A new statistical approach for modeling the bladder cancer and leukemia patients data sets: Case studies in the medical sector

1.   Introduction

It is a well-recognized fact that statistical methodologies are effectively used in the analysis of healthcare data sets. For example, (i) the number of admitted patients to the hospital during a week, month, or year, (ii) the survival times of the patients, (iii) the death rate of the patients, (iv) the survival rate of the patients, (v) the recovered rate of the patients, (iv) the number of discharged patients during a week, month, or year. In the healthcare sector, several studies about the implementation of statistical methods have appeared. For detailed information, please see El-Morshedy et al. ^[1], de Villiers et al. ^[2], Eliwa et al. ^[3], Sivaparthipan et al. ^[4], Urlacher ^[5], Sandberg et al. ^[6], Altun et al. ^[7], Altaf-Ul-Amin et al. ^[8], Eliwa et al. ^[9], Ratnovsky et al. ^[10], Onchonga et al. ^[11], El-Morshedy et al. ^[12], El-Morshedy et al. ^[13], and Illescas et al. ^[14].

Due to the recognized importance of statistical methodologies and models in the healthcare sector, numerous statistical models have been introduced and implemented. For example, (i) Jones et al. ^[15] used a bivariate version of the power generalized Weibull (PG-Wei) distribution for survival analysis, (ii) Looha et al. ^[16] introduced a mixture of the Weibull model for analyzing the survival times of male and female patients, (iii) Kumar et al. ^[17] used a generalized version of the Log-Weibull (L-Wei) distribution for modeling the medical data sets. (iv) Liu et al. ^[18] proposed a new statistical distribution for analyzing the survival times of the COVID-19 infected patients. (v) Omer et al. ^[19] implemented a mixture of the generalized modified Weibull (GM-Wei) distribution for analyzing a medical data set related to leukemia patients. (vi) Mohammed et al. ^[20] proposed a new logarithmic version of the Weibull distribution for analyzing the bladder cancer data, and (vii) Klakattawi ^[21] implemented a new extended Weibull (NEx-Wei) distribution for conducting the survival analysis of the cancer patient's data set.

To further improve the fitting power of the statistical distributions, numerous approaches have appeared in the literature. For example, (i) a new family of heavy-tailed (NFHT) models introduced by Arif et al. ^[22]. (ii) a new generalized-$X$ family studied by Wang et al. ^[23], (iii) a new extended alpha power transformed (NEAPTrans) family introduced by Ahmad et al. ^[24], (iv) a class of claim distributions of Ahmad et al. ^[25], and (v) a new flexible exponentiated-$X$ (NFEx-$X$) family of Arif et al. ^[26]. For the latest review about the recent development of statistical methods; see Ahmad et al. ^[27]

In this paper, we also introduce a new approach for data modeling in the medical sector. The new approach is called, a new modified-$G$ (for 'NModi-$G$') family of distributions.

Definition: A random variable Y has the NModi-$G$ family, if its DF $G\left(y; \sigma, \boldsymbol{\phi}\right)$ is given by

where $\sigma \geq 1, \sigma \leq -1, y \in \mathbb{R}$, and $F\left(y; \boldsymbol{\phi}\right)$ is a baseline DF. The proofs in Propositions 1 and 2, we show that $G\left(y; \sigma, \boldsymbol{\phi}\right)$ is a compact DF.

Proposition 1. Using the DF $G\left(y; \sigma, \boldsymbol{\phi}\right)$ given in Eq (1.1), we have to prove that $\lim_{y \to -\infty} G\left(y; \sigma, \boldsymbol{\phi}\right) = 0$ and $\lim_{y \to \infty} G\left(y; \sigma, \boldsymbol{\phi}\right) = 1.$

Proof.

Also, we have

Proposition 2. The DF $G\left(y; \sigma, \boldsymbol{\phi}\right)$ in Eq (1.1), is differentiable and right continuous.

Proof.

From the mathematical results proved in Propositions 1 and 2, we can see that $G\left(y; \sigma, \boldsymbol{\phi}\right)$ is a valid DF.

