When a Variable’s Level of Measurement Isn’t Obvious

by Karen Grace-Martin

A central concept in statistics is level of measurement of variables. It’s so important to everything you do with data that it’s usually taught within the first week in every intro stats class.

But even something so fundamental can be tricky once you start working with real data. The same variable can be considered to have different levels of measurement in different situations. It sounded like an absolute in that intro stats class because your wise professor didn’t want to confuse beginning students.

But now that you’re a more sophisticated student of data analysis, I will show you how the same variable can be considered to have different levels of measurement. But first, let me review some definitions.

A Review of the Levels of Measurement of Variables

Nominal:

Unordered categorical variables. These can be either binary (only two categories, like gender: male or female) or multinomial (more than two categories, like marital status: married, divorced, never married, widowed, separated). The key thing here is that there is no logical order to the categories.

Ordinal:

Ordered categories. Still categorical, but in an order. Likert items with responses like: “Never, Sometimes, Often, Always” are ordinal.

Interval:

Numerical values without a true zero point. The idea here is the intervals between the values are equal and meaningful, but the numbers themselves are arbitrary. 0 does not indicate a complete lack of the quantity being measured. IQ and degrees Celsius or Fahrenheit are both interval.

Ratio:

Numerical values with a true zero point.

Interval and Ratio variables can be further split into two types: discrete and continuous. Discrete variables, like counts, can only take on whole numbers: number of children in a family, number of days missed from work. Continuous variables can take on any number, even beyond the decimal point.

Not always obvious is that these levels of measurement are not only about the variable itself. Also important are the meaning of the variable within the research context and how it was measured.

An Example: Age

A great example of this is a variable like age. Age is, technically, continuous and ratio. A person’s age does, after all, have a meaningful zero point (birth) and is continuous if you measure it precisely enough. It is meaningful to say that someone (or something) is 7.28 year old.

That said, you may not be able to treat it as continuous in your analysis. It depends on how you measured it and whether there are qualitative implications about age in your research context.  Here are 5 examples in which Age has another level of measurement:

Age as Ordinal

For example, it’s not uncommon to give people age categories as possible responses on a survey. Common reasons are that people don’t want to reveal their actual age or because they don’t remember the actual age at which some event occurred.

I worked with a client whose dependent variable was the age at which adult smokers started smoking. It would have been great to get an accurate date on which each person smoked their first cigarette, but it’s a big burden on respondents to ask them a very specific number from a long time ago.

Rather than have respondents guess inaccurately or leave the answer blank, the researchers gave them a series of ordered age categories: 0 to 10, 11-12, 13-15, 16-17, etc. They gave up precision to gain accuracy.

Ordinal response variables require a model like an Ordinal Logistic Regression.

Age as Discrete Counts

Likewise, a continuous variable may be rendered discrete because of the way people think about and measure it.

For example, consider the example of age measured in days on which germinated seeds of a specific species begin to sprout leaves. Most will do so within a few days, and it may range from 2-9 days.

In this context, age is definitely a discrete count—the number of days. If it is used as an outcome variable, a Poisson (or related) regression would be appropriate, not a linear model.

Age as Multinomial

Sometimes numerical variables are rendered categorical due to the lack of values.

In one study I analyzed, the key independent variable was the age of a witness in a trial. While technically, ages are continuous, in this study there were only four values: 49, 69, 79 and 89.

So even though one could use statistics that treated this variable as continuous, they don’t make a lot of sense. In a linear model, if you treat this age variable as a numerical predictor, the model will fit a regression line across these four ages. If you treat it as categorical, the line will fit, and allow you to compare, the mean of Y at each age.

The effect of age in this context is better measured through a difference in the mean of Y at two different ages than through a slope–the difference in Y for each one year increase.

Now if your multinomial age variable is the response, you’ll need a multinomial logistic regression.

Age as Binary Categories

In a similar example, a researcher was studying math abilities in first grade children. The key independent variable was whether the child had reached a specific cognitive developmental milestone and the dependent variable was math score. Age was a control variable and it was mildly related to, but not confounded, with attainment of the milestone.

Because each child was asked how old they were, it was measured in whole years. It would have been ideal to collect more specific data on ages—such as their birth dates from their parents or school records. For whatever reason, it wasn’t possible.

So the only two values for age were 6 and 7.  So just like in the last example, it only made sense to treat this predictor variable as categorical in the analysis.

If you had a binary outcome variable, you’d most likely need a binary logistic regression.

Age as Binary Categories (another one)

In a study comparing the work-life balance of men and women, the outcome variable was number of hours worked per week. One key predictor for women, but not men, was the age of their youngest child.

There is a qualitative difference between a 5 year old, who may only be eligible for part-time kindergarten  and a 6 year old, who is old enough to go to full-time school.

This qualitative difference exists in this context between 5 and 6 that doesn’t exist at other one-year age differences*. This qualitative difference is in fact the most important feature of the youngest child’s age. Treating age as continuous actually ignores this important qualitative difference.

Notice that both of these binary examples are very different situation from doing a median split on a continuous variable.

That kind of categorizing isn’t a good idea because you’re throwing away good information based on an arbitrary cutoff.

*It also doesn’t exist in other contexts. The difference between ages 5 and 6 wouldn’t be important if you’re studying drug use or retirement planning.

{ 8 comments… read them below or add one }

laura

hi Karen, great article. Am trying to figure out what my variables are for a correlational study. Emotional support and Reoffense rates are my variables. Trying to determine the negative relationship between the two, i believe emotional support will decrease the rate of recidivism. Would either of my variables be interval or ratio?

Reply

kemi

is drug dosage a ration or interval measure?

thanks for your response in advance

Reply

Karen Grace-Martin

Hi Kemi,

As I mentioned in the article, it depends on more than just the variable. In your design there may be only two drug dosages or there may be an infinite number.

Reply

Verena Richter

Dear Karen,

your article was very helpful. I am trying to figure out the levels of measurement for some variables I am using. First I have militarized interstate disputes ( I count how many there are during a year-I have determined this to be ratio), second GDP and military spending (again I think ratio), the one I can’t figure out is my independent variable time (years) years and Militarized interstate disputes will be analyzed together. I would be very thankful for your help.
Thank you
Verena

Reply

Karen Grace-Martin

Hi Verena,

Years is very similar to the variable I used in the article: Age. Technically it’s ratio, but it all depends on how detailed the data was collected and how it’s going to be used.

Reply

Laura

Thanks for Your article! I am a beginner and I am trying to make a little analysis where dependent variable is monthly wage. In the data set it is measured in intervals (up to 370eur, 371-450, 451-700, 701-1400, 1401 and more). These are quite huge intervals and I am not sure how to approach them. Would You have any suggestion?

Reply

Karen Grace-Martin

Hi Laura,

The fact that they’re huge intervals won’t affect how you treat them. Those are definitely ordinal categories.

Reply

henia heller

thanks for the insightfull article. I will share it with my new group graduates

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