Parameter estimation of Weibull distribution for the number of days between drug administration and early onset adverse event

ABSTRACT The Weibull distribution is applied to the number of days between the start date of drug administration and the date of occurrence of an adverse event. The tendency of occurrence of adverse events can be clarified by estimating the two- or three-parameter Weibull distribution, using the data regarding the number of days. Our purpose is to estimate the parameters of the Weibull distribution with high accuracy, even in low-reported adverse events, such as new drugs, polypharmacy and small clinical trials. Furthermore, the two-sample Kolmogorov – Smirnov test (two-sided) is used to examine whether the tendency of occurrence of adverse events is different for two Weibull distributions estimated from two drugs with similar efficacy. We used discrete data derived from FDA Adverse Event Reporting System (FAERS), as the FAERS data are presented in years, months and days without hours and minutes. Because this study focuses on early onset adverse events, data may be contained 0 days. The discreteness of the data and the fact that it may include zero make this distribution different from the general Weibull distribution, which is defined for continuous data greater than zero. We search for the optimal parameter estimation method for the Weibull distribution under these two conditions, and verify its effectiveness using Monte Carlo simulations and FAERS data. Because the results obtained from FAERS data may differ depending on data handling, we describe the of data handling technique and the sample code that can reproduce the results.


Introduction
The FDA Adverse Event Reporting System (FAERS) (Food and Drug Administration 2021) presents considerable data regarding the drug name, start date of drug administration, name of the adverse event, and date of occurrence of the adverse event. FAERS data have been analyzed by various researchers to obtain new insights (Chen et al. 2013;Huang et al. 2013). For example, 27 cases of olaparib-induced anaemia were reported between the second quarter of 2015 (2015Q2) and 2017Q1. The number of days between olaparib administration and occurrence of anaemia was 0, 76, 120, 27, 41, 392, 50, 21, 52, 235, 70, 51, 87, 57, 14, 63, 38, 18, 5, 3, 37, 28, 42, 20, 193, 51, 18 (day). This data was obtained as discrete data and contained 0 days. This study aims to estimate the parameters of the Weibull distribution with high accuracy, even if the adverse events are low-reported, similar to the above-mentioned example. For high-reported adverse events, the parameter estimation of the Weibull distribution is performed using the data up to the date of low-reported adverse events, and the data not used for parameter estimation are used to verify the validity. Because FAERS data were presented as 'anaemia' not 'anemia,' we used the notation of 'anaemia.' The results obtained from FAERS data may differ depending on data handling. Our data handling technique is presented in Section 5. The approval from an institutional review board is not required, as the FAERS is an unlinkable anonymized database open to the public.
A Weibull distribution is considered as suitable in analyzing various medical data (Sverdlov et al. 2014;Wu 2015). Previous studies have obtained new insights by analyzing data about adverse events (Liu 2016;Nam et al. 2017). Some of these studies utilized the parameter estimation of the Weibull distribution for the number of days between the start date of drug administration and the date of occurrence of the adverse event (Ando et al. 2019;Kan et al. 2021). These studies assumed a twoparameter Weibull distribution, instead of the three-parameter Weibull distribution; moreover, the two-parameter was estimated using the maximum likelihood method (MLE). Generally, if the sample size is large, the MLE can estimate the parameters of a distribution with high accuracy. However, if the sample size is small or the number of parameters is large, the population parameters may not be accurately estimated. When the data is contained 0 days, it is better to use a two-parameter Weibull distribution with the location parameter set to 0. When the data is not contained 0 days, it is considered better to use a three-parameter Weibull distribution. However, it is necessary to register a statistical analysis method before the start of a clinical trial (Berry et al. 2010). After the data are obtained, it is not possible to determine the parameter estimation method for the Weibull distribution. Therefore, a clinical trial plan using a three-parameter Weibull distribution is recommended. If the data is contained 0 days in the clinical trial, the two-parameter Weibull distribution can be accommodated by setting the location parameter in the planned estimation method to 0, as mentioned earlier. We use the following four proposed methods for parameter estimation using a three-parameter Weibull distribution: a shape parameter estimation method proposed by Ogura et al. (2020) and discussed by Sugiyama and Ogura (2022), using the minimum-variance linear estimator with a hyperparameter (MVLE-H); the weighted MLE (w-MLE) method proposed by Cousineau (2009), wherein three weights are added to the MLE; the Bayesian likelihood (BL) method proposed by Hall and Wang (2005), wherein the MLE is multiplied with an empirical prior; and the location and scale parameter free maximum likelihood estimators (LSPF-MLE) method proposed by Nagatsuka et al. (2013), which estimates the shape parameter using the independent statistics of the location and scale parameters.
The discreteness of the data and possibility of obtaining zero values distinguished this distribution from the general Weibull distribution, which is defined for greater than zero and continuous data. Because this study focuses on early onset adverse events, we searched for the optimum estimation method for data contained 0 days. We will determine the optimal estimation method for late-onset adverse events in our future work. The tendency of occurrence of adverse events can be clarified through high accuracy estimation of the parameters of the Weibull distribution. The tendencies were classified into the three categories. First, when the shape parameter is less than 1, many adverse events occur immediately after drug administration. Second, when the shape parameter is equal to 1, the occurrence of the adverse event stays constant regardless of the number of days elapsed. Third, when the shape parameter is greater than 1, many adverse events occur after a number of days have passed from drug administration. Obtaining the tendency of occurrence of adverse events is very useful in the clinical setting.
Additionally, it is also important to clarify whether two drugs with similar efficacy have different tendencies of occurrence of adverse events. Previous studies have discussed whether the shape parameter of the Weibull distribution estimated for each drug is less than or greater than 1 (Kan et al. 2021). The two-sample Kolmogorov -Smirnov (K-S) test (two-sided) (Kolmogorov 1933;Smirnov 1939) is used to examine whether the tendency of occurrence of the adverse events is different for two Weibull distributions estimated using FAERS data for two drugs. Generally, statistical tests are performed using samples, but we are interested in whether the two estimated Weibull distributions are different. Therefore, we conduct a test using the two cumulative distribution functions of the two estimated Weibull distributions. In this study, we utilize a distribution different from the general Weibull distribution due to the discreteness of the data and possibility of obtaining zero values. Therefore, before performing the two-sample K-S test (two-sided) using FAERS data, we use Monte Carlo simulations (MCSs) to verify that when the null hypothesis is true, the probability that the null hypothesis is rejected is lower than the significance level.
In Section 2, we describe the MLE, MVLE-H, w-MLE, BL and LSPF-MLE methods for estimating the three-parameter Weibull distribution, and derive a two-parameter Weibull distribution estimation method with the location parameter set to 0. In Section 3, we present a method for comparing the estimated two Weibull distributions using the two-sample K-S test (two-sided). In Section 4, we use MCSs to search for an optimum estimation method and verify the performance of the two-sample K-S test (two-sided). In Section 5, we present an attempt to estimate the parameters using FAERS data. In Section 6, conclusions of this study are presented.

