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Inspection plan for COVID19 patients for Weibull distribution using repetitive sampling under indeterminacy
BMC Medical Research Methodology volumeÂ 21, ArticleÂ number:Â 229 (2021)
Abstract
Background
This research work is elaborated investigation of COVID19 data for Weibull distribution under indeterminacy using time truncated repetitive sampling plan. The proposed design parameters like sample size, acceptance sample number and rejection sample number are obtained for known indeterminacy parameter.
Methods
The plan parameters and corresponding tables are developed for specified indeterminacy parametric values. The conclusion from the outcome of the proposed design is that when indeterminacy values increase the average sample number (ASN) reduces.
Results
The proposed repetitive sampling plan methodology application is given using COVID19 data belong to Italy. The efficiency of the proposed sampling plan is compared with the existing sampling plans.
Conclusions
Using the tables and COVID19 data illustration, it is concluded that the proposed plan required a smaller sample size as examined with the available sampling plans in the literature.
Background
It is broadly established that a huge number of COVID19 cases are unnoticed worldwide. A rudimentary measure of population occurrence is the small part of positive cases for a given date in any country. On the other hand, this is subject to largely found that bias since tests are normally only ordered from suggestive cases, whereas a large proportion of infected people might show little symptoms or sometimes no symptoms for more details see [1]. Most governments are applying the mechanism of test randomly selected individuals to estimate the true disease occurrence in inhabitants in a particular locality. Nevertheless, when the disease occurrence is low and difficult to acquire from each patient/person by tests, under such situations we may use an acceptance sampling plan under indeterminacy. The health practitioners are paying attention to estimate the average number of deaths or ratio of deaths to the total number of COVID19 death cases on daily basis, for the coming days, next week or month, etc. Reference [2]. In such a case, the health practitioners are paying attention to test the null hypothesis that the average number of deaths or ratio of deaths to the total number of COVID19 death cases on daily basis is equal to the specified average number of deaths due to COVID19 against the alternative hypothesis that the average number of deaths due to COVID19 varies significantly. In this situation for testing of the hypothesis, practically it is difficult to record the average number of death for the whole year, whereas it is easy to record the daily basis and the average number of deaths can be obtained from the randomly selected days. The null hypothesis may be rejected if the daily average number of deaths due to COVID19, state acceptance number of days, is more than or equal to the specified average number of deaths due to COVID19 throughout the given number of days.
Many researchers have done studies on the time truncated life test for various distributions. Some of them are [3] developed the acceptance sampling plan for life tests: loglogistic models. Reference [4] derived acceptance sampling based on truncated life tests for generalized Rayleigh distribution. Reference [5] developed the acceptance sampling plans based on the generalized BirnbaumSaunders distribution. Reference [6, 7] constructed the acceptance sampling plans for BirnbaumSaunders and Burr XII distributions. References [8, 9] constructed acceptance sampling plans for extended exponential and generalized inverted exponential distributions. The details about the acceptance sampling plans can be seen in [10, 11]. The generalization of a single acceptance sampling plan namely repetitive sampling plan, [12] derived the decision rule of the repetitive acceptance sampling plan. The method of repetitive group acceptance sampling plan (RGASP) was first proposed by [13] for an attribute. Reference [14, 15] constructed the RASP for inverse Gaussian distribution and Burr type XII. Reference [16] developed generalized inverted exponential distributions. References [17,18,19] studied the repetitive sampling plan under different situations.
More details about the neutrosophic logic, their measure of determinacy, and indeterminacy are given by [20]. Numerous authors studied the neutrosophic logic for various real problems and showed its efficiency over fuzzy logic, for more details refer [21,22,23,24,25,26]. The idea of neutrosophic statistics was given using the idea of neutrosophic logic, [27,28,29]. The neutrosophic statistics give information about the measure of determinacy and measure of indeterminacy, see [30]. The neutrosophic statistics reduce to classical statistics if no information is recorded about the measure of indeterminacy. References [31,32,33] proposed the acceptance sampling plans using neutrosophic statistics [34]. proposed the timetruncated group plans for the Weibull distribution. Reference [35] worked on neutrosophic Weibull and neutrosophic family of Weibull distribution.
