# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 6 Real Data Illustrations
### Illustration 1: Concrete Fatigue Life Dataset  
The dataset under analysis represents the fatigue life of concrete specimens, measured in cycles (multiplied by \(10^{-3}\)). The dataset is defined by the applied stress-causing failure, with ratios of 0.95, 0.90, and 0.825. A lower ratio is expected to correspond to a longer number of cycles until failure. Each ratio is accompanied by 15 observations, as outlined in ([46], p. 88). The concrete fatigue life is designated as the response variable, and the ratio is identified as the covariate. Initially, we provide an exploratory analysis of the response variable. Table 7 presents some descriptive statistics, including SD, CSk, the coefficient of variation (CV), and coefficient of kurtosis (CK). Additionally, Figure 1 displays the corresponding histogram and boxplot. From Table 7 and Figure 1, it is evident that the dataset exhibits considerable variability (CV = 131.401%), positive skewness (CSk = 1.414), heavy tails (CK = 0.771), and the presence of outliers (case nos. 40, 41, 42, 43, 44, and 45). In this study, a regression structure is incorporated into the WEI4 (with \(q=0.5\)) and WEI5 models for both \(\lambda\) and \(\
u\), as follows:  
\[\lambda_{i}=\exp\big{(}\mathbf{x}_{i}^{\top}\boldsymbol{\beta}\big{)},\quad \
u_{i}=\exp(\mathbf{x}_{i}^{\top}\boldsymbol{\zeta}),\quad i=1,\ldots,45,\]  
where \(\boldsymbol{\beta}=(\beta_{0},\beta_{1})^{\top}\), and \(\boldsymbol{\zeta}=(\zeta_{0},\zeta_{1})^{\top}\) represent the regression coefficients, and \(\mathbf{x}_{i}^{\top}=(1,x_{1i})\) denotes the respective design matrix, with \(x_{1i}\) being the \(i\)-th observation of the covariate ratio. Table 8 presents the fitting of these models, including parameter estimation using the ML method, the respective standard errors, the associated \(Z\)-test \(p\)-value, the AIC, and the \(p\)-value from the Kolmogorov-Smirnov (KS), Anderson-Darling (AD), and Cramer von Mises (CVM) goodness-of-fit tests for the CS and RQ residuals. It is important to note that the theoretical distributions for CS and RQ residuals are exponential and standard normal, respectively. In this table, it can be observed that for modeling \(\lambda\), both regression \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(n\) & **min** & **Max** & **Mean** & **Median** & **SD** & **CSk** & **CV** & **CK** \\ \hline
45 & 37 & 5598 & 1216.58 & 342 & 1598.60 & 1.41 & 131.40 & 0.77 \\ \hline \end{tabular}
\end{table} 
Table 7: Descriptive statistics of the concrete fatigue life dataset. Figure 1: Histogram with the density kernel estimation (**a**) and boxplot (**b**) for the concrete fatigue life dataset. will be used to illustrate the reparameterized Weibull model in the context of censored data. It comprises information on 26 patients with stage III and IV ovarian carcinoma who were treated with radical surgery and postoperative chemotherapy. The covariates used are the minimal residual disease (residual), which indicates a tumor mass of less than 2 cm in diameter, and the patient's age in years. Figure 4 displays the Kaplan-Meier (K-M) estimator for both residual and age. Descriptively, it is evident that both covariates have an impact on patient survival. In addition, and similar to the last application, in Figure 5, the QQ plot and simulated envelope for the RQ residuals are provided, suggesting that this kind of residual is reasonably fitted to the standard normal distribution. Table 9 shows the AIC for the WEI, WEI2, and WEI3 models in the ovarian dataset with different combinations of covariates in \(\lambda\) and \(\
u\). As discussed previously, the AIC is the same for the three models if covariates are not included in \(\
u\) and differ if \(\
u\) is modeled. The results also suggest that the best combination is to include the covariates residual plus age in \(\lambda\), but not in \(\
u\). Table 10 shows the estimate, the corresponding SE for the WEI2 model (it could have been any of them), and the \(p\)-value from the KS, AD, and CVM goodness-of-fit tests for RQ residuals.