While the survival function focuses on reporting the “non-occurrence” of the event (for example, the patient has not died), the risk function focuses on the “occurrence” of the event. This is very interesting because it allows us to pose answers to questions such as, for example, “at what point am I going to have a ‘spike’ in hospital discharges?” Curiously enough, this function is hardly ever reported, and, as we have seen, it provides more interesting information in the area of clinical studies.

**Challenges of these analyses**

In conclusion, survival analyses allow to study the time passed until the occurrence of a certain event. Although basic analyses are very intuitive to interpret, they can become something very complex. This makes them a real challenge in contexts such as:

**The need to incorporate different covariates:** One of the possible solutions in this case is modeling by Cox regression (a review of this model can be found in Fox and Weisberg, 2018). These analyses make it possible to study the dependence of survival time considering a series of predictor variables, such as the randomization group, patients’ age, the severity of the pathology, etc.**When the event is time-dependent**: We may be studying an event of interest that is much more likely to occur at the start of the follow-up period than at the end of the study when considering certain risk factors. In this situation, we should bear in mind that the Cox proportional hazards model may require some extensions (Kleinbaum and Klein, 2011).**A patient may experience more than one event**: This is very common when, for example, we are studying relapses, since a patient may have more than one (Baethge and Schlattmann, 2004). In these cases, the biases associated with survival-dependent censoring must be corrected (Gómez, 2012; Ruth et al., 2022).**Large volumes of data**: Sometimes, the analysis incorporates a great number of variables. When the number of variables is very large, the analysis faces the challenge of high dimensionality. The use of machine learning tools becomes then an optimal alternative (an example of such applications can be found in Gong et al., 2018).

As open-source software recommendations, the R packages *survival* and *survminer*, as well as Python’s scikit-survival, are versatile tools that allow for the elaboration of both basic as well as more advanced survival analysis.

**References**

Baethge, C., & Schlattmann, P. (2004). A survival analysis for recurrent events in psychiatric research. *Bipolar Disorders*, *6*(2), 115-121.

Fox, J., & Weisberg, S. (2002). Cox proportional-hazards regression for survival data. *An R and S-PLUS companion to applied regression*, *2002*.

Gómez, G., & Serrat, C. (2014). Correcting the bias due to dependent censoring of the survival estimator by conditioning. *Statistics*, *48*(2), 295-314.

Gong, X., Hu, M., & Zhao, L. (2018). Big data toolsets to pharmacometrics: application of machine learning for time‐to‐event analysis. *Clinical and translational science*, *11*(3), 305-311.

Kassambara, A., Kosinski, M., & Biecek, P. (2021). *survminer: Drawing Survival Curves using ‘ggplot2’*. R package version 0.4.9, https://CRAN.R-project.org/package=survminer.

Kleinbaum, D. G., & Klein, M. (2012). Extension of the Cox proportional hazards model for time-dependent variables. In *Survival analysis* (pp. 241-288). Springer, New York, NY.

Pölsterl, S. (2020). scikit-survival: A Library for Time-to-Event Analysis Built on Top of scikit-learn. *Journal of Machine Learning Research, 21*(212), 1–6.

Prinja, S., Gupta, N., & Verma, R. (2010). Censoring in clinical trials: review of survival analysis techniques. *Indian journal of community medicine: official publication of Indian Association of Preventive & Social Medicine*, *35*(2), 217.

Ruth, D. M., Wood, N. L., & VanDerwerken, D. N. (2022). Fully nonparametric survival analysis in the presence of time-dependent covariates and dependent censoring. *Journal of Applied Statistics*, 1-15.

Therneau, T. (2022). *A Package for Survival Analysis in R*. R package version 3.3-1, https://CRAN.R-project.org/package=survival.

Turkson, A. J., Ayiah-Mensah, F., & Nimoh, V. (2021). Handling Censoring and Censored Data in Survival Analysis: A Standalone Systematic Literature Review. *International Journal of Mathematics and Mathematical Sciences*, *2021*.

Xu, S., Shetterly, S., Powers, D., Raebel, M. A., Tsai, T. T., Ho, P. M., & Magid, D. (2012). Extension of Kaplan-Meier methods in observational studies with time-varying treatment. *Value in Health*, *15*(1), 167-174.