Archive for the ‘natural law’ Category

Externalities are to markets as anomalies are to scientific laws

October 28, 2011

Economic externalities are to efficient markets as any consistent anomaly is relative to a lawful regularity. Government intervention in markets is akin to fudging the laws of physics to explain the wobble in Uranus’ orbit, or to explain why magnetized masses would not behave like wooden or stone masses in a metal catapult (Rasch’s example). Further, government intervention in markets is necessary only as long as efficient markets for externalized forms of capital are not created. The anomalous exceptions to the general rule of market efficiency have long since been shown to themselves be internally consistent lawful regularities in their own right amenable to configuration as markets for human, social and natural forms of capital.

There is an opportunity here for the concise and elegant statement of the efficient markets hypothesis, the observation of certain anomalies, the formulation of new theories concerning these forms of capital, the framing of efficient markets hypotheses concerning the behavior of these anomalies, tests of these hypotheses in terms of the inverse proportionality of two of the parameters relative to the third, proposals as to the uniform metrics by which the scientific laws will be made commercially viable expressions of capital value, etc.

We suffer from the illusion that trading activity somehow spontaneously emerges from social interactions. It’s as though comparable equivalent value is some kind of irrefutable, incontestable feature of the world to which humanity adapts its institutions. But this order of things plainly puts the cart before the horse when the emergence of markets is viewed historically. The idea of fair trade, how it is arranged, how it is recognized, when it is appropriate, etc. varies markedly across cultures and over time.

Yes, “’the price of things is in inverse ratio to the quantity offered and in direct ratio to the quantity demanded’ (Walras 1965, I, 216-17)” (Mirowski, 1988, p. 20). Yes, Pareto made “a direct extrapolation of the path-independence of equilibrium energy states in rational mechanics and thermodynamics” to “the path-independence of the realization of utility” (Mirowski, 1988, p. 21). Yes, as Ehrenfest showed, “an analogy between thermodynamics and economics” can be made, and economic concepts can be formulated “as parallels of thermodynamic concepts, with the concept of equilibrium occupying the central position in both theories” (Boumans, 2005, p. 31).  But markets are built up around these lawful regularities by skilled actors who articulate the rules, embody the roles, and initiate the relationships comprising economic, legal, and scientific institutions. “The institutions define the market, rather than the reverse” (Miller & O’Leary, 2007, p. 710). What we need are new institutions built up around the lawful regularities revealed by Rasch models. The problem is how to articulate the rules, embody the roles, and initiate the relationships.

Noyes (1936, pp. 2, 13; quoted in De Soto 2000, p. 158) provides some useful pointers:

“The chips in the economic game today are not so much the physical goods and actual services that are almost exclusively considered in economic text books, as they are that elaboration of legal relations which we call property…. One is led, by studying its development, to conceive the social reality as a web of intangible bonds–a cobweb of invisible filaments–which surround and engage the individual and which thereby organize society…. And the process of coming to grips with the actual world we live in is the process of objectivizing these relations.”

 Noyes (1936, p. 20, quoted in De Soto 2000, p. 163) continues:

“Human nature demands regularity and certainty and this demand requires that these primitive judgments be consistent and thus be permitted to crystallize into certain rules–into ‘this body of dogma or systematized prediction which we call law.’ … The practical convenience of the public … leads to the recurrent efforts to systematize the body of laws. The demand for codification is a demand of the people to be released from the mystery and uncertainty of unwritten or even of case law.” [This is quite an apt statement of the largely unstated demands of the Occupy Wall Street movement.]

  De Soto (2000, p. 158) explains:

 “Lifting the bell jar [integrating legal and extralegal property rights], then, is principally a legal challenge. The official legal order must interact with extralegal arrangements outside the bell jar to create a social contract on property and capital. To achieve this integration, many other disciplines are of course necessary … [economists, urban planners, agronomists, mappers, surveyers, IT specialists, etc]. But ultimately, an integrated national social contract will be concretized only in laws.”

  “Implementing major legal change is a political responsibility. There are various reasons for this. First, law is generally concerned with protecting property rights. However, the real task in developing and former communist countries is not so much to perfect existing rights as to give everyone a right to property rights–‘meta-rights,’ if you will. [Paraphrasing, the real task in the undeveloped domains of human, social, and natural capital is not so much the perfection of existing rights as it is to harness scientific measurement in the name of economic justice and grant everyone legal title to their shares of their ownmost personal properties, their abilities, health, motivations, and trustworthiness, along with their shares of the common stock of social and natural resources.] Bestowing such meta-rights, emancipating people from bad law, is a political job. Second, very small but powerful vested interests–mostly repre- [p. 159] sented by the countries best commercial lawyers–are likely to oppose change unless they are convinced otherwise. Bringing well-connected and moneyed people onto the bandwagon requires not consultants committed to serving their clients but talented politicians committed to serving their people. Third, creating an integrated system is not about drafting laws and regulations that look good on paper but rather about designing norms that are rooted in people’s beliefs and are thus more likely to be obeyed and enforced. Being in touch with real people is a politician’s task. Fourth, prodding underground economies to become legal is a major political sales job.”

