Multivariate survival data

This section describes the typical notation of survival analysis for information on the time taken for the event to occur and the censored time is considered for N samples and J events. In a typical case of multivariate survival analysis, T_j denotes the true occurrence time and C_j denotes the censored time for the event j = 1,2,3,…j. Let T_j = (t_1,j,…,t_n,j,…,t_N,j) be a vector of the occurrence times where t_n,j is the true time-to-event of the event j for sample n. Let C_j = (c_1,j,…,c_n,j,…,c_N,j) be a vector of the censored times, where c_n,j is the censored time of the event j for the sample n. Given T_j and C_j, only (X_j, Δ_j) is observed, where X_j = (x_1,j,…,x_n,j,…,x_N,j), Δ_j = (δ_1,j,…,δ_n,j,…,δ_N,j), x_n,j = min (t_n,j,c_n,j) and δ_n,j = I(t_n,j ≤ c_n,j) for n = 1,2,3,…N. Note that T_j,C_j,X_j,Δ_j are vectors and T, C, X, Δ are matrices where T = (T₁,…,T_J), C = (C₁,…,C_J), X = (X₁,…,X_J), Δ = (Δ₁,…,Δ_J). That is, the event j has a set of observed times and censoring indicators represented as (X_j,Δ_j), or (X_j +) if Δ_j is zero, or (X_j) if Δ_j is one [7]. Here, the representation of survival data (X_j,Δ_j) is used. Note that the true correlation of times to two events only depends on the occurrence times T, and the correlation cannot be completely measured because only X and Δ are given.

Density estimation

As the covariance cannot be calculated if the true values are not given, it is difficult to directly estimate the partial correlation. To solve this problem, the multivariate survival analysis based on the optional Polya tree (OPT) Bayesian estimator [11] is applied here to estimate the joint probability density function of censored times to events. This enables the handling of bi-dimensional survival data.

Consider the calculation for the probability density with survival times. Let f_i,j(t_i,t_j) be the joint probability density function of T_i and T_j for the true times to the occurrence of the events i and j, respectively. We estimate f_i,j by recursive binary splits of the bi-dimensional region through an optional Polya tree approach [12]. Let A be a region in the sample space, A₁₁, A₁₂ be regions partitioned by the T_i-axis, and A₂₁, A₂₂ be regions partitioned by the T_j-axis. Additionally, let Φ(A) be the likelihood for the region A, mathematically defined by where B(·) is a beta function and Φ₀(A) is the likelihood when all sample points are uniformly distributed, and N(A) is NP_A, where N is the total number of observations and P_A is the probability mass obtained in region A by Kaplan-Meier’s survival estimator. If , the probability density of A is calculated as , where |A| is the area of A. If not, the probability density is considered to have a non-uniform distribution, and the region is split into the sub-region. If A is split into A₁₁ and A₁₂. Otherwise, A is split into A₂₁ and A₂₂.

The estimator provides block-wise and uniformly distributed probability density based on a non-parametric Bayesian estimation. The details of this method are described in [11].

Covariance matrix estimation

The covariance matrix of the times to multiple events can be obtained by the probability density function. Let M be the covariance matrix of the times to multiple events and m_i,j be the element of the matrix for entries i and j where i = 1,2,3,…,J and j = 1,2,3,…,J. If we know the joint probability density function f_i,j for times to two events, m_i,j is explicitly calculated as

Despite the clear and straightforward calculation, it is difficult to estimate the covariance given X and Δ and not T. Let be the joint probability density function estimated by Bayesian estimator where (X_i,Δ_i) and (X_j,Δ_j) are given. The covariance matrix estimated by the above equation using instead of f_i,j is not semi-positive definite because of the limitation of the probability estimation. The joint probability density function is estimated using only two events, without the conditional consideration of other events. The inconsideration causes inconsistencies in the probability density for events. The marginal probability density function of times to the event i can be approximated from ,

These derived marginal probability densities are similarly distributed but not identical. The marginal probability density is not consistently measured by estimating the joint probability density.