For $y \in \mathbb{R}$, corresponding to $G\left(y; \sigma, \boldsymbol{\phi}\right)$, the PDF (probability density function) $g\left(y; \sigma, \boldsymbol{\phi}\right)$, is given by

where $\frac{d}{d y} F\left(y; \boldsymbol{\phi}\right) = f\left(y; \boldsymbol{\phi}\right).$

In link to $G\left(y; \sigma, \boldsymbol{\phi}\right)$ and $g\left(y; \sigma, \boldsymbol{\phi}\right)$, the SF (survival function) $S\left(y; \sigma, \boldsymbol{\phi}\right) = 1- G\left(y; \sigma, \boldsymbol{\phi}\right)$, HF (hazard function) $h\left(y; \sigma, \boldsymbol{\phi}\right) = \frac{g\left(y; \sigma, \boldsymbol{\phi}\right)}{1-G\left(y; \sigma, \boldsymbol{\phi}\right)},$ and cumulative HF (CHF) $H\left(y; \sigma, \boldsymbol{\phi}\right) = - \log \left[1-G\left(y; \sigma, \boldsymbol{\phi}\right)\right],$ are given by

and

respectively.

By implementing the DF $G\left(y; \sigma, \boldsymbol{\phi}\right)$ in Eq (1.1), we introduce an updated version of the Weibull distribution, called a NModi-Weibull distribution. The NModi-Weibull distribution can be considered a special member of the NModi-$G$ family. In Section 2, the expressions for the DF, PDF, SF, HF, and CHF of the NModi-Weibull distribution are obtained.

2.   A NModi-Weibull distribution

Consider the DF $F\left(y; \boldsymbol{\phi}\right)$ of the Weibull distribution is given by

with PDF $f\left(y; \boldsymbol{\phi}\right)$ given by

where $\boldsymbol{\phi} = \left(\theta, \delta\right).$

Using Eq (2.1) in Eq (1.1), we get the DF of the NModi-Weibull distribution. Let Y have the NModi-Weibull distribution with parameters $\delta > 0, \theta > 0, \sigma \geq 1,$ and $\sigma \leq -1$, then, its DF $G\left(y; \sigma, \boldsymbol{\phi}\right)$ is given by

For $y > 0$, the PDF of the NModi-Weibull distribution is given by

Different plots of $g\left(y; \sigma, \boldsymbol{\phi}\right)$ of the NModi-Weibull distribution are provided in Figure 1. The plots of $g\left(y; \sigma, \boldsymbol{\phi}\right)$ are obtained for (i) $\theta = 1.2, \delta = 1.8, \sigma = 1.2$ (red curve), (ii) $\theta = 3.8, \delta = 0.1, \sigma = 1.4$ (green curve), (iii) $\theta = 2.9, \delta = 0.3, \sigma = 3.2$ (black curve), and (iv) $\theta = 0.5, \delta = 1.3, \sigma = 3.4$ (blue curve).

From the plots of $g\left(y; \sigma, \boldsymbol{\phi}\right)$ in Figure 1, we can see that the shape of $g\left(y; \sigma, \boldsymbol{\phi}\right)$ of the NModi-Weibull can be (i) positively skewed (red curve), (ii) left skewed (green curve), (iii) symmetrical (black curve), or (iv) decreasing (blue curve).

Furthermore, the SF, HF, and CHF of the NModi-Weibull distribution are given by

and

respectively.

Different behaviors of $h\left(y; \sigma, \boldsymbol{\phi}\right)$ of the NModi-Weibull distribution are shown in Figure 2. The visual behaviors of $h\left(y; \sigma, \boldsymbol{\phi}\right)$ are obtained for (i) $\theta = 1.2, \delta = 1.8, \sigma = 1.2$ (blue curve), (ii) $\theta = 0.5, \delta = 1.3, \sigma = 3.2$ (red curve), and (iii) $\theta = 0.9, \delta = 0.6, \sigma = 1.7$ (green curve).

From the visual behaviors of $h\left(y; \sigma, \boldsymbol{\phi}\right)$, it can be observed that the shape of the HF of the NModi-Weibull distribution can be (i) decreasing (blue curve), (ii) increasing (red curve), or (iii) bathtub shaped (green curve).

