TX_1~ABS:AT/ADD:TX_2~ABS:AT 24 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 ReseaRch aRticle The Estimation of (Covid-19) Cases in Kurdistan Region Using Nelson Aalen Estimator Sami A. Obed1*, Parzhin A. Mohammed1, Dler H. Kadir1,2 1Department of Statistics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq, 2Department of Business Administration, Cihan University-Erbil, Kurdistan Region, Iraq ABSTRACT It is described how the Nelson–Aalen estimator may be used to control the rate of a nonparametric estimate of the cumulative hazard rate function based on right censored as well as left condensed survival data, furthermore how the Nelson–Aalen estimator can be utilized to estimate various amounts. This technique is mostly applied to survival data and product quality data similar to the incorporated relative mortality in a multiplicative model with outer rates and the cumulative infection rate in a straightforward epidemic model. It is shown that tallying measures produce a structure that permits to a brought together treatment of all these different conditions, and the main little and massive sample properties of the assessor are summarized. This estimator is a weighted average of the Nelson-Aalen reliability estimates over 2  time periods. The suggested estimator’s suitability and utility in model selection are reviewed. Morover, a real-world dataset is evaluated to demonstrate the proposed estimator’s suitability and utility. This work proposes a simple and nearly unbiased estimator to fill this gap. The information was gathered from the Ministry of Health’s website between October 1, 2020, and February 28, 2021. The results of the Nelson Allen Estimator demonstrated that the odds of surviving were higher during a short period of time after being exposed to the virus. As time passes, the possibilities become slimmer. The closer the estimate comes to value 1 from 0.5 upward, the greater the chances of surviving the infection. Keywords: Covied-19, death, estimator, nelson Allen estimator, survival analysis INTRODUCTION Nelson-Aalen estimation is useful to separate how a assumed population develops through time. This strategy is frequently applied to survival data and product feature information. This makes the virus lethal since it can be encountered by someone who is infected without showing signs. Many papers/articles had been written on the novel coronavirus (Covid-19) within a relatively short time after its existence.[1] This research will answer questions such as: What are the affected people’s odds of survival? What is the survival probability’s time frame? How can a person survive a viral outbreak, particularly in the impacted areas? The Nelson-Aalen estimator, or more generally visualizing the hazard function over time, is not a very popular approach to survival analysis. That is because in evaluation to the survival function clarification of the curves is not so simple and intuitive. However, the risk function is of great significance for more advanced methods to survival analysis, for example, the Cox regression. That is why it is vital to appreciate the concept. Moreover, it will try to provide some insights about it. We can say that the cumulative hazard functions (CHFs).[2] The Nelson-Aalen estimation is a method for estimating a public’s hazard function without considering that the data originates from a specific distribution. The Nelson-Aalen approach produces the CHF from the hazard function, which is then used to obtain the survival function. It is a non-parametric approach of generating results due to the lack of parameters necessary in this model. This research will serve as a foundation for health policy formulation, resulting in fewer untimely deaths and an overall improvement in quality of life.