OptinMon

A normally distributed variable can have values without limits in both directions on the number line. While most variables have practical limitations, most of the time, this assumption of infinite tails is quite reasonable as there is no real boundary.

Air temperature is an example of a variable that can extend far from its mean in either direction.

But for other variables, there is a practical beginning or ending point. Age is left-bounded. It starts at zero.

The number of wins that a baseball team can have in a season is bounded on the upper end by the number of games played in a season.

The temperature of water as a liquid is bound on the low end at zero degrees Celsius and on the high end at 100 degrees Celsius.

There are two types of bounded data that have direct implications for how to work with them in analysis: censored and truncated data. Understanding the difference is a critical first step when working with these variables.

Understanding Censored and Truncated Data

Censored Data

Censored data have unknown values beyond a bound on either end of the number line or both. It can exist by design. When the data is observed and reported at the boundary, the researcher has made the decision to restrict the range of the scale.

An example of a lower censoring boundary is the recording of pollutants in our water. The researcher may not care about (or instruments may not be able to detect) the level of pollutants if it falls below a certain threshold (e.g., .005 parts per million). In this case, any pollutant level below .005 ppm is reported as “<.005 ppm.”

An upper censor could be placed on temperature in a science experiment. Once the temperature goes above x degrees the scientist doesn’t care. So s/he measures it as “>x”.

Data can be censored on both ends as well. Income could be reported as “<$20,000” if the actual is below $20,000 and reported as “ >$200,000” if above that level.

There are potential censored data not created by design. Test scores or college admission tests are examples of censored data not created by design, but by the actual bounds.  A student cannot score above 100% correct no matter how much better they know the topic than other students. These are bounded by actual results.

Truncated Data

Truncation occurs when values beyond a boundary are either excluded when gathered or excluded when analyzed. For example, if someone conducting a survey asks you if you make more than $100,000, and you answer “yes” and the surveyor says “thanks but no thanks”, then you’ve been truncated.

Or if a number of arrests is measured from police records, then everyone with 0 arrests will, by definition, be excluded from the sample.

Excluding cases from a data set at a preset boundary has the same effect. Creating models on middle income values would involve truncating income above and below specific amounts.

So to summarize, data are censored when we have partial information about the value of a variable—we know it is beyond some boundary, but not how far above or below it.

In contrast, data are truncated when the data set does not include observations in the analysis that are beyond a boundary value. Having a value beyond the boundary eliminates that individual from being in the analysis.

In truncation, it’s not just the variable of interest that we don’t have full data on. It’s all the data from that case.

Jeff Meyer is a statistical consultant with The Analysis Factor, a stats mentor for Statistically Speaking membership, and a workshop instructor. Read more about Jeff here.

Go to the next article or see the full series on Easy-to-Confuse Statistical Concepts

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Principal Component Analysis is really, really useful.

You use it to create a single index variable from a set of correlated variables.

In fact, the very first step in Principal Component Analysis is to create a correlation matrix (a.k.a., a table of bivariate correlations). The rest of the analysis is based on this correlation matrix.

You don’t usually see this step — it happens behind the scenes in your software.

Most PCA procedures calculate that first step using only one type of correlations: Pearson.

And that can be a problem. Pearson correlations assume all variables are normally distributed. That means they have to be truly (more…)

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Outliers are one of those realities of data analysis that no one can avoid.

Those pesky extreme values cause biased parameter estimates, non-normality in otherwise beautifully normal variables, and inflated variances.

Everyone agrees that outliers cause trouble with parametric analyses. But not everyone agrees that they’re always a problem, or what to do about them even if they are.

Sometimes a nonparametric or robust alternative is available — and sometimes not.

There are a number of approaches in statistical analysis for dealing with outliers and the problems they create. It’s common for committee members or Reviewer #2 to have very strong opinions that there is one and only one good approach.

Two approaches that I’ve commonly seen are: 1) delete outliers from the sample, or 2) winsorize them (i.e., replace the outlier value with one that is less extreme).

The problem with both of these “solutions” is that they also cause problems — biased parameter estimates and underweighted or eliminated valid values. (more…)

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An incredibly useful tool in evaluating and comparing predictive models is the ROC curve.

Its name is indeed strange. ROC stands for Receiver Operating Characteristic. Its origin is from sonar back in the 1940s. ROCs were used to measure how well a sonar signal (e.g., from an enemy submarine) could be detected from noise (a school of fish).

ROC curves are a nice way to see how any predictive model can distinguish between the true positives and negatives. (more…)

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In a previous post we discussed the difficulties of spotting meaningful information when we work with a large panel data set.

Observing the data collapsed into groups, such as quartiles or deciles, is one approach to tackling this challenging task.  We showed how this can be easily done in Stata using just 10 lines of code.

As promised, we will now show you how to graph the collapsed data. (more…)

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Panel data provides us with observations over several time periods per subject. In this first of two blog posts, I’ll walk you through the process. (Stick with me here. In Part 2, I’ll show you the graph, I promise.)

The challenge is that some of these data sets are massive. For example, if we’ve collected data on 100,000 individuals over 15 time periods, then that means we have 1.5 million cells of information.

So how can we look through this massive amount of data and observe trends over the time periods that we have tracked? (more…)

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