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Surveys often ask for time intervals, with start and end dates:
<ul>
<li>When did you buy the product? When did you finish it?</li>
<li>When did the patient start and end each line of therapy?</li>
<li>When did respondents start and end different programs?</li>
</ul>
One thing we can do is to look at the data. Another is to look at how 
survival data is summarized in clinical research...
<img src="/images/blog/2018-07-18-kaplan/intervals-toyota.png" style="max-width: 540px; margin-top:20px"/>

<h2 id="visualizing-intervals">Visualizing intervals</h2>
The chart above is visualization of production runs for a number of Toyota models
using Protobi's <code>IntervalChart</code>. 
Each line represents a single model, with start and stop dates:
<ul>
<li>The Toyota Land Cruiser was first produced in 1951, and is still in production today, 67 years later.</li>
<li>The Toyota Stout was first produced in 1954, but stopped in 1989 after 35 years.</li>
</ul>
Some runs are complete, these we show these with round endpoints.
Some runs are still continuing at the time of this analysis, these are shown cut off with a straight end.
Here the runs are sorted in ascending order of start date. These could be
ordered by end date, duration run, etc.
Scanning a chart like this is useful to just see what the data looks like.
<h2 id="distribution-of-durations-to-date">Distribution of durations to-date</h2>
We can calculate the duration for each automobile model, and display it 
as a statistical distribution. 
Here we can differentiate between production runs that are complete (for whom the data is final)
versus those intervals that are still ongoing.
<img src="/images/blog/2018-07-18-kaplan/production-duration.png" style="max-width: 540px"/>
Here we can clearly see that many production runs are still ongoing. So 
a simple mean or median of run lengths to-date would seriously under-report the expected production runs.
For right censored data, a useful way to summarize the data is the <a href="/post/kaplan-meier-survival-estimator">Kaplan-Meier Survival Estimator</a>

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