ch04levelsofmeasurement.Rmd

# Levels of measurement {#ch-levelsofmeasurement}

## Introduction

In Chapter \@ref(ch-research), we were already introduced to variables: properties that can take different values. As we know, a variable's value is a way of indicating a property or quality of an object or person. If we are dealing of a dependent variable, this value may also be called a *score* or *response*, often represented with the symbol $Y$. The way in which a property is expressed in a measurable value is called the variable's *level of measurement*; thus, level of measurement is an inherent property of the variable itself! We distinguish four levels of measurement, in order of increasing informativeness: nominal, ordinal, interval, ratio. For the former two levels of measurement, only discrete categories are distinguished, with or without ordering. The latter two levels of measurement use numerical values, with or without a zero point. We will discuss the levels of measurement in more detail below. Insight into a variable's level of measurement is important for interpreting scores for that variable, and -- as we will see later -- for choosing the correct statistical test to answer a research question.

## Nominal {#sec:nominal}

We speak of a nominal variable (or a nominal level of measurement) when a property is categorized into separate (discrete) categories that have no order between them. Well-known examples include a participant's nationality, a car's make, the colour of someone's eyes, the flavour of a tub of ice cream, one's living arrangements (with one's family, with housemates, living independently, living with a partner, other), etc. Scores may only be used to distinguish between the categories (a statement like, "vanilla is different from strawberry" does make sense). We can, indeed, count how often each category occurs, but there is no interpretable order (the statement, "vanilla is larger than strawberry" does not make sense), and we can also not do any arithmetic on the values measured for a nominal variable. For instance, we can determine the most frequently occurring nationality, but we cannot calculate the average nationality.


## Ordinal {#sec:ordinal}

We speak of an ordinal variable (or an ordinal level of measurement) when a property is categorized into separate categories that do have an *order* or ranking between them. However, in the case of an ordinal variable, we do not know anything about the distance between the various categories. Well-known examples include level of education (primary education, secondary education, bachelor's degree, master's degree/PhD, ...), answer on a scale question (*agree, do not know, disagree*), position within a ranking, order of elimination in a talent show, clothing size (XS, S, M, L, XL, …), or military rank (soldier, major, general, …). Here, as well, we can count how often each category occurs, and we can even sensibly interpret the rank order (whoever is eliminated last has performed better than whoever is eliminated first, size L is greater than size M, a general outranks a major). However, we still can do no arithmetic on the values measured for an ordinal variable. We may determine the bestselling clothing size, but we cannot calculate the average clothing size sold[^fn04-1].

## Interval {#sec:interval}

We speak of an interval variable (or an interval level of measurement) when a property is expressed as a number on a continuous scale for which there is *no zero point*. Because of the scale, we know what the distances or intervals are between the various values of an interval variable. Well-known examples include temperature in degrees Celsius (the zero point is arbitrary) or calendar year (ditto for this zero point). We can count how often each category occurs, we can sensibly interpret the rank order (in our Gregorian calendar, the year 1999 preceded the year 2000), and we can also sensibly interpret the intervals (the interval between 1918 and 1939 is just as long as that between 1989 and 2010). We may, indeed, do arithmetic on the values of an interval variable, but the only operations that make sense are addition and subtraction. These are enough to calculate an average, e.g., the average year in which the individuals in the sample obtained their first mobile phone.


## Ratio {#sec:ratio}

The fourth and highest level of measurement is the ratio level. We speak of a ratio variable (or a ratio level of measurement) when a property is expressed as a number on a continuous scale for which there is, indeed, a *zero point*. Because of the scale, we know what the distances or intervals are between the various values of a ratio variable. In addition, because of the zero point, we know what the proportions or ratios are between the various values (hence the name of this level). Well-known examples include temperature in degree Kelvin (measured from absolute zero), response time[^fn04-2] in thousandths of a second (ms), your height in centimetres (cm), your age in years, the number of errors made on a test, etc. When we are dealing with a ratio variable, we can count how often each category occurs, we can sensibly interpret the rank order (someone whose height is 180 cm is taller than someone whose height is 179 cm), we can sensibly interpret intervals (the increase in age between 12 and 18 is two times as large as that between 9 to 12), and we can also sensibly interpret proportions between the values (the age of 24 is *twice* as great as the age of 12). We may do arithmetic on the values of a ratio variable, which includes not just addition and subtraction, but also division and multiplication. Here, as well, it is possible to calculate an average, e.g., the average age at which the individuals in the sample obtained their first mobile phone.


## Ordering of levels of measurement {#sec:orderinglevelsofmeasurement}

In the above, we have discussed the levels of measurement in order of increasing informativeness or strength. A nominal variable contains the least amount of information and is considered the lowest level of measurement, while a ratio variable contains the greatest amount of information and is considered the highest level of measurement.

It is always possible to reinterpret data measured at a high level of measurement as if they had been measured at a lower level. For instance, if, for each individual in a sample, we had measured their monthly income at a ratio level (in €), we would be able to make an ordinal variable out of this with no problem (e.g. *less than average, average to twice the average, more than twice the average*). This would mean discarding information: the original measurements in terms of € contain more information than the classification into three ordered categories derived from it.

Of course, the opposite is not possible: a variable at a low level of measurement cannot be reinterpreted at a higher level. We would have to add, after the fact, information that we did not collect during the original measurement of this variable. It is therefore imperative to observe the relevant variables at the correct level of measurement. Supposed we wanted to compare body height  in adult men and women. If we measure body height at an ordinal level (having defined three categories, *short, medium, and tall*, equally for all individuals), this means that we cannot calculate the average body height, and we can also not use any statistical test that would refer to the average body height. This does not have to be a problem, but it is a good idea to think through the consequences of using a particular level of measurement in advance of the actual measurement.

[^fn04-1]: If half of our respondents answers *agree*, and the other half answers *disagree*, it does not make sense to conclude that the responses are *neutral* on average.

[^fn04-2]: The zero point is the moment in time when the event begins that the participant is to respond to.