MI, MN, OH, WI) at the third development stage (November 19, 2020 through January 31, 2021).
Conclusions
This study is among the first to explore the potential of the MKS test applied for change point detection of COVID19 cases. The MKS test is characterized by several advantages, including high computational efficiency, easy implementation, the ability to identify the change of direction, and no assumption for data distribution. However, due to its conservative nature in change point detection and moderate agreement with other methods, we recommend using the MKS test primarily for initial pattern identification and data pruning, especially in large data. With modification, the method can be further applied to other health data, such as injuries, disabilities, and mortalities.
Background
The Coronavirus Disease 2019 (COVID19) pandemic has disrupted every aspect of human society. Because of the highly infectious nature of the disease, state governments in the United States (US) have implemented social distancing measures (e.g., closure of nonessential businesses, regional lockdown, and facecovering mandates) to contain the virus spread and flatten the epidemic curve (epi curve) [1]. However, since these statelevel measures have differed in the strength and timeline of policy enforcement, it is intractable to rely on a simple rubric to evaluate the policy effectiveness. An alternative step is via analyzing the time series of the COVID19 cases, which can eventually assist stakeholders with proactive health policymaking, such as determining the optimal timing to relieve social distancing.
One critical variable in the time series analysis is the change point, also called the inflection point, which is the point where a sudden change occurs in chronologically ordered observations. The change point detection has been long employed in statistical theory [2], but its applications to COVID19 are relatively underexplored. For example, when modeling COVID19 cases, the majority of studies have defined change points as key dates of policy interventions or social events [1, 3]. Other studies have employed parametric models, such as the linear regression model [4, 5] and the Bayesian model [6, 7] to derive change points. However, most of these parametric models require that the observations are normally distributed and that the trend line cannot have extreme variability. In situations where the observations show large variability over time and the trend line cannot be well fitted, parametric models become less reliable. These situations are not uncommon in fitting the COVID19 epi curve, as the disease progression has a considerable degree of uncertainties and variability [1].
To overcome the limitations of the parametric model, we have applied a nonparametric model, called the MannKendallSneyers (MKS) test, to change point detection in the COVID19 epi curve. The MKS test, developed from a prototype model by Mann [8], is used to detect the monotonic trends (e.g., upward, downward) and their corresponding change points in time series data. The model has been primarily employed in earth science research to characterize the fluctuation of climatic and environmental variables, such as rainfall, air temperature, and surface runoff [9,10,11]. Recently, some COVID19 studies have used the MannKendall (MK) test, which is an earlier version of the MKS test, for trend detection [12, 13]. While the MK test is useful in detecting monotonic trends, it cannot detect changes in the trends and the corresponding change points, making it less useful for disease tracking and monitoring in the mid to long term. The MKS test, as a sequential extension of the MK test [14], fills this gap. It can become a valuable tool for longterm disease monitoring and can thus support public health decisionmaking.
The contributions of the paper are as follows.

The paper is the first to apply the MKS test to COVID19 time series analysis.

The paper identifies six change point patterns for state COVID19 cases.