[2] The cumulative hazard is one of the most reliable parameters in the analysis of time-to-event data with Corresponding Author: Sami A. Obed, Department of Statistics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq E-mail: muhsin.hmeadi@cihanuniversity.edu.iq Received: July 7, 2021 Accepted: September 6, 2021 Published: September 30, 2021 DOI: 10.24086/cuesj.v5n2y2021. pp 24-31 Copyright © 2021 Sami A. Obed, Parzhin A. Mohammed, Dler H. kadir, this is an open-access article distributed under the Creative Commons Attribution License (CC BY-NC-ND 4.0). Cihan University-Erbil Scientific Journal (CUESJ) Obed, et al.: The Estimation of (Covid-19) Cases in KRG 25 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 independently right-censored and left-truncated survival times. Cumulative transition intensity is another name for it. The well-known Nelson–Aalen estimator is frequently used to estimate it nonparametric ally. In this context time-simultaneous confidence bands are perhaps the best interpretive tool for accounting for related estimating uncertainty in this situation.[3] METHODOLOGY This section studied some basic concepts of survival analysis; survival function, hazard function, and some tests and methods used to analyze survival data. Coronavirus is a virus whose genome consists of a single strand of ribonucleic acid. This is to distinguish this disease from other coronavirus outbreak in the past or future. This was a result of genetic comparison of the animal and the infected person. Numerous review methods and strategies were designed during the course of this study to collect data/ materials relevant to the study in order to gain a thorough understanding of the subject matter.[4] A survival analysis can be used to evaluate not just the likelihood of manufacturing equipment failure based on the number of hours it has been in operation, but also to distinguish between different operating situations. If the chance varies depending on whether the equipment is operated outdoors or indoors, for example.[4] As a result, it is important to remember that the overall quality of the sample frames and methods utilized determines the trustworthiness of the survey estimates. In general, the broader the sample frame, the greater the result’s quality and validity.[5] NELSON–AALEN ESTIMATOR The Nelson–Aalen estimator is a nonparametric estimator which might be used to estimate the cumulative hazard rate function from censored survival information the Nelson– Aalen estimator might be applied and exemplify its use in one specific case. Moreover, it is shown how checking measures give a system, which permits to a bound together treatment of every one of these different circumstances, and summarizes the main properties of the Nelson–Aalen estimator.[2] The Nelson-Aalen analysis strategy has a place with the enlightening strategies for survival analysis, for example, life table analysis and Kaplan-Meier investigation. The Nelson- Aalen approach can rapidly provide you with a bend of combined peril and gauge the hazard capacities dependent on unpredictable time intervals.[6] An “occasion” can be the failure of a non-repairable component, the death of a human being, or any occurrence for which the experimental unit remains in the “failed” state (e.g., death) from the point at which it changed on.[4] SURVIVAL FUNCTION ESTIMATOR[5] Let t 1 < t 2 <… < t k represent the observed death times in a sample of n subjects from a homogeneous population with survival function S(t). Consider S(t) as a discrete function with probability mass at each t i ,; i =1;…; k. Therefore it can be written that S q q q qti i j j i ( ) � �� � �� �� �� � � �� � ��1 1 1 11 2 1 (1) where q j is the probability of subject death in the interval [t j -1, t i ]conditional of being alive at t j -1 that is, q j can be written as q p Tj � �( [ t j -1,t T tj j) / ≥ -1) (2) Suppose that d i deaths occurs at t i and there are n i subjects at risk at t i , KME is obtained from (1) and (2) as ( ) < < − = = −∏ ∏ . ( ) / / (1 i i i i i KM t i ii t t i t t n d d S n n ) (3) It is a staircase function of t with a jump at each failure time. The NAE is given by H d n R t H d ntj k k j t jk j k kk j ( ) , exp� � � � �� � �� ��� ��1 1 (4) Where H(t) is the CHF and is also a staircase function of t. It is noted that the above definitions extend the original definitions of the NAE to the censored observations.