Computational and Mathematical Methods in Medicine

Computational and Mathematical Methods in Medicine / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 6634887 | https://doi.org/10.1155/2021/6634887

Muhammad Aslam, G. Srinivasa Rao, Muhammad Saleem, Rehan Ahmad Khan Sherwani, Chi-Hyuck Jun, "Monitoring Mortality Caused by COVID-19 Using Gamma-Distributed Variables Based on Generalized Multiple Dependent State Sampling", Computational and Mathematical Methods in Medicine, vol. 2021, Article ID 6634887, 17 pages, 2021. https://doi.org/10.1155/2021/6634887

Monitoring Mortality Caused by COVID-19 Using Gamma-Distributed Variables Based on Generalized Multiple Dependent State Sampling

Academic Editor: Nadia A. Chuzhanova
Received31 Dec 2020
Revised15 Mar 2021
Accepted08 Apr 2021
Published29 Apr 2021

Abstract

More recently in statistical quality control studies, researchers are paying more attention to quality characteristics having nonnormal distributions. In the present article, a generalized multiple dependent state (GMDS) sampling control chart is proposed based on the transformation of gamma quality characteristics into a normal distribution. The parameters for the proposed control charts are obtained using in-control average run length (ARL) at specified shape parametric values for different specified average run lengths. The out-of-control ARL of the proposed gamma control chart using GMDS sampling is explored using simulation for various shift size changes in scale parameters to study the performance of the control chart. The proposed gamma control chart performs better than the existing multiple dependent state sampling (MDS) based on gamma distribution and traditional Shewhart control charts in terms of average run lengths. A case study with real-life data from ICU intake to death caused by COVID-19 has been incorporated for the realistic handling of the proposed control chart design.

1. Introduction

One of the important techniques for improving manufactured product quality and for reducing the manufacturing costs is statistical quality control (SQC). Since the pioneer work by Shewhart A. Walter during 1920s in Bell Telephone Laboratories, wide varieties of control chart techniques have been constructed and extensively implemented in SQC. The main feature of control charting is to identify the amount of assignable cause(s) and hence rectify it by taking necessary action on the production process before sending the outcome of the products into the market. This control charting helps to avoid nonconforming products from being manufactured by the company. More details about Shewhart control charts can be seen in Montgomery [1].

Usually, control charts are being designed and operating under the assumption of the normality for the variable of interest. Nevertheless, these assumptions may not be true for various realistic situations and other distributions away from normality had been considered and discussed by many authors in the literature (e.g., see [25]). The waiting time of an event, for example, can be represented by a gamma distribution as in [6]. Numerous researchers concentrate on quality characteristic understudy which follows a nonnormal distribution or transformed into normality to apply Shewhart type control charts. For skewed data, the gamma distribution is widely used. The works on the control charts for the gamma distribution are presented by Al-Oraini and Rahim [7], Jearkpaporn et al. [8], Sheu and Lin [9], Aslam et al. [10], and Zhang et al. [11]. Santiago and Smith [5] used transformation given by Johnson and Kotz [12] and Nelson [13]. Mohammed [14], Mohammed and Laney [15], and Aslam et al. [16] discussed the application of the -chart.

Several researchers have developed diversified sampling designs to obtain more efficient control charts. Recently, researchers focused on multiple dependent state (MDS) sampling in the creation of a control chart. Wortham and Baker [17] proposed the MDS sampling in quality control charts. MDS design is more competent than the existing single sampling plans because it considers the previous lot information along with the current lot to make a decision whether the process is under control or not (see [18]). Aslam et al. [19] developed a control chart for gamma distribution using MDS sampling. The control chart scheme using MDS sampling was studied by different authors for various schemes (see [16, 19, 2029, 30, 31]).

More recently, Raza and Aslam [32], Rao et al. [33], Rao et al. [34], and Aslam et al. [35, 36] formulated generalized MDS (GMDS) sampling for various schemes. GMDS is more flexible and efficient than MDS to design the control chart using the gamma distribution. The aim of this article is to construct a gamma control chart for monitoring the process mean based on GMDS sampling design. The application of the proposed chart will be given using the COVID-19 mortality data. It is expected that the proposed chart will perform better than the existing Shewhart control chart and control charts using MDS in terms of average run length and standard deviation of run length.