3.   Mathematical properties

Here, we derive some mathematical properties of the NModi-$G$ distributions. These properties include (i) the quantile function (QF), (ii) the identifiability property (IP), (iii) the $r^{th}$ moment, and moment generating function (MGF).

3.1.   The quantile function

The QF of the NModi-$G$ distributions can be derived as

where $k$ is the solution of $\left(\sigma - 1\right) k + k^{2} -\sigma u.$

3.2.   The identifiability property

This subsection offers the derivation of the identifiability property of the NModi-$G$ distributions using the additional parameter $\sigma.$ Suppose that the parameter $\sigma_{1}$ has the DF $G\left(y; \sigma_{1}, \boldsymbol{\phi}\right)$ and the parameter $\sigma_{2}$ has the DF $G\left(y; \sigma_{2}, \boldsymbol{\phi}\right)$. The parameter $\sigma$ is said to be identifiable, if $\sigma_{1} = \sigma_{2}.$ Consider

Using Eq (1.1) in Eq (3.1), we have

3.3.   The $r^{th}$ moment

This subsection considers the derivation of the $r^{th}$ moment of the NModi-$G$ distributions. Suppose $Y$ has the NModi-$G$ distributions, then the $r^{th}$ moment of $Y$, say $\mu_{r}^{\prime},$ is derived as

Using Eq (1.2) in Eq (3.2), we get

where

and

The MGF $M_{t} (y)$ of the NModi-$G$ distributions is given by

4.   Estimation and simulation

In this section, we discuss and implement the method of ML (maximum likelihood) to obtain the ML estimators (MLEs) $\left(\hat{\sigma}_{MLE}, \hat{\boldsymbol{\phi}}_{MLE}\right)$ of the parameters $\left(\sigma, \boldsymbol{\phi}\right)$ of the NModi-$G$ distributions. Furthermore, the performances of $\hat{\sigma}_{MLE}$ and $\hat{\boldsymbol{\phi}}_{MLE}$ are evaluated using a simulation study.

4.1.   Estimation

The method of ML estimation is one of the prominent methods to obtain the MLEs. The estimators obtained via implementing the ML approach enjoy numerous properties such as (i) efficiency, (ii) consistency, and (iii) asymptotic normality. Therefore, we use the ML approach to obtain the estimators of the NModi-$G$ distributions with PDF $g\left(y; \sigma, \boldsymbol{\phi}\right)$.

Let $Y_{1}, Y_{2}, ..., Y_{s}$ be a collection of random samples, say $s$, taken from the NModi-$G$ distributions. Corresponding to $g\left(y; \sigma, \boldsymbol{\phi}\right)$, the LikF (likelihood function), say $\lambda \left(\sigma, \boldsymbol{\phi}| y_{1}, y_{2}, ..., y_{s} \right)$, is given by

Using Eq (1.2) in Eq (4.1), we get

Corresponding to $\lambda \left(\sigma, \boldsymbol{\phi}| y_{1}, y_{2}, ..., y_{s} \right)$ presented in Eq (4.2), the log LikF, say $\lambda \left(y_{1}, y_{2}, ..., y_{s} | \sigma, \boldsymbol{\phi}\right)$, is given by

In link to $\lambda \left(\sigma, \boldsymbol{\phi}| y_{1}, y_{2}, ..., y_{s} \right),$ the partial derivates are given by

and

By solving $\frac{\partial}{\partial \sigma} \lambda \left(\sigma, \boldsymbol{\phi}| y_{1}, y_{2}, ..., y_{s} \right) = 0$ and $\frac{\partial}{\partial \boldsymbol{\phi}} \lambda \left(\sigma, \boldsymbol{\phi}| y_{1}, y_{2}, ..., y_{s} \right) = 0$, we obtain the MLEs $\left(\hat{\sigma}_{MLE}, \hat{\boldsymbol{\phi}}_{MLE}\right)$ of the parameters $\left(\sigma, \boldsymbol{\phi}\right)$.

