Computational Evaluation For The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application

Azam Zaka; Riffat Jabeen

doi:10.12688/f1000research.146106.1

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Research Article

Computational Evaluation For The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application

[version 1; peer review: 3 approved with reservations]

Azam Zaka¹, Riffat Jabeen ²

PUBLISHED 27 Jun 2024

Author details Author details

¹ Statistics, COMSATS University Islamabad, Lahore, Lahore, 54000, Pakistan
² Statistics, Government College of Science, Lahore, Lahore, 54000, Pakistan

Azam Zaka
Roles: Methodology, Writing – Original Draft Preparation

Riffat Jabeen
Roles: Conceptualization, Supervision, Validation

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background

Statistical process control (SPC) is utiled in manufacturing and service applications. It is used to monitor the behavior of the process, also to discover and solve the issues in internal systems. SPC allows immediate corrective actions to prevent defects and maintain process stability. There are two types of control charts are available in the literature: variable control charts and attribute control charts.

Methods

The use of attribute control charts for two positively skewed distributions such as power function distribution and weighted power function distribution is discussed in this research paper. We compared the behavior of both types of control charts and presented the results in form of tables.

Results

We conclude that control charts under a weighted power function distribution perform better than control charts using a power function distribution. Furthermore, these results will help in the early detection of errors that enhance the process reliability of the telecommunications and financing industries.

Keywords

power function distribution, weighted power function distribution, attribute control chart, truncated life test, average run length, simulation

Corresponding author: Riffat Jabeen

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2024 Zaka A and Jabeen R. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Zaka A and Jabeen R. Computational Evaluation For The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application [version 1; peer review: 3 approved with reservations]. F1000Research 2024, 13:700 (https://doi.org/10.12688/f1000research.146106.1) First published: 27 Jun 2024, 13:700 (https://doi.org/10.12688/f1000research.146106.1) Latest published: 27 Jun 2024, 13:700 (https://doi.org/10.12688/f1000research.146106.1)

1. Introduction

Statistical process control is used as an effective tool to monitor the manufacturing process and prevent products from being out of the given provisions. Two different types of control charts are available in the literature: variable control charts and attribute control charts. When numerical measurements of the variable of interest from any process are available, the use of a variable control chart is preferred. However, there are many processes that have a variable of interest in the form of attributes and are not measurable. In such situations, it is preferable to use attribute control charts instead of variable control charts.

Attribute control charts are used to monitor a process with two possible outcomes: conforming and nonconforming. The basic condition for using these charts was the assumption of normality. However, in real-life situations, whenever a process is working, the outcome may not follow a normal distribution.

It is possible that the shape of the distribution is skewed. For such cases, many studies, such as Chang and Bai,¹ Lin and Chou,² and Rao and Subbaiah,³ have been conducted. Work on attribute control charts considering different skewed distributions has also been done by Aslam et al.,⁴ Karagöz,⁵ and Noiplab and Mayureesawan.⁶ We can also refer to Zaka et al.,⁷ who modified the attribute control chart for the shape of the process using the power function and survival-weighted power function distribution.

The shape of any distribution can be studied by estimating its shape parameter. Several estimators have been developed to estimate the shape parameters of any distribution. Zaka et al.⁷ introduced and studied an attribute control chart for the shape of the process using the power function and survival-weighted power function distribution. Dallas⁸ introduced the power function distribution (PFD) as a powerful tool to study many processes in reliability and health sciences, which was further assumed and modified to increase its applicability in unequal probability theory by Zaka et al.⁹ as the weighted power function distribution (WPFD). Jabeen and Zaka¹⁰ proposed modified control charts based on shape parameters.

We introduce the PFD and WPFD in Section 2. In Section 3, we propose an attribute control chart for PFD and WPFD. A discussion of the use of the proposed control chart is presented in Section 4.

2. Methods

2.1. Distribution function for PFD and WPFD

The weighted power function distribution (WPFD) was used by Zaka et al⁹ by modifying the existing Power function distribution (PFD). The cumulative distribution function and probability density fuction for PFDand WPFD is given asintroduced

(1)

F (x_{1}) = {(\frac{x_{1}}{β})}^{γ} and f (x_{1}) = \frac{γ {x_{1}}^{γ - 1}}{β^{γ}}, 0 < x_{1} < β, and β, γ > 0 .

(2)

F (x_{2}) = {(\frac{x_{2}}{β})}^{2 γ} and f (x_{2}) = \frac{2 γ {x_{2}}^{2 γ - 1}}{β^{2 γ}}, 0 < x_{2} < β, and β, γ > 0 .

where

β and γ

are the scale and shape parameters.