2. Design of Control Chart for Gamma Distribution Based on GMDS Sampling

The proposed control chart for a gamma distribution using gamma to normal transformation is discussed. Let be a random variable from a gamma distribution with shape parameter and scale parameter . The cumulative distribution function (cdf) of the gamma distribution is given by

Wilson and Hilferty [37] recommended that if follows a gamma distribution with specific parameters, then the transformed variable can be distributed approximately as normal with mean and variance , where

The proposed gamma control chart using GMDS sampling comprises the two pairs of control chart limits. The inner lower control limit (LCL) and upper control limit (UCL) are denoted by subscript 1, and the outer lower control limit (LCL) and upper control limit (UCL) are denoted by subscript 2. The four control limits are given bywhere and are the chart constants to be found when the in-control ARL is approximately equal to preassigned value . The convenient form of the above control limits is given as follows: , , , and , where

The operation of the proposed control chart using GMDS scheme is described as follows:(1)Obtain quality measurement from the manufacturing process, and denote the quality characteristic by . Compute the transformed variable as (2)The process can be considered under control if , and the process can be considered out-of-control if . Or else, go to Step 3(3)The process can be considered under control whenever out of proceeding subgroups have been declared as under control, that is, ; otherwise, the output of the product can be considered out-of-control and go back to Step 1

The probability of declaring as in-control for the proposed control chart when the process is actually in-control is given as follows:where

Therefore, the in-control average run length (ARL) when the process is under control is given by

Assume the gamma scale parameter has been changed from to , where is the shift value.

The probability of process is declared as in-control while the scale parameter which has been changed can be obtained as follows:where

The out-of-control average run length (ARL) when the process is out-of-control is given as

The proposed control chart parameters and along with ARL1 are obtained using the following algorithm:(1)Decide the predetermined in-control ARL as (2)Fix the known values for and (3)Obtain the ARL0 using Equation (14), which consists of chart parameters and (4)Determine the most possible values of chart parameters and such that, (5)In the above step, we get more values of and to satisfy the condition. Choose the best values of and for which the value of is almost equal to (6)Using the best parametric values of and determined in the previous step, work out the using Equation (18) and hence obtain standard deviation (SD) of run-length (SDRL) for various shift () values

The R codes to find the design parameters of the control chart are given in the appendix.

3. Numerical Results and Discussion

The performance of the proposed gamma control chart using GMDS sampling is considered based on ARL, such as ARL0 and ARL1. These ARL values are used to know the effectiveness of the developed control chart. The developed chart is said to be efficient if it shows larger in-control ARL and smaller out-of-control ARL. Using the aforementioned algorithm in Section 2, the chart coefficients and are obtained. The out-of-control ARLs and SDRL are computed for a choice of shift values, from 1.0 to 2.0 with an interval of 0.1 and 2.0 to 4.0 with an interval of 0.5. The values of considered are 4, 5, and 6 and , 10, and 20. Table 1 is for and , Table 2 is for and , Table 3 is for and , Table 4 is for and , Table 5 is for and , and Table 6 is for and .


, , , , , , , , ,
ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0370.05369.55370.01369.51370.02369.52370.00369.50370.01369.51370.04369.54370.04369.54370.03369.53370.02369.52
1.1206.61206.11207.63207.13208.25207.75205.73205.23206.33205.83207.32206.82200.51200.01203.00202.50204.91204.41
1.2100.3699.86102.57102.07104.12103.6299.9399.43100.88100.38103.04102.5493.7793.2796.9796.47100.3799.87
1.350.2349.7352.5152.0154.5654.0649.9649.4550.9950.4953.6853.1845.0744.5647.8747.3651.6251.12
1.427.1726.6729.0028.5031.1030.6026.8626.3627.8227.3130.4629.9623.4822.9725.6325.1329.0328.53
1.516.0015.5017.3716.8619.2718.7715.6815.1716.4915.9918.8318.3313.4412.9315.0414.5317.8617.35
1.610.219.7011.2210.7112.8712.369.919.4010.6010.0812.5612.058.437.929.639.1211.9111.39
1.77.006.487.767.249.158.646.746.227.326.808.958.435.755.236.676.158.507.98
1.85.114.585.695.176.876.354.904.375.384.856.736.214.223.694.944.416.425.90
1.93.933.394.403.865.394.873.763.224.173.645.314.783.292.743.873.345.094.56
2.03.162.613.543.004.403.873.032.483.392.844.343.812.692.133.182.634.183.65
2.51.661.041.861.262.301.731.631.011.841.242.301.731.560.931.801.212.271.70
3.01.280.601.410.761.671.061.290.611.420.771.681.071.270.591.420.771.671.05
3.51.140.401.230.541.400.741.160.431.250.551.400.751.160.431.250.561.400.75
4.01.080.301.150.411.250.561.100.331.160.421.260.571.100.331.160.431.260.57


ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0370.01369.51370.02369.52370.02369.52370.01369.51370.00369.50370.05369.55370.01369.51370.04369.54370.01369.51
1.1176.21175.71178.51178.01179.79179.29167.40166.90168.84168.33171.65171.15161.99161.49163.28162.77167.95167.45
1.266.3265.8268.9968.4971.5071.0057.8457.3460.0259.5264.2263.7153.3952.8955.4654.9661.1960.69
1.327.0126.5128.9528.4531.6131.1121.8921.3823.5623.0527.1126.6119.5119.0021.1320.6325.4224.92
1.412.6512.1413.9113.4016.1215.619.899.3710.9810.4713.5313.028.708.199.799.2812.6412.13
1.56.866.347.677.159.378.855.354.826.085.567.857.334.754.225.484.967.376.85
1.64.253.724.804.276.085.563.392.843.903.365.164.633.062.523.593.054.904.37
1.72.942.393.342.804.323.792.431.862.812.263.733.192.251.682.652.093.593.05
1.82.231.662.541.983.302.751.911.322.211.642.912.361.821.222.131.552.832.27
1.91.821.222.071.482.662.101.621.001.861.262.401.831.570.941.811.212.351.78
2.01.560.941.771.162.251.671.430.791.631.022.061.481.410.761.610.992.031.44
2.51.130.381.220.511.400.751.110.351.190.481.350.681.120.371.200.481.340.68
3.01.040.201.080.301.160.431.040.201.070.281.140.401.040.211.080.281.140.39
3.51.020.121.030.191.070.281.020.131.030.181.060.251.020.131.030.181.060.25
4.01.010.081.020.121.030.181.010.081.010.121.030.171.010.091.010.121.030.17


ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0370.02369.52370.04369.54370.05369.55370.01369.51370.02369.52370.05369.55370.04369.54370.05369.55370.05369.55
1.1119.55119.05120.53120.02121.50120.99113.23112.73114.29113.79118.35117.84108.88108.38109.14108.64115.61115.11
1.225.9225.4126.7426.2429.4428.9322.7622.2624.2623.7628.1827.6721.5721.0722.4421.9327.1626.65
1.37.937.418.317.799.949.437.416.907.517.009.549.036.375.856.986.469.248.73
1.42.902.363.763.224.694.163.312.763.492.944.574.042.942.393.322.784.493.96
1.52.071.502.271.702.842.292.011.432.171.602.822.261.871.282.131.552.822.26
1.61.521.001.671.052.051.461.510.881.641.022.051.471.460.821.641.032.071.49
1.71.330.661.380.731.641.031.280.601.380.731.661.051.270.591.400.751.691.07
1.81.180.461.230.541.420.771.170.441.240.551.440.791.170.451.260.571.460.81
1.91.110.341.150.411.280.601.110.341.160.431.300.621.110.351.170.451.310.64
2.01.060.261.100.331.190.481.070.271.110.341.200.491.070.281.120.361.210.51
2.51.010.091.010.121.030.181.010.091.020.121.030.191.010.101.020.131.040.19
3.01.000.031.000.041.010.071.000.031.000.051.010.081.000.041.000.051.010.08
3.51.000.011.000.021.000.031.000.011.000.021.000.031.000.011.000.021.000.03
4.01.000.001.000.011.000.011.000.011.000.011.000.011.000.011.000.011.000.01


ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0500.03499.53500.04499.54500.05499.55500.01499.51500.02499.52500.05499.55500.04499.54500.01499.51500.05499.55
1.1270.28269.78271.57271.07273.31272.81263.92263.42266.57266.07268.09267.59259.99259.49262.25261.75264.22263.71
1.2126.46125.96123.19122.69126.16125.66117.99117.49117.61117.11121.35120.84112.09111.58113.21112.71117.87117.37
1.361.1360.6357.0556.5560.1559.6554.2253.7253.0952.5957.0856.5849.4948.9950.1849.6854.9454.43
1.432.0831.5828.9928.4831.5931.0927.2926.7926.5126.0029.7729.2724.1823.6824.7824.2828.5428.03
1.518.4117.9116.3715.8618.4017.8915.2414.7314.8414.3317.3116.8013.2912.7813.8213.3116.6016.09
1.611.5010.9910.209.6811.7511.249.398.889.238.7111.0910.588.157.638.618.0910.6710.15
1.77.747.236.916.398.127.606.315.796.285.757.707.185.494.975.895.377.456.93
1.85.565.045.024.505.995.464.564.034.604.075.715.194.013.474.353.825.565.04
1.94.233.693.873.334.654.123.502.963.583.044.483.943.122.573.422.884.393.86
2.03.362.823.132.583.783.242.832.272.922.373.663.122.562.002.822.273.613.07
2.51.701.091.691.082.011.421.560.931.661.052.011.421.510.871.661.052.021.44
3.01.290.621.320.651.510.881.250.561.330.661.520.891.240.551.340.671.540.91
3.51.150.411.180.461.300.621.140.391.190.481.310.641.140.401.200.491.320.65
4.01.090.301.110.351.190.481.080.301.120.371.200.491.090.311.130.381.210.50


ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0500.01499.51500.02499.51500.05499.55500.04499.54500.03499.53500.05499.55500.05499.55500.05499.55500.05499.55
1.1221.13220.63222.17221.66223.48222.98215.54215.04217.15216.65218.55218.05208.30207.80209.08208.58214.77214.27
1.271.1670.1672.2171.7175.7575.2569.5269.0268.1567.6572.5372.0362.5262.0163.2062.7070.1669.66
1.325.3124.8126.1225.6229.1128.6124.8224.3224.0823.5827.6127.1021.1920.6821.9521.4526.5526.04
1.411.0610.5511.5711.0613.5913.0810.7610.2510.6010.0912.8912.389.058.539.679.1612.4311.92
1.55.895.376.215.697.557.035.675.145.735.207.226.704.824.295.294.767.016.49
1.63.212.883.903.364.824.303.522.983.643.104.664.133.072.533.432.894.584.05
1.72.502.052.762.213.442.892.491.932.632.073.362.822.241.672.521.963.342.79
1.82.101.322.151.572.662.101.951.362.081.502.632.071.811.212.031.452.632.07
1.91.591.151.791.182.191.611.631.021.751.152.181.611.550.931.741.132.201.62
2.01.440.911.560.941.881.291.440.801.550.921.891.301.400.751.550.931.911.32
2.51.120.361.150.421.270.591.110.361.160.431.290.611.120.361.170.451.300.62
3.01.040.201.060.241.110.341.040.201.060.251.110.361.040.211.070.271.120.36
3.51.010.121.020.151.050.221.020.131.020.161.050.231.020.131.030.171.050.23
4.01.010.081.010.101.020.151.010.081.010.101.020.151.010.081.010.111.020.15


ARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL

1.0500.04499.54500.04499.54500.05499.55500.05499.55500.05499.55500.05499.55500.04499.54500.05499.55500.05499.55
1.1153.53153.03154.54154.04155.79155.29143.46142.96145.72145.22151.15150.65138.53138.03139.18138.68147.91147.41
1.230.9330.4232.1131.6035.5535.0527.6027.1028.8728.3633.8033.3025.7425.2326.6726.1732.6432.13
1.38.237.729.468.9511.4410.938.087.068.477.9510.8910.387.196.677.857.3310.5510.04
1.43.292.754.123.585.194.673.583.043.773.235.024.493.172.633.583.044.934.40
1.52.121.552.411.843.052.512.111.542.291.713.012.461.961.382.241.663.002.45
1.61.691.081.731.132.151.571.550.931.691.082.151.571.500.871.691.082.171.59
1.71.360.701.420.771.701.101.310.631.410.761.721.111.290.621.430.781.741.14
1.81.200.491.250.561.460.811.180.461.260.571.470.841.180.471.280.601.490.86
1.91.120.361.160.431.310.631.110.361.170.451.320.651.120.371.190.471.330.67
2.01.070.281.110.341.210.501.070.281.120.361.220.521.080.291.130.381.230.53
2.51.010.091.010.121.030.191.010.101.020.131.040.201.010.101.020.141.040.20
3.01.000.031.000.051.010.081.000.041.000.051.010.081.000.041.000.051.010.08
3.51.000.011.000.021.000.031.000.011.000.021.000.031.000.021.000.021.000.04
4.01.000.011.000.011.000.011.000.011.000.011.000.021.000.011.000.011.000.02