4.2.   Simulation study

A Monte Carlo simulation study (MCSS) is implemented to review/evaluate the performance of the estimation method. Here, we provide MCSS to evaluate the MLEs $\hat{\sigma}_{MLE}$ and $\hat{\boldsymbol{\phi}}_{MLE}$. To carry out the MCSS, random samples (RSs) from the PDF $g\left(y_{i}; \sigma, \boldsymbol{\phi} \right)$ are generated using the formula

where $k$ is the solution of $\left(\sigma - 1\right) k + k^{2} -\sigma u.$

The MCSS is conducted for (a) $\theta = 1.6, \delta = 0.5, \sigma = 1.3$, and (b) $\theta = 1.5, \delta = 0.8, \sigma = 1.2$. Using the function in Eq (4.3) and the above specific values of $\theta, \delta$, and $\sigma$, RSs of sizes $s = 25, 50, 75, ..., 500$ are generated for the MCSS.

After obtaining the RSs, two statistical measures, such as (a) MSE (mean square error) and (b) Bias are selected to evaluate $\hat{\sigma}_{MLE}$ and $\hat{\boldsymbol{\phi}}_{MLE}$. These two quantities are, respectively, computed as

and

where $\boldsymbol{\Xi} = \left(\sigma, \boldsymbol{\phi} \right)$.

Corresponding to (a) $\theta = 1.6, \delta = 0.5, \sigma = 1.3$, and (b) $\theta = 1.5, \delta = 0.8, \sigma = 1.2$, the MCSS results for the NModo-Weibull model are presented in Tables 1 and 2, respectively. Furthermore, corresponding to Tables 1 and 2, visual illustrations of the MCSS results are presented in Figures 3 and 4, respectively.

5.   Modeling the medical data sets

This section illustrates the applicability of the NModi-Weibull distribution in the medical sector. We apply the NModi-Weibull to two medical data sets. The very first data set refers to the bladder cancer patients' data; see Lee et al. ^[28]. While the second data set refers to the lifetimes of leukemia patients; see El-Gohary et al. ^[29]. The key measures of the first data set are given by minimum = 0.080, $Q_{1}$ = 3.348, $Q_{2}$ = 6.395, $Q_{3}$ = 11.838, mean/average = 9.366, maximum = 79.050, variance = 110.425, skewness = 3.286569, kurtosis = 18.48308, and range = 8.97. For the second data set, the key measures are given by Minimum = 1.150, $Q_{1}$ = 8.025, $Q_{2}$ = 12.220, $Q_{3}$ = 15.562, mean/average = 11.370, maximum = 18.520, variance = 23.19414, skewness = -0.4879275, kurtosis = 2.271123, and range = 17.37. It is important to note that $Q_{1}$ refers to the $1^{st}$ quartile, $Q_{2}$ refers to the $2^{nd}$ quartile or median, and $Q_{3}$ refers to the $3^{rd}$ quartile.

Using the bladder cancer and leukemia patients' data sets, the comparison of the NModi-Weibull distribution is made with the (i) Weibull distribution, (ii) novel exponent power-Weibull (NEPow-Weibull) distribution, and (iii) an exponential T-$X$ exponentiated exponential (ET-$X$EE) distribution. The DFs of these models are given by

and

respectively.

The next step is to choose certain statistical measures to show the best model for the bladder cancer and leukemia patients' data sets. To figure out the best model for medical data sets, three statistical measures (i) Cramer-von Mises (CM), (ii) Anderson-Darling (AD), and (iii) Kolmogorov–Smirnov (SK) test with the p-value are considered.

After carrying out the numerical analysis using the bladder cancer and leukemia patients' data sets, the results of the competing models are presented in Tables 3–6. Tables 3 and 4 offer the MLEs and statistical measures of the fitted models using the bladder cancer data, respectively. Whereas, Tables 5 and 6 offer the MLEs and statistical measures of the fitted models using the leukemia data, respectively. From the numerical evaluations presented in Tables 4 and 6, we can see the NModi-Weibull distribution is the best choice to apply for modeling the medical data sets.

In addition to the numerical evaluation presented in Tables 4 and 6, a visual evaluation of the NModi-Weibull distribution is also presented; see Figures 5 and 6. From the plots in Figures 5 and 6, we can see that the NModi-Weibull distribution closely follows the fitted PDF, DF, PP (probability-probability), SF, and QQ (quantile-quantile) functions.

