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How do we describe the distribution of time intervals when some aren't yet complete?
The Kaplan–Meier Survival Estimator is a non-parametric curve that describes the 
empirical survival function given observed interval to-date. 
Importantly 
it is designed to handle "censored" data where the intervals are observed
before they are known to be complete.
<img src="/images/blog/2018-07-18-kaplan/kaplan-meier-global-toyota.png" style="max-width: 540px"/>

<h2 id="background">Background</h2>
Survival data arises arises in many scenarios:
<ul>
<li>Lifespan of mutual funds, given many are still active</li>
<li>Duration of therapy when many patients are still being treated</li>
<li>Production span of cars when many are still produced</li>
</ul>
The challenge is that in many samples we have start dates
but many intervals are still continuing. Observed durations
are are clipped at the time of the survey. So a simple average or even median of observed durations
when those durations 
may seriously underestimate the total time interval.
Survival data is well established in clinical trials and other settings.
<h2 id="kaplan-meier-estimator">Kaplan-Meier Estimator</h2>
The Kaplan-Meier Survival Estimator is a simple robust approach to reporting 
empirical survival distributions with censored data.
The estimator is given by:
 <img src="/images/blog/2018-07-18-kaplan/kaplan-meier-formula.png" style="max-width: 200px; margin:auto; display: block"/>
<ul>
<li>ni is number of intervals known to have survived to time ti</li>
<li>di is number of intervals known to have stopped at time ti</li>
</ul>
<h2 id="kaplan-meier-estimator-in-protobi">Kaplan-Meier Estimator in Protobi</h2>
Protobi can calculate Kaplan-Meier survival curves from start dates and end dates.
Above is the distribution of production run lengths for specific auto models.
One curve shows the global overall curve and one shows just Toyota models.
Closed circles show termination events. Open circles show right-censored events
where an interval is observed but still continuing.
Below the chart is a table showing the median survival time, here defined as the time 
of the first event where empiric survival falls below 50%.
Closed circles show termination events. Open circles show right-censored events

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