[1] CENSORING CATEGORIES FOR THE NELSON-AALEN ANALYSIS[3] There are some types of censoring of survival data: 1. Left censoring: We know that an event occurred at t * t when it is reported at time t=t(i) (i) 2. Right censoring: When an occasion is reported at time t=t(i), we see that it happened at time t * t(i) if it happened at all 3. Interval censoring: We know that an event occurred during [t(i-1); t(i)] when it is recorded at time t=t(i) 4. Exact censoring: We know that an incident happened exactly at t=t(i) because it is reported at time t=t(i) (i). INDEPENDENT CENSORING FOR THE NELSON-AALEN METHOD The Nelson-Aalen technique requires that the explanations be unbiased. Second, the censoring should be consistent – if two random persons are enrolled in the research at time t-1 and the other is censored at time t while the other lives, each must have the same chance of surviving at time t.[7] Independent censorship can be divided into four categories: • Simple Type I: All individuals are censored at the same time, or individuals are followed for a set period of time • Progressive Type I: At the same time, everybody is censored (for illustration, when the study terminates). • Type II: The research will attempt until all of the n measures have been validated • Random: The period when censoring occurs is unrelated to the time it takes to survive.[8] Obed, et al.: The Estimation of (Covid-19) Cases in KRG 26 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 THE NELSON-AALEN METHOD AND THE CHF The Nelson-Aalen methodology allows for population comparisons based on their hazard curves. When studying CHFs, the Nelson-Aalen estimator should be used instead of the Kaplan-Meier estimator. The Kaplan-Meier estimator should be used when evaluating cumulative survival functions. The CHF is a function that calculates the probability of: H(T) = � �T TI (5) d r i i With d i being the number of observation dropping at time Ti and r i the number of observation at hazard (still in the study) at time Ti.[9] RESULTS AND DISCUSSION Descriptive Statistics Table 1 gives the results of Nelson Allen Estimator from the first of October (2020) till 28th of February (2021) applied to a data set of size 61716 cases, and shows that the most of the percentage of disease in this study are on October (2020) (46.1%) and deaths (28.9%). After that on November (2020) the percent estimate to life (33.6%) and (27.9%) was death. On the other hand, the highest estimated time until the death on December (2020) and equal (37.5%). The lowest estimated time for death which is equal to (2.1%) on February (2021). Application Nelson Aalen Survival Estimate Formula/Notations e t m t� � � � 0 1 � Nelson Aalen survival estimate (SNA(t) �t t m � � � 0 1 Estimate of the CHF. m day of observation t m date/time death observed d m number of deaths at tm n m number of people available to die at tm �m m m d n � hazard rate or probability that a life fails at tm. ANALYSIS AND INTERPRETATION OF DATA A value of 0.5 in the Nelson Aalen survival model suggests an equal chance of life and death. This survival model was chosen to analyze the data since it is a non-parametric technique with a non-informative censoring estimator assumption. From 0.5 to 1, the closer the value is to 1, the better the chance of survival, and vice versa. Nelson Aalen survival analysis of the data is shown in Table 2 Calculation of Nelson Aalen Survival Analysis. In the Kurdistan