6.   Robust estimation approach

When a dataset is impure with a single or few outliers, it causes a serious problem in estimating the parameters. To overcome this problem, we use the robust estimation (RoE) method. The RoE is an important technique for analyzing the data sets that contain outliers.

The robust estimators (RoEs) have a large family such as the least absolute deviation (LAD), the least quantile of squares, the least median of squares, and M-estimation methods. For a brief discussion about the RoEs, we refer to Fang et al. ^[30], Kantar et al. ^[31], and Almetwally et al. ^[32].

In this section, we implement the LAD method to obtain the RoEs of the proposed model. Consider a sample of size $s$ from the NModi-Weibull distribution, then, the RoEs of the NModi-Weibull distribution can be derived as follows

where $Q_{i}(\theta, \delta, \sigma)$ is the quantile function of the NModi-Weibull distribution and $\xi _{i}$ is the error term with mean equals zero and known variance.

For the RoE method, it is necessary to scale the invariant error ($\varsigma _{i}$) as follows $\varsigma _{i} = \frac{\xi _{i}}{\Lambda },$ where $\xi _{i}$ denotes the $i^{th}$ residual, and $\Lambda = \frac{\text{Median}\left\{ \left\vert \xi _{i}-\text{Median(}\xi_{i}\text{)}\right\vert \right\} }{0.6745}.$ The constant 0.6745 is chosen so that the RoEs become asymptotically unbiased for a normal error case. The LAD method is obtained by minimizing $\varsigma _{i},$ where LAD$(\theta, \delta, \sigma) = \min \left\vert \varsigma _{i}\right\vert$.

After differentiating Eq (6.1) with respect to the parameters $\theta, \delta$ and $\sigma$, we get three nonlinear equations that cannot be solved analytically. Hence, iterative procedures like the Newton-Raphson algorithms can be utilized to solve for the solution of the $\widehat{\theta }_{\text{LAD}}, \; \widehat{\delta }_{\text{LAD}},$ and $\widehat{\sigma }_{\text{LAD}}$ numerically.

6.1.   Robust estimation using the bladder cancer patients data

As we discussed earlier that the RoE methods can be implemented when there are outliers in the data. In this paper, we have analyzed two data sets; see Section A. The first data set (i.e., the bladder cancer patients' data) has some outliers. Whereas, the second data set (i.e., the lifetimes of leukemia patients) has no outliers. Therefore, we provide the RoE of the NModi-Weibull distribution using the bladder cancer patients data set, only.

The initial density shape is listed using the non-parametric kernel density estimation approach in Figure 7. From Figure 7, we can see that the shape of the density is asymmetric. The normality condition is checked via the QQ plot; see Figure 7. The outliers can also be spotted using the box plot; see Figure 7. Henceforth, we can say that there are outliers in the data.

Based on the LAD approach, the RoEs of the NModi-Weibull parameters are $\hat{\theta} = 2.3178, \hat{\delta} = 0.00127,$ and $\hat{\sigma} = 4.50187$ with P-value $= 0.71279.$ It is noted that the LAD approach is the best as compared to the MLE technique due to its high p-value.

7.   Concluding remarks

Statistical distributions have proven great applicability in healthcare-related sectors. For modeling and describing the medical data sets, numerous statistical distributions have been introduced and implemented. Keeping in view the recognized importance of the statistical methods in the healthcare sector, a new modified-$G$ family was introduced in this paper. Certain mathematical properties of the NM-$G$ distributions were derived. A special sub-case of the NM-$G$ distributions by taking the Weibull distribution as a baseline model was discussed. The special model of the NM-$G$ distributions was named a NModi-Weibull distribution. A simulation study based on the NModi-Weibull distribution has also been carried out. At last, the importance of the NModi-Weibull distribution was illustrated by modeling two data sets (the bladder cancer patient's data and the lifetimes of leukemia patient's data) taken from the healthcare sector. Using these data sets, it is observed that the NModi-Weibull distribution was the best competitor for dealing with the medical data sets.

Conflict of interest

The authors declare there is no conflict of interest.