We used model identification criteria such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and log-likelihood to assess and compare the performance of each model. AIC and BIC provide a trade-off between model complexity and goodness-of-fit. A lower AIC/BIC value indicates a better-fitting model with fewer parameters [16] .

AIC:

$\text{AIC}=-2\times \log \left(L\right)+2k$ (5)

BIC:

$\text{BIC}=-2\times \log \left(L\right)+k\times \log \left(n\right)$ (6)

where:

Parameter estimates for the shape and scale parameters of the Weibull distribution, as well as for covariates in the AFT and Cox models, were obtained using MLE. These estimates are provided along with their corresponding standard errors (SE), Z-values, and P-values in Table 2 .

For each estimated parameter, Z-statistics were used to test the null hypothesis that the coefficient is equal to zero (no effect).

P-values were calculated to assess the statistical significance of the model parameters. A P-value less than the chosen significance level ( $\alpha =0.05$) indicates that the corresponding parameter is statistically significant.

To assess the goodness-of-fit for the Weibull distribution, we used the Anderson-Darling (AD) test. The AD test evaluates how well the data follow the specified Weibull distribution. A Lower AD values depicts a better fit of the Weibull distribution to the data.

The parameter estimates, standard errors, Z-values, P-values, and goodness-of-fit statistics (AIC, BIC, and AD value) are reported in Table 2 for each model. Additionally, model fit statistics (AIC, BIC) were used to compare the models, with the lowest AIC and BIC indicating the best-fitting model for each gender and age group.

Akaike information criteria (AIC), Bayesian information criteria) and the log-likelihood criteria (LL) were used to compare the performance of the models in modeling the state holding time [17] - [19] .

In this study, the natural logarithm (logarithm base e) was used for all logarithm functions. The significance level, $\alpha =0.05$, was chosen as it is a commonly accepted threshold in statistical hypothesis testing, ensuring a 5% probability of rejecting a true null hypothesis.”

The hazard function, hazard rate, and failure rate refer to the probability of failure in an infinitesimally small time. It is the probability of surviving for an additional time $\delta t$ having survived for $t$ hours. It is a measure of risk, the greater the hazard the greater the chances of failure.

Let $r\left(t\right)$ be the hazard function for a given distribution, then:

$\begin{array}{c}r\left(t\right)=\frac{\text{P}\left(X\in \left(t,t+\delta t\right)\right)}{\left(X>t\right)}=\frac{\text{P}\left(X\in \left(t,t+\delta t\right),X>t\right)}{\text{P}\left(X>t\right)}\\ =\frac{\text{P}\left(X\in \left(t,t+\delta t\right)\right)}{\text{P}(X>t)}=\frac{f\left(t\right)\text{d}t}{1-F\left(t\right)}=\frac{f\left(t\right)\text{d}t}{S\left(t\right)}\end{array}$ (7)

where $S\left(t\right)=1-F\left(t\right)$; $S\left(t\right)$ is the survival function.

The Weibull distribution has two parameters, namely $\lambda$, the shape parameter, and $\theta$, the scale parameter. The probability density function of the Weibull distribution is given by:

$f\left(x\right)=\{\begin{array}{ll}\frac{\lambda x^{\lambda -1}}{{\theta }^{\lambda }}\text{exp}\left[-{\left(\frac{x}{\theta }\right)}^{\lambda }\right],\hfill & \text{for}x>0,\theta >0,\lambda >0\hfill \\ 0,\hfill & \text{elsewhere}\hfill \end{array}$ (8)

The survival function is given by:

$S\left(t\right)=\exp \left[-{\left(\frac{x}{\theta }\right)}^{\lambda }\right]$ (9)

The hazard function is given by:

$h\left(t\right)=\frac{f\left(x\right)}{S\left(t\right)}=\frac{\frac{\lambda x^{\lambda -1}}{{\theta }^{\lambda }}\exp \left[-{\left(\frac{x}{\theta }\right)}^{\lambda }\right]}{\exp \left[-{\left(\frac{x}{\theta }\right)}^{\lambda }\right]}=\frac{\lambda x^{\lambda -1}}{{\theta }^{\lambda }}=\left(\frac{\lambda }{{\theta }^{\lambda }}\right)x^{\lambda -1}$ (10)

The hazard function depends on x. For $\lambda >1$ hazard rate will increase and for $\lambda <1$ hazard rate decreases, for $\lambda =1$ the hazard rate will be constant in a special case of exponential distribution. The hazard function,

$h\left(t\right)=\left(\frac{1}{\theta }\right)$, which is a constant similar to the hazard function of the exponential

distribution. Therefore, exponential distribution is a special case of Weibull distribution and can be used where exponential distribution is used as long as $\lambda =1$. Weibull distribution is therefore able to care of the varying hazard rates at varied stages of the disease progression accommodating the effect of the covariates, which indirectly or directly affects the failure rates. However, it is evident the hazard function of the Weibull distribution is limited to constant, increasing, and decreasing hazard rates, it cannot accommodate any other form of hazard rates.