To avoid this effect, the covariance matrix is empirically calculated by estimating the true times to censored events rather than directly calculating from the joint probability density. The true times are estimated by the expectation of conditional probability for each sample. Clearly, if the time-to-event for a sample is not censored, the true time is equivalent to the observed time. However, when the time-to-event for a sample is censored, only the fact that the true time is larger than the observed time is given, and the true time is unknown. The true times of event occurrences are estimated by the expectation of the conditional probability given the observed times for the sample n and the event j,

Then, a set of the estimated times to the event are obtained as . The expectation is approximated from a set of expectations for the marginal probability distribution derived from the joint conditional probability distribution for pairs of events. Let be the estimated marginal probability density function for event j obtained from . Then, the expected time can be calculated as

where . However, it is difficult to directly calculate the expectation of the conditional probability distribution. Therefore, the Monte Carlo simulation is used to approximate the above expectation. Following this, the covariance matrix is estimated by matrix multiplication of the estimated times and the average of the estimated times to each event as follows: where and 1 denotes a vector that consists of N elements of ones.

Inverse covariance estimation

The typical algorithm for inverse covariance estimation, graphical lasso, is briefly described. Inverse covariance estimation was designed to detect the partial correlation of fully observed data. The algorithm is based on the maximization of L₁-penalized Gaussian log-likelihood of the observations with respect to the mean parameter [5, 6]: where Θ is the inverse matrix of non-negative definite covariance. To solve this problem, the lasso regression was utilized. One column of the empirical covariance matrix of data is considered as response variables. And, columns excluding the column index corresponding to the response is considered as independent variables. It is iteratively calculated by permuting the target column and completed if it is converged.

For multivariate survival data, if all the times to events are fully observed, the above algorithm can estimate the inverse covariance matrix of the data. However, the covariance matrix of the times to events cannot be calculated completely because of the censoring. Censoring causes distortion of times to events and replaces the actual times of occurrence of events with censored times. Therefore, the log-likelihood of the estimated times to events is used here instead of the observations,

The partial correlation of censored events is determined by the graphical lasso with the covariance matrix of the estimated times to events depending on the non-negative penalty ρ. If ρ is zero, all the absolute values of partial correlation between events will be greater than 0. As ρ increases from 0, less non-zero partial correlations are detected. Finally, the network based on non-zero partial correlation can be constructed for multiple censored events.

Additionally, the penalty parameter ρ can be selected through cross-validation to obtain a single network [13]. We briefly introduce the cross-validation for the graphical lasso. By partitioning the entire sample into k-fold, we can find an appropriate ρ that maximizes the summation of the log-likelihood of each fold. Let and be the empicial covariance matrix of train set and validation set of fold i, respectively. Then, the log-likelihood can be calculated by where denotes the covariance matrix estimated by the graphical lasso with ρ. Then, the penalty parameter is obtained by . The details are described in [13]. The single network can be constructed through the above process.

Data for the case study

The proposed method was applied in a case study to a real dataset to demonstrate its effectiveness. The National Health Insurance Sharing Service (NHISS) in South Korea has provided the National Sample Cohort (NSC), which contains medical information of one million people extracted by random sampling from 2002 to 2015 for research purposes [14, 15]. For all samples, personal information, such as date of birth and place of residence, is masked to prevent identification of individuals. The NCS consists of about 2 billion medical events such as the date of diagnosis with disease codes, the month of death, and health screenings.

Here, the first diagnosis record for categorized disease codes was considered as an event of interest. Disease codes have been categorized via the Korean Standard Classification of Diseases 7th Revision (KCD-7) modified from the International Statistical Classification of Diseases and Related Health Problems 10th Revision (ICD-10) code [16].

The data can be found on the NHISS website (https://nhiss.nhis.or.kr). Details for the data used in the case study were described in S1 File. These data were accessed with IRB-2018-0110 approval from Korea University Institutional Review Board. We declare no conflict of interest with the NHISS.

Simulation studies

This section presents the performance of CNE recorded through simulation experiments. The main purpose of the simulation is to show the accuracy with which CNE detects non-zero partial correlations through the censored times to events. The simulation follows three steps:

1. Configure a network structure that represents the relation between events. Note that nodes and edges in the network represent events and non-zero partial correlations, respectively.
2. Generate the time-to-event data which follow the configured network and randomly censor occurrence times. The values of the data must be positive because they represent time, and the inverse covariance matrix of the data follows the configured network. Accordingly, the true times to events are not independent to each other. On the other hand, the censored time C_j is generated independently to T_j because the occurrence of censoring is irrelevant to the event of interest. It reflects that events can be censored by different reasons in real data and is a general way to simulate censoring in related works [11]. Note that the proposed method does not assume the independence of censoring.
3. Select the non-zero partial correlations of times to the occurrence of events given the survival data, X,Δ. The edge selection performance is evaluated from the area under the curve (AUC) and true positive rate (TPR) for false positive rates (FPRs) of 0.05 and 0.1 from the receiver operating characteristic (ROC) curve. The undirected edges that represent the true times being partially correlated are considered as conditional positives.