The mean of PFD and WPFD are respectively

μ_{1} = \frac{βγ}{γ + 1} and μ_{2} = \frac{2 βγ}{2 γ + 1}

2.2. Average Run Length ARL for proposed control monitoring using attribute control chart for PFD and WPFD

The control limits for attribute control charts to monitor the shape parameter of weighted Power fuction distribution is illustrated in from Zaka et al.⁷ and similarly the control limits for attribute control charts is given by Aslam et al.¹¹ , the proposed control limits are given as:

UCL = n p_{0} + L \sqrt{n p_{0} (1 - n p_{0})}

LCL = max [0, n p_{0} - L \sqrt{n p_{0} (1 - n p_{0})}] .

where p₀ is the failure probability, n is the sample size, and L is the control coefficient. Thus, it is obtained from (1) and (2) as follows:

P_{o 1} = {(\frac{t_{o 1}}{β_{o 1}})}^{γ_{01}}, P_{o 1} = {(\frac{a μ_{o 1}}{β_{o 1}})}^{γ_{01}}; where μ_{o 1} = \frac{β_{o 1} γ_{o 1}}{γ_{o 1} + 1}

P_{o 2} = {(\frac{t_{o 2}}{β_{o 2}})}^{2 γ_{02}}, P_{o 2} = {(\frac{a μ_{o 2}}{β_{o 2}})}^{2 γ_{02}}; where μ_{o 2} = \frac{2 β_{o 2} γ_{o 2}}{2 γ_{o 2} + 1}

where

β_{o 1}, β_{o 2}

are scale parameters and

γ_{01,} γ_{02}

are the shape parameters for the two proposed distributions.

The control limits are presented as

UCL = \bar{D} + L \sqrt{\bar{D} (1 - \frac{\bar{D}}{n})}

LCL = max [0, \bar{D} - L \sqrt{\bar{D} (1 - \frac{\bar{D}}{n})}]

where

\bar{D}

is the average number of failures over the subgroups.

We can find the ARL using the following formulae

{ARL}_{oj} = \frac{1}{1 - p_{in}^{oj}}

where the probability of the control chart for being in the state of control is

p_{in}^{oj} = P (LCL \leq D \leq UCL | P_{oj}) = \sum_{d = ⌊ LCL ⌋ + 1}^{⌊ UCL ⌋} (\begin{matrix} n \\ d \end{matrix}) {P_{oj}}^{d} {(1 - P_{oj})}^{n - d}

2.1.1. The probabilities of failure for PDF in out of control process are

→ When the shape parameter varies → $P_{11} = {(\frac{a \frac{β_{o 1} γ_{11}}{γ_{11} + 1}}{β_{o 1}})}^{γ_{11}}$

→ When the scale parameter varies → $P_{21} = {(\frac{a \frac{β_{11} γ_{o 1}}{γ_{o 1} + 1}}{β_{11}})}^{γ_{o 1}}$

→ All the parameters vary → $P_{31} = {(\frac{a \frac{β_{11} γ_{11}}{γ_{11} + 1}}{β_{11}})}^{γ_{11}}$

2.1.2. The probabilities of failure for WPFD in out of control process are

→ When the shape parameter varies → $P_{12} = {(\frac{a \frac{2 β_{o 2} γ_{12}}{2 γ_{12} + 1}}{β_{o 2}})}^{2 γ_{12}}$

→ When the scale parameter varies → $P_{22} = {(\frac{a \frac{2 β_{12} γ_{02}}{2 γ_{02} + 1}}{β_{12}})}^{2 γ_{02}}$

→ All the parameters vary → $P_{32} = {(\frac{a \frac{2 β_{12} γ_{12}}{2 γ_{12} + 1}}{β_{12}})}^{2 γ_{12}}$

We used $γ_{11}, γ_{12}$ when the shape parameters varied from $γ_{o 1,} γ_{02}$ for the first and second distributions, respectively. We also used $β_{11}, β_{12}$ when the shape parameters varied from $β_{o 1}, β_{o 2}$ for the first and second distributions, respectively.

