A New Alpha Power Weibull Model for Analyzing Time-to-Event Data: A Case Study from Football

Shengjie, Gao; Craig, Alisa; Mekiso, Getachew Tekle

doi:https://doi.org/10.1155/2022/7257264

Mathematical Problems in Engineering

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Robust Statistical Modeling and Machine Learning with Applications in Data Science

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Research Article | Open Access

Volume 2022 | Article ID 7257264 | https://doi.org/10.1155/2022/7257264

A New Alpha Power Weibull Model for Analyzing Time-to-Event Data: A Case Study from Football

Gao Shengjie,¹Alisa Craig,²and Getachew Tekle Mekiso³

Academic Editor: Saima K Khosa

Received28 Nov 2021

Revised21 Dec 2021

Accepted21 Jan 2022

Published09 Feb 2022

Abstract

Statistical methodologies have wider applications in exercise science, sports medicine, sports management, sports marketing, sports science, and other related sciences. These methods can be used to predict the winning probability of a team or individual in a match, the number of minutes that an individual player will spend on the ground, the number of goals to be scored by an individual player, the number of red/yellow cards that will be issued to an individual player or a team, etc. Keeping in view the importance and applicability of the statistical methodologies in sport sciences, healthcare, and other related sectors, this paper introduces a novel family of statistical models called new alpha power family of distributions. It is shown that numerous properties of the suggested method are similar to those of the new Weibull-X and exponential type distributions. Based on the novel method, a special model, namely, a new alpha power Weibull distribution, is studied. The new model is very flexible because the shape of its probability density function can either be right-skewed, decreasing, left-skewed, or increasing. Furthermore, the new distribution is also able to model real phenomena with bathtub-shaped failure rates. Finally, the usefulness/applicability of the proposed distribution is shown by analyzing the time-to-event datasets selected from different football matches during 1964–2018.

1. Introduction

In the practice of healthcare [1], reliability [2], education [3], hydrology [4], management [5], metrology [6], and sports sciences [7], statistical modeling and predicting real-life events are very crucial. Numerous statistical models (traditional/classical and modified distributions) have been suggested to model data in these sectors.

In the recent era, the development of generating new families of statistical models has received considerable attention [8]. Recently, Alzaatreh et al. [9] introduced an interesting method, namely, the T-X approach, to update the distributional flexibility of the existing models. The DF (distribution function) of the T-X method is given bywhere the function fulfills some certain conditions (for details, see [9]). In equation (1), is a baseline DF with parameter vector and is the probability density function (PDF) of the parent model with parameter vector . Corresponding to , the PDF is given by

Most of the exponential type distributions belong to the class defined in equation (1). Some implemented functions of are provided in Table 1.

Recently, Ahmad et al. [19] implemented the T-X approach and introduced an interesting method, namely, a new Weibull-X (NWei-X) family, to update the distributional flexibility of the classical or modified distributions. They introduced the NWei-X family by using in equation (1) with , where is the PDF of the one-parameter Weibull model with parameter vector . The DF of the NWei-X family is given bywhere is a parameter vector. The expression can also be written aswhere is the CHF (cumulative hazard function) associated with DF .

In terms of CHF , the DF of the exponential type distributions is defined aswhere must fulfil certain conditions as given below:(i) is a nonnegative, differentiable, and increasing function of y.(ii), and .

The traditional exponential, Rayleigh, Weibull, and other extended lifetime distributions belong to the class defined in equation (5) (see [22]). In link to equation (5), the PDF is given bywhere .

Here, we introduce an additional parameter in equation (5), by replacing the exponent term with to propose a very flexible family by the DF

For , the DF in equation (7) becomes similar to equation (5). The function fulfills the conditions given in (i) and (ii). As it is quite clear thatfor clarification, let

Hence, using the results obtained in equations (9) and (10), we observed that the function defined in equation (7) is a proper DF. The expression in equation (7) is very interesting and can be useful to generate new statistical models belonging to the T-X family of distributions.

Now, we introduce the proposed family called new alpha power (NAPow) family by using in equation (7). The DF of the NAPow family is given by

The DF in equation (11) can also be written as

Corresponding to , the PDF , SF (survival function) , and HF (hazard function) are given byrespectively.