Region, there were 61171 confirmed cases and 2476 deaths, according to Table 3. The total reported case number is the total number of people who have been infected, died, or who have been recovered/discharged. Out of the 61171 cases reported, 28234 were found to be infected in October. The remaining 28% of cases were those that had a resolution this month. In Figure 1, observation/investigation of cases was approved out from October 1 to February 28; 2021. The descriptive statistics is concerned largely with summary calculations and graphical displays of results/data to derive reasonable decisions. The horizontal axis shows the time to event, and the vertical axis shows the number of Cases and Deaths In this plot, drops in the survival curve occur whenever the covid-19 takes effect in a patient. Thus, any point on the survival curve shows the probability that a patient on a given treatment will not have experienced relief by that time. Explanation: In Figure 2, the survival estimate and the range of time it applies have been clearly stated for better understanding. In the same vein, the value of m which ∑ µt=1 applies has also been shown. That is if there are no lives remaining to be censored among the observed lives when the investigation ended, and the estimation of the last value of the estimate will be zero which means sure death at that time. From the analysis carried out in Table 3, at the initial day of the investigation, the estimate shows the value of one (1). From the analysis, it is evident that as time passes by, the estimate figure activates to reduce, meaning that more Table 1 : Descriptive statistics for qualitative variables Months Status % October (2020) Disease 46.1 Deaths 28.9 November (2020) Disease 33.6 Deaths 27.9 December (2020) Disease 11.6 Deaths 37.5 January (2021) Disease 4.4 Deaths 3.4 February (2021) Disease 4.6 Deaths 2.1 October (2020) November(2020) December (2020) January (2021) Deaths 717 691 929 86 Cases 28234 20591 6768 2703 0 5000 10000 15000 20000 25000 30000 35000 Figure 1: Monthly Deaths and Cases Growth Rate Obed, et al.: The Estimation of (Covid-19) Cases in KRG 27 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 Day of observation Cases Death Hazard rates Cumulative hazard rates Estimation 1 684 21 0.0307 0.0307 0.97 2 640 21 0.0328 0.06351 0.938 3 494 16 0.0324 0.0959 0.909 4 630 19 0.0302 0.12606 0.882 5 839 31 0.0369 0.16301 0.85 6 793 30 0.0378 0.20084 0.818 7 886 29 0.0327 0.23357 0.792 8 793 31 0.0391 0.27266 0.761 9 838 12 0.0143 0.28698 0.751 10 769 19 0.0247 0.31169 0.732 11 589 24 0.0407 0.35244 0.703 12 720 26 0.0361 0.38855 0.678 13 900 23 0.0256 0.41411 0.661 14 912 20 0.0219 0.43604 0.647 15 974 28 0.0287 0.46478 0.628 16 932 18 0.0193 0.4841 0.616 17 722 23 0.0319 0.51595 0.597 18 797 32 0.0402 0.5561 0.573 19 1032 26 0.0252 0.5813 0.559 20 1010 24 0.0238 0.60506 0.546 21 1078 15 0.0139 0.61897 0.538 22 1308 17 0.013 0.63197 0.532 23 1053 20 0.019 0.65096 0.522 24 860 26 0.0302 0.6812 0.506 25 828 25 0.0302 0.71139 0.491 26 1283 23 0.0179 0.72932 0.482 27 1597 24 0.015 0.74434 0.475 28 1297 27 0.0208 0.76516 0.465 29 1285 18 0.014 0.77917 0.459 30 1002 25 0.025 0.80412 0.447 31 689 24 0.0348 0.83895 0.432 32 1054 28 0.0266 0.86552 0.421 33 1137 26 0.0229 0.88839 0.411 34 1235 23 0.0186 0.90701 0.404 35 962 28 0.0291 0.93612 0.392 36 1195 23 0.0192 0.95536 0.385 37 997 33 0.0331 0.98846 0.372 38 771 21 0.0272 1.0157 0.362 39 787 21 0.0267 1.04238 0.353 40 900 24 0.0267 1.06905 0.343 41 765 27 0.0353 1.10434 0.331 42 772 22 0.0285 1.13284 0.322 43 1170 29 0.0248 1.15763 0.314 44 686 28 0.0408 1.19844 0.302 Table 2: Results of Estimation of Covid-19 cases (Contd...) Obed, et al.: The Estimation of (Covid-19) Cases in KRG 28 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 Day of observation Cases Death Hazard rates Cumulative hazard rates Estimation 45 573 27 0.0471 