Continuous, Censored and Truncated Data: what are the differences and do you need to care?

Whenever I work with someone whose statistical or econometric experience has been more practical than theoretical, two things happen. The first is that the poor person inexplicably develops a twitch whenever I launch into an enthusiastic tangent that requires a sheet of graph paper and extensive hand waving.

The other thing that inevitably happens is that the digression comes to an end and the question is asked “but does that matter in practice?”

When it comes to model section, the difference between data types really does matter. You may make choices one way or another, but understanding the differences (both obvious and subtle) lets you make those choices understanding that you do have them.

This post is a cliff-notes version of the issue. Maybe you’ve heard of these differences in data types and just need a memory jog. Maybe you’ve not heard of them at all and want somewhere simple to start.

Continuous data is pretty simple: it’s data that can lie anywhere on the real line with a positive probability. That is, it can be anywhere from very large negative numbers to very large positive numbers. The normal distribution is an example of continuous data.

Truncated data, on the other hand, is data which is continuous but has the added complication of only being observed above or below a certain point. The classic example suggested by Greene is income [1]. One example would be if we only surveyed the income of those earning above the tax-free threshold: then we would have truncated data.

Censored data is similar. It’s an issue not of observation but in the way the data is sampled. Some parts of the distribution are obscured, but not ignored. The survey may, for example, interview all income levels, but only record those above the tax free threshold and describe the rest as “under the tax threshold” rather than giving the income in dollar terms. In this case all parts of the distribution are reported on, but the level of information differs above or below a threshold.

Most people are aware of issues modelling categorical data using techniques designed for continuous data. However, censored and truncated data also need special treatment. A lot of the data we deal with has a natural truncation point: distance isn’t negative, prices are not (well, hardly ever) negative. Recognising that you may be dealing with truncated or censored data is an important part of initial data analysis. For a thorough discussion, see W.H. Green’s chapter on the subject here.

In practice, continuous data methodologies may work quite well for these types of data as long as there isn’t a large amount of data sitting at or near the truncation or censoring point (which is often zero).

Test scores are something I’ve worked a lot with. In my experience, once the proportion of test scores began to approach around 20% zeros, I needed to switch over to models designed for the issue. In the 10%-20% range I will often try a few of different models to see which is most appropriate. That’s just a general rule of thumb- your mileage may vary.

Hand waving and furious graph-paper drawing aside: yes in this case knowing the differences does matter in practice.

Notes:

[1] W. H. Green, Econometric Analysis, is a classic text and here I’m looking at p. 756 in the fifth edition. There are three copies of this book living in my house. Definitely worth the investment if you are looking for either a classic text covering everything econometrics or a useful TV stand. What can I say? We were young and poor and a matched set of texts made up for deficits in our furniture budget. I’ve owned this book for nearly twenty years and I still use it- even long after we can afford furniture.

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