The Weibull maximum likelihood estimator is given by:

$f_{\lambda ,\beta }\left(x\right)=\{\begin{array}{ll}\frac{\beta }{\lambda }{\left(\frac{x}{\lambda }\right)}^{\beta -1}\text{exp}\left[-{\left(\frac{x}{\lambda }\right)}^{\beta }\right],\hfill & \text{for}x\ge 0\hfill \\ 0,\hfill & \text{elsewhere}\hfill \end{array}$ (11)

where $\beta >0$ and $\lambda >0$.

The likelihood function is given by:

$L_{\hat{x}}\left(\lambda ,\beta \right)=\underset{i=1}{\overset{N}{{\displaystyle \prod }}}{f}_{\lambda ,\beta }\left(x_i\right)=\underset{i=1}{\overset{N}{{\displaystyle \prod }}}\frac{\beta }{\lambda }{\left(\frac{x_i}{\lambda }\right)}^{\beta -1}\exp \left[-{\left(\frac{x_i}{\lambda }\right)}^{\beta }\right]$ (12)

Expanding the product gives:

$L_{\hat{x}}\left(\lambda ,\beta \right)=\frac{{\beta }^N}{{\lambda }^{N\beta }}\exp \left(-\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left(\frac{x_i}{\lambda }\right)}^{\beta }\right)\underset{i=1}{\overset{N}{{\displaystyle \prod }}}{x}_i^{\beta -1}$ (13)

Taking the first derivative of the likelihood function with respect to the parameters and equating it to zero gives the maximum likelihood estimators.

$\frac{\text{d}l}{\text{d}\lambda }=-N\beta \frac{1}{\lambda }+\beta \underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }\frac{1}{{\lambda }^{\beta +1}}=0$ (14)

$\frac{\text{d}l}{\text{d}\beta }=\frac{N}{\beta }-N\ln \left(\frac{x_i}{\lambda }\right){\text{e}}^{\beta \ln \left(\frac{x_i}{\lambda }\right)}+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\ln x_i=0$ (15)

From (14), the derivative of the likelihood function with respect to $\lambda$ is:

$\frac{\text{d}l}{\text{d}\lambda }=-N\beta \frac{1}{\lambda }+\beta \underset{i=1}{\overset{n}{{\displaystyle \sum }}}\frac{x_i^{\beta }}{{\lambda }^{\beta +1}}=-\beta \frac{N}{\lambda }+\frac{\beta }{\lambda }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\frac{x_i^{\beta }}{{\lambda }^{\beta }}=0$ (16)

Dividing through by $\left(\frac{\lambda }{N\beta }\right)$, we get:

$\frac{\text{d}l}{\text{d}\lambda }=-1+\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\frac{x_i^{\beta }}{{\lambda }^{\beta }}=0$ (17)

This implies:

$\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\frac{x_i^{\beta }}{{\lambda }^{\beta }}=1$ (18)

Therefore, solving for ${\lambda }^{\beta }$, we get:

$\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }={\lambda }^{\beta }$ (19)

Thus,

${\left({\lambda }^{\beta }\right)}^{\frac{1}{\beta }}={\left(\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }\right)}^{\frac{1}{\beta }}$ (20)

The maximum likelihood estimate of $\lambda$ is given by:

$\hat{\lambda }={\left(\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }\right)}^{\frac{1}{\beta }}$ (21)

Similarly,

${\lambda }^{\text{*}}={\left(\frac{1}{N}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{{\beta }^{\text{*}}}\right)}^{\frac{1}{{\beta }^{\text{*}}}}$ (22)

Substituting ${\lambda }^{\text{*}}$ in Equation (15), we get:

${\beta }^{*}={\left[\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i^{{\beta }^{*}}\ln x_i}{{{\displaystyle \sum }}_{i=1}^n{x}_i^{{\beta }^{*}}}-\overline{\ln x}\right]}^{-1}$ (23)

Using the Newton-Raphson algorithm, the equation is numerically solvable.

