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According to Stevens (1946) ratio level variables require an absolute zero. The implication, I believe, is that 0 = an absence of the thing being measured.

As such, can ratio level scales have negative measurements? We can manipulate the measurements mathematically to get negative numbers, but can we have actual negative measurements?

An internet search results in competing arguments, and I cannot find a peer-reviewed academic work that provides an answer.

Against negative numbers: http://www.statisticshowto.com/ratio-scale/ https://www.dummies.com/education/science/biology/levels-of-measurement-for-biostatistics-data/

For negative numbers (they all seem to use bank balances): http://web.pdx.edu/~newsomj/pa551/lecture1.htm https://www.spss-tutorials.com/measurement-levels/

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    $\begingroup$ A good example of a ratio scale is temperature. It makes no sense to say 100 degrees F is twice as hot as 50 degrees F. In Celsius that would be 38 and 10, which don't have a 2:1 ratio. But absolute temp is allegedly calibrated so that 0 means no molecular activity. So on an absolute scale, it is meaningful to say one temp is twice as hot as another. On an absolute temperature scale you can't have less than 'no molecular activity' (ignoring quantum stuff), so negative values wouldn't make sense. // I don't know about bank balances: \$20 is twice \$10, but a negative bank balance can happen. $\endgroup$
    – BruceET
    Commented Sep 13, 2018 at 23:19
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    $\begingroup$ @Bruce Temperature is a good example for the many subtleties it reveals. It took hundreds of years of theory and experiment to determine a suitable way to define and measure temperature: that is, to construct an appropriate scale of measurement. The quantum mechanical issues are no less intriguing, because at absolute zero there is molecular motion (even in classical QM where relativistic effects are ignored). $\endgroup$
    – whuber ♦
    Commented Sep 14, 2018 at 3:50

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By definition, ratio scales include an absolute zero point. If it's possible to have negative quantities than it's not an absolute zero.

For example, in the physical realm, you can have 0 or 3 apples but you can't physically have -1 apples. Because the concept you are measuring is the existence of apples it's possible to use a ratio scale.

But if the thing you were measuring is the abstract concept of possession of apples, you can have -1 apples. But this means it's no longer on the ratio scale but is instead on the interval scale. Since it's possible to owe 10 apples or 10M apples, it's no longer possible to define an absolute zero. Any choice of zero becomes arbitrary.

As another example, the Kelvin temperature system is a ratio scale and 0°K is representative of an absolute physical inexistence of kinetic energy. While -273.15°C and -459.67°F are equal to 0°K, °C/°F are considered interval scales. That's because they are actually representing the concept of existence of energy in relationship to some criteria, like the freezing temperature of water. This again makes zero an arbitrary point which means it's impossible to define an absolute zero.

Without an absolute zero, ratios of values have no meaning.

In Stevens, S.S., 1946. On the theory of scales of measurement. he says:

If, in addition, a constant can be added (or a new zero point chosen), it is proof positive that we are not concerned with a ratio scale.

More importantly, though, Steven's framework is a tool for thinking, not an absolute truth. You can read about how it fails in real data in Velleman, P.F. and Wilkinson, L., 1993. Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), pp.65-72.:

Many of the discussions of scale types, and virtually all of the mathematical results, treat them as absolute categories. Data are expected to fit into one or another of the categories. A failure to attain one level of measurement is taken as a demotion to the next level. However, real data do not follow the requirements of many scale types.

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  • $\begingroup$ Counts--such as apple counts--are not on a ratio scale, at least not according to Stevens' definitions. Your remark suggesting "it's impossible to define an absolute zero" for temperature is strange, because it seems to contradict your previous argument. $\endgroup$
    – whuber ♦
    Commented Jan 11, 2020 at 21:37

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