Parameter estimation methods
The probability density and cumulative distribution functions of the three-parameter Weibull distribution are shown as follows: where m > 0, η > 0 and γ < x are the shape, scale and location parameters, respectively. In a twoparameter Weibull distribution, we substitute γ ¼ 0 into (1) and (2). Let x 1 ; . . . ; x n be random samples from the three-parameter Weibull distribution, and let x i ð Þ denote the i-th order statistic,0 � x 1 ð Þ � � � � � x n ð Þ . Random samples from the general Weibull distribution do not contain equal signs, but the discrete data in this study may contain equal signs. In this study, we replace 0 with a small value of 0.001, because the parameter estimation of the Weibull distribution cannot be estimated when x 1 ð Þ ¼ 0. We describe the following five existing parameter estimation methods: MLE, MVLE-H, w-MLE, BL and LSPF-MLE methods.

MLE
In a three-parameter Weibull distribution, the likelihood function of (1) is shown as follows: The log-likelihood function is further presented as follows: The MLE is obtained by setting the partial differentiation of log L with respect to m, η and γ being equal to 0, and presented as follows: Three parameters are estimated by solving three simultaneous equations; however, these equations may not be solved them when n is small. In a two-parameter Weibull distribution, we obtain the following two equations by substituting γ ¼ 0 into (5) and (6): Two simultaneous equations can be solved even when n is small (Lehmann and Casella 1998).