The log-likelihood of the Weibull distribution is given by:

$f\left(x;\alpha ,\beta \right)=\frac{\beta }{\alpha }{\left(\frac{x_i}{\alpha }\right)}^{\beta -1}\exp \left[-{\left(\frac{x_i}{\alpha }\right)}^{\beta }\right]$ (24)

The likelihood function is:

$L\left(x;\alpha ,\beta \right)=\underset{i=1}{\overset{N}{{\displaystyle \prod }}}\frac{\beta }{\alpha }{\left(\frac{x_i}{\alpha }\right)}^{\beta -1}\exp \left[-{\left(\frac{x_i}{\alpha }\right)}^{\beta }\right]$ (25)

$n\log \left(\beta \right)-n\beta \log \left(\alpha \right)+\left(\beta -1\right)\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\log \left(x_i\right)-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }$ (26)

$\frac{\partial L\left(\alpha ,\beta \right)}{\partial \alpha }=\frac{n\beta }{\alpha }-\frac{\beta }{{\alpha }^2}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }$ (27)

$\frac{\partial L\left(\alpha ,\beta \right)}{\partial \beta }=\frac{n}{\beta }-n\log \left(\alpha \right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\log \left(x_i\right)-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }\log \left(\frac{x_i}{\alpha }\right)=0$ (28)

$\frac{n}{\beta }-n\ln \left(\alpha \right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\log \left(x_i\right)-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }\log \left(\frac{x_i}{\alpha }\right)=0$ (29)

Taking the second derivative:

$\frac{{\partial }^{2}L\left(\alpha ,\beta \right)}{\partial {\alpha }^2}=\frac{n\beta }{{\alpha }^2}+\frac{\beta \left(\beta +1\right)}{{\alpha }^2}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }$ (30)

$\frac{{\partial }^{2}L\left(\alpha ,\beta \right)}{\partial \alpha \partial \beta }=\frac{n}{\alpha }-\frac{1}{\alpha }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }{\log }^2\left(\frac{x_i}{\alpha }\right)$ (31)

$\frac{{\partial }^{2}L\left(\alpha ,\beta \right)}{\partial {\beta }^2}=\frac{n}{{\beta }^2}+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }{\left[\log \left(\frac{x_i}{\alpha }\right)\right]}^2$ (32)

From (27),

$-\frac{n\beta }{\alpha }+\frac{\beta }{\alpha }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }=0$ (33)

which simplifies to:

$\frac{n\beta }{\alpha }=\frac{\beta }{\alpha }\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }$ (34)

$n=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\left(\frac{x_i}{\alpha }\right)}^{\beta }=\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }}{{\alpha }^{\beta }},n=\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }}{{\alpha }^{\beta }}$ (35)

$\hat{\alpha }={\left(\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }\right)}^{\frac{1}{\beta }}$ (36)

$L\left(\beta \right)=n\log \left(\beta \right)-n\log \left(\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{x}_i^{\beta }\right)+\left(\beta -1\right)\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\log \left(x_i\right)-n$ (37)

Differentiating Equation (37) with respect to $\beta$:

$\frac{\partial L\left(\beta \right)}{\partial \beta }=\frac{n}{\beta }-n\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }}{{{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }+{{\displaystyle \sum }}_{i=1}^n\log \left(x_i\right)}$ (38)

Obtaining the second derivative:

$\frac{{\partial }^{2}L\left(\beta \right)}{\partial {\beta }^2}=\frac{n}{{\beta }^2}-n\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }\log \left(x_i\right)-{\left({{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }\log \left(x_i\right)\right)}^2}{{\left({{\displaystyle \sum }}_{i=1}^n{x}_i^{\beta }\right)}^2}$ (39)

Cox proportional hazard model is semi-parametric, its baseline hazard function $h_0\left(t\right)$ And the probability distribution of the survival times is not specified.

The hazard function at time $t$ is given by:

$h\left(t;Z\right)=h_0\left(t\right){\text{e}}^{{\beta }^{I}Z}$ (40)

which can be expanded as:

$h\left(t;Z\right)=h_0\left(t\right){\text{e}}^{\left({\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q\right)}$ (41)

Taking the natural logarithm:

${\log }_{\text{e}}h\left(t\right)={\log }_{\text{e}}{h}_0\left(t\right)+{\log }_{\text{e}}{\text{e}}^{\left({\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q\right)}$ (42)

Simplifying further:

${\log }_{\text{e}}h\left(t\right)={\log }_{\text{e}}{h}_0\left(t\right)+{\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q$ (43)

Cox proportional hazard model is semi-parametric, its baseline hazard function. The baseline hazard function, $h_0\left(t\right)$, is unspecified, and the probability distribution of the survival times is not explicitly defined.