2.3. Case 1 (when we assume shift in scale parameter)

When the parameter is shifted from $β_{1} = C β_{0}$ , the probability in the control process is

p_{in}^{1 j} = P (LCL \leq D \leq UCL | P_{1 j}) = \sum_{d = ⌊ LCL ⌋ + 1}^{⌊ UCL ⌋} (\begin{matrix} n \\ d \end{matrix}) {P_{1 j}}^{d} {(1 - P_{1 j})}^{n - d}

The ARL for the shifted process is given as

{ARL}_{1 j} = \frac{1}{1 - p_{in}^{1 j}}

2.4. Case 2 (when we assume a shift in the shape of a parameter)

We adopt $γ_{1} = C γ_{0}$ . The probability of in-control for the shifted process is

p_{in}^{2 j} = P (LCL \leq D \leq UCL | P_{2 j}) = \sum_{d = ⌊ LCL ⌋ + 1}^{⌊ UCL ⌋} (\begin{matrix} n \\ d \end{matrix}) {P_{2 j}}^{d} {(1 - P_{2 j})}^{n - d}

The ARL to compare the efficiency of the process after the shift is

{ARL}_{2 j} = \frac{1}{1 - p_{in}^{2 j}}

2.5. Case 3 (when we assume a shift in the scale of a parameter)

When the scale and shape parameters are shifted as $β_{1} = C β_{0}$ and $γ_{1} = C γ_{0}$ we get the probability

p_{in}^{3 j} = P (LCL \leq D \leq UCL | P_{3 j}) = \sum_{d = ⌊ LCL ⌋ + 1}^{⌊ UCL ⌋} (\begin{matrix} n \\ d \end{matrix}) {P_{3 j}}^{d} {(1 - P_{3 j})}^{n - d}

The ARL for the shifted process is

{ARL}_{3 j} = \frac{1}{1 - p_{in}^{3 j}}

Zaka et al.⁷ provided the simulation steps to conduct the monitoring of the shape parameter of weighted power function distribution. By considering these steps we assumed different observations of the ARL, shape parameter, and a. For these observations of ARL, we took the values of the shape and scale parameters of the control chart and sample size (n) in such a way that the ARL from equation (ARL₀) was close to R₀. Using equation (ARL₁) and the values fixed in equation (2), we found ARL₁ using different values of c, $γ_{0}$ , R₀, a, and n, which are in Tables 1-4.

Table 1. Average run length for the attribute control chart setting $R_{0} = 300$ and sample size=5.

For n=5	PFD	WPFD	PFD	WPFD
	$γ_{o 1} = 0.5$	$γ_{o 2} = 0.5$	$γ_{o 1} = 1$	$γ_{o 2} = 1$
Shift in parameter	$R_{0} = 300$ d=0.306388 r=2.843	$R_{0} = 300$ d=0.143885 r=3.977	$R_{0} = 300$ d=0.333058 r=3.627	$R_{0} = 300$ d=0.20458 r=3.951
1.0	300.002	300.004	300.029	300.016
0.95	244.815	220.501	233.793	216.646
0.9	199.177	161.995	181.846	156.447
0.85	161.511	118.963	141.156	112.988
0.8	130.495	87.326	109.325	81.619
0.75	105.015	64.079	84.461	58.981
0.7	84.137	47.005	65.071	42.646
0.6	53.189	25.277	38.248	22.354
0.5	32.800	13.595	22.124	11.789
0.4	19.583	7.331	12.533	6.287
0.3	11.188	3.985	6.906	3.423
0.2	5.992	2.214	3.666	1.942
0.1	2.877	1.312	1.858	1.215

Table 2. Average run length for the attribute control chart setting $R_{0}$ =300 and sample size=10.

For n=10	PFD	WPFD	PFD	WPFD
Shift in parameter	$γ_{o 1} = 0.5$	$γ_{o 2} = 0.5$	$γ_{o 1} = 1$	$γ_{o 2} = 1$
Shift in parameter	$R_{0} = 300$ d=0.18446 r=3.063	$R_{0} = 300$ d=0.13733 r=3.108	$R_{0} = 300$ d=0.2342 r=2.808	$R_{0} = 300$ d=0.1408 r=5.408
1.0	300.012	300.005	300.123	300.025
0.95	216.107	201.670	207.981	201.823
0.9	155.594	135.888	144.323	135.926
0.85	111.980	91.820	100.310	91.691
0.8	80.567	62.252	69.852	61.984
0.75	57.956	42.376	48.753	42.019
0.7	41.690	28.985	34.120	28.589
0.6	21.596	13.815	16.890	13.444
0.5	11.237	6.816	8.531	6.530
0.4	5.910	3.543	4.447	3.349
0.3	3.183	2.001	2.443	1.882
0.2	1.808	1.292	1.476	1.234
0.1	1.164	1.029	1.068	1.017

3. Results and Discussion

To study the proposed control charts (Section 3), we assume the shape parameter as $γ_{o 1} = 0.5 and γ_{o 2} = 1 .$ We also assumed a target ARL value of 300 and simultaneously for the in-control process. We assume a shift of 0.1 in the shape parameter for PFD and WPFD, and n = 5, 10, and 20. The results are presented in Tables 1-3 and Figures 1-2. We observed a rapid decrease in ARL for WPFD compared to PFD, with a decrease in C and an increase in n. We see that for $γ_{o 1} = 0.5$ to $γ_{o 1} = 1$ , there is an early detection in out of control ARLs for both the PFD and WPFD control chart. We also observed that with an increase in the size of the sample and the value of the shape parameter, the WPFD proved to be more reliable for use instead of PFD.