Due to an extra parameter in the place of the exponent term in equation (7), the NAPow family delivers greater flexibility.

2. A New Alpha Power Weibull Model

Here, we discuss a special member of the NAPow family called new alpha power Weibull (NAPow-Weibull) model. The NAPow-Weibull model is introduced by using the DF of the Weibull (two-parameter) distribution as a parent model. A random variable Y has the Weibull model, if its DF is given bywhere . Corresponding to , the PDF is given by

Corresponding to and , the HF and CHF are given byrespectively.

Using and in equation (12), we get the DF of the NAPow-Weibull model, given bywhere .

Corresponding to , the SF is given by

A visual display of and for , and is presented in Figure 1.

In link to , the PDF is given by

In Figure 2, a visual illustration of is presented for (i) (red line), (ii) (green line), (iii) (blue line), and (iv) (gold line).

Furthermore, in link to , the HF is given by

Some graphical sketches of are provided in Figure 3. The plots of in Figure 3 are obtained for (i) (blue line), (ii) (green line), and (iii) (red line).

3. Estimation

Here, we derive the estimators of the parameter of the NAPow-Weibull model with PDF . Consider a RS (random sample) say from PDF . Then, corresponding to , the log likelihood function is

The partial derivatives of are given byrespectively.

The estimators of , and can be obtained by solving , , and , respectively.

4. Data Analyses

Here, we show the applicability of the NAPow-Weibull model by analyzing two datasets. These datasets are taken from the sports sciences.

The first dataset consists of seventy-eight (78) observations and represents the waiting time duration till the first goal was scored in different football matches during 1964–2018. The first sports dataset is given by 14.00, 14.00, 13.00, 13.00, 12.00, 12.00, 12.00, 12.00, 12.00, 12.00, 12.00, 11.00, 11.00, 11.00, 10.80, 10.69, 10.12, 10.00, 10.00, 10.00, 10.00, 9.90, 9.60, 9.55, 9.00, 9.00, 9.00, 9.00, 8.70, 8.30, 8.10, 8.10, 8.00, 8.00, 8.00, 8.00, 8.00, 8.00, 8.00, 7.80, 7.70, 7.69, 7.66, 7.42, 7.30, 7.27, 7.22, 7.00, 7.00, 7.00, 7.00, 7.00, 7.00, 6.00, 6.00, 6.00, 6.00, 6.00, 6.00, 6.00, 5.00, 5.00, 4.00, 4.00, 4.00, 3.90, 3.69, 3.60, 3.57, 3.55, 3.17, 3.00, 3.00, 2.80, 2.80, 2.56, 2.20, 2.10.

The second dataset consists of seventy-six (78) observations and is also taken from the sports sciences. The second dataset also represents the waiting time duration till the first goal was scored in different football matches. The second dataset is given by 2.3520, 2.4640, 2.8672, 3.1360, 3.1360, 3.3600, 3.3600, 3.5504, 3.9760, 3.9984, 4.0320, 4.1328, 4.3680, 4.4800, 4.4800, 4.4800, 5.6000, 5.6000, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 7.8400, 7.8400, 7.8400, 7.8400, 7.8400, 7.8400, 8.0864, 8.1424, 8.1760, 8.3104, 8.5792, 8.6128, 8.6240, 8.7360, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 9.0720, 9.0720, 9.2960, 9.7440, 10.0800, 10.0800, 10.0800, 10.0800, 10.6960, 10.7520, 11.0880, 11.2000, 11.2000, 11.2000, 11.2000, 11.3344, 11.9728, 12.0960, 12.3200, 12.3200, 12.3200, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 14.5600, 14.5600.

The complete information about these datasets can retrieved at https://en.wikipedia.org/wiki/Fastest_goals_in_association_football.

We fit the NAPow-Weibull and three other models (competitive models) to these datasets. The SFs of the competitive models (i) Weibull (Wei), (ii) exponentiated Weibull (Exp-Wei), and (iii) Kumaraswamy Weibull (Ku-Wei) are given by respectively.

