To measure an unknown mineral, you try to scratch the polished surface of one of the standard minerals with it; if it scratches the surface, the unknown is harder. Notice that I can get two different unknown minerals with the same measurement that are not equal to each other, and that I can get minerals that are softer than my lower bound or harder than my upper bound. There is no origin point and operations on the measurements make no sense [e.g., if I add 10 talc units I do not get a diamond].
A common use we see of ordinal scales today is to measure preferences or opinions. You are given a product or a situation and asked to decide how much you like or dislike it, how much you agree or disagree with a statement, and so forth. The scale is usually given a set of labels such as “strongly agree” through “strongly disagree,” or the labels are ordered from 1 to 5.
Consider pairwise choices between ice cream flavors. Saying that “Vanilla” is preferred over “Rotting Leather” in our taste test might well be expressing a universal truth, but there is no objective unit of “likeability” to apply. The lack of a unit means that such things as opinion polls that try to average such scales are meaningless; the best you can do is a histogram of respondents in each category.
Many times an ordinal scale can be made more precise. Dr. Wilbur Scofield developed the Scofield scale in 1912 to measure the strength of peppers. Before the Scofield scale, peppers were informally rated by the personal preferences of cook book authors. There is a long joke [//www.dabearz.com/forums/f56/yankee-chili-cookoff-33291/] about a New Yorker volunteering to be a judge at a Texas chili competition. While the two native judges can report on the ingredients, the New Yorker is dying from chemical burns.
Likewise, temperature scales on an air conditioner are shown as cool to warm instead of a degree on a Celsius temperature scale.
Another problem is that an ordinal scale may not be transitive. Transitivity is the property of a relationship in which if R[a, b] and R[b, c] then R[a, c]. We like this property, and expect it in the real world where we have relationships like “heavier than,” “older than,” and so forth. This is the result of a strong metric property.
The most common example is the finger game “scissors, paper, stone” in which two players make their moves simultaneously [see The Official Rock Paper Scissors Strategy Guide by Douglas and Graham Walker]. There are also Efron dice and other recreational mathematic puzzles that are nontransitive.
This problem tends to show up with human preferences in the real world. Imagine an ice cream taster who has just found out that the shop is out of vanilla and he is left with squid, wet leather, and wood as flavor options. He might prefer squid over wet leather, prefer wet leather over wood, and yet prefer wood over squid ice cream, so there is no metric function or linear ordering at all. Again, we are into philosophical differences, since many people do not consider a nontransitive relationship to be a scale.
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Scientific Foundations
I. Scott MacKenzie, in Human-computer Interaction, 2013
4.4.4 Ordinal data
Ordinal scale measurements provide an order or ranking to an attribute. The attribute can be any characteristic or circumstance of interest. For example, users might be asked to try three global positioning systems [GPS] for a period of time and then rank the systems by preference: first choice, second choice, third choice. Or users could be asked to consider properties of a mobile phone such as price, features, cool-appeal, and usability, and then order the features by personal importance. One user might choose usability [first], cool-appeal [second], price [third], and then features [fourth]. The main limitation of ordinal data is that the interval is not intrinsically equal between successive points on the scale. In the example just cited, there is no innate sense of how much more important usability is over cool-appeal or whether the difference is greater or less than that between, for example, cool-appeal and price.
If we are interested in studying users’ e-mail habits, we might use a questionnaire to collect data. Figure 4.6 gives an example of a questionnaire item soliciting ordinal data. There are five rankings according to the number of e-mail messages received per day. It is a matter of choice whether to solicit data in this manner or, in the alternative, to ask for an estimate of the number of e-mail messages received per day. It will depend on how the data are used and analyzed.
Figure 4.6. Example of a questionnaire item soliciting an ordinal response.
Ordinal data are slightly more sophisticated than nominal data since comparisons of greater than or less than are possible. However, it is not valid to compute the mean of ordinal data.
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Test Theory: Applied Probabilistic Measurement Structures
H. Scheiblechner, in International Encyclopedia of the Social & Behavioral Sciences, 2001
3 The Construction of Interval Scales for Subjects and Items: the Additive Conjoint ISOP [ADISOP] and the Complete ADISOP [CADISOP]
The axioms of the ISOP model are sufficient for the construction of two separate unidimensional ordinal scales on the sets of subjects and items. We now strengthen the ISOP model by additional axioms in order to derive ordered metric/interval scales for subjects and items. First the ADISOP model is considered which is an ISOP model with a cancellation axiom added that relates the subject and item factors.
Definition. A probabilistic pair comparison system 〈A×Q, Fvi[x]〉, where Fvi[x] is a family of cumulative distribution functions defined on A×Q, is an ADISOP iff the axioms of the ISOP model and the following axiom are satisfied.
Axiom Co. Validity of cancellation of subjects and items up to order o.
[Similar structures are the PACOME models, Falmagne 1979]. An explanation of cancellation up to order o is given in Scheiblechner [1999] and Scott [1964]. It means that a reduction of the item parameter by an amount δ can be compensated by an increase in the subject parameter by the same amount δ. The subject and item parameters ‘cancel.’ Or if we attribute real values to subjects and items then the order of their sum is isotonic in the order of their distribution functions. For example, if the ‘ability’ parameter of subject v plus the ‘easiness’ parameter of item i is larger than or equal to the sum of the ability and easiness parameter of subject w and item j, then the distribution functions are isotonic in the sum, i.e., Fvix≿Fwjxand the reverse.
If the system 〈A×Q, P〉 is an ADISOP model then the order of the CDFs is identical to the order of the sums of the subject and item parameters, i.e., there exist real functions ϕA, ϕQ on A and Q and ψA ×Q[v, i]=ϕA[v]+ϕQ[i] on A×Q such that for all [v, i], [w, j]∈A×Q
ϕA[v]+ϕQ[i]Fwj[x]
for somexor⇔Fvix≺Fwjx
The scales ϕA, ϕQ are unique up to positive linear transformations, i.e., they are interval scales if Axiom Co is valid up to arbitrary order o, and they are ordered metric scales if Co is valid up to the finite order [min{n, k}−1] where n and k are the number of elements of A and Q, respectively. Ordered metric scales are unique up to monotonic transformations which leave invariant the order of the [finite number of] intervals. Ordered metric scales are approximations to interval scales. The cumulative response distribution functions on the product set of subjects and items
Fvi[x]=F[ϕ[x];ϕA[v]+ϕQ[i]]
are antitonic [do not increase] in ψA×Q[v, i]=ϕA[v]+ϕQ[i] and isotonic [do not decrease] in ϕ[x], where ϕ[x] is an arbitrary strictly isotonic function of x [i.e., is an ordinal scale]. The functions ϕA[v] and ϕQ[i] are interval scales [ordered metric scales] with an arbitrary origin [additive constant] and a common unit [scale factor].
The ADISOP model can further be strengthened to a model where the response category parameter also is additive to the item and subject parameter. The axiom W3 strengthens Co and extends cancellation from subjects and items to include subjects and instrumental variable [response category] and items and response variable.
Definition. A complete ADISOP [CADISOP] model is an ADISOP with axiom [W3] [and RS] replacing axiom Co:
Axiom W3. Weak instrumental variable independence holds if: Fvi[x]