Following this, the survival data were generated based on the information of correlated events and censored times. Accordingly, the estimation of non-zero partial correlation was performed with the survival data. For all edges, the penalty parameter ρ_j was measured when the estimated absolute of the inverse covariance became greater than 0. The predicted positive was determined through ρ_j. Three types of well-known networks, scale-free, random, small-world networks, were used in order to demonstrate that the proposed method is suitable for detecting partially correlated neighbors.

In addition, the proposed method was compared with the graphical lasso estimations based on the survival estimators of Dabrowska [17] and Lin-Ying [18]. Both are conventional approaches to estimating probability density functions for bivariate survival data. The upper bound performance was measured by the graphical lasso based on uncensored true times T. Additionally, the lower bound performance was measured based on the observed time X regardless of the censoring. The baseline was measured by the correlation coefficient of the observed times in uncensored samples, which have the censoring indicator of 1. The performance was investigated with repetitions and varied by changing the number of samples and events.

In addition to precedent simulation, the variation of performance was investigated by varying the censoring rate of events in each network. The simulation was performed repetitively, and the average, minimum, and maximum performances were specified.

Simulation data generation

In this simulation, the detection performance was evaluated by comparing the edges estimated from time-to-event data with the true edges. To perform the simulation, the censored times to events were generated. The data have N samples and J events where N = 100,200,1000 and J = 20,50,100 and were generated through four steps.

First, an undirected network was generated and the network structure was considered. Edges of the network were considered as the conditional positive that should be detected, and nodes indicated events. The scale-free networks, random networks, and small-world networks were considered for this simulation.

Second, the true time matrix T was generated. The generation of the true times follows the simulation by Peng [1], but there is a difference in that all elements must be positive. The inverse covariance matrix should reflect the topology of the configured network. Essentially, the true time matrix T was determined by the undirected network structure. Let P denote the initial inverse covariance matrix and P(i,j) denote the element of P for entries of i,j = 1,…,J. Additionally, let Σ denote the covariance matrix of the true times and Σ(i,j) denote its element. To generate true times that reflect the network structure, P was set as follows:

where i~j and i≁j denote the existence and non-existence, respectively, of an edge between nodes i and j, and is a correlation constant. In this simulation, was used. To ensure the non-negative definiteness of the covariance matrix, p_i,j was rescaled by dividing itself by except for the diagonal element. Then, the rescaled matrix was symmetrized to be diagonally dominant. Let A be the symmetrized matrix and A(i,j) be its element. Then, the covariance matrix Σ which follows the preconfigured network is obtained by

For the specific case in which there are some isolated nodes in the network when , the rows and columns corresponding to indices of isolated nodes were first eliminated from the initial inverse covariance matrix. Following the above procedure, these rows and columns were inserted into the original position of Σ. Then, the non-negative true time-to-event data T were generated as the following log of the multivariate Gaussian distribution, where . Note that the true time-to-event data matrix T has N rows and J columns.

Third, the censored times C were generated. The censored times are assumed to be completely independent of true times and network structure. To reflect the irregular effect of censoring, C_j has no correlation with the corresponding true times and other censored times. The censored times were generated as following the exponential distribution: where the mean of censored times to each event is . Different λ values were used for simulation.

Lastly, the survival data were composed. The observed time-to-event matrix X was obtained by min(T_j,C_j), and the censored time matrix Δ was obtained by I(T_j ≤ C_j). The censored rate which represents the rate of censored samples can be represented by ; it changes based on the λ value. The censored rate moves closer to one if λ becomes larger.

Simulation results

To show the proposed method is effective under diverse conditions, its performance when selecting partially correlated events was investigated. All simulation studies were performed under the scale-free, random, and small-world networks.

