level of measurement

Level is a term in statistics that is confusing because it has multiple meanings in different contexts (much like alpha and beta).

There are three different uses of the term Level in statistics that mean completely different things. What makes this especially confusing is that all three of them can be used in the exact same research context.

I’ll show you an example of that at the end.

So when you’re talking to someone who is learning statistics or who happens to be thinking of that term in a different context, this gets especially confusing.

Levels of Measurement

The most widespread of these is levels of measurement. Stanley Stevens came up with this taxonomy of assigning numerals to variables in the 1940s. You probably learned about them in your Intro Stats course: the nominal, ordinal, interval, and ratio levels.

Levels of measurement is really a measurement concept, not a statistical one. It refers to how much and the type of information a variable contains. Does it indicate an unordered category, a quantity with a zero point, etc?

It is important in statistics because it has a big impact on which statistics are appropriate for any given variable. For example, you would not do the same test of association between two nominal variables as you would between two interval variables.

Levels of a Factor

A related concept is a Factor. Although Factor itself has multiple meanings in statistics, here we are talking about a categorical predictor variable.

The typical use of Factor as a categorical predictor variable comes from experimental design. In experimental design, the predictor variables (also often called Independent Variables) are generally categorical and nominal. They represent different experimental conditions, like treatment and control traditions.

Each of these categorical conditions is called a level.

Here are a few examples:

In an agricultural study, a fertilizer treatment variable has three levels: Organic fertilizer (composted manure); High concentration of chemical fertilizer; low concentration of chemical fertilizer.

In a medical study, a drug treatment has four levels: Placebo; low dosage; medium dosage; high dosage.

In a linguistics study, a word frequency variable has two levels: high frequency words; low frequency words.

Although this use of level is very widespread, I try to avoid it personally. Instead I use the word “value” or “category” both of which are accurate, but without other meanings.

Level in Multilevel Models

A completely different use of the term is in the context of multilevel models. Multilevel models is a term for some mixed models. (The terms multilevel models and mixed models are often used interchangably, though mixed model is a bit more flexible).

Multilevel models have two or more sources of random variation.  A two level model has two sources of random variation and can have predictors at each level.

A common example is a model from a design where the response variable of interest is measured on students. It’s hard though, to sample students directly or to randomly assign them to treatments, since there is a natural clustering of students within schools.

So the resource-efficient way to do this research is to sample students within schools.

Predictors can be measured at the student level (eg. gender, SES, age) or the school level (enrollment, % who go on to college).  The dependent variable has variation from student to student (level 1) and from school to school (level 2).

We always count these levels from the bottom up. So if we have students clustered within classroom and classroom clustered within school and school clustered within district, we have:

• Level 1: Students
• Level 2: Classroom
• Level 3: School
• Level 4: District

So this use of the term level describes the design of the study, not the measurement of the variables or the categories of the factors.

Putting them together

So this is the truly unfortunate part. There are situations where all three definitions of level are relevant within the same statistical analysis context.

I find this unfortunate because I think using the same word to mean completely different things just confuses people. But here it is:

Picture that study in which students are clustered within school (a two-level design). Each school is assigned to use one of three math curricula (the independent variable, which happens to be categorical).

So, the variable “math curriculum” is a factor with 3 levels (ie, three categories). Because those three categories of “math curriculum” are unordered, “math curriculum” has a nominal level of measurement. And since “math curriculum” is assigned to each school, it is considered a level 2 variable in the two-level model.

See the rest of the Confusing Statistical Terms series.