MVLE-H method
In a three-parameter Weibull distribution, Ogura et al. (2020) presents an unbiased estimator of 1=m as: where and u is the standard uniform distribution. By substituting 1= x 1 ð Þ À γ (10), it is converted to an equation that does not include γ, as follows: where c > 0 is a hyperparameter. The shape parameter is expressed as cov log x i ð Þ À γ var log x i ð Þ À γ In a two-parameter Weibull distribution, we substitute γ ¼ 0 into (10) and (14)-(16) and obtain the following: The shape parameter can be estimated by substituting (17) and (18) into (13).

W-MLE method
The weights of W 1 and W 2 for every n and the weight of W 3 for every n and shape parameter are posted on the w-MLE website (Cousineau 2021). In a three-parameter Weibull distribution, the shape and location parameters are estimated by minimizing the following: Using m ¼m and γ ¼γ by minimizing wMLE m; γ ð Þ, the scale parameter can be estimated as η ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi In a two-parameter Weibull distribution, we substitute γ ¼ 0 into (21). Thus, the shape parameter is estimated by minimizing the following: Using m ¼m by minimizing wMLE m ð Þ, the scale parameter can be estimated as η ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi

BL method
In a three-parameter Weibull distribution, the BL method multiplies the MLE using an empirical prior. The shape and location parameters are estimated by maximizing the following: Using m ¼m and γ ¼γ by maximizing BL1 m; γ ð Þ, the scale parameter can be estimated by maximizing the following: In a two-parameter Weibull distribution, we substitute γ ¼ 0 into (23). Thus, the shape parameter is estimated by maximizing the following: Using m ¼m by maximizing BL1 m ð Þ, the scale parameter can be estimated by maximizing the following:

LSPF-MLE method
In a three-parameter Weibull distribution, the LSPF-MLE method of the shape parameter is estimated by maximizing the following equation: . . . ; n À 1). If z i ð Þ ¼ 0, we replace it a with the small value z i ð Þ ¼ 0:001 in this study. Using m ¼m by maximizing LSPF m ð Þ, the temporary location and scale parameters can be estimated using . The location and scale parameters are , respectively. In a two-parameter Weibull distribution, the scale parameter is estimated by η ¼ 1

Modified w-MLE, BL, and LSPF-MLE methods
The MVLE-H method estimates only the shape parameter; however, Ogura et al. (2020) improved the w-MLE, BL and LSPF-MLE methods by utilizing the shape parameter estimated through the MVLE-H method. These methods treat m in (13) as if it was known. These methods are referred to as modified w-MLE, BL and LSPF-MLE methods, to distinguish them from the original three methods. In a three-parameter Weibull distribution, m is considered as known. Therefore, only γ is estimated using wMLEm; γ ð Þ in (21) of the modified w-MLE method. Only γ is estimated using BL1m; γ ð Þ in (23) of the modified BL method. The LSPF m ð Þ is not used in (27) of the modified LSPF-MLE method. In a two-parameter Weibull distribution, because γ ¼ 0 and m is considered as known, only the scale parameter is estimated. The scale parameters are estimated as η ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi for the w-MLE method, maximization of (26) for the BL method, and η ¼ 1

Two-sample K-S test
The two-sample K-S test (two-sided) is used to examine whether the two drugs have different occurrence trends in for adverse events. Let m 1 andη 1 be the two-parameter Weibull distribution, estimated using the first drug of sample size n 1 . Similarly, let m 2 andη 2 be the two-parameter Weibull distribution, estimated from the second drug of sample size n 2 . Using the two cumulative distribution functions of the two Weibull distributions, the statistic D is defined as follows: For sample sizes of n 1 � 25 and n 2 � 25, the critical values of D at the significance level α were posted by Rohlf and Sokal (2012). For sample sizes of n 1 > 25 or n 2 > 25, the null hypothesis is rejected at significance level α when the following is satisfied: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi n 1 þ n 2 n 1 n 2 r ; where c α ð Þ ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi

