# Breast Cancer Survival Analysis Model
## 4 Discussion
### SemiMarkov Chain  
The fundamental model was used to estimate the time-dependent transition rate from one state to another by evaluating each patient's transfer of the previous state, where the rate value relies on the time spent in the present state using a semi-Markov process-transfer rate. According to Figure 9, the probability of metastasis from stages 1 to 2 is higher than that of stages 3 and 4, indicating that BC is unlikely to deteriorate fast. When BC is at stage 1 it does not metastasize but only when it reaches stage 2. As a result, the probability of BC metastasis is significantly higher than that of BC metastasis in stages 3 and 4. By year 5, the probability of BC metastasis in stages 3 and 4 is higher than that of metastasis in stage 1, suggesting that BC would deteriorate with time and progress to the third stage. It can be seen that the probability of reverting to stage 1 is relatively low but the probability of metastasis transcending to stage 4 is rather high. After stage 4, the probability of metastasis is approximately the same in stages 1, 2, and 3. This indicates that if BC is discovered near the end of the stage, the probability of metastasis in other stages is low. As can be seen in \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Stage**} & **Five-Year Survival Rate** & **Five-Year Survival Rate (No Time** & **Five-Year Survival Rate (Time** \\  & **(Department of Health)** & **Difference in Transition Matrix)** & **Difference in Transition Matrix)** \\ \hline Stage 0 & 97.7\% & 99.6\% & 85.2\% \\ Stage 1 & 95.2\% & 93.9\% & 96.0\% \\ Stage 2 & 92.3\% & 91.2\% & 37.3\% \\ Stage 3 & 89.7\% & 85.3\% & 85.4\% \\ Stage 4 & 25.7\% & 25.3\% & 18.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison of survival rates. the graph, the majority of the transition probabilities is high at the beginning and decreases with greater duration in a given condition. A simple linear regression technique is employed to calculate the influence of variables on each transition. The first stage was to construct a univariate model for each covariate, resulting in 48 sets of transformation-specific parameters. These sets include 12 types \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{5}{c}{**(a) Comparison of Actual and Predicted Survival Rates**} \\ \hline \hline
**Stage** & \multicolumn{2}{c}{**Actual Survival Rate**} & \multicolumn{2}{c}{**Predicted Survival Rate**} \\ \hline Stage 0 & 99.9\% & & 99.6\% \\ Stage 1 & 99.1\% & & 93.9\% \\ Stage 2 & 98.9\% & & 91.2\% \\ Stage 3 & 55.6\% & & 85.3\% \\ Stage 4 & & 75.0\% & & 25.3\% \\ \hline \hline \multicolumn{5}{c}{**(b) Five Years of Actual and Predicted Survival Rates**} \\ \hline \hline
**Fifth Year Forecast** & **Stage 0** & **Stage 1** & **Stage 2** & **Stage 3** & **Stage 4** & **Deaths** \\ \hline Stage 0 & 0.635 & 0.172 & 0.125 & 0.064 & 0 & 0.004 \\ \hline Stage 1 & 0.216 & 0.213 & 0.301 & 0.209 & 0 & 0.061 \\ \hline Stage 2 & 0.164 & 0.251 & 0.203 & 0.294 & 0 & 0.088 \\ \hline Stage 3 & 0.1673 & 0.191 & 0.271 & 0.222 & 0.002 & 0.147 \\ \hline Stage 4 & 0 & 0 & 0 & 0 & 0.254 & 0.746 \\ \hline Deaths & 0 & 0 & 0 & 0 & 0 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison of five years actual and predicted survival rates. Figure 9: Probability of breast cancer metastasis rate. (4 BC stages, multiplied by 3 potential stages of metastasis) and 4 covariates of the treatment methods. A negative sign of \(\beta\) in the parametric column indicates that the risk of metastasis is higher in the base group than in the other groups, whereas a positive number of \(\beta\) indicates that the risk of metastasis is higher in the non-base group. As demonstrated in Table 13, hormonal therapy had no significant influence on metastatic risk, as only three cases of metastasis were affected by variations in the patients' hormonal medication. Surgical therapy, on the other hand, had a substantial influence on metastasis.