## 2 Methods
### The VILIAlert System Architecture
We add the assumption that if TV\({}_{15}\) is above 8 mL/kg for 4 blocks, the TV\({}_{15}\) is expected to remain above 8 mL/kg throughout all 24 blocks of APA and such blocks are referred to as \"non-defective blocks\" (NDB) as these are the genuine times when we expect the patient at risk of VILI and when the alerts should be triggered. If they do not follow these criteria, then they are referred to as \"defective blocks\" (DB) as these are the times when the alerts are not considered genuine as the patient should indeed be receiving mechanical ventilation and there is no risk of the patient experiencing VILI. The ideal situation is to have the APA corresponding to a genuine alert that consists of only NDB. There are however many external factors that can influence the effect of mechanical ventilation such as the clinicians making a change to the patient's treatment or a change occurring between control and support modes of the VILIAlert system. With the 24 blocks collectively making up six hours of time, it is possible that external factors can have an impact on the TV\({}_{15}\) values and result in lower than 8 mL/kg scores, which would result in having DBs occurring during the APA. The coding of the data from continuous TV\({}_{15}\) readings to a discrete variable in binary form, opens up the opportunity to consider the implementation of survival analysis on the data to estimate the performance of the VILIAlert system. Survival analysis is an approach found in the statistical literature most commonly used to model time until an event occurs (survival time). As the name suggests, the original use of survival analysis was in medical research, where it was used to consider for example the event of interest for patients recovering from a disease. The time until an event occurs is known as survival time and on visual representation would typically be skewed in nature with graphs for survival times generally showing a high peak at the beginning before a long tail to the right, as fewer people experience the event as time progresses. Survival analysis is also known to accommodate the scenario of censoring when the event of interest may not have been observed for some of the individuals in the study. In such situations, survival analysis considers all the information known up to the time when censoring occurs [20]. In the context of the VILIAlert system, the event of interest is the occurrence of the DB and our aim is to predict the survival time at the point (i.e., at which APA block position) when the TV\({}_{15}\) metric is going to drop to below 8 mL/kg upon alert generation. Therefore, the time to event is the position of the first APA block, which is marked as DB, with the longer times to event indicating better performance of the alerting algorithm. The genuine alerts will be those cases where all APA blocks are marked as NDBs and considered censored. It is important to highlight that although from the performance analytics perspective the longer follow-up times indicate better performance of the alerting algorithm, for the patient it means poorer quality of mechanical ventilation due to longer exposure of the patient to high tidal volumes, thus increasing their risk of VILI. The raw VILIAlert data is initially analysed using the very established Kaplan-Meier (KM) method [21] to estimate the overall survival function. Let \(DFS(t)\) denote the overall defect free survival probability at timestamp \(t\), i.e., the probability that the APA consists of NDB blocks only (i.e., having \(TV_{15}>8\) mL/kg) for up to at least the time of interest \(t\), where \(t=i\times 15\) min, and \(1\leq i\leq 24\) is the position of the block in APA.