Parameter estimation of Weibull distribution
This study focuses on early onset adverse events. Therefore, it was assumed that the data was contained 0 days. We searched for the optimum estimation method for the two-parameter Weibull distribution. The settings in MCSs included the sample size n ¼ 5; 10; 15; 20; 25; 30, the shape parameter m ¼ 0:5; 0:6; 0:7; 0:8; 0:9; 1:0; 1:1; 1:2; 1:3; 1:4; 1:5, the scale parameter η ¼ 100; 200, and the location parameter γ ¼ 0. The random samples were rounded down to the nearest whole number. When the random samples truncated to the nearest integer amounted to 0, we replaced it with a small value of 0.001, as the two-parameter cannot be estimated in a Weibull distribution. The MCSs were repeated 10,000 times and evaluated using bias and root-mean-square error RMSE ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi variance þ bias 2 p . We used the software R version 4.1.1 (R Core Team 2021) for MCSs. The MCSs were conducted using the following procedure: Step 1. Generate random samples x 1 ; . . . ; x n ð Þ from g x; m; η; γ ð Þ.
Step 2. Round down x i to the nearest whole number.
Step 3. If x i becomes 0 by step 2, replace 0 with 0.001.
Step 4. Estimate the two-parameter using each method.
Step 6. Calculate the bias and RMSE of the shape and scale parameters. Table 1 lists the bias and RMSE of the shape parameter, estimated using the five methods (MLE, MVLE-H, BL, w-MLE and LSPF-MLE methods) for n ¼ 15; 20; 25; 30 and m ¼ 0:5; 1:0; 1:5. Table 2 lists the bias and RMSE of the scale parameter, estimated using the seven methods (MLE, modified BL, BL, modified w-MLE, w-MLE, modified LSPF-MLE and LSPF-MLE methods) for n ¼ 15; 20; 25; 30 and m ¼ 0:5; 1:0; 1:5. The bias and RMSE of the shape and scale parameters for the remaining settings are presented in Supplementary Material A. The RMSE of the shape parameter estimated using the MVLE-H method was the smallest value in many settings, and the difference between it and the smallest value was small, even when it was not the smallest value. The RMSEs of the scale parameter, estimated using the modified BL and modified LSPF methods, provided similar values and showed the smallest value or the second smallest value, for all settings. The bias and RMSE of the shape parameter estimated using the MLE were large when n was small; whereas those of the shape parameter estimated using the MVLE-H methods were small even when n was small. Further, the histograms of the shape parameter estimated using the five methods are presented in Supplementary Material B. The dashed lines in the histograms present population shape parameters. The mode of the shape parameter estimated using the MVLE-H method was often concentrated near the dashed line. Therefore, we recommend using the shape parameter estimated using the MVLE-H method and the scale parameter estimated using the modified BL method or modified LSPF method.
Step 3. If x i or y i becomes 0 by step 2, replace 0 with 0.001.
Step 5. Calculate the D in (28) using the two-parameter estimated in step 4.
Step 6. Judge whether the inequality in (29) is satisfied with α ¼ 0:05 (two-sided) and count the number of times it is satisfied.
Step 8. Divide the count in step 6 by 10,000 and compare it to the significance level of 0.05.
When the probability obtained in step 8 was less than 0.05, this two-sample K-S test (two-sided) was considered as strictly adhering to the significance level. Because the recommended estimation method was presented in Section 4.1, the MCSs of the two-sample K-S test (two-sided) were executed using the shape parameter estimated using the MVLE-H method and scale parameter estimated using the modified BL method. Table 3 lists the probability of the inequality in (29) was satisfied with α ¼ 0:05 (two-sided). It was confirmed that the significance level of the two-sample K-S test (two-sided) was strictly adhered to, in all settings.