The paper develops an openaccess tool for model implementation.
Methods
The nonparametric MKS test [15], oftentimes called the sequential MannKendallSneyers test, has been applied to the change point detection for longterm time series data (e.g., hydrological changes, climatic changes). According to the Centers for Disease Control and Prevention (CDC) report, both social distancing and mass gathering can potentially lead to an abrupt change in regional COVID19 cases, albeit in different directions [16]. Then, we have evaluated the potential of the MKS test for change point detection in shortterm time series data, the COVID19 cases of infection.
In this section, we first articulate the MKS test. Then, we use an example to demonstrate the model implementation.
Method description
The MKS test applied to the COVID19 time series data can be completed in three major steps.
Step 1: Deriving test statistics (S _{k})
We have treated new weekly cases as an independent observation in a 45week time series data. Under the null hypothesis that the development of new cases remains stable, for each state, we have a time series of the weekly new cases: Xâ€‰=â€‰{x_{1},â€‰x_{2},â€‰x_{3}â€¦x_{N}Â }, where n is the total number of weeks under observation (Nâ€‰=â€‰45 in our case study). m_{i} (iÂ =â€‰1, 2, â€¦, N) represents the total number of elements x_{j} preceding x_{i}Â (jâ€‰<â€‰i) where x_{j}â€‰<â€‰x_{i}.
Based on m_{i}, the test statistic S_{k} derives the cumulative m_{i} for each week, as shown in Eq. (1).
The mean of S_{k} can be derived by Eq. (2).
The variance of S_{k} can be derived by Eq. (3).
Step 2: Deriving two sequences (U _{f} and U _{b})
Next, we derive two sequences, the forward sequence U_{f} and the backward sequence U_{b}, based on the three variables (S_{k}, E(S_{k}), andÂ VAR(S_{k})) in Eqs. (1) through (3). Specifically, the forward sequence U_{f} of the time series is derived by Equation [4].
Then, we reverse the sequence of the original time series X and term it X_{r}. An intermediate sequence U_{fr} is derived by applying Eq. (4) to the reversed time series X_{r}. We reverse the sequence of the values in U_{fr} (i.e., the first value appears the last, and vice versa). We generate the backward sequence U_{b} by adding a negative sign to the reversed values.
Step 3: Deriving change points
Lastly, we identify the change points of the time series X based on the two generated sequences (U_{f} and U_{b}). We first identify the initial set of the change points as the points of intersection between the two sequences. Previous studies show that it is uncertain to recognize all of these change points as abrupt changes, as a change point can be induced by a sudden shift of the mean value over two stable periods [17]. These outlier points could be reevaluated by using additional detection methods, such as the double mass curve [18]. To avoid miscounting the change points while making the proposed method more applicable, we employ a statistical filterâ€”the points of intersection falling beyond the 95% confidence intervals (CIs), which correspond to Zscoresâ€‰=â€‰Â±1.96, are rejected. This filter has been used in relevant MKS studies [19]. It is worth noting that the MKS test can also identify the monotonic trend or the change of directionâ€”if a point of intersection is between the Zscores of 0 and 1.96, the change is upward; if the point is between the Zscores of âˆ’â€‰1.96 and 0, the change is downward.
Model implementation
In this section, we take the state of Virginia as an example to further elaborate on the model implementation. The MKS test can be implemented in Microsoft Excel by calling embedded functions. The datasets and codes are available on GitHub (https://github.com/peterbest52/mks).
Data cleaning
Daily confirmed cumulative COVID19 case data between March 22, 2020 and January 31, 2021 (in a total of 45â€‰weeks) were obtained from the USAFacts website (https://usafacts.org/data/). Then, we aggregated the data on a weekly basis, generating a 45week time series for each state representing new weekly cases. Lastly, to demonstrate the method, we extracted the data for Virginia as the time series X.
MKS test
For time series X, we derived m_{i}, the cumulative times that the case value of the current week is larger than that of each preceding week. Following this step, S_{k} was derived as the cumulative m_{i} (iâ€‰=â€‰1, 2, â€¦, k), according to Eq. (1); then, the mean value of S_{k} or E(S_{k}) and the variance of S_{k} or VAR(S_{k}) were derived by Eqs. (2) and (3), respectively. It is worth noting that, since k is the only independent variable in Eqs. (2) and (3), E(S_{k}) and VAR(S_{k}) are the same for all states in this study. Based on Eq. (4), we derived the forward sequence U_{f} for Virginia (solid line in Fig.Â 1).
Then, we reversed the time series X and derived X_{r}. We derived the intermediate sequence U_{fr} by applying Eq. (4) to X_{r}. Lastly, we derived the backward sequence U_{b} (dashed line in Fig. 1) by first reversing the sequence of values in U_{fr} and then adding a negative sign to these values.
Change point detection
The forward sequence (U_{f}) and the backward sequence (U_{b}) were plotted as the solid line and dashed line, respectively (Fig. 1). The points of intersection between the two sequences became the initial set of the change points. The thresholds of 95% CIs (Zscoresâ€‰=â€‰Â± 1.96) were set as the statistical filter. Only change points within the thresholds were retained. Specifically, in the case of Virginia, three points of intersection were initially detected. Week 4 (Point A in Fig. 1) and Week 43 (Point C in Fig. 1) were identified as the final change points with statistical confidence. Week 8 (Point B in Fig. 1) was excluded (Zscoreâ€‰=â€‰2.72), as it fell beyond the thresholds. Since both Point A and Point C were between Zscores of 0 and 1.96, these changes were upward.
Results
By applying the MKS test to weekly new COVID19 cases in 50 states, we identified that 30 states (60.0%) have at least one change point within the 95% CIs. For the unqualified states, most of them have no change points within the 95% CIs but have at least one change point beyond the 95% CIs. Only the state of Vermont has no change points either within the 95% CIs or beyond, meaning that there is no abrupt case decrease or increase during the entire study period.
To characterize the temporal distribution of these change points, we further divided the study period into three disease development stages, namely, Weeks 1â€“10 (March 23 through May 31, 2020), Weeks 11â€“30 (June 1 through November 19, 2020), and Weeks 31â€“45 (November 19, 2020 through January 31, 2021). These three stages were determined by the three clusters of chronologically ordered change points, as shown in Fig.Â 2. Based on the three development stages, we then mapped out the emergence of the change point for each state, as shown in Fig.Â 3.
