# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 3 Interpretation of the Coefficients
### The Bi-Univocity among the Four Parameterizations When Modeling Only \(\lambda\)  
To date, there is no discussion in the literature about the connection between the different parameterizations of the Weibull distribution when incorporating regression structures into these parameterizations. The following theorem establishes the biunivocal relationship between the WEI, WEI2, WEI3, WEI4, and WEI5 models when modeling only \(\lambda\). **Theorem 1**: _Let \(Y_{1},\ldots,Y_{n}\) be independent random variables, such as \(Y_{i}\sim\text{WEI}(\lambda_{i},\
u)\), where \(\lambda_{i}=\exp(-\mathbf{x}_{i}^{\top}\boldsymbol{\beta})\), \(i=1,\ldots,n\) and a constant shape parameter \(\
u\). Consider alternative parameterizations for the Weibull model, such as \(Y_{i}\sim\text{WEI2}(\lambda_{i}^{\star},\tau),Y_{i}\sim\text{WEI3}(\lambda_{i }^{\star},\tau),Y_{i}\sim\text{WEI4}(\lambda_{i}^{\star},\tau)\) or \(Y_{i}\sim\text{WEI5}(\lambda_{i}^{\star},\tau)\), with \(\lambda_{i}^{\star}=\exp(\mathbf{x}_{i}^{\top}\boldsymbol{\delta})\) and a non-model \(\tau\). If intercept terms are considered in \(\mathbf{x}_{i}\), then the elements of \((\boldsymbol{\delta}^{\top},\tau)\) can be obtained uniquely from \((\boldsymbol{\beta}^{\top},\
u)\)._  
Proof. : Equating the hazard functions for the WEI2, WEI3, WEI4, and WEI5 models with those associated with the WEI model, we obtain \(\lambda_{i}=k(\lambda_{i}^{\star},\tau)\) and \(\
u=\tau\), where \(k(\lambda_{i}^{\star},\tau)=\left[\lambda_{i}^{\star}\right]^{-1/\tau}\), for the WEI2 model, \(k(\lambda_{i}^{\star},\tau)=\lambda_{i}^{\star}\times\left[\tau\{\Gamma(1+1/ \tau)\}^{\top}\right]^{-1}\), for the WEI3 model, \(k(\lambda_{i}^{\star},\tau)=\lambda_{i}^{\star}\times-\log(1-q)\), for the WEI4 model and \(k(\lambda_{i}^{\star},\tau)=\lambda_{i}^{\star}\times\left[1-\frac{1}{\tau} \right]^{1/\tau}\), for the WEI5 model. Considering the partitions \(\mathbf{x}_{i}^{\top}=(1,\mathbf{x}_{i}^{\bigtriangleup\top})\), \(\boldsymbol{\beta}^{\top}=(\beta_{0},\boldsymbol{\beta}^{\bigtriangleup\top})\) and \(\boldsymbol{\delta}^{\top}=(\delta_{0},\boldsymbol{\delta}^{\bigtriangleup\top})\), i.e., \(\mathbf{x}_{i}^{\bigtriangleup\top},\boldsymbol{\beta}^{\bigtriangleup\top}\) and \(\boldsymbol{\delta}^{\bigtriangleup\top}\) are related to the non-intercept terms of \(\mathbf{x}_{i}^{\top},\boldsymbol{\beta}^{\top}\) and \(\boldsymbol{\delta}^{\top}\), respectively. With this, the equation \(\lambda_{i}=k(\lambda_{i}^{\star},\tau)\) assumes the following forms for each case:  
* WEI2 model: \[\exp\Bigl{(}-\beta_{0}-\mathbf{x}_{i}^{\bigtriangleup\top}\boldsymbol{\beta}^ {\bigtriangleup}\Bigr{)}=\exp\Biggl{(}\frac{\delta_{0}}{\tau}+\frac{\mathbf{x}_ {i}^{\bigtriangleup\top}\boldsymbol{\delta}^{\bigtriangleup}}{\tau}\Biggr{)}.\] In this case, we conclude that \(\delta_{0}=-\tau\beta_{0}\) and \(\boldsymbol{\beta}^{\bigtriangleup}=-\boldsymbol{\delta}^{\bigtriangleup}/\tau\). * WEI3 model: \[\exp\Bigl{(}-\beta_{0}-\mathbf{x}_{i}^{\bigtriangleup\top}\boldsymbol{\beta}^ {\bigtriangleup}\Bigr{)}=\exp\Bigl{(}\delta_{0}-\log\tau-\tau\log\Gamma(1+1/ \tau)+\mathbf{x}_{i}^{\bigtriangleup\top}\boldsymbol{\delta}^{\bigtriangleup }\Bigr{)}.\] Here, it can be concluded that \(\delta_{0}=-\beta_{0}+\log\tau+\tau\log\Gamma(1+1/\tau)\) and \(\boldsymbol{\beta}^{\bigtriangleup}=\boldsymbol{\delta}^{\bigtriangleup}\). * WEI4 model: \[\exp\Bigl{(}-\beta_{0}-\mathbf{x}_{i}^{\bigtriangleup\top}\boldsymbol{\beta}^ {\bigtriangleup}\Bigr{)}=\exp\Bigl{(}\delta_{0}+\log(-\log(1-q))+\mathbf{x}_ {i}^{\bigtriangleup\top}\boldsymbol{\delta}^{\bigtriangleup}\Bigr{)},\] where we conclude that \(\delta_{0}=-\beta_{0}-\log(-\log(1-q))\) and \(\boldsymbol{\beta}^{\bigtriangleup}=\boldsymbol{\delta}^{\bigtriangleup}\).