The hazard function at time $t$ is given by:

$h\left(t;Z\right)=h_0\left(t\right){\text{e}}^{{\beta }^{I}Z}$ (44)

Expanding this:

$h\left(t;Z\right)=h_0\left(t\right){\text{e}}^{\left({\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q\right)}$

Taking the natural logarithm:

${\log }_{\text{e}}h\left(t\right)={\log }_{\text{e}}{h}_0\left(t\right)+{\log }_{\text{e}}\left({\text{e}}^{{\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q}\right)$ (45)

Simplifying further:

${\log }_{\text{e}}h\left(t\right)={\log }_{\text{e}}{h}_0\left(t\right)+{\beta }_1{Z}_1+{\beta }_2{Z}_2+\cdots +{\beta }_q{Z}_q$ (46)

Equation (46) is analogous to the ordinary multiple regression equation, with ${\log }_{\text{e}}{h}_0\left(t\right)$ representing the Y-axis intercept, which is the baseline or underlying hazard function, the probability of dying when all the explanatory variables are zero.

The hazard function of any two objects at any point in time is assumed to be proportional, hence, the name proportional hazard (PH) models; if one object carries twice the risk of death as the other one at the initial stage or time, then at all later times the risk will remain twice as high.

The corresponding survival function is given by:

$S\left(t|x\right)=S_0\left(t\right)\exp \left(\underset{i=1}{\overset{p}{{\displaystyle \sum }}}{\beta }_i{\chi }_i\right)$ (47)

This is known as the Cox regression model. It is a semiparametric model because it does not assume a specific distribution for the baseline hazard function, $h_0\left(t\right)$, making that part non-parametric. However, it assumes a parametric form for the effect of the predictor variables on the hazard, thereby creating the parametric component of the model. Thus, the model consists of two parts:

A non-parametric part and a parametric part, which is why it is referred to as a semi-parametric model.

The hazard ratio (HR) between two individuals with different covariates, $X$ and $X^{\text{*}}$, is given by:

$\text{HR}=\frac{h\left(t\right)\exp \left({\beta }^{l}X\right)}{h\left(t\right)\exp \left({\beta }^l{X}^{*}\right)}=\exp \left(\underset{i=1}{\overset{p}{{\displaystyle \sum }}}{\beta }^l\left(x_i-x_i^{*}\right)\right)$ (48)

The hazard ratio is independent of time, which is why the model is referred to as the proportional hazards model. The semi-parametric proportional hazard can be made fully parametric by parameterizing the baseline hazard function according to a specific model for the distribution of survival times of exponential, Weibull, and Gompertz distributions. The aspect of proportionality can be tested by plotting the Kaplan-Meier survival curves together if they happen to cross the implication is that the assumption may be violated. A complementary loglog plot which involves plotting the logarithm of the negative logarithm of the estimated survival function against the logarithm of survival time yields parallel curves across the groups for proportional hazards. For a baseline hazard function that is constant over time it is assumed to follow the Exponential distribution and the hazard function for the exponential distribution is used as the baseline hazard function $h_0\left(t\right)$.

The hazard function for the Weibull distribution can be used, the shape of its hazard depends on the shape parameter, and it can take a variety of forms depending on the shape parameter. When the risk or the hazard is expected to increase or decrease within a short time and then become constant.

They are full parametric models that directly accommodate the multiplicative effects of explanatory variables of survival times and hence do not rely on proportional hazards. Since the role of the explanatory variable is to accelerate (or decelerate) the time to failure, the model is referred to as the accelerated failure time (AFT) of the accelerated Failure model).

Under AFT models we measure the direct effect of the explanatory variable on the survival time instead of the hazard as done in the PH model where the effect of the explanatory variable is on the hazard function which can be increasing, decreasing constant, unimodal, or bathtub type; no direct effect on the survival time.

$S_0\left(t\right)$ is the baseline survival function.

