**4. Conclusions**

In this paper, we have extended the applications of the sine-modified Lindley (S-ML) distribution developed by [4] to model biomedical data. The distribution yields the benefits of both the modified Lindley and S-G distributional functionalities. It was used to investigate the distribution of tumor size, patients diagnosed with cancer's survival durations, and medications provided. The AIC, BIC, and test statistics such as *A*∗, *w*∗, and *Dn* with their associated *p*-values are used to select the best-fitting model. These metrics are supported by a visual representation of how well the S-ML model fits the data, such as a box plot or a TTT plot. We believe the findings are superior to other competing distributions for modeling biomedical data and can be used to model a range of other biological data. We have also included the data sets and R codes for all of the figures in the paper, as well as all of the estimations, and the tests carried out. We refer readers to the Appendix A for these R codes.

**Author Contributions:** Conceptualization, L.T., V.G. and C.C.; methodology, L.T., V.G. and C.C.; software, L.T., V.G. and C.C.; validation, L.T., V.G. and C.C.; formal analysis, L.T., V.G. and C.C.; investigation, L.T., V.G. and C.C.; resources, L.T., V.G. and C.C.; data curation, L.T., V.G. and C.C.; writing—original draft preparation, L.T., V.G. and C.C.; writing—review and editing, L.T., V.G. and C.C.; visualization, L.T., V.G. and C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We would like to thank the two referees for the constructive comments on the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

In this section, we have included the code to analyze data set 1, using the software R. The codes for the graphs in the data analysis are also plotted.

*Appendix A.1. Data Sets*

```
Data set 1
```
(2.15, 2.20, 2.55, 2.56, 2.63, 2.74, 2.81, 2.90, 3.05, 3.41, 3.43, 3.43, 3.84, 4.16, 4.18, 4.36, 4.42, 4.51, 4.60, 4.61, 4.75, 5.03, 5.10, 5.44, 5.90, 5.96, 6.77, 7.82, 8.00, 8.16, 8.21, 8.72, 10.40, 13.20, 13.70) **Data set 2**

(12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48, 52, 53, 54, 54, 55, 56, 57, 58, 58,59, 60, 60, 60, 60, 61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76, 81, 83, 84, 85, 87, 91, 95, 96,98, 99, 109, 110, 121, 127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258, 258, 263, 297, 341, 341, 376) **Data set 3**

(0.96, 1.06, 1.09, 1.16, 1.19, 1.20, 1.32, 1.33, 1.40, 1.42, 1.46, 1.49, 1.51, 1.52, 1.54, 1.57, 1.59, 1.68, 1.70, 1.70, 1.76, 1.76, 1.77, 1.80, 1.81, 1.86, 1.89, 1.89, 1.94, 2.20, 2.20, 2.22, 2.36, 2.36, 2.39, 2.41, 2.45, 2.69, 2.71, 2.73, 2.77, 2.80, 2.83, 2.87, 2.94, 2.98, 3.03, 3.04, 3.19, 3.31, 3.57, 3.73, 4.17, 4.27, 4.30, 4.36, 4.45, 4.79, 4.85, 4.97, 5.26, 5.33, 5.53, 5.55, 5.91, 6.25, 6.31, 7.62, 7.84, 8.49, 8.63, 8.99, 9.94, 10.43, 10.86, 11.18)

*Appendix A.2. Graphics for the PDF of S-ML Distribution*

```
x= 0:10
f= function(x,p) #defining the pdf of S-ML model
{
(((pi/2)*(p/(1+p))*exp(-2*p*x))*(((1+p)*exp(p*x))+(2*p*x)-1)*sin((pi/2)*
(1+(exp(-p*x)*((p*x)/(1+p))))*exp(-p*x)))
}
```

```
curve(f(x,p= 5),col="yellow",xlab="x", ylim=c(0,1),ylab="pdf",lwd=2 )
curve(f(x,p= 15),col="pink", lwd=2, add= TRUE)
curve(f(x,p= 50),col="purple", lwd=2, add= TRUE)
curve(f(x,p=100),col="orange", lwd=2, add= TRUE)
```

```
legend("topright",legend=c(expression(paste(beta," = ",5)),
expression(paste(beta," = ",15)),
expression(paste(beta, " = ",50)),
expression(paste(beta, " = ",100))),
ncol=1, col=c("yellow","pink","purple", "orange"),
lwd=c(2,2,2,2), cex=c(1,1,1,1),text.width = 0.1, inset=0.011, bty ="n")
### In the same way, we can plot the pdf for other beta~values.
```
*Appendix A.3. Parameter Estimate along with GOF Metrics*

```
install.packages (c("EstimationTools", "MASS", "plyr" ))
library(EstimationTools)
library(MASS)
library(plyr)
```

```