Table 3. Average run length for the attribute control chart setting $R_{0} = 300$ and sample size=20.

	PFD	WPFD	PFD	WPFD
Shift in parameter	$γ_{o 1} = 0.5$	$γ_{o 2} = 0.5$	$γ_{o 1} = 1$	$γ_{o 2} = 1$
Shift in parameter	$R_{0} = 300$ d=0.12797 r=2.972	$R_{0} = 300$ d=0.155871 r=3.011	$R_{0} = 300$ d=0.21199 r=3.41	$R_{0} = 300$ d=0.26795389 r=3.4
1.0	300.028	300.003	300.003	300.000
0.95	185.935	179.961	183.113	182.826
0.9	116.035	108.906	112.665	112.212
0.85	72.994	66.579	69.958	69.450
0.8	46.344	41.185	43.899	43.411
0.75	29.738	25.827	27.883	27.454
0.7	19.320	16.455	17.960	17.604
0.6	8.537	7.086	7.847	7.626
0.5	4.088	3.383	3.752	3.627
0.4	2.193	1.862	2.034	1.968
0.3	1.380	1.241	1.311	1.281
0.2	1.068	1.031	1.048	1.040
0.1	1.002	1.000	1.001	1.000

Figure 1. ARLs for attribute control charts of PFD and WPFD for $γ = 0.5$ and $K = 300 .$

Figure 2. ARLs for attribute control charts of PFD and WPFD under $γ =$ 1 and $R_{0} = 300 .$

3.1. Simulated data

In this section, we generated data from the PFD and WPFD. Let n=10 and R₀=300, $γ_{oi} = 0.5$ and $β_{oi} = 1$ , where i=1, 2. We generated the first 10 observations from the process under control and the next 20 observations from the shifted process using shift = 0.85, as presented in Table 4. We used the life test termination times t₀₁=0.31259* $μ_{o 1}$ and t₀₂=0.2235101* $μ_{o 2}$ . Figures 3 and 4 show the 30^th observation (10^th observation after the shift) using the WPFD control chart. We conclude that WPFD is more efficient than the PFD control chart.

Table 4. Simulated data from PFD and WPFD.

PFD	WPFD
6	3
2	2
4	5
5	4
2	6
1	3
3	4
4	2
5	5
7	5
3	6
2	2
4	3
5	2
4	1
2	3
1	5
4	5
5	5
6	2
7	3
6	4
3	6
5	7
4	1
4	5
6	4
3	2
2	5
5	4
7	8
3	2
5	5
4	5
4	2
3	3
2	5
1	1
5	3
4	2

Figure 3. Simulation for PFD.

Figure 4. Simulation for WPFD.

3.2. Real Life data

We used survival time data for a group of patients given by Bekker et al.¹² The data consisting of survival times (in years) for 46 patients were: .0470, .1150, .1210, .1320, .1640, .1970, .2030, .2600, .2820, .2960, .3340, .3950, .4580, .4660, .5010, .5070, .5290, .5340, .5400, .6410, .6440, .6960, .8410, .8630, 1.0990, 1.2190, 1.2710, 1.3260, 1.4470, 1.4850, 1.5530, 1.5810, 1.5890, 2.1780, 2.3430, 2.4160, 2.4440, 2.8250, 2.8300, 3.5780, 3.6580, 3.7430, 3.9780, 4.0030, 4.0330.

We used the maximum likelihood estimator (MLE) to estimate the PFD and WPFD parameters. The statistics used to show the application of the control charts are shown in Figures 5 and 6. Zaka et al.⁹ showed that the above data follow the WPFD with the shape parameter $γ_{o 1} = 0.3087394$ . We also assumed that $a_{o 1} = 0.35548$ . We obtained the control coefficient of the proposed chart as $k_{o 1} = 2.978$ , n=15, and p₀₁ = 0.465220. From Figure 5, we observe that the control chart based on the PFD shows an in-control situation. All the points lie between the control limits.

Figure 5. Real life analysis for PFD.

Figure 6. Real life analysis for WPFD.

