# Measurement Theory: Fallacies and Transformations (Nominal, Ordinal, Interval, Ratio)

The Winter 1993 issue of "The Statistical Consultant", published by the Section of Statistical Consulting Education of the American Statistical Association, contains excerpts from an email exchange led by Warren Sarle and Paul Velleman. The discussion started from the notion that data conform to "levels of measurement" based on S.S. Stevens' classification of Nominal, Ordinal, Interval, Ratio. Debate hinged on the extent to which analysis of such data should be controlled by the "permissible transformations" that maintain the mathematical properties of numbers reported on each level. Jack Stenner forwarded a copy of this discussion (provided by Donald Burdick) for comment.

Dear Jack:

Thank you for the copy of "The Statistical Consultant" with its discussion on Measurement Theory. It highlights an essential difference between a Rasch, theory-based approach and the usual statistical, description-based approach.

They argue about the observed "levels" of measurement and "permissible" transformations, as though the data is forced onto us with all its attributes. In nature, all things we observe are nominal. It is we who choose to order (i.e., count) them, in some way according to some theory we propose. A further step is to transform these orderings into linear measures which are more useful to us. If we construct a theory with a useful "zero" location (as opposed to zero difference), then we can measure away from that zero point and so construct a ratio scale. To say that degrees Kelvin are on a ratio scale, but degrees Celsius are not, is fallacious. They are both ratio scales, but with "zeros" defined according to different theories of heat. Frankly, I would like a temperature scale with its zero at room temperature, then "twice as hot", indicating twice as far from room temperature, would be meaningful to me.

Over the course of time, we have developed all types of measurement and counting devices to assist us in summarizing the common features of nominal nature. How these summaries relate to the linear (or ratio) relationships we require for quantitative understanding (in the context of any particular theory) is not initially clear in any situation. How does reaction time function as an indicator of physical condition? Or weight of food relate to overall health?

It is not a matter of identifying permissible transformations of time or weight. First, it is a matter of reordering the observed values so that more "time" or "weight" indicates better condition or health. Then it is a matter of linearizing so that the values provide a basis for inference. As linear measures they may have low precision and poor quality (fit) in the context of our theory, no matter how sophisticated the electronic devices employed in the data collection. Of course, it is Rasch that provides us with the linearizing mechanism.

John Michael Linacre

Note: Thomas O'Neill summarizes this as:
1. the answer to the question being nominal,
2. the scoring of the answer to the question as 0,1, ... being ordinal,
3. the creation of abstract units (Rasch) from the scored answers as being interval (additive, linear),
4. the creation of a meaningful zero on the scale of abstract units as being ratio.

Measurement theory: fallacies and transformations (Nominal, Ordinal, Interval, Ratio). Linacre JM. … Rasch Measurement Transactions, 1994, 8:1 p.340

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