To provide evidence for the superiority of the proposed method, the selecting power of the upper bound, lower bound, baseline, and other competitive methods were investigated via simulation repetitions. Clearly, if the true times to events are provided, the partial correlation can be explicitly estimated by graphical lasso. Thus, the case when times are fully observed, i.e., T is given, was set as the ‘upper bound,’ while the case when the censored time X was provided without consideration of censoring was set as ‘lower bound.’ The case when estimating correlated edges by the absolute value of the correlation coefficient only by the samples in which both events are not censored was considered as ‘baseline.’ The estimators by Dabrowska and Lin and Ying were also adopted.

Fig 1 shows the simulation results for the network estimation in which there are 100 events corresponding to nodes and 1,000 samples for the events of interest. For generating the censored times, λ = 1 was used, and the censored rate ranged between about 50–70%. Fig 1A shows the scale-free network that has 99 edges corresponding to partially correlated pairs for the case in which some hub nodes of a comparatively larger degree exist. Fig 1B shows the performance of estimation for the scale-free networks. On average, the proposed method achieved an AUC of 0.96 with a TPR of 0.88 controlled at an FPR of 0.05, and a TPR of 0.92 controlled at an FPR of 0.1, with 10 repetitions. Additionally, the average AUCs of the upper bound, lower bound, baseline, Dabrowska estimator, and Lin-Ying estimator were 0.98, 0.8, 0.8, 0.81, and 0.46, respectively. An improvement of about 0.15 AUCs is shown in comparison to the lower bound, baseline, and competing method. The proposed method shows that the difference in performance is 0.02 AUCs in comparison to the upper bound despite the data being censored. The proposed method also shows stability with a standard error of about 0.01 for the AUCs. Fig 1C shows the random network, which has some isolated nodes and edges generated by the probability of 0.02. Fig 1D shows the performance of selecting a partial correlation for the random networks. On average, the proposed method also achieved a remarkable performance of 0.95 AUCs with a 0.81 TPR controlled at 0.05 FPR, and 0.87 TPR controlled at 0.1 FPR. Fig 1E shows the small-world network with the rewiring probability of 0.15. Fig 1F shows the ROC curve for network estimation for the small-world network in the presence of censoring. The proposed method achieved 0.98 AUCs with a 0.9 TPR controlled at 0.05 FPR, and 0.94 TPR controlled at 0.1 FPR. Additionally, networks estimated through cross-validation showed 0.94 TPR with 0.19 FPR, 0.75 TPR with 0.02 FPR, and 0.97 TPR with 0.22 FPR in Fig 1B, 1D and 1F, respectively. The cross-validation for the penalty parameter provided fairly dense networks in the scale-free and small-world network settings and sparse network in the random network setting. In all the methods except the Lin-Ying estimator, the network estimations are more effective in the scale-free and small-world networks than in random networks.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Topologies of the undirected network settings and the performance for detecting correlated events.

‘Upper bound’ denotes the case of given true time and ‘Lower bound’ denotes the network estimation using the censored times regardless of censoring. ‘Baseline’ denotes the selection by the absolute value of the correlation coefficient. ‘Dabrowska’ and ‘Lin-Ying’ indicate competing estimators for CNE. (A) Scale-free networks with 100 nodes and 99 edges. (B) Network estimation performance where the true times follow the network (A) and the censoring parameter λ = 1. (C) Random networks with 100 nodes and the probability of 0.02 to generate edge containing some isolated nodes. (D) Network estimation performance when the true times follow the network (C). (E) Small-world networks with 100 nodes and the rewiring probability of 0.15. (F) ROC curve for estimating the true edges of the network (E). * represents network estimation with cross-validation.

https://doi.org/10.1371/journal.pone.0239760.g001

In addition to the above evaluation, we investigated which edges were found by estimators. In Fig 2, network skeletons correspond to networks in Fig 1A, 1C and 1E, respectively. Red lines indicate edges found by estimators and grey lines indicate that they were not found. The edge was selected at an FPR of 0.05. The proposed method located subsets that consist of edges connected to hub nodes more effectively than other estimators (Fig 2A). The proposed method also effectively found subsets that consisted of a few nodes and edges, as shown in Fig 2B. Fig 2C shows that the proposed method adequately estimated large clusters in the small-world setting.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Edges detected by each estimator.