FAERS data
We downloaded the data files (faers_ascii_AAAAQB.zip) from the US FDA Adverse Event Reporting System, on December 7, 2021 (AAAA and B represent the year and quarter, respectively). We demonstrated the parameter estimation of the Weibull distribution using two FAERS data. The two FAERS data were contained 0 days. Therefore, we replaced 0 with a small value of 0.001, and estimated the shape and scale parameters, with the location parameter set to 0. We searched for drugs using 'prod_ai'. FAERS data may not include the start date of drug administration or the date of occurrence of the adverse event, or the days may be unknown, even if they were presented up to the years and months. We used only data presented up to years, months and days, for the start date of drug administration and the date of occurrence of the adverse event. Cases in which the adverse event occurred more than 1 day after the end date of drug administration were excluded because of the high possibility of other effects. However, data for which the end date of drug administration was not specified were not excluded, considering the possibility that it was not stated because drug administration was continued. For data in which the same drug was administered in two or more periods in the same case, we used the data pertaining the period in which the adverse event occurred. However, when the end date was not specified in two or more periods, the start date that was closer to the date of occurrence of the adverse event was used. When the period of drug administration was divided into two owing to dose change (when the end date of drug administration in the first period or the next day was the start date of drug administration in the second period), the two periods were merged. Even when 'caseid' was different, there were data that corresponded to the start date of the drug administration, date of occurrence of the adverse event, and the subject background (country, age, weight and . We used only one case because these data were likely to be duplicated. We used the software R to estimate the shape parameters using the five methods (MLE, MVLE-H, BL, w-MLE and LSPF-MLE methods) and scale parameters using the seven methods (MLE, modified w-MLE, w-MLE, modified BL, BL, modified LSPF-MLE and LSPF-MLE methods). The two-parameter estimation results of the Weibull distribution using the three FAERS data are summarized in Table 4. The sample code is presented in Supplementary Material C.

Example 1
The first case we considered was the olaparib-induced anaemia. Olaparib is the first oral Poly-ADP ribose polymerase (PARP) inhibitor for germline breast cancer susceptibility gene (BRCA)-mutated cancer such as ovarian cancer. The maximum dosage was administered twice in a day, through 300 mg tablets and resulted in an objective clinical response in patients with ovarian cancer (Mateo et al. 2016). The phase III trial showed that the incidence of grade ≥3 anaemia was about 20% in patients with ovarian cancer (Montemorano et al. 2019). Therefore, anaemia displays the common doselimiting toxicity of olaparib (Guo et al. 2018). Olaparib-induced anaemia often leads to dose discontinuation, interruption and modification in the clinical setting. Therefore, early detection and effective management of anaemia are crucial for the safe use of olaparib.
In the FAERS data, adverse event reports caused by olaparib administration were reported after 2015Q2. The data between 2015Q2 and 2017Q1 for 27 cases introduced in Section 1, and the number of days pertained to 0, 76, 120, 27, 41, 392, 50, 21, 52, 235, 70, 51, 87, 57, 14, 63, 38, 18, 5, 3, 37, 28, 42, 20, 193, 51, 18 (day). The number of the olaparib-induced anaemia was 230 between 2015Q2 and 2021Q3. Using the data between 2015Q2 and 2021Q3, the shape and scale parameters estimated using the MLE were 0.539 and 42.618, respectively. When data of high-reported adverse events were used, the shape and scale parameters estimated using the MLE were considered to be close to population parameters. Because we focused on the data of low-reported adverse events, we omitted the other four methods that require a long calculation time when using the data of high-reported adverse events. Using the data between 2015Q2 and 2017Q1, the shape parameter estimated using the MVLE-H method was 0.788, which was the closest value to 0.539. Although the scale parameter estimated using the w-MLE method was 40.961, the shape parameter estimated using the w-MLE method was very small. Therefore, we exclude the w-MLE method from further discussion. The scale parameters estimated using the modified BL and modified LSPF-MLE methods were 59.689, which were the closest values to 42.618. The same method recommended for MCSs was recommended.

Example 2
The second case we considered was the niraparib-induced anaemia. Niraparib is another PARP inhibitor, which has been recently approved for maintenance treatment with cancer, such as recurrent epithelial ovarian cancer that has achieved complete or partial response to platinum-based chemotherapy. The maximum dosage of 300 mg per day, resulted in an objective clinical response in patients with ovarian cancer (Zhou et al. 2017). The phase III trial showed that the incidence of grade ≥3 anaemia was approximately 31% in patients with ovarian cancer (González-Martín et al. 2019). Anaemia is also a side-effect of common dose-limiting toxicity of niraparib. The PARP inhibition potency and trapping ability of niraparib and olaparib are similar in cell-free systems established using recombinant proteins. Therefore, various studies have been conducted to compare the effectiveness and safety of olaparib and niraparib (Murai et al., 2012;Sun et al. 2018).
In the FAERS data, adverse event reports caused by niraparib administration were reported after 2017Q2. The data between 2017Q2 and 2018Q1 for 24 cases were as follows: 9, 20, 19, 0, 72, 22, 4, 0, 1, 114, 17, 3, 8, 0, 11, 41, 25, 19, 0, 27, 3, 25, 0, 32 (day). The number of the niraparib-induced anaemia was 186 between 2017Q2 and 2021Q3. Using the data between 2017Q2 and 2018Q1, the shape parameter estimated using the MVLE-H method was 0.414. The scale parameters estimated using the modified BL and modified LSPF-MLE methods were 10.750. The shape and scale parameters in Example 2 were less than those in Example 1. Thus, compared to Example 1, the probability of occurrence of anaemia is high immediately after the niraparib administration, but it declines significantly as the days progress. It was examined in Section 5.3 whether there was a statistically significant difference between the two Weibull distributions estimated in Examples 1 and 2.