FigureÂ 4 shows the change points detected by the MKS test for the 30 states with at least one change point within the 95% CIs. Among these states, we identified that a single change point exists for 25 states, two change points exist for 4 states (i.e., LA, OH, VA, and WA), and three change points exist for one state (i.e., GA). Then, we further derived 6 change patterns based on the emergence and direction of the change point at the three stages, as shown in TableÂ 1.
Discussion
Two epidemiologic patterns can be identified in Table 1. First, the downward changes at the first stage (Pattern 4) appear only in Northeastern states (e.g., CT, MA, NJ, NY), as confirmed in Fig. 3a. This pattern can be explained by the immediate state policy actions on social distancing in this region during the early outbreak. After COVID19 was declared a national emergency by the presidential proclamation on March 1, 2020 [20], most Northeastern states enforced social distancing regulations in late March and early April, including the closure of nonessential businesses and schools [21]. These policies largely restricted facetoface interactions, slowed the virus diffusion, and eventually, suppressed the epi curves. Second, the upward changes at the third stage appear mostly in the Western states (e.g., AZ, CA, CO, NM, WA, WY) and the Midwestern states (e.g., IL, IN, MI, MN, OH, WI), as shown in Fig. 3c. This result is consistent with the observation that most Western and Midwestern states experienced an abrupt case surge in the late summer and fall [22]. The rising trend could be linked to their less restrictive reopening policies, especially reopening indoor dining without a statewide facecovering mandate [23].
To further validate the MKS test, we compared it with two other change point detection methods, the pruned exact linear time (PELT) method and the regressionbased method (TableÂ 2), both of which are commonly used for detecting multiple change points in time series data. Specifically, the PELT method searches for change points by minimizing a cost function over possible numbers and locations of change points, and it implements an efficient pruning to increase the computational efficiency [24, 25]. The regressionbased method analyzes the time series using a regression model with multiple segments, where the coefficients shift from one stable regression relationship to another. It implements a dynamic programming approach to find segments that can minimize the residual sum of squares [26, 27]. We implemented the PELT method using the â€˜changepointâ€™ package in R [25] and the regressionbased method using the â€˜strucchangeâ€™ package in R [28].
The validation tested if the MKSidentified change points can be confirmed by the two other methods. A confirmation is accepted if an MKSidentified change point is validated by another method within a twoweek window. The comparison results are shown in Table 2. Based on the 36 MKSidentified change points, the MKStest reaches 41.7% agreement (15/36) with the PELT method and 47.2% agreement (17/36) with the regressionbased method. It is also worth mentioning that the other two methods identified at least one change point for every state, even when there is no obvious change of direction. The comparison results signify that the MKS test is a relatively conservative method for change point detection, as it can only detect abrupt changes and can thus avoid falsepositive results.
Conclusions
To sum up, the MKS test has several advantages in change point detection. First and foremost, it is characterized by high computational efficiency and easy implementation. Users can easily implement this method in Microsoft Excel without any prior statistical knowledge or modeling skills. Second, the method can detect the change of direction, whereas some other methods (e.g., PELT) can only identify the existence of a change without specifying the direction. Third, since the MKS test is a nonparametric model, it can be applied to time series data where the distribution is not normal or has extreme variability. However, due to its conservative nature and moderate agreement with the other slower but more sensitive methods, we recommend using the MKS test primarily for initial pattern identification and data pruning, especially in large data. For example, to identify the change points in a long sequence of COVID19 infection data, we can first use the MKS test to narrow down the time window where changes are likely to occur, and then use a second method (which has aÂ higher computational cost but isÂ more sensitive) to reconfirm the change pattern. In addition, as the conservativeness of the MKS test can be easily modified by adjusting the width of the statistical filter, future studies should examine how the quality of the results derivedÂ from the MKS test may vary as a function of the statistical filter.
This pilot study is the first to implement the MKS test for COVID19 studies. An openaccess tool is developed to facilitate the model implementation. With further validation and modification, the method can be applied to other health data, such as injuries, disabilities, and mortalities. By identifying key time points where chronologically ordered observations have a drastic change, the method can eventually contribute to revealing the etiology of these health outcomes and supporting public health decisionmaking.
Availability of data and materials
The data and codes for the study can be accessed on Github [https://github.com/peterbest52/mks].
Abbreviations
 CDC:

Centers for Disease Control and Prevention
 CI:

Confidence interval
 COVID19:

Coronavirus Disease 2019
 CP:

Change point
 MKS:

MannKendallSneyers
 PELT:

Pruned exact linear time
 US:

United States
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XC contributed towards conceptualization, writing the initial draft, and revising the draft. HW contributed towards conceptualization, methodology, and writing the initial draft. WL contributed towards visualizing the results and revising the draft. RX contributed towards methodology and revising the draft. All authors read and approved the manuscript.
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Chen, X., Wang, H., Lyu, W. et al. The MannKendallSneyers test to identify the change points of COVID19 time series in the United States. BMC Med Res Methodol 22, 233 (2022). https://doi.org/10.1186/s12874022017146
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DOI: https://doi.org/10.1186/s12874022017146
Keywords
 MannKendallSneyers
 Epi curve
 Time series
 Nonparametric
 COVID19
 Change point detection