To figure out the flexibility/applicability of the fitted models (NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei) using these two datasets, certain IC (information criteria) such as (i) AIC [23], (ii) BIC [24], (iii) CAIC [25], and (iv) HQIC [26] are considered. In addition to the IC, three g-o-f measures (goodness-of-fit measures) such as (i) Cramer–von Mises (CM) test statistic, (ii) Anderson–Darling (AD) test statistic, and (iii) Kolmogorov–Smirnov (KS) test statistic are also considered. Besides these seven statistical quantities, the value of the fitted models is also derived.

4.1. Data Analysis Using the First Dataset

In this section, we apply the NAPow-Weibull in comparison with the Wei, Exp-Wei, and Ku-Wei models to the first dataset taken from the sports sciences. The summary measures of the first sports dataset are given by minimum = 2.100, 1st quartile = 6.000, median = 7.900, mean = 7.729, 3rd quartile = 10.000, maximum = 14.000, variance = 9.428, standard deviation = 3.070, and range = 11.900. Corresponding to the first sports dataset, the plots of hist (histogram), PB (boxplot), and TTT (total time test) are provided in Figure 4.

In link to the first sports dataset, (i) the values of , and of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models are obtained in Table 2, (ii) the values of the IC measures of the fitted models are reported in Table 3, and (iii) the values of the g-o-f measures are provided in Table 4.

Furthermore, for the first sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP (probability-probability), and (iv) QQ (quantile-quantile) plots of the NAPow-Weibull model are sketched in Figure 5. Corresponding to the first sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained using the functions and , respectively. The mathematical expressions of and are given byrespectively.

4.2. Data Analysis Using the Second Dataset

In this section, we again implement the NAPow-Weibull distribution in comparison with the Wei, Exp-Wei, and Ku-Wei models to the second sports dataset. The summary measures of the second sports dataset are given by minimum = 2.352, 1st quartile = 6.720, median = 8.680, mean = 8.472, 3rd quartile = 11.116, maximum = 14.560, variance = 10.792, standard deviation = 3.285, and range = 12.208. Corresponding to the first sports data, the plots of hist, PB, and TTT are presented in Figure 6.

Corresponding to the second sports dataset, (i) the values of , and of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei distributions are provided in Table 5, (ii) the values of the IC measures of the fitted models are presented in Table 6, and (iii) the values of the g-o-f measures are reported in Table 7.

In link to the second sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP, and (iv) QQ plots of the NAPow-Weibull model are presented in Figure 7. Corresponding to the second sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained via the expressions and , respectively. For the second sports dataset, the mathematical expressions of and are, respectively, given by

4.3. Discussion

In this section, the NAPow-Weibull model was applied to two datasets taken from the sports sciences. The first dataset consists of seventy-eight observations and was taken from the sports sciences. The NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to these data. Based on the IC measures, it is observed that the NAPow-Weibull was the best model. Furthermore, it is also observed that the Wei model was the second-best competitor in terms of the IC measures, whereas the Exp-Weibull model was observed to be the second-best model based on the value.

The second dataset consists of seventy-six observations and was also taken from the sports sciences. Again, the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to the second sports dataset. Based on the IC and g-o-f measures, it is again observed that the NAPow-Weibull was the best competitor than the Wei, Exp-Wei, and Ku-Wei distributions.

5. Final Remarks

In this article, a new family, that can be used as an alternative to a NWei-X family, is introduced. Numerous distributional properties of the new family are obtained. There are several advantages of using the proposed family. First, a very simple approach was proposed to modify the existing distributions by adding an extra parameter. Second, the proposed method can be useful to introduce new distributions belonging to T-X family. Third, the existing distributions having a closed-form of DF were extended. Furthermore, a special model (NAPow-Weibull) of the new alpha power family was studied. Some graphical behaviors of the PDF and HF of the NAPow-Weibull model were obtained. Finally, two applications were discussed to illustrate the NAPow-Weibull distribution. Based on the findings using these two datasets, it is shown that the NAPow-Weibull model was an appropriate model for dealing with the data in sports and other related sciences.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Gao Shengjie and Alisa Craig were responsible for conceptualization and methodology. Gao Shengjie supervised the study. Gao Shengjie, Alisa Craig, and Getachew Tekle Mekiso were responsible for original draft preparation and review and editing. Alisa Craig and Getachew Tekle Mekiso were responsible for formal analysis and software.

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Copyright © 2022 Gao Shengjie et al. This is an open access article distributed under 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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