Two-sample K-S test
The two-sample K-S test (two-sided) was used to examine whether there was a significant difference between the tendency of occurrence of olaparib-induced anaemia in Section 5.1, and the tendency of occurrence of niraparib-induced anaemia in Section 5.2. The two-sample K-S test (two-sided) was executed using the shape parameter estimated using the MVLE-H method and the scale parameter estimated using the modified BL method. In the FAERS data between 2015Q2 and 2017Q1 in Section 5.1, the sample size was n 1 ¼ 27 and the two-parameters were estimated to be m 1 ¼ 0:788 and η 1 ¼ 59:689. In the FAERS data between 2017Q2 and 2018Q1 in Section 5.2, the sample was size was n 2 ¼ 24 and the two-parameters were estimated to be m 1 ¼ 0:414 and η 1 ¼ 10:750. The D in (28) was calculated to be 0.404. The right side of (29) was calculated as 0.381 using c α ð Þ ¼ 1:358. The null hypothesis was rejected at the significance level of 0.05 because the inequality in (29) was satisfied. Therefore, the tendencies of occurrence of olaparib-and niraparib-induced anaemia were different. Furthermore, we compared the two Weibull distributions estimated using the data from entire all period. In the FAERS data between 2015Q2 and 2021Q3 in Section 5.1, the sample size was n 1 ¼ 230 and the two-parameters were estimated to be m 1 ¼ 0:539 and η 1 ¼ 42:618. In the FAERS data between 2017Q2 and 2021Q3 in Section 5.2, the sample was size was n 2 ¼ 186 and the two-parameters were estimated to be m 1 ¼ 0:378 and η 1 ¼ 25:667. The D in (28) was calculated to be 0.147. The right side of (29) was calculated as 0.134 using c α ð Þ ¼ 1:358. The null hypothesis was rejected at the significance level of 0.05 because the inequality in (29) was satisfied. This result for the two-sample K-S test (twosided), in which the null hypothesis was rejected, was the same as that of low-reported adverse events.
Although the probabilities of occurrence of olaparib-and niraparib-induced anaemia were reported, these probabilities were for the entire period of the clinical trial or the monthly (Berek et al. 2018;González-Martín et al. 2019;Montemorano et al. 2019). From their results, it might be difficult to notice that the probability of occurrence of anaemia immediately after niraparib administration was very high. Conversely, examining the result of the two-sample K-S test (two-sided), it was evident that the probability of occurrence of anaemia immediately after niraparib administration was higher than that of anaemia induced immediately after olaparib administration. Furthermore, it became clear that the probability of occurrence of anaemia after long-term olaparib administration was higher than that of the anaemia induced after long-term niraparib administration.

Conclusions
We attempted to clarify the tendency of occurrence of an adverse event, using the number of days between drug administration and the date of occurrence of the adverse event. In this study, the data differed from the general Weibull distribution in two ways. First, the data was discrete. Second, the data could be contained 0 days. We searched for an optimal estimation method using the generated random samples that fit these two characteristics. The MCS results showed that it was optimal to estimate the shape parameter using the MVLE-H method and the scale parameter using the modified BL or modified LSPF methods. Additionally, using the two-sample K-S test (two-sided), we presented the method to examine whether the tendency of occurrence of adverse events was different for two Weibull distributions estimated from two drugs with similar efficacy. In the two FAERS data, if the conventional method was used, it might only be possible to discuss that the shape parameter of the Weibull distribution was less than 1 for both olaparib-and niraparib-induced anaemia. Using the twosample K-S test (two-sided), it was found that the tendencies of occurrence of olaparib-and niraparibinduced anaemia were different. These research results can offer contributions to the clinical setting.