The existing sampling plans based on classical statistics and fuzzy philosophies do not give information about the measure of indeterminacy. Reference [36] worked on the single sampling plan using a fuzzy approach. Reference [37] discussed the effect of sampling error on inspection using a fuzzy approach. Reference [38] proposed a single plan using fuzzy logic. Reference [39] proposed the improved sampling plan using fuzzy logic. For details, the reader may refer to [40, 41]. To the best of our knowledge, there is no work on a timetruncated sampling plan for Weibull distribution under indeterminacy. In this paper, a repetitive acceptance sampling plan for Weibull distribution under indeterminacy is developed to testing the daily average deaths. We are anticipated the proposed sampling plan shows a fewer sample size as compared with the existing sampling plans for testing the daily average deaths.
In Section 2, we present an introduction of a repetitive acceptance sampling plan for Weibull distribution under indeterminacy. In Section 3, the proposed repetitive acceptance sampling plan under indeterminacy is compared with the single sampling plan proposed by [42]. The proposed sampling plan is illustrated using COVID19 data belong to Italy, which was recorded from 1 April to 20 July 2020 in Section 4. Finally, the conclusions and future research works are established in Section 5.
Methods
The repetitive acceptance sampling plan depends upon the truncated life test procedure is developed by [43,44,45]. The operational steps of this test are given as follows:

Step 1: Draw a sample of size n from the lot. These samples can be put on a life test for a fixed time t_{0}. Specify the averageÂ Î¼_{0} and indeterminacy parameter I_{N}Ïµ[I_{L},â€‰I_{U}].

Step 2: Accept H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} if the daily average deaths in c_{1} days are more than or equal toÂ Î¼_{0} (i.e.,Â Î¼_{0}â€‰â‰¤â€‰c_{1}). If daily average deaths in c_{2} days are less than toÂ Î¼_{0} (i.e., Î¼_{0} >c_{2}) then we reject H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} and terminate the test, where c_{1â‰¤} c_{2}.

Step 3: If c_{1}< Î¼_{0} â‰¤c_{2} then go to step 1 and repeat the above experiment.
The above procedure of repetitive acceptance sampling plan (RASP) mainly depends on four characteristics those are n, c_{1}, c_{2} and I_{N}, where I_{N}Ïµ[I_{L},â€‰I_{U}] is considered as the specified parameter and set according to the uncertainty level. RASP is nothing but the generalization of an ordinary single sampling plan under uncertainty. If c_{1}Â =â€‰c_{2} in RASP, it ultimately reduces to a single sampling plan under uncertainty. Suppose that t_{0}â€‰=â€‰aÎ¼_{0} be the time in days, where a is the termination ratio. The lot acceptance probability is to be determined with the help of operating characteristic (OC) function for details see [13] and it is given by
Here P_{a}(p) is the probability of accepting H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} and P_{r}(p) is the probability of rejecting H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N}, which are given by
and
where p is the probability of unreliability.
Therefore eq. (1) becomes
The Weibull distribution under neutrosophic statistics is developed by [42] for the design of the sampling scheme plan for testing the average wind speed under an indeterminate environment.
