The following statement might surprise you, but it’s true.

To run a linear model, you don’t need an outcome variable Y that’s normally distributed. Instead, you need a dependent variable that is:

- Continuous
- Unbounded
- Measured on an interval or ratio scale

The normality assumption is about the errors in the model, which have the same distribution as Y|X. It’s absolutely possible to have a skewed distribution of Y and a normal distribution of errors because of the effect of X. (more…)

**by Jeff Meyer**

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*.

by Jeff Meyer

In a previous post we explored bounded variables and the difference between truncated and censored. Can we ignore the fact that a variable is bounded and just run our analysis as if the data *wasn’t* bounded? (more…)