# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 6 Applications
### Guinea Pigs Data  
The set of data [53] shows survival rates for Guinea pigs infected with virulent tubercle bacilli. One of the reasons guinea pigs were chosen for this investigation is that they are thought to be highly susceptible to human tuberculosis. The data set given as follows: 0.10, 0.33, 0.440, 0.56, 0.59, 0.72, 0.74, 0.77, 0.920, 0.93, 0.96, 1.00, 1.00, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.160, 1.200, 1.21, 1.22, 1.22, 1.24, 1.30, 1.34, 1.360, 1.390, 1.44, 1.46, 1.53, 1.59, 1.60, 1.63, 1.63, 1.680, 1.71, 1.72, 1.76, 1.83, 1.950, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.40, 2.45, 2.51, 2.53, 2.54, 2.54, 2.780, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.580, 5.55. By using equations Equations (22)-(25) to obtain the MLEs of the model parameters and the results are reported in Table 10. The estimated density with histogram of waiting times data for \(TLMW\) distribution and other distributions are displayed in Figure 11. The estimated cumulative and the fitted \(TLMW\) distribution and other distributions for the Guinea Pigs data set are displayed in Figure 12. The estimated PP and QQ plot for \(TLMW\) distribution are displayed for Guinea Pigs data in Figure 13.  
Now, applying the formal goodness-of-fit tests in order to verify which distribution fits better to these data set. The measures AKIC, BIC, CAKIC, HQIC and the KSD with PVKS (See Table 11) are considered. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Models** & & \(\alpha\) & \(\beta\) & \(\theta\) & \(\lambda\) & \(\delta\) \\ \hline \multirow{2}{*}{TLMW} & estimates & 0.2497 & 0.2004 & 1.2916 & 2.7723 & \\  & SE & 0.9087 & 0.7554 & 0.7622 & 1.5924 & \\ \hline \multirow{2}{*}{KW} & estimates & 2.4052 & 0.7823 & 0.9365 & 1.2522 & \\  & SE & 1.1901 & 0.1516 & 0.3952 & 0.8752 & \\ \hline \multirow{2}{*}{WL} & estimates & 11.2147 & 2.8862 & 0.2501 & 0.6035 & \\  & SE & 27.0764 & 2.3091 & 0.2095 & 2.0505 & \\ \hline \multirow{2}{*}{MOAPEW} & estimates & 0.0106 & 0.6961 & 0.9662 & 0.4619 & 0.0332 \\  & SE & 0.0144 & 0.3549 & 0.1980 & 0.4283 & 0.0079 \\ \hline \multirow{2}{*}{EGAPEx} & estimates & 2.8371 & 3.2962 & 16.2875 & 0.1090 & \\  & SE & 0.9711 & 1.0053 & 2.9542 & 0.0265 & \\ \hline \end{tabular}
\end{table} 
Table 10: MLE for parameters of TLMW and comparative models for Guinea Pigs data. Figure 10: ACF test plots for parameters of \(TLMW\) model for COVID-19 data. The 95% CIs for the parameters \(\alpha\), \(\beta\), \(\theta\) and \(\lambda\) are presented for MLE and Bayesian estimation methods in Table 12. Figure 14 displayed contour plot of parameters for \(TLMW\) model to check uniqueness and max-value of likelihood for Guinea Pigs data. While, Figure 15 confirmed convergence of MCMC results, which show the MCMC samples as a pairs plot, with the trace plot, and marginal frequency of proposed distribution as normal distribution for each parameter on the diagonal for parameters of \(TLMW\) model for Guinea Pigs data. Figure 16 discussed ACF test. Auto-correlation for MCMC is the correlation between a iteration series with a lagged version of itself. The ACF starts at a lag of 0, which is the correlation of the iteration series with itself and therefore results in a correlation of 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Models** & **AKIC** & **BIC** & **CAKIC** & **HQIC** & **KSD** & **PVKS** \\ \hline TLMW & 196.1265 & 205.2332 & 196.7235 & 199.7519 & 0.0885 & 0.6253 \\ KW & 196.2290 & 205.3357 & 196.8260 & 199.8544 & 0.0922 & 0.5737 \\ WL & 196.8496 & 205.9562 & 197.4466 & 200.4750 & 0.0955 & 0.5277 \\ MOAPEW & 210.8167 & 222.2000 & 211.7258 & 215.3484 & 0.1359 & 0.1398 \\ EGAPEx & 196.3202 & 205.4268 & 196.9172 & 199.9456 & 0.0907 & 0.5948 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Discrimination criteria and KS test of the TLMW model parameters for Guinea Pigs data. Figure 11: Pdf for different models for Guinea Pigs data. Figure 12: Cdf for different models for Guinea Pigs data.