Suppose that f(x_{N})â€‰=â€‰f(x_{L})â€‰+â€‰f(x_{U})I_{N}; I_{N}Ïµ[I_{L},â€‰I_{U}] be a neutrosophic probability density function (npdf) having determinate partÂ f(x_{L}), indeterminate part f(x_{U})I_{N} and indeterminacy intervalÂ I_{N}Ïµ[I_{L},â€‰I_{U}]. Note that x_{N}Ïµ[x_{L},â€‰x_{U}] be a neutrosophic random variable follows the npdf. The npdf is the generalization of pdf under classical statistics. The proposed neutrosophic form of f(x_{N})Ïµ[f(x_{L}),â€‰f(x_{U})] reduces to pdf under classical statistics when I_{L} =0. Based on this information, the npdf of the Weibull distribution is defined as follows.
where Î± and Î² are scale and shape parameters, respectively. Note here that the proposed npdf of the Weibull distribution is the generalization of pdf of the Weibull distribution under classical statistics. The neutrosophic form of the npdf of the Weibull distribution reduces to the Weibull distribution when I_{L} =0. The neutrosophic cumulative distribution function (ncdf) of the Weibull distribution is given by
The neutrosophic mean of the Weibull distribution is given by.
The null and alternative hypotheses for the daily average deaths are stated as follows:
Where Î¼_{N} is a true daily average death and Î¼_{0N} is the specified daily average deaths. Suppose that t_{0N}â€‰=â€‰aÎ¼_{0N} be the time in days, where a is the termination ratio. The probability of the item will fail before it reaches the experiment time t_{0N} is defined as follows:
where Î¼_{N}/Î¼_{0N} is the ratio of true average daily wind speed to specified average daily wind speed. Suppose that \(\tilde{\alpha }\) and \(\tilde{\beta }\) be typeI and typeII errors. The medical practitioners are interested to apply the proposed plan for testing H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} such that the probability of accepting H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} when it is true should be larger than \(1\tilde{\alpha }\) at Î¼_{N}/Î¼_{0N} and the probability of accepting H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} when it is wrong should be smaller than \(\tilde{\beta }\) at Î¼_{N}/Î¼_{0N}â€‰=â€‰1. In order to find the design parameters n, c_{1}, c_{2} and I_{N} for the proposed RASP, we consider two points on the OC function. In our approach, the quality level mainly depends on the ratio Î¼_{N}/Î¼_{0N}. This ratio is helpful for the producer to improve the lot quality. From in producer point of view, the probability of acceptance should be at least \(1\tilde{\alpha }\) at acceptable quality level (AQL), p_{1N}. So, the producer demands the lot should be accepted at various levels of Î¼_{N}/Î¼_{0N}. Similarly, from in consumer point of view the lot rejection probability should not be exceeded \(\tilde{\beta }\) at limiting quality level (LQL), p_{2N}. The design parameters are determined by satisfying the following two inequalities
where p_{1N} and p_{2N} are defined by
The estimated designed parameters of the proposed plan should be minimizing the average sample number (ASN) at an acceptable quality level. The ASN for the proposed plan with fraction defective (p) is derived to be
Therefore, the design parameters for the proposed plan with minimum sample size will be obtained by solving the below optimization technique
The values of the designed parameters n, c_{1} and c_{2} for various values of \(\tilde{\beta }\) =0.25, 0.10, 0.05, 0.01; \(\tilde{\alpha }=0.10\); aâ€‰=â€‰0.5 and 1.0, Î¼_{N}/Î¼_{0N} =1.1, 1.2, 1.3, 1.4, 1.5, 1.8, 2.0 and I_{N} =0.0, 0.02, 0.04 and 0.05 when shape parameter Î²â€‰=â€‰1, 2 and 3 are given in Tables 1, 2, 3, 4, 5 and 6. Tables 1 and 2 are shown for the exponential distribution case. For exponential distribution, it can be seen that the values of ASN decrease as the values of a increases from 0.5 to 1.0. On the other hand for other the same parameters, the values of n decreases as the values of Î² increases. Note here that the indeterminacy parameter I_{N} also plays a significant role in minimizing the sample size. As indeterminacy parameter I_{N} increases the ASN values are decreasing.