From Figure 6, the WPFD control chart is presented with shape parameter $γ_{o 2} = 0.3087394$ . The plotting statistics are shown in Figure 6, which are used to apply the proposed control chart. The data are known to follow the WPFD with shape parameter $γ_{o 2} = 0.308739, a_{o 2} = 0.262064$ and control coefficients $k_{o 2} = 2.869$ , n=15, and p₀₂ = 0.423427. From Figure 6, we see that all points are in the control area, but the 30^th observation shows an out-of-control situation.

We see that it is better to use the WPFD-based control chart to control the process than the PFD control chart.

4. Conclusion

In this paper, we proposed attribute control charts assuming that the lifespan of the products under study follows the PFD and WPFD simultaneously when the shape parameter is unknown. We used the average run length (ARL) as the performance criterion to assess the competence of the proposed idea. The results of the ARLs are presented in the form of tables using different parameter values and sample sizes. The paper is concluded as follows.

• We observed that the WPFD is a more applicable control chart because of its narrower control limits than the PFD control chart.
• We also observed that the detection ability of the proposed control charts increased with increasing sample size.
• A real-life application and simulation study were conducted to demonstrate the applicability of the proposed control charts, which showed the outstanding execution of the suggested idea.
• The proposed control charts can be extended using other sampling schemes, such as repetitive sampling and multiple dependent state sampling, in future work.
• The proposed chart under WPFD efficiently detected the shift in the process.
• We suggest further applications of the proposed work in Engineering and Medical Sciences. In addition, more probability distributions could be used to construct the ACC under TTLT.

Data Availability

Underlying Data

All underlying data are available as part of the article and no additional source data are required.

References

1. Chang YS, Bai DS: Control charts for positively-skewed populations with weighted standard deviations. Qual. Reliab. Eng. Int. 2001; 17(5): 397–406. Publisher Full Text
2. Lin YC, Chou CY: Non-normality and the variable parameters X control charts. Eur. J. Oper. Res. 2007; 176(1): 361–373. Publisher Full Text
3. Rao KS, Subbaiah KV: Optimal operating policies of X-bar control chart for non-normal variables with generalised Pareto in-control times. International Journal of Productivity and Quality Management. 2010; 6(1): 91–111. Publisher Full Text
4. Aslam M, Arif OH, Jun CH: A control chart for gamma distribution using multiple dependent state sampling. Industrial Engineering and Management Systems. 2017; 16(1): 109–117. Publisher Full Text
5. Daragöz K: Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process. Math. Sci. 2018; 12(4): 249–262. Publisher Full Text
6. Noiplab T, Mayureesawan T: Modified EWMA control chart for skewed distributions and contaminated processes. Thailand Statistician. 2019; 17(1): 16–29.
7. Zaka A, Naveed M, Jabeen R: Performance of attribute control charts for monitoring the shape parameter of modified power function distribution in the presence of measurement error. Qual. Reliab. Eng. Int. 2022; 38: 1060–1073. Publisher Full Text
8. Dallas C: Characterization of Pareto and power function distribution. Ann. Inst. Stat. Math. 1976; 28: 491–497. Publisher Full Text
9. Zaka A, Akhter AS, Jabeen R: Asymmetry approach to study for chemotherapy treatment and devices failure times data using modified power function distribution with some modified estimators. Int. J. Comput. Sci. Math. 2022; 15(2): 99–116. Publisher Full Text
10. Jabeen R, Zaka A: Error detection through modified phase II process monitoringunder different classical estimators. Journal of the National Science Foundation Sirilanka. 2023; 51(3): 465–477. Publisher Full Text
11. Aslam M, Jun CH: Attribute control charts for the Weibull distribution under truncated life tests. Qual. Eng. 2015; 27(3): 283–288. Publisher Full Text
12. Bekker A, Roux J, Mostert P: A generalization of the compound Rayleigh distribution: using Bayesian methods on cancer survival times. Communication in Statistics Theory and Methods. 2000; 29: 1419–1433. Publisher Full Text

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Version 1

VERSION 1 PUBLISHED 27 Jun 2024

Author details Author details

¹ Statistics, COMSATS University Islamabad, Lahore, Lahore, 54000, Pakistan
² Statistics, Government College of Science, Lahore, Lahore, 54000, Pakistan

Azam Zaka
Roles: Methodology, Writing – Original Draft Preparation

Riffat Jabeen
Roles: Conceptualization, Supervision, Validation

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Published: 27 Jun 2024, 13:700

https://doi.org/10.12688/f1000research.146106.1

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© 2024 Zaka A and Jabeen R. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zaka A and Jabeen R. Computational Evaluation For The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application [version 1; peer review: 3 approved with reservations]. F1000Research 2024, 13:700 (https://doi.org/10.12688/f1000research.146106.1)

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ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions

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Reviewer Report 29 Aug 2024

Shakila Bashir, Forman Christian College, Lahore, Punjab, Pakistan

Approved with Reservations

https://doi.org/10.5256/f1000research.160146.r297249

The authors utilized attribute control charts for two positively skewed distributions—the power function distribution (PFD) and the weighted power function distribution (PWFD). They compared the performance of both types of control charts, presenting their findings in the form of tables. ... Continue reading

The authors utilized attribute control charts for two positively skewed distributions—the power function distribution (PFD) and the weighted power function distribution (PWFD). They compared the performance of both types of control charts, presenting their findings in the form of tables. Their conclusion indicates that control charts based on the weighted power function distribution outperform those using the power function distribution. Additionally, the authors suggest that their proposed results can aid in the early detection of errors, thereby enhancing process reliability in the telecommunications and finance industries.
While the proposed PWFD model appears to be superior to the PFD in attribute control charts and in detecting early shifts, I have some reservations about the findings, such as:

Grammatical errors are present throughout the article, and the first sentence of the abstract's methods section needs correction.
The PWFD model shows better performance, but there should be a valid discussion explaining why the weighted model produces superior results. The assignment of weights to models in specific situations could be explored here, with reference to applications or real-life examples.
The distinction between cases 2.3 and 2.5 needs to be made clearer. Currently, they appear similar, especially in the titles and text. Although a formula is provided, further clarification is required.
Figures 1 and 2 need to be clearer. Comparisons should be made separately for 'n' and model types. If presented on the same graph, the lines should be more distinct. The current color scheme makes it difficult to differentiate between PFD under n = 5 and WPFD under n = 10.
The title of the article, "Computational Evaluation for The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application," is not fully aligned with the content. The introduction and findings focus on SPC to monitor manufacturing processes and prevent products from failing, which are terms typically used in reliability engineering or quality inspection. These concepts should be related to the survival times of patients, which is the subject of the paper, not products. Additionally, the concept of truncated life tests is not clearly described, leading to a title that contradicts the findings of the paper and does not match the real-life example discussed.
The introduction section and body of the article too need to be revised according to the finding of the paper and also provide the motivation of proposing this work.
The conclusion is not comprehensive and needs to be refined to align more closely with the findings.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematical Statistics, Distribution Theory, Probability Theory, reliability Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report 29 Aug 2024

Alaa El-Alosey, Tanta University, Tanta, Egypt

Approved with Reservations

https://doi.org/10.5256/f1000research.160146.r309911

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature ... Continue reading

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature review. Based on that, it is suggested to link the motivation of this work to the existing gaps of the existing works.
2. In Page 3 lines 14 and 17, Zaka et al. repeated with same meaning. Provide space between words for example Page 3 lines 28 (asintroduced).
2. The reason(s) for considering the shape parameter for monitoring purposes should be clearly stated.
3. References to the related conclusions in section 1 are needed. All the formulas/equations should be indexed
4. Why ARL0=300?
5. What could the possible areas for future studies?
6. Many of the references are fairly old. The authors should check the last 2 years or so of this journal and ones that publish related papers to see if there are any more recent papers that should be cited.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Statistics, Probability theory, Inference, Data analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report 27 Jul 2024

Ahmed Z. Afify, Benha University, Banha, Al Qalyubia Governorate, Egypt

Approved with Reservations

https://doi.org/10.5256/f1000research.160146.r297242

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature ... Continue reading

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature review. Based on that, it is suggested to link the motivation of this work to the existing gaps of the existing works
2. The reason(s) for considering the shape parameter for monitoring purposes should be clearly stated.
3. References to the related conclusions in section 1 are needed. All the formulas/equations should be indexed
4. Please check the spaces between words and symbols; more places no space between them,
5. Why ARL0=300?
6. What could the possible areas for future studies?
7. Many of the references are fairly old. The authors should check the last 2 years or so of this journal and ones that publish related papers to see if there are any more recent papers that should be cited.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Statistics, Probability theory, Inference, Data analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

CITE

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VERSION 1 PUBLISHED 27 Jun 2024

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Version 1 27 Jun 24	read	read	read

Ahmed Z. Afify, Benha University, Banha, Egypt
Alaa El-Alosey, Tanta University, Tanta, Egypt
Shakila Bashir, Forman Christian College, Lahore, Pakistan