(A) Scale-free network setting. (B) Random network setting. (C) Small-world network setting. Red lines indicate well-estimated edges controlled at FPR 0.05 in the network skeleton.

https://doi.org/10.1371/journal.pone.0239760.g002

To show the general case of network estimation, the methods were evaluated via ten repetitions by varying the simulation settings in the scale-free and random network settings (Table 1). The number of samples was considered as 100, 200, and 1000, while the number of events was considered as 20, 50, and 100, respectively. In this simulation, we set λ = 1, and both networks were used. In the scale-free network, there are J nodes whose degree distribution follows a power law and J − 1 edges. The random networks were generated with J nodes and the probability of 2/J that a node is connected to another node. The simulation results are shown in Table 1. We investigated the AUC and TPR at FPRs of 0.05 and 0.1 with respect to each method and summarized the averages and standard errors. The proposed method achieved the most powerful selection compared with baseline and competitive methods in all N and J. There is a tendency for the overall methods to show lower performance if the insufficient sample size is given, and the ratio of the dimension and sample number is low. In other words, the network estimation becomes challenging when N ≪ J [1] and it also affects the estimation of the probability density function with multivariate survival data [11]. In particular, the Lin-Ying estimator yields a negative probability density and leads to significantly lower performance.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Summary of simulations with various sample numbers and event numbers.

Each cell of the table consists of the average and standard error of AUC (top) and TPR at FPR of 0.05 (middle) and 0.1 (bottom) for simulations repeated ten times.

https://doi.org/10.1371/journal.pone.0239760.t001

The above simulation was carried out on 24 CPU cores, which are Intel (R) Xeon (R) E5-2630 v2 @ 2.60GHz and 128GB RAM. The average computation time of each simulation is summarized in Table 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Average computation time (second) of censored network estimation for three types of networks according to the number of nodes and samples.

https://doi.org/10.1371/journal.pone.0239760.t002

In addition, the selection power of the proposed method and the lower bound were investigated by changing the parameters for generating the censored times with respect to 1,000 samples and 20 events. The censored times were generated based on the exponential distribution with parameter λ. If λ changes, the censored rate of an event also changes. We took λ = 2⁻⁵, 2⁻⁴,…, 2², and the simulation was repeated ten times per λ value. Fig 3A and 3C show the ranges of the censored rates for the scale-free and random networks. When the expectations of the true and censored times are identical, the censored rate was about 50–70%. Fig 3B and 3D show changes in the AUCs for the network estimation based on λ in the scale-free and random networks. The AUCs of both methods were over 0.9 up to λ = 2⁻². From λ = 2⁻¹, performance differences begin to emerge. When λ = 2⁰, the AUCs of the proposed method and lower bound were 0.96 and 0.84, respectively. When λ = 2¹, 2², the lower bound performance was under 0.5, which is as weak as random selection. In contrast, the AUCs of the proposed method were 0.72–0.97 (B) and 0.56–0.96 (D) for the scale-free and random networks, respectively, despite the censored rate being 75–93%.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Censored rate and AUC for the change of censored time parameter λ.

(A) Variation of the censored rate in the scale-free network simulation. (B) Variation of AUC in the scale-free network simulation. (C) Variation of the censored rate in the random network. (D) Variation of AUC in the random network. Points represent the average values, upper bars represent the maxima, and lower bars represent the minima.

https://doi.org/10.1371/journal.pone.0239760.g003

Lastly, we investigated the performance of three network estimation methods with the covariance matrix measured from multivariate censored data with 1,000 samples and 100 events. The compared methods are sparse columnwise inverse operator (SCIO) [19], sparse partial correlation estimation with degree-based weights (SPACE) [1], and graphical lasso (GLasso). Fig 4 shows the ROC curves for the network estimation in each network setting. AUCs were evaluated as 0.942, 0.921, 0.946 in the scale-free network setting, 0.95, 0.961, 0.96 in the random network setting, and 0.963, 0.974, 0.977 in the small-world setting (SCIO, SPACE, GLasso). There were slight performance gap between the three methods, but the methods showed similar estimation performance.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Comparison of network estimation methods after the covariance matrix measure.

(A) Scale-free network. (B) Random network. (C) Small-world network.

https://doi.org/10.1371/journal.pone.0239760.g004

Application studies

The proposed method was applied to a case study to investigate the relationship with the next disease after the initial diagnosis of ‘acute upper respiratory infections’ for the newborn baby dataset. It is important to detect connections to other diseases because the ‘acute upper respiratory infections’ are a common disease experienced by many newborns [20]; in the dataset, 99.49% of newborns had been diagnosed with them. Additionally, the number of samples is sufficient to apply the method.