Results
A comparative study is carried out between the proposed sampling plans with the existing sampling plans available in the literature with respect to the sample size in this section. We know the cost of the study is always directly proportional to the sample size, a plan is said to be economical if it requires a smaller number of samples for testing the hypothesis about the daily new deaths from COVID19. The proposed repetitive sampling plan under uncertainty/indeterminacy for Weibull distribution is the generalization of the testing average wind speed using sampling plan for Weibull distribution under indeterminacy plan developed by [42]. The comparison for the proposed and the existing sampling plan for Weibull distribution under indeterminacy plan developed by [42] are displayed in TablesÂ 7 and 8 for \(\tilde{\alpha }=0.10;\beta =2\) at aâ€‰=â€‰0.5 and 1.0. The developed sampling plan reduces to the existing sampling plan when c_{1}Â =â€‰c_{2}Â =â€‰c. From Tables 7 and 8, it is noticed that the values of the sample size required for testing H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰Î¼_{0N} smaller for the proposed sampling plan as compared with the existing sampling plan developed by [42]. For example, when Î¼_{N}/Î¼_{0N} =1.1 and a =0.5 from TableÂ 7, it can be seen that ASNâ€‰=â€‰491.58 from the plan proposed sampling plan whereas existing sampling plan sample size nâ€‰=â€‰617 when I_{N} =0.02,Â Î²â€‰=â€‰2 and aâ€‰=â€‰0.5. Hence, the proposed sampling plan is more economical than the existing sampling plan.
Discussions
At this juncture, application of the proposed methodology will be illustrated using COVID19 data belong to Italy of 111â€‰days that are recorded from 1 April to 20 July 2020. The data are available at https://covid19.who.int/. This data is made up of the ratio of daily new deaths (i.e. daily number of deaths over new cases). The data is reported in TableÂ 9. We have taken this data from [46] and they studied applications of COVID19 data for Kumaraswamy inverted ToppLeone distribution. Coronavirus disease (COVID19) is an infectious disease caused by a newly discovered coronavirus. A large number of people affected by the COVID19 virus and it are infected at random and uncertain, the COVID19 data follows a certain statistical distribution under neutrosophic statistics. The World health organization and different countriesâ€™ health administrators are involved to check the daily affected cases, recovered cases and deaths under indeterminacy. It is found that the COVID19 data follows the Weibull distribution with shape parameter \(\hat{\beta}=\) 2.2222 with the standard error (SE) as 0.1596 and scale parameter \(\hat{\alpha}=0.1880\) with SE value as 0.00845. The KolmogorovSmirnov test and it p value are Dâ€‰=â€‰0.0684 and pâ€‰=â€‰0.6766. The goodness of fit of the Weibull distribution is highlight by depicts the histogram and quantilequantile (QQ) plot in Fig.Â 1. We also applied various life distributions to fit the COVID19 data set for the intention of comparative study. We have considered here the existing three models like odds Weibull distribution (OWD), NadarajahHaghighi distribution (NHD) and Exponentiated NadarajahHaghighi distribution (ENHD) for the same data. For more details please refer to [47].
Pdf and cdf of Weibull distribution are respectively
and \(F(x)=1{e}^{{\left(\frac{x}{\alpha}\right)}^{\beta }}\);Â xâ€‰>â€‰0, Î±â€‰>â€‰0, Î²â€‰>â€‰0
Pdf and cdf of odds Weibull distribution (OWD) are respectively (suggested by [48])
and \(F(x)=1{\left[1+{\left({e}^{{\left(\frac{x}{\theta}\right)}^{\alpha }}1\right)}^{\beta}\right]}^{1}\);Â xâ€‰>â€‰0, Î±â€‰<â€‰0, 0â€‰<â€‰(Î²,â€‰Î¸).
Pdf and cdf of NadarajahHaghighi distribution (NHD) are respectively (see [49])
and \(F(x)=1{e}^{1{\left(1+\lambda x\right)}^{\alpha }}\);Â xâ€‰>â€‰0, Î±â€‰>â€‰0, Î»â€‰>â€‰0.