1.24556 0.288 46 571 28 0.049 1.2946 0.274 47 659 23 0.0349 1.3295 0.265 48 645 25 0.0388 1.36826 0.255 49 576 22 0.0382 1.40646 0.245 50 530 21 0.0396 1.44608 0.235 51 567 23 0.0406 1.48664 0.226 52 361 20 0.0554 1.54204 0.214 53 495 17 0.0343 1.57639 0.207 54 488 21 0.043 1.61942 0.198 55 500 21 0.042 1.66142 0.19 56 435 25 0.0575 1.71889 0.179 57 379 17 0.0449 1.76375 0.171 58 403 26 0.0645 1.82826 0.161 59 259 14 0.0541 1.88232 0.152 60 304 14 0.0461 1.92837 0.145 61 415 14 0.0337 1.9621 0.141 62 389 23 0.0591 2.02123 0.132 63 331 17 0.0514 2.07259 0.126 64 248 9 0.0363 2.10888 0.121 65 236 8 0.0339 2.14278 0.117 66 298 9 0.0302 2.17298 0.114 67 377 10 0.0265 2.19951 0.111 68 372 12 0.0323 2.23176 0.107 69 356 9 0.0253 2.25704 0.105 70 287 13 0.0453 2.30234 0.1 71 248 11 0.0444 2.3467 0.096 72 197 6 0.0305 2.37715 0.093 73 183 9 0.0492 2.42633 0.088 74 167 9 0.0539 2.48022 0.084 75 201 9 0.0448 2.525 0.08 76 222 3 0.0135 2.53851 0.079 77 238 11 0.0462 2.58473 0.075 78 215 6 0.0279 2.61264 0.073 79 215 8 0.0372 2.64985 0.071 80 156 4 0.0256 2.67549 0.069 81 118 8 0.0678 2.74329 0.064 82 165 3 0.0182 2.76147 0.063 83 278 5 0.018 2.77945 0.062 84 163 8 0.0491 2.82853 0.059 85 150 3 0.02 2.84853 0.058 86 164 3 0.0183 2.86683 0.057 87 128 4 0.0313 2.89808 0.055 88 118 9 0.0763 2.97435 0.051 89 132 2 0.0152 2.9895 0.05 Table 2: (Continued) (Contd...) Obed, et al.: The Estimation of (Covid-19) Cases in KRG 29 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 Day of observation Cases Death Hazard rates Cumulative hazard rates Estimation 90 140 3 0.0214 3.01093 0.049 91 123 3 0.0244 3.03532 0.048 92 153 1 0.0065 3.04185 0.048 93 122 4 0.0328 3.07464 0.046 94 68 2 0.0294 3.10405 0.045 95 97 1 0.0103 3.11436 0.044 96 102 6 0.0588 3.17319 0.042 97 107 6 0.0561 3.22926 0.04 98 104 3 0.0288 3.25811 0.038 99 80 4 0.05 3.30811 0.037 100 140 5 0.0357 3.34382 0.035 101 139 3 0.0216 3.3654 0.035 102 68 6 0.0882 3.45364 0.032 103 126 2 0.0159 3.46951 0.031 104 88 0 0 3.46951 0.031 105 158 1 0.0063 3.47584 0.031 106 69 4 0.058 3.53381 0.029 107 78 3 0.0385 3.57227 0.028 108 43 2 0.0465 3.61878 0.027 109 110 3 0.0273 3.64606 0.026 110 97 2 0.0206 3.66668 0.026 111 100 4 0.04 3.70668 0.025 112 68 3 0.0441 3.75079 0.023 113 77 4 0.0519 3.80274 0.022 114 85 2 0.0235 3.82627 0.022 115 49 1 0.0204 3.84668 0.021 116 58 1 0.0172 3.86392 0.021 117 94 1 0.0106 3.87456 0.021 118 67 2 0.0299 3.90441 0.02 119 70 5 0.0714 3.97584 0.019 120 73 2 0.0274 4.00324 0.018 121 68 1 0.0147 4.01794 0.018 122 48 1 0.0208 4.03877 0.018 123 50 2 0.04 4.07877 0.017 124 107 2 0.0187 4.09747 0.017 125 54 2 0.037 4.1345 0.016 126 48 3 0.0625 4.197 0.015 127 56 3 0.0536 4.25058 0.014 128 51 1 0.0196 4.27018 0.014 129 24 4 0.1667 4.43685 0.012 130 38 2 0.0526 4.48948 0.011 131 70 2 0.0286 4.51805 0.011 132 64 3 0.0469 4.56493 0.01 133 69 0 0 4.56493 0.01 134 59 0 0 4.56493 0.01 Table 2: (Continued) (Contd...) Obed, et al.: The Estimation of (Covid-19) Cases in KRG 30 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 Day of observation Cases Death Hazard rates Cumulative hazard rates Estimation 135 98 3 0.0306 4.59554 0.01 136 53 1 0.0189 4.61441 0.01 137 106 3 0.0283 4.64271 0.01 138 72 4 0.0556 4.69827 0.009 139 138 2 0.0145 4.71276 0.009 140 115 1 0.0087 4.72145 0.009 141 117 2 0.0171 4.73855 0.009 142 131 0 0 4.73855 0.009 143 77 1 0.013 4.75153 0.009 144 150 2 0.0133 4.76487 0.009 145 157 1 0.0064 4.77124 0.008 146 129 2 0.0155 4.78674 0.008 147 176 3 0.017 4.80379 0.008 148 199 6 0.0302 4.83394 0.008 149 202 0 0 4.83394 0.008 150 78 0 0 4.83394 0.008 151 237 0 0 4.83394 0.007 Table 2: (Continued) Table 3: Reported cases Covid-19 monthly in Kurdistan Region Months Cases Deaths October (2020) 28234 717 November (2020) 20591 691 December (2020) 6768 929 January (2021) 2703 86 February (2021) 2875 53 deaths are being noted. This indicates that there are balances of