The NCS contains a table composed of the diagnostic date and primary sick symbol for one million samples. Among all the samples, we focused on a dataset of newborns born between 2002–2006 to identify the date of the first diagnosis. In particular, 13,735 babies who were first diagnosed with the ‘acute upper respiratory infections’ were selected to explicitly identify time-to-event data. Accordingly, the time of the dataset was measured by a period from the ‘acute upper respiratory infections’ to diagnoses for other diseases. The dates of the first diagnosis were collected by tracking the medical records of the babies up to 2015. If there is no record for the diagnosis of a categorized disease until 2015, the event is considered as censored. Additionally, only 36 disease categories with observation rates exceeding ten percent were considered instead of the entire disease data because disease categories with low observation rates were rarely connected to the others. The disease code, title of the disease, and censored rates of each of the diseases are listed in Table 3. ‘Other acute lower respiratory infections’ were the most observed and ‘disorders of skin appendages’ were the least observed cases in the dataset.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Summary of disease codes with an observation rate of over 10% for newborn babies who were first diagnosed for acute upper respiratory infections.

https://doi.org/10.1371/journal.pone.0239760.t003

Using the proposed method, the correlations between diseases were analyzed based on the times to the first diagnosis. The intensively correlated diseases were estimated by cutting the top five percent partial correlation among all possible pairs. The interrelation was visualized as an undirected network. The relation and each disease were represented by edges and nodes, respectively. In this network, the size of each node indicates its degree, and three of the most highly correlated edges have been marked with a bold line.

Fig 5A shows the correlation network estimated by the proposed method based on times to the first diagnoses of diseases after ‘acute upper respiratory infections’ in babies. To compare the results, another disease pair was estimated by the co-occurrence of diseases within the observation period [21], as shown in Fig 5B. The co-occurrence network indicates that the pair of diseases were frequently observed within the study period. On the other hand, the correlation network represents an analytical relation for the diagnostic time points.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Disease networks estimated from the first diagnosis of the newborn baby dataset in Korea.

(A) Correlation network estimated by CNE. (B) Co-occurrence network within the observation period.

https://doi.org/10.1371/journal.pone.0239760.g005

Using CNE and co-occurrence, 32 pairs of interrelated diseases were found within 36 disease codes. For example, three of the most intensive correlations, (‘influenza and pneumonia,’ ‘chronic lower respiratory diseases’), (‘other intestinal diseases,’ ‘symptoms and signs involving the digestive system and abdomen’), and (‘injuries to the wrist and hand,’ ‘injuries to the ankle and foot’), were found in the correlation network. The most intensive correlation has been explicitly demonstrated through research on the effect of influenza and pneumonia on lower respiratory infection [22, 23]. The second highly correlated pair corresponds to symptoms of the digestive system due to intestinal problems, leading to hospital visits [24]. The third intensive pair can be explained by the association of injuries to the wrist and hand and injuries to the ankle and foot [25]. The correlation tends to be affected by times to events rather than the observation rate of each disease. The above example is clearly a plausible case based on advanced research. In contrast, the above example could not be found in the conventional co-occurrence network; instead, (‘other acute lower respiratory infections,’ ‘other diseases of upper respiratory tract’), (‘other acute lower respiratory infections,’ ‘intestinal infectious diseases’), and (‘other acute lower respiratory infections’, ‘dermatitis and eczema’) were representatively found. The first case is marginally reliable, and the third case could be supported by the conventional concept of inclusion of the lower respiratory illness in atopic diseases in infants [26], whereas the second case was difficult to demonstrate based on existing clinical research. In addition to the example, there are several improbable cases in the co-occurrence network, such as (‘disorders of conjunctiva,’ ‘chronic lower respiratory diseases’) and (‘intestinal infectious diseases,’ ‘disorders of ocular muscles, binocular movement, accommodation and refraction’). Additionally, the co-occurrence network showed that the co-occurrence tends to be strictly dependent to the observation rate rather than times to events. This implies that the correlation network based on times to events is more appropriate to reveal a potential relationship between diseases. Therefore, we suggest a researchable hypothesis for a new discovery for probable relationships of diseases in the correlation network.

This work was carried out on the remote windows server of Intel(R) Xeon(R) CPU E5-2690 v4 A 2.60GHz and 3.00GB RAM provided by NHISS. The computation time was 73,170 seconds for 13,735 samples and 36 events.