Pdf and cdf of Exponentiated NadarajahHaghighi distribution (ENHD) are respectively (see [49])
and \(F(x)={\left(1{e}^{1{\left(1+\lambda x\right)}^{\alpha }}\right)}^{\theta }\);Â xâ€‰>â€‰0, Î±â€‰>â€‰0, Î»â€‰>â€‰0, Î¸â€‰>â€‰0.
We have estimated the parameters and good fit for the COVID19 data for WD, OWD, NHD and ENHD, and are reported in Table 10 and depicted in Fig.Â 2. From Table 10 and Fig. 2 it is noticed that WD shows less AIC, BIC and 2logLL, moreover OWD and NHD are not fitted for COVID19 data. Hence, Weibull distribution shows a good fit for the COVID19 data belongs to Italy. The plan parameters for this shape parameter are shown in Tables 11 and 12. For the proposed plan, the shape parameter is \({\hat{\beta}}_N=\left(1+0.04\right)\times 2.2222\approx 2.31\) when I_{U} =0.04.
Suppose that a quality medical practitioner would like to use the proposed repetitive sampling plan for Weibull distribution under indeterminacy to ensure the mean ratio of daily new deaths at least 60â€‰days using the truncated life test for 60â€‰days. Let the producerâ€™s risk be 10% at Î¼_{N}/Î¼_{0N} =1.1 and the consumerâ€™s risk is 10%. From Table 11, with aâ€‰=â€‰1.0, \(\tilde{\beta }=0.10\) and \(\tilde{\alpha }=0.10\) for the repetitive sampling plan, it could be found that the plan parameters are c_{1}â€‰=â€‰58 c_{2}â€‰=â€‰66 and ASNâ€‰=â€‰191.67. Therefore, the plan could be implemented as follows: selecting a random sample of 114 patients from the arrived lot of patients, and doing the truncated life test for 60â€‰days. The proposed sampling plan will be implemented as: accept the null hypothesis H_{0}â€‰:â€‰Î¼_{N}â€‰=â€‰0.1665 if the average ratio of daily new deaths in 60â€‰days is less than 58, the ratio of daily deaths, but the lot should be rejected as soon as the ratio of daily new deaths exceeds 66. Otherwise, the experiment would be repeated. Table 9 shows the 56 ratios of daily new deaths before the average ratio of daily new deaths of 0.1665. Therefore, the quality medical practitioners would have accepted the arrived lot of patients.
Conclusions
An elaborated investigation of COVID19 data for Weibull distribution under indeterminacy using time truncated repetitive sampling plan is studied. The proposed design parameters are obtained for known values of the indeterminacy parameters. The plan parameters and corresponding tables are developed for the industrial purposes at specified indeterminacy parametric values. The proposed sampling plan is compared with the existing sampling plans. The result shows that the proposed repetitive sampling plan is more economical than the existing sampling plan. The proposed sampling plan saves time; labor and amount for experimentation, the proposed plan is recommended to apply for testing the average number of deaths due to COVID19. Also, noticed that if the indeterminacy values increase then the average sample number is decreased. The developed repetitive sampling plan procedure is illustrated with COVID19 data belong to Italy as an application. The proposed sampling plan can be implemented in various industries covering the packing industry, medical sciences, food industries and electronic industries. Further research can be established to extend our study to group sampling plans, multiple dependent state sampling plans, and multiple dependent state repetitive sampling plans.
Availability of data and materials
The data is given in the paper.
Abbreviations
 ASN:

Average sample number
 RASP:

Repetitive acceptance sampling plan
 OC:

Operating characteristic
 QQ:

Quantilequantile
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Rao, G.S., Aslam, M. Inspection plan for COVID19 patients for Weibull distribution using repetitive sampling under indeterminacy. BMC Med Res Methodol 21, 229 (2021). https://doi.org/10.1186/s12874021013877
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DOI: https://doi.org/10.1186/s12874021013877