Does Ordinal level Measurement Use Continuous Variables

A central concept in statistics is a variable's level of measurement. 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 practitioner 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 Variables' Levels of Measurement

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 will estimate means 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.

Interpreting Linear Regression Coefficients: A Walk Through Output

Learn the approach for understanding coefficients in that regression as we walk through output of a model that includes numerical and categorical predictors and an interaction.

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Source: https://www.theanalysisfactor.com/level-of-measurement-not-obvious/

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