Does an ordinal scale have a zero point?

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 (http://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.

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B9780123747228000049

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.

Does an ordinal scale have a zero point?

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.

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B9780124058651000042

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)<ϕA(w)+ϕQ(j)⇔Fvi(x)>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)

Axiom RS. Restricted solvability. Certain restrictions apply to the solution of stochastic dominances.

The weak instrumental variable independence axiom W3 is an independence axiom like W1 and W2 and makes the factor of the coding of the response variable X independent of the experimental factor of subjects and of the factor of items (for each value x of the instrumental variable the biorder of subjects and items is the same). It makes the coding of the instrumental variable another additive factor. Cancellation is no longer restricted to subjects and items but includes the response variable factor (an increase of the response category parameter by an amount c can be compensated by an equal increase in the subject or item parameter). Axiom RS grants that the system of equations and inequalities representing stochastic dominances has a solution under certain restrictions (e.g., if the CDF of subject v at item i and category x is between the CDFs of subjects u and w at item j and category x, then there is a subject s whose CDF at j and x′ is equal to the CDF at (v, i, x); similarly for the item and category factors (Roberts 1979 p. 218).

If the pair comparison system 〈A×Q, P〉 is CADISOP then there exist scales ϕA on A, ϕQ on Q and ϕX on X such that for all (v, i, x), (w, j, x′)∈A×Q×X

ϕX(x)−(ϕA(v)+ϕQ(i))⩽ϕX(x′)−(ϕA(w)+ϕ(j))⇔Fvi(x)⩽Fwj(x′).

The scales are interval scales with a common unit (scale factor). The ‘system characteristic function’

Fvi(x)=F(ϕX(x)−(ϕA(v)+ϕQ(i)))

is strictly isotonic in its argument (Scheiblechner 1999).

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B0080430767006604

Tony Russell-Rose, Tyler Tate, in Designing the Search Experience, 2013

Range sliders

In the previous examples, the facet values are categorical in nature—qualitative data organized on a nominal or ordinal scale. But facets often need to display quantitative data, such as price ranges, product sizes, date ranges, and so on. In such cases, a range slider is often a more suitable display mechanism. An example can be found at Molecular’s Wine Store (Figure 7.13).

Does an ordinal scale have a zero point?

Figure 7.13. Range sliders at Molecular.

This example shows the use of sliders for quantitative data such as Price, Expert Score, and User Rating and for interval data such as Vintage. Note also that this example uses single-ended sliders for the first three but a double-ended slider for the latter. The rationale here is that most users would be interested only in a maximum value for price or a minimum value for Expert Score and User Rating. However, they might be interested in both a start date and an end date for a particular range of vintages.

This example also illustrates how sliders can be complemented with additional information, such as a histogram showing the distribution of record counts across the range. This option helps users understand the information landscape within each facet, guiding them toward more meaningful and productive selections.

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B9780123969811000070

Corporatism

G. Lehmbruch, in International Encyclopedia of the Social & Behavioral Sciences, 2001

2 Typological Approaches to the Study of Corporatist Interest Intermediation

Political scientists employed this more recent version of the concept of corporatism to explain empirical observations that did not square with the established interpretation of interest politics often called ‘pluralist.’ In the tradition of the ‘group approach,’ political interest groups were defined as ‘shared attitude groups that make claims upon other groups in the society,’ eventually ‘through or upon any of the institutions of government’ (Truman 1951, p. 37), and hence the ‘access’ of interest groups to government (or, with a somewhat pejorative connotation, ‘pressure politics’) was a key analytical question. However, such an approach was not particularly suited to explain the emergence, in different countries, of ‘social contracts’ between government and organized interests for voluntary incomes policy as an instrument of Keynesian economic policy. Since in these instances government was making claims upon interest groups the vectors of influence were inverted.

Generalizing from such observations, the concept of corporatism describes a type of policy formation where organized interests are encouraged to (explicitly or preemptively) coordinate their activities with government and to foster its policy objectives. Thus the analytical problem is not access but a process of exchange in which interest associations pledge organizational compliance and are often rewarded with specific organizational privileges. As a complementary terminological consequence of this theoretical reorientation of interest group research, the established notion of ‘interest representation’ was substituted by ‘interest intermediation’ (Schmitter 1979, p. 93, endnote 1).