The effect on the survival time is indirect through the hazard function, and the survival function is given by:

$S\left(t|x\right)=S_0\left(\frac{t}{n\left(x\right)}\right)$ (49)

where

$n\left(x\right)=\exp \left({\alpha }_1{x}_1+{\alpha }_2{x}_2+\cdots +{\alpha }_p{x}_p\right)$ (50)

is the acceleration factor.

Here, $S_0\left(t\right)$ represents the baseline survival function.

The ratio of two survival times is always constant. In disease modeling the proportion of patients surviving in any consecutive disease stages is a constant.

The hazard function for an individual with covariates $X_1,X_2,\cdots ,X_p$ is given by:

$h\left(\frac{t}{x}\right)=\left[\frac{1}{n\left(x\right)}\right]h_0\left(\frac{t}{n\left(x\right)}\right)$ (51)

where $n\left(x\right)$ represents the acceleration factor based on the covariates.

In this model, the effect of the covariate is assumed to be constant and multiplicative on the time scale. The acceleration factor gives the impact of the covariates on survival.

The log-linear form of the model with respect to time is given by:

$\log T_1=\mu +{\alpha }_1{x}_{1i}+{\alpha }_2{x}_{2i}+\cdots +{\alpha }_p{x}_{pi}+\sigma {\epsilon }_i$ (52)

where $\mu$ is the intercept (the y-axis intercept), $\sigma$ is a scale parameter, and ${\epsilon }_i$ is a random variable assumed to follow a particular distribution for the acceleration failure rate. The survival function of $T_i$ can be expressed in terms of the survival function of ${\epsilon }_i$. If ${\epsilon }_i$ follows an extreme value distribution, then $T_i$ follows an exponential distribution. The Gumbel survival function is given by:

$S_{{\epsilon }_i}\left(\epsilon \right)=\exp \left(-\exp \left(\epsilon \right)\right)$ (53)

Consider a survival time $T$ with a Weibull distribution $\text{W}\left(\lambda ,\alpha \right)$ where $\lambda$ is the scale parameter and $\alpha$ is the shape parameter.

The hazard function for the $i^{\text{th}}$ individual is given by:

$h\left(t\right)=\frac{1}{\eta \left(x\right)\alpha }\lambda t^{\lambda -1}$ (54)

where

$\eta \left(x\right)=\text{exp}\left({\alpha }_1{x}_1+{\alpha }_2{x}_2+\cdots +{\alpha }_p{x}_p\right)$

The survival function for the $i^{\text{th}}$ individual is:

$S_i\left(t\right)=\exp \left[-\exp \left(\frac{\mu -{\alpha }_1{X}_{1i}-\cdots -{\alpha }_p{X}_{pi}}{\sigma }\right)t^{\frac{1}{\sigma }}\right]$ (55)

Simplified, the survival function becomes:

$S_i\left(t\right)=\exp \left[-\exp \left(\frac{\log t-\mu -{\alpha }_1{X}_1-\cdots -{\alpha }_p{X}_p}{\sigma }\right)\right]$

The hazard function is given by:

$h_i\left(t\right)=\frac{1}{\sigma }{t}^{\frac{1}{\sigma }-1}\exp \left(\frac{\mu -{\alpha }_1{X}_{1i}-\cdots -{\alpha }_p{X}_{pi}}{\sigma }\right)$ (56)

The cumulative hazard is:

$H_i\left(t\right)=-\log S_i\left(t\right)=\exp \left(\frac{\log t-\mu -{\alpha }_1{X}_{1i}-\cdots -{\alpha }_p{X}_{pi}}{\sigma }\right)$ (57)

The median survival time is:

$t\left(50\right)=\exp \left(\sigma \log \left(\log 2\right)+\mu +{\alpha }^{\top }{x}_i\right)$ (58)

The probability density function (PDF) is: $f_i\left(t\right)=S_i\left(t\right)\times h_i\left(t\right)$ and,

$\begin{array}{l}=\exp \left[-\exp \left(\frac{\log t-\mu -{\alpha }_1{X}_1-\cdots -{\alpha }_p{X}_p}{\sigma }\right)\right]\\ \begin{array}{c}\times \frac{1}{\sigma }{t}^{\frac{1}{\sigma }-1}\exp \left(\frac{\mu -{\alpha }_1{X}_{1i}-\cdots -{\alpha }_p{X}_{pi}}{\sigma }\right)\end{array}\end{array}$ (59)

The Weibull distribution is unique in that it satisfies both the proportional hazards (PH) and accelerated failure time (AFT) assumptions, making it versatile for survival analysis.