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Reviewer Report

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29 Aug 2024 | for Version 1

Shakila Bashir, Forman Christian College, Lahore, Punjab, Pakistan

2 Views Cite this report Responses(0)

Approved With Reservations

The authors utilized attribute control charts for two positively skewed distributions—the power function distribution (PFD) and the weighted power function distribution (PWFD). They compared the performance of both types of control charts, presenting their findings in the form of tables. Their conclusion indicates that control charts based on the weighted power function distribution outperform those using the power function distribution. Additionally, the authors suggest that their proposed results can aid in the early detection of errors, thereby enhancing process reliability in the telecommunications and finance industries.
While the proposed PWFD model appears to be superior to the PFD in attribute control charts and in detecting early shifts, I have some reservations about the findings, such as:

Grammatical errors are present throughout the article, and the first sentence of the abstract's methods section needs correction.
The PWFD model shows better performance, but there should be a valid discussion explaining why the weighted model produces superior results. The assignment of weights to models in specific situations could be explored here, with reference to applications or real-life examples.
The distinction between cases 2.3 and 2.5 needs to be made clearer. Currently, they appear similar, especially in the titles and text. Although a formula is provided, further clarification is required.
Figures 1 and 2 need to be clearer. Comparisons should be made separately for 'n' and model types. If presented on the same graph, the lines should be more distinct. The current color scheme makes it difficult to differentiate between PFD under n = 5 and WPFD under n = 10.
The title of the article, "Computational Evaluation for The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application," is not fully aligned with the content. The introduction and findings focus on SPC to monitor manufacturing processes and prevent products from failing, which are terms typically used in reliability engineering or quality inspection. These concepts should be related to the survival times of patients, which is the subject of the paper, not products. Additionally, the concept of truncated life tests is not clearly described, leading to a title that contradicts the findings of the paper and does not match the real-life example discussed.
The introduction section and body of the article too need to be revised according to the finding of the paper and also provide the motivation of proposing this work.
The conclusion is not comprehensive and needs to be refined to align more closely with the findings.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematical Statistics, Distribution Theory, Probability Theory, reliability Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Respond to this report

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Reviewer Report

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29 Aug 2024 | for Version 1

Alaa El-Alosey, Tanta University, Tanta, Egypt

0 Views Cite this report Responses(0)

Approved With Reservations

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature review. Based on that, it is suggested to link the motivation of this work to the existing gaps of the existing works.
2. In Page 3 lines 14 and 17, Zaka et al. repeated with same meaning. Provide space between words for example Page 3 lines 28 (asintroduced).
2. The reason(s) for considering the shape parameter for monitoring purposes should be clearly stated.
3. References to the related conclusions in section 1 are needed. All the formulas/equations should be indexed
4. Why ARL0=300?
5. What could the possible areas for future studies?
6. Many of the references are fairly old. The authors should check the last 2 years or so of this journal and ones that publish related papers to see if there are any more recent papers that should be cited.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Statistics, Probability theory, Inference, Data analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Respond to this report

Responses (0)

Back to all reports

Reviewer Report

4 Views

27 Jul 2024 | for Version 1

Ahmed Z. Afify, Benha University, Banha, Al Qalyubia Governorate, Egypt

4 Views Cite this report Responses(0)

Approved With Reservations

I have some comments to improve the current version of the paper. The author should addressed the following comments carefully.

1. In the introduction section, the authors have already done a good job on the literature review. Based on that, it is suggested to link the motivation of this work to the existing gaps of the existing works
2. The reason(s) for considering the shape parameter for monitoring purposes should be clearly stated.
3. References to the related conclusions in section 1 are needed. All the formulas/equations should be indexed
4. Please check the spaces between words and symbols; more places no space between them,
5. Why ARL0=300?
6. What could the possible areas for future studies?
7. Many of the references are fairly old. The authors should check the last 2 years or so of this journal and ones that publish related papers to see if there are any more recent papers that should be cited.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Statistics, Probability theory, Inference, Data analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Respond to this report

Responses (0)

[1] 1. Chang YS, Bai DS: Control charts for positively-skewed populations with weighted standard deviations. Qual. Reliab. Eng. Int. 2001; 17(5): 397–406. Publisher Full Text

[2] 2. Lin YC, Chou CY: Non-normality and the variable parameters X control charts. Eur. J. Oper. Res. 2007; 176(1): 361–373. Publisher Full Text

[3] 3. Rao KS, Subbaiah KV: Optimal operating policies of X-bar control chart for non-normal variables with generalised Pareto in-control times. International Journal of Productivity and Quality Management. 2010; 6(1): 91–111. Publisher Full Text

[4] 4. Aslam M, Arif OH, Jun CH: A control chart for gamma distribution using multiple dependent state sampling. Industrial Engineering and Management Systems. 2017; 16(1): 109–117. Publisher Full Text

[5] 5. Daragöz K: Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process. Math. Sci. 2018; 12(4): 249–262. Publisher Full Text

[6] 6. Noiplab T, Mayureesawan T: Modified EWMA control chart for skewed distributions and contaminated processes. Thailand Statistician. 2019; 17(1): 16–29.