A corporatist exchange relationship between government and organized interests tends to benefit from specific structural properties of organizations. In Schmitter's well-known definition of corporatism, a limited number of associations enjoy a monopoly in the representation of specific interests, membership may be compulsory, and leadership is clearly centralized. This enables a corporatist association to obligate the organization in its relationship with government and other associations and permits the government to hold it responsible for obtaining the compliance of the membership. Pluralism, on the other hand, is characterized by a multiplicity of associations that may compete for turf in representing specific interests, in which membership is voluntary, and leadership positions may be subject to internal contestation. A crucial element of Schmitter's influential contribution was thus the conceptualization of ‘corporatism’ as one building block of a comparative typology of organized interests.

From the original fourfold typology with its strongly deductive flavor, subsequent research has only retained the distinction of ‘corporatism’ and ‘pluralism,’ while the constructs of ‘monism’ and ‘syndicalism’ turned out to be of limited relevance for empirical research. In this typology, ‘pluralism’ carries a specific connotation derived from earlier US research on interest associations. In particular the emphasis of the ‘group approach’ (as it was developed by David Truman 1951) on society as a ‘mosaic of overlapping groups of various specialized sorts,’ with an inescapable ‘proliferation of associations’ that may eventually split and compete among each others, formed a striking contrast not only to the organizational monopolies of older ‘state corporatism’ but also to the all-encompassing ‘peak associations’ of some European democracies.

The concept of ‘corporatism’ should not be construed as an alternative to the theory of ‘pluralist democracy’ or ‘polyarchy’ with which it is, on the contrary, largely compatible. Rather, the typological distinction from ‘pluralism’ aimed specifically at modifying the ‘pluralist’ paradigm of interest group politics: ‘pluralism’ was no longer treated as a general paradigm but rather as a special type of interest group activity. The swift adoption of the typology in political science was not least due to its innovative contribution to comparative empirical research, where it facilitated the development of measurement concepts that so far had largely been missing.

2.1 The Measurement of Corporatism

In empirical research, the constructs of corporatism and pluralism should be considered as extreme types permitting the location of individual cases on (nominal or ordinal) scales. The typology has served not only to describe individual associations but also the structure of ‘interest systems’ (Schmitter 1979) of whole countries. Indeed, within particular countries the organizational properties of the most important associations and of their relationship with government and among each other are often characterized by a considerable degree of ‘organizational isomorphism’ (as defined by DiMaggio and Powell 1991). The location of countries on a corporatism scale has, in particular, become a preferred instrument of comparative research on public policies for the investigation of the impact of structures of interest intermediation on policy outcomes.

Operationalization of these typological constructs is subject to the availability of standardized indicators and data. The degree of organizational centralization of labor unions or of the collective bargaining system are most easy to observe and have in particular been employed by economists focusing on the impact of collective bargaining on macroeconomic performance (Calmfors and Driffill 1988). To describe more complex patterns of interest intermediation, some political scientists have preferred to elaborate multidimensional indexes for measuring the strength of corporatism in specific countries, focusing either on associational properties (Schmitter 1981) or on associational participation in policy making (Lehmbruch 1984).

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B0080430767011256

Influences of architectural and implementation choices on CyberInfrastructure quality—a case study

Emilia Farcas, ... Celal Ziftci, in Software Quality Assurance, 2016

13.2.3 Metrics

In general, various measurement scales can be applied for evaluating quality characteristics: (i) nominal scale (i.e., classification); (ii) ordinal scale (i.e., classification into ordered categories with little to no information on the magnitude of the differences between elements) such as the five-point scales; (iii) interval scale (i.e., indicates the exact difference between measurement points, requiring a well-defined unit of measurement for the scale and allowing the mathematical operations of addition and subtraction); and (iv) ratio scale (i.e., an interval scale with a nonarbitrary zero point, allowing all mathematical operations including division and multiplication).

Well-known metrics for assessing maintainability include the number of bugs or defect rate, lines of code, function points, Halstead volume (Halstead, 1977), and McCabe’s Cyclomatic Complexity (McCabe, 1976). Some metrics are often combined by using weights, which are determined statistically. More details on metrics can be found in Kan (2002).

Such metrics provide insight into quality aspects of a system, but they are limited by focusing on syntactic aspects that can be measured automatically, whereas many quality aspects are semantic in nature. In general, metrics are neither sufficient not necessary to indicate quality. Furthermore, the variety of languages, frameworks, and tools available make such metrics irrelevant in practice when comparing different systems. Therefore, in this chapter, we focus on evaluating quality by manual inspection of the design and code, and we typically use ordinal scales. While such analysis can be subjective (e.g., depending on the expertise of each reviewer), it covers both syntactic and semantics aspects with more insightful outcomes that can be leveraged for refactoring existing code or developing new systems.