We pointed out the following several noteworthy comments from Tables 16 for the developed control charts:(1)The out-of-control ARL and SDRL values decline speedily when the shift () of the manufacturing process increases(2)It is detected that the chart coefficient shows an increasing tendency for increased value of for a fixed value of when other parametric combinations are fixed(3)From the tables, it is noticed that ARL1 and SDRL values decrease when values increase. In addition, ARL1 and SDRL values increased with the increase of value (i.e., -2 to -0). It also observed the same inclination over the other parametric combinations and and 500(4)It is interesting to observe from the results that the values of ARL1 and SDRL are small for -2 and these values are increasing from -2 to for fixed values of . In addition, noticed that ARL1 and SDRL values are large at as compared to the values at -1 and -2 (we know that if , the developed plan becomes MDS design). Hence, it is concluded from the results that gamma control chart using GMDS sampling is an enormous amount of accurate than gamma control chart using MDS sampling

4. Comparison with Existing Charts

In this part, a comparison is made between the developed control chart and the existing Shewhart type control chat and MDS control chart for gamma distribution. Also, the application of developed control chart and its dominance over available control chart schemes studied using real data set is presented. In addition, through a simulation study, the supremacy of the developed control chart when compared with the existing control charts is examined. The performance of the developed control chart is studied through ARL values and we know that a control chart with smaller ARL values is more desirable. In this investigation, we studied when and ; the shape parameter of gamma distribution is given as , 10, and 20 to compare the developed gamma control chart under GMDS with the existing MDS and Shewhart type control chart at various shift values. These comparisons are presented in Table 7 for and and in Table 8 for and at various shape parameters of the gamma distribution.


Proposed

MDS

Shewhart
Proposed

MDS

Shewhart
Proposed

MDS

Shewhart

1.0370.05370.02370.96370.01370.02370.96370.02370.05370.96
1.1206.61208.25217.16176.21179.79188.41119.55121.50144.07
1.2100.36104.12114.7366.3271.5080.5425.9229.4446.78
1.350.2354.5663.7827.0131.6138.477.939.9419.11
1.427.1731.1038.4412.6516.1220.972.904.699.59
1.516.0019.2724.986.869.3712.772.072.845.64
1.610.2112.8717.304.256.088.501.522.053.75
1.77.009.1512.642.944.326.071.331.642.73
1.85.116.879.642.233.304.601.181.422.15
1.93.935.397.631.822.663.641.111.281.78
2.03.164.406.221.562.253.001.061.191.55
2.51.662.303.071.131.401.651.011.031.11
3.01.281.672.071.041.161.271.001.011.02
3.51.141.401.631.021.071.121.001.001.00
4.01.081.251.401.011.031.061.001.001.00


Proposed

MDS

Shewhart
Proposed

MDS

Shewhart
Proposed

MDS

Shewhart

1.0500.01500.05500.94500.04500.05500.93500.05500.05500.04
1.1263.92268.09283.07215.54218.55244.74143.46151.15185.05
1.2117.99121.35144.6069.5272.53100.6327.6033.8057.41
1.354.2257.0878.2624.8227.6146.638.0810.8922.64
1.427.2929.7746.1510.7612.8924.793.585.0211.04
1.515.2417.3129.455.677.2214.782.113.016.33
1.69.3911.0920.093.524.669.661.552.154.12
1.76.317.7014.472.493.366.801.311.722.96
1.84.565.7110.911.952.635.081.181.472.29
1.93.504.488.541.632.183.981.111.321.88
2.02.833.666.901.441.893.241.071.221.61
2.51.562.013.291.111.291.721.011.041.12
3.01.251.522.171.041.111.291.001.011.03
3.51.141.311.691.021.051.141.001.001.01
4.01.081.201.441.011.021.071.001.001.00

It is noticed that from the results on the basis of Tables 7 and 8, the developed gamma control charts show smaller quantity ARL1 values as compared with the MDS and Shewhart type control charts at various shifts () values and various parametric values studied in this article. At a glance, when , and from Table 7, for the developed control chart whereas for MDS scheme and from the Shewhart type control chart. Similarly, for , , and from Table 8, we sense that the developed control chart gives while for the MDS control chart and from the Shewhart type control chart. The graphical presentation is given to show the performance of developed control chat over the existing MDS and Shewhart type control charts along with various shift values (see Figures 1 and 2). From these two figures, it is articulated that the developed gamma control chart based on GMDS is certified extra sensitive as compared to the MDS and the Shewhart-type control charts. To draw attention to this conclusion, a real data illustration and a simulation study are also carried out in the following subsections.