[7] 7. Zaka A, Naveed M, Jabeen R: Performance of attribute control charts for monitoring the shape parameter of modified power function distribution in the presence of measurement error. Qual. Reliab. Eng. Int. 2022; 38: 1060–1073. Publisher Full Text

[8] 8. Dallas C: Characterization of Pareto and power function distribution. Ann. Inst. Stat. Math. 1976; 28: 491–497. Publisher Full Text

[9] 9. Zaka A, Akhter AS, Jabeen R: Asymmetry approach to study for chemotherapy treatment and devices failure times data using modified power function distribution with some modified estimators. Int. J. Comput. Sci. Math. 2022; 15(2): 99–116. Publisher Full Text

[10] 10. Jabeen R, Zaka A: Error detection through modified phase II process monitoringunder different classical estimators. Journal of the National Science Foundation Sirilanka. 2023; 51(3): 465–477. Publisher Full Text

[11] 11. Aslam M, Jun CH: Attribute control charts for the Weibull distribution under truncated life tests. Qual. Eng. 2015; 27(3): 283–288. Publisher Full Text

[12] 12. Bekker A, Roux J, Mostert P: A generalization of the compound Rayleigh distribution: using Bayesian methods on cancer survival times. Communication in Statistics Theory and Methods. 2000; 29: 1419–1433. Publisher Full Text

PFD	WPFD
6	3
2	2
4	5
5	4
2	6
1	3
3	4
4	2
5	5
7	5
3	6
2	2
4	3
5	2
4	1
2	3
1	5
4	5
5	5
6	2
7	3
6	4
3	6
5	7
4	1
4	5
6	4
3	2
2	5
5	4
7	8
3	2
5	5
4	5
4	2
3	3
2	5
1	1
5	3
4	2

PFD	WPFD
6	3
2	2
4	5
5	4
2	6
1	3
3	4
4	2
5	5
7	5
3	6
2	2
4	3
5	2
4	1
2	3
1	5
4	5
5	5
6	2
7	3
6	4
3	6
5	7
4	1
4	5
6	4
3	2
2	5
5	4
7	8
3	2
5	5
4	5
4	2
3	3
2	5
1	1
5	3
4	2

Computational Evaluation For The Survival Times Of Patients Under Truncated Life Tests: Simulation And Application

Abstract

Background

Methods

Results

Keywords

1. Introduction

2. Methods

2.1. Distribution function for PFD and WPFD

(1)

(2)

2.2. Average Run Length ARL for proposed control monitoring using attribute control chart for PFD and WPFD

2.3. Case 1 (when we assume shift in scale parameter)

2.4. Case 2 (when we assume a shift in the shape of a parameter)

2.5. Case 3 (when we assume a shift in the scale of a parameter)

Table 1. Average run length for the attribute control chart setting R0=300 and sample size=5.

Table 2. Average run length for the attribute control chart setting R0=300 and sample size=10.

3. Results and Discussion

Table 3. Average run length for the attribute control chart setting R0=300 and sample size=20.

Figure 1. ARLs for attribute control charts of PFD and WPFD for γ=0.5 and K=300.

Figure 2. ARLs for attribute control charts of PFD and WPFD under γ=1 and R0=300.

3.1. Simulated data

Table 4. Simulated data from PFD and WPFD.

Figure 3. Simulation for PFD.

Figure 4. Simulation for WPFD.

3.2. Real Life data

Figure 5. Real life analysis for PFD.

Figure 6. Real life analysis for WPFD.

4. Conclusion

Data Availability

Underlying Data

References

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Table 1. Average run length for the attribute control chart setting $R_{0} = 300$ and sample size=5.

Table 2. Average run length for the attribute control chart setting $R_{0}$ =300 and sample size=10.

Table 3. Average run length for the attribute control chart setting $R_{0} = 300$ and sample size=20.

Figure 1. ARLs for attribute control charts of PFD and WPFD for $γ = 0.5$ and $K = 300 .$

Figure 2. ARLs for attribute control charts of PFD and WPFD under $γ =$ 1 and $R_{0} = 300 .$

PFD	WPFD
6	3
2	2
4	5
5	4
2	6
1	3
3	4
4	2
5	5
7	5
3	6
2	2
4	3
5	2
4	1
2	3
1	5
4	5
5	5
6	2
7	3
6	4
3	6
5	7
4	1
4	5
6	4
3	2
2	5
5	4
7	8
3	2
5	5
4	5
4	2
3	3
2	5
1	1
5	3
4	2