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B9780128023013000132

Implementing Risk Management

Jack Freund, Jack Jones, in Measuring and Managing Information Risk, 2015

Multiplying red times yellow

Last but not least, most people would laugh at the idea of multiplying red times yellow, yet that is exactly equivalent to multiplying 3 times 2 when the numbers are based on an ordinal scale. We don’t need to get into the mathematical details of why this is a bad idea, but it is an incredibly common mistake in risk rating systems that often contribute to inaccurate assessments and poorly informed decisions.

TALKING ABOUT RISK

For a great read on the problems associated with doing math on ordinal scales, we suggest you read “The Failure of Risk Management” by Douglas Hubbard. In his book, Hubbard describes the reasons why this approach is worse than useless in many instances.

Obviously, the problems we’ve described above can negatively impact the quality of risk management in an organization. Perhaps not surprisingly, they can also affect compliance because the black-and-white veneer of compliance sits atop an underlying (and sometimes overlooked) need to prioritize compliance gaps and optimize gap mitigation choices. Consequently, if the “R” in an organization’s GRC implementation is bad enough, the organization may be checking compliance boxes but it may not be addressing the most important gaps first, or optimizing its gap mitigation choices. When this is the case, an argument can be made that the organization is not fully realizing even the “C” in GRC.

We would also like to point out that the problems we are describing here are to some degree less a function of poorly designed GRC products and more a matter of the inconsistent and sometimes illogical use of nomenclature in our profession combined with organizations simply not giving enough thought to these kinds of risk management considerations. Although it’s true that many (if not most) GRC products would benefit from refinements that better facilitate and guide a more mature implementation, their clientele have to set the bar higher before that’s likely to happen. It is also crucial to recognize that GRC is a set of processes and not a technology. The technology simply facilitates the processes, and both the processes and technology must support the organization’s risk management objectives, or else it’s all wasted time, money, and energy.

View chapterPurchase book

Read full chapter

URL: https://www.sciencedirect.com/science/article/pii/B9780124202313000142

Reliability

Duane F. Alwin, in Encyclopedia of Social Measurement, 2005

Scaling of Variables

The discussion to this point has assumed interval-level measurement of continuous latent variables and the use of standard Pearson-based covariance approaches to the definition of statistical associations. Observed variables measured on ordinal scales are not continuous (i.e., they do not have origins or units of measurement), and therefore should not be treated as if they are continuous. This does not mean that where the observed variables are categorical the underlying latent variable being measured cannot be assumed to be continuous. Indeed, the tetrachoric and polychoric approaches to ordered dichotomous and ordinal polytomous data assume there is an underlying continuous variable Y*, corresponding to the observed variable Y, that is normally distributed. The use of these approaches is somewhat cumbersome and labor-intensive because it requires that the estimation of the model be done in two steps. First, one estimates the polychoric or tetrachoric correlations for the observed data, which often takes a substantial amount of time, especially when the number of categories is large. In the second step, one estimates the parameters of the CTST model using maximum likelihood or weighted least squares. There are two basic strategies for estimating polychoric/tetrachoric correlations. One is to estimate the polychoric/tetrachoric correlations and thresholds jointly from all the univariate and bivariate proportions in a multiwave contingency table. This approach is computationally very complex and not generally recommended. The other, which almost always produces identical results, is to estimate the thresholds from the univariate marginals and the polychoric correlations from the bivariate marginals. In 1994, Jöreskog presented a procedure for estimating the asymptotic covariance matrix of polychoric correlations, which requires the thresholds to be equal.

Another approach to examining measurement errors, which also assumes continuous latent variables, appropriate for categoric data, is based on item response theory (IRT). Test psychologists and others have used IRT models to describe item characteristic curves in a battery of items. One specific form of the IRT model, namely Rasch models, has been suggested as one approach to modeling measurement errors. Duncan's work illustrates how the Rasch model can be applied to dichotomous variables measured on the same occasion. These approaches have not explicitly included parameters for describing reliability as defined here.

Is there a zero in ordinal scale?

The difference between an ordinal and ratio scale is that the ordinal scale has an absolute zero point.

Which scale has a zero point?

Moreover, the distance between two variables in a ratio scale is also equal in distance. In addition, a ratio scale has a true zero point, meaning the value of zero is not arbitrary.

Does interval scale include 0?

An interval scale is any range of values that have a meaningful mathematical difference but no true zero. These include everyday measurement systems like Fahrenheit and Celsius, which have set interval variables (degrees), but arbitrary zero values.