surviving the disease at the first period of the infection, provided the infected persons take adequate medical aid. Invariably, as shown in Table 2, the more the virus stays in the body, the lower the probability of survival or the chance of recovery from the disease. It’s worth noting that the survival rate follows a specific pattern. As the value of m increases, the estimate values drop. This is due to the fact that the number of cases and deaths included in the study follow the same trend. Both the number of cases and the number of deaths recorded increased during the study period. In practice, if the results of this study are used to predict a future epidemic of this virus, an average of 2476 infected people out of a 61171infected people are predicted to die, because the survival chance for that duration is 0.97. This projection will be accurate or true if interventions targeted at reducing risk factors and ensuring adequate access to health care are not implemented properly and efficiently. When life’s quality is poor, it will be brief. Figure 2: The Nelson Aalen estimate of S(t) ( ) 1 0 0.97 0 1 0.93 1 2 0.90 2 3 0.882 3 4 0.850 4 5 0.818 5 6 0.792 6 7 0.761 7 8 . . . . . . 0.009 142 143 0.009 143 144 0.009 144 m m m m m m m m m m < ≤ < ≤ < ≤ < ≤ < ≤ < ≤ < ≤ < ≤ < ≤ < ≤ NA tS 145 0.008 145 146 0.008 146 147 0.008 147 148 0.008 148 149 0.008 149 150 0.007 150 151 m m m m m m m                        < ≤  < ≤  < ≤  < ≤  < ≤  < ≤   < ≤                           Obed, et al.: The Estimation of (Covid-19) Cases in KRG 31 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 24-31 CONCLUSION AND RECOMMENDATION Conclusion During conducting the Nelson Allen Estimator from Covied-19 cases and according to the results from the practical part the following conclusions have been drawn: 1. The number of Novel Coronavirus cases reported was unusually high. It was in the thousands. This demonstrates how easily the virus can be spread from one person to another or from an animal to a person 2. If prevention efforts were not made in a timely manner, the number could rise to a million. In this study, it was discovered that some people may be infected with this virus without realizing it because the symptoms of infection are not visible 3. These individuals infect others without even realizing it. According to the findings, those who have recently been exposed to the disease have a better chance of surviving if medical assistance is provided without severe or time- consuming requirements. Recommendation This study suggests that those who are unsure if they have contracted the disease due to a lack of symptoms should consult a doctor right once for tests. Every country’s government should establish a free testing center for such validation. Making it free and easily available will encourage everyone who is sick to do a confirmatory test. This will help to slow the disease’s spread. Since the virus first appeared in Wuhan, this study suggests using the WUHAN preventive approach to keep the infection from spreading or infecting humans. WUHAN Prevention Concept • Wash your hands/body regularly • Use nose cover/mask • Have your hotness/coldness of your body checked • Avoid unnecessary crowd • Never touch sensitive parts of your body with unclean hands/materials/equipment. REFERENCES 1. D. Abbas. Analysis of breast cancer data using Kaplan-Meier survival analysis. Journal of Kufa for Mathematics and Computer, vol. 1, no. 6, pp. 7-14, 2012. 2. T. D.Bluhmki. The wild bootstrap for multivariate Nelson-Aalen estimators. Lifetime Data Analysis, vol. 25, no. 1, pp. 97-127, 2019. 3. R. L. Cao. Presmoothed Kaplan-Meier and Nelson-Aalen estimators. Journal of Nonparametric Statistics, vol. 17, no. 1, pp. 31-56, 2005. 4. C. A. El-Nouty. The presmoothed Nelson-Aalen estimator in the competing risk model. 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