## 2 Methods
### The VILIAlert System Architecture
Using the KM method this probability is estimated as follows:  
\[\widehat{DFS}(t)=\prod_{n:t_{n}\leq t}\biggl{(}1-\frac{d(t_{n})}{r(t_{n})} \biggr{)}, \tag{1}\]  
where \(d(t_{n})\) denotes the number of APA with DB event occurrence at time \(t_{n}\) and \(r(t_{n})\) denotes the number of APA at risk prior to time \(t_{n}\) (i.e., the number of APA that are not censored nor containing DB prior to time \(t_{n}\)). The influence of other factors, the covariates, and their influence on the estimated survival curves can also be investigated using KM methods. Previous studies reported that female patients are under increased risk of receiving poorer ventilatory treatments in comparison to males [22], hence the introduced KM approach can be used to investigate the performance of the VILIAlert system in terms of these covariates. Gender can be investigated using the KM methods where each gender is considered separately, and the separate survival distributions compared for significant differences. The log-rank test may then be used to test for statistical significance of the difference in survival distributions. The results will provide information to help understand how, if at all, the patient gender is associated with the risk of receiving injurious ventilation associated with VILI and if this is an increased risk. The KM approach described above can be seen as a static prediction as the overall survival estimates are made using the time of diagnosis (the time of generating the alert from the baseline reference) but it is unable to precisely capture how the prognosis changes over time [23]. In order to overcome this issue and process mine the changing defect free survival likelihood with increasing duration of follow-up after triggering the alert, we use the conditional survival function. This estimates the probability that APA will remain defect free until at least the time of interest \(t_{y}\) given that it was previously defect free until the time \(t_{x}\), where \(t_{x}<t_{y}\). In the text to follow, we will refer to this probability as the conditional defect free survival probability (\(CDFS\)), estimated using the previously described KM survival estimates (1), as follows [24; 25]:  
\[CD\widehat{FS\left(t_{y}\right|}t_{x})=\widehat{\frac{DFS\left(t_{y}\right)}{DFS \left(t_{x}\right)}}. \tag{2}\]  
The Greenwood method can be used to calculate the 95% confidence intervals (95% CI) around the CDFS estimates [24; 26; 27]:  
\[95\%\ CI=CD\widehat{FS\left(t_{y}\right|}t_{x})^{\exp\left(\pm 1.96\sqrt{ \widehat{V}\left[\log\left(CD\widehat{FS\left(t_{y}\right|}t_{x}\right)\right) \right]}\right)}\, \tag{3}\]  
where  
\[\widehat{V}\left[\log\left(\widehat{CDFS\left(t_{y}\right|}t_{x}\right) \right)\right]=\frac{1}{\log\left(\widehat{CD\widehat{FS\left(t_{y}\right|}t_ {x}\right)}\right)}\sum_{x\leq t_{k}\leq y}\frac{d_{k}}{r_{k}(r_{k}-d_{k})}\.\]  
Similar to the DFS, the use of the KM approach to estimate the \(CDFS\) allows us to analyse the effect of the gender covariate on the conditional survival probabilities. As the measure of effect size of conditional defect free survival differs, we use the standardised difference (\(d_{FM}\)) method by [PERSON] and co-workers [28] for our study defined as:  
\[d_{FM}=\frac{P_{F}-P_{M}}{\sqrt{\left[P\left(1-P\right)\right]}}, \tag{4}\]  
where \(P_{F}\) and \(P_{M}\) are the \(\widehat{CD\widehat{FS\left(t_{y}\right|}t_{x})}\) of the female and male patients, respectively, while \(P\) represents their weighted mean [29]. To interpret the \(d_{FM}\) values, the following guidelines are used from the literature [28; 30; 31]:  
* \(|d_{FM}|<0.1\) suggests very small differences;
* \(0.1\leq|d_{FM}|<0.3\) suggests small differences;
* \(0.3\leq|d_{FM}|<0.5\) suggests moderate differences, and;
* \(0.5\leq|d_{FM}|\) suggests considerable differences between the two genders. The use of survival analytics techniques for performance estimation is best illustrated with the following use-case scenario.