## 2 Methodology
### Survival Analysis Methods for Measuring Product Lifespan  
The presented study uses survival analysis to analyze the expected time duration when interest in new products exists, which represents the product lifespan. In the field of survival analysis length of time taken is referred to event time [27]--product usage time in our case. Survival analysis was originally developed in the medical field, as a means of analyzing the time between medical intervention and death. Over the past few decades, the field was expanded to include other events as well as events that occur multiple times for a given individual [28]. Survival analysis has wide applications in the field of marketing, including customer-relationship management (CRM), marketing-campaign management, and trigger-event management [29]. If we denote the time taken for an event to occur as \(T\), we can construct a frequency histogram and model a series of events as a function of time. The probability distribution function for \(T\) can be denoted by \(f(t)\). The cumulative distribution function can be denoted by \(F(t)\). This provides the following equation:  
\[F(t)=p(T\leq t)\]  
Using the above approach, we can represent survival as a function of time \(S(t)\) such that: for \(t=0\), \(S(t)=1\) for the specific time that a failure occurs, the value of \(S(t)\) is zero [30]. In some cases, the time to failure will not be observable and only partial observation will be possible. In this case, we consider a specific 'censoring time' \(c\). The survival function is then denoted as:  
\[S(t)=P(T>t)=1-F(t)\]  
Instantaneous hazard or conditional failure rate is the instantaneous rate at which a randomly selected individual--who is known to be alive at time \((t-1)\) and will die at time \(t\)[31]. Mathematically, instantaneous hazard is equal to the number of failures between time \(t\) and time \(t+\Delta(t)\), divided by the size of the population at risk (at time \(t\)), divided by \(\Delta(t)\). This gives us the proportion of the present population at time \(t\) that fail, per unit of time, represented by the equation:  
\[h(t)=\lim_{\Delta t\to 0}\frac{P(t<T\leq t+\Delta(t)|T>t)}{\Delta(t)}=\frac{f(t)}{S(t)}\]  
Widely used, the Kaplan-Meier method is used to estimate time-related events [27]. Most commonly, it is used in biostatistics to analyze death as outcome. However, in more recent years, the technique has seen adoption in the fields of social sciences and industrial statistics. For example, in economics, we might measure how long people tend to remain unemployed after being let go by an employer; in engineering, we might measure how long a certain mechanical component tends to last before mechanical failure takes place. The survival function is theoretically a smooth curve, but it can be estimated using the Kaplan-Meier (KM) curve. Plotting the Kaplan-Meier estimate entails a series of horizontal steps of declining magnitude that, for a sufficiently large sample approach, estimate the true survival function for the given population. When applying this approach, survival-function value between successive sampled observations is presumed constant [32]. An important advantage of the Kaplan-Meier curve is its ability to take into account censored data loss within the sample before the final outcome is observed. In cases where no truncation or censoring occurs, the Kaplan-Meier curve is equivalent to empirical distribution [33; 34]. As mentioned, survival analysis has wide applications for marketing, including CRM, marketing-campaign management, and trigger-event management [29]. Depending on the business setting, e.g., contractual versus noncontractual, different techniques can be applied [29]. For example, a goal might be to analyze the performance of a marketing campaign (while in progress), and how different customer features affect its performance.