 De Soto continues (p. 159), intending to refer only to real estate but actually speaking of the need for formal legal title to personal property of all kinds, which ought to include human, social, and natural capital:

  “Without succeeding on these legal and political fronts, no nation can overcome the legal apartheid between those who can create capital and those who cannot. Without formal property, no matter how many assets they accumulate or how hard they work, most people will not be able to prosper in a capitalist society. They will continue to remain beyond the radar of policymakers, out of the reach of official records, and thus economically invisible.”

Boumans, M. (2005). How economists model the world into numbers. New York: Routledge.

De Soto, H. (2000). The mystery of capital: Why capitalism triumphs in the West and fails everywhere else. New York: Basic Books.

Miller, P., & O’Leary, T. (2007, October/November). Mediating instruments and making markets: Capital budgeting, science and the economy. Accounting, Organizations, and Society, 32(7-8), 701-34.

Mirowski, P. (1988). Against mechanism: Protecting economics from science. Lanham, MD: Rowman & Littlefield.

Noyes, C. R. (1936). The institution of property. New York: Longman’s Green.

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Rasch Measurement as a Basis for a New Standards Framework

October 26, 2011

The 2011 U.S. celebration of World Standards Day took place on October 13 at the Fairmont Hotel in Washington, D.C., with the theme of “Advancing Safety and Sustainability Standards Worldwide.” The evening began with a reception in a hall of exhibits from the celebrations sponsors, which included the National Institute for Standards and Technology (NIST), the Society for Standards Professionals (SES), the American National Standards Institute (ANSI), Microsoft, IEEE, Underwriters Laboratories, the Consumer Electronics Association, ASME, ASTM International, Qualcomm, Techstreet, and many others. Several speakers took the podium after dinner to welcome the 400 or so attendees and to present the World Standards Day Paper Competition Awards and the Ronald H. Brown Standards Leadership Award.

Dr. Patrick Gallagher, Under Secretary of Commerce for Standards and Technology, and Director of NIST, was the first speaker after dinner. He directed his remarks at the value of a decentralized, voluntary, and demand-driven system of standards in promoting innovation and economic prosperity. Gallagher emphasized that “standards provide the common language that keeps domestic and international trade flowing,” concluding that “it is difficult to overestimate their critical value to both the U.S. and global economy.”

James Shannon, President of the National Fire Protection Association (NFPA), accepted the R. H. Brown Standards Leadership Award in recognition for his work initiating or improving the National Electrical Code, the Life Safety Code, and the Fire Safe Cigarette and Residential Sprinkler Campaigns.

Ellen Emard, President of SES, introduced the paper competition award winners. As of this writing the titles and authors of the first and second place awards are not yet available on the SES web site (http://www.ses-standards.org/displaycommon.cfm?an=1&subarticlenbr=56). I took third place for my paper, “What the World Needs Now: A Bold Plan for New Standards.” Where the other winning papers took up traditional engineering issues concerning the role of standards in advancing safety and sustainability issues, my paper spoke to the potential scientific and economic benefits that could be realized by standard metrics and common product definitions for outcomes in education, health care, social services, and environmental resource management. All three of the award-winning papers will appear in a forthcoming issue of Standards Engineering, the journal of SES.

I was coincidentally seated at the dinner alongside Gordon Gillerman, winner of third place in the 2004 paper competition (http://www.ses-standards.org/associations/3698/files/WSD%202004%20-%203%20-%20Gillerman.pdf) and currently Chief of the Standards Services Division at NIST. Gillerman has a broad range of experience in coordinating standards across multiple domains, including environmental protection, homeland security, safety, and health care. Having recently been involved in a workshop focused on measuring, evaluating, and improving the usability of electronic health records (http://www.nist.gov/healthcare/usability/upload/EHR-Usability-Workshop-2011-6-03-2011_final.pdf), Gillerman was quite interested in the potential Rasch measurement techniques hold for reducing data volume with no loss of information, and so for streamlining computer interfaces.

Robert Massof of Johns Hopkins University accompanied me to the dinner, and was seated at a nearby table. Also at Massof’s table were several representatives of the National Institute of Building Sciences, some of whom Massof had recently met at a workshop on adaptations for persons with low vision disabilities. Massof’s work equating the main instruments used for assessing visual function in low vision rehabilitation could lead to a standard metric useful in improving the safety and convenience of buildings.

As is stated in educational materials distributed at the World Standards Day celebration by ANSI, standards are a constant behind-the-scenes presence in nearly all areas of everyday life. Everything from air, water, and food to buildings, clothing, automobiles, roads, and electricity are produced in conformity with voluntary consensus standards of various kinds. In the U.S. alone, more than 100,000 standards specify product and system features and interconnections, making it possible for appliances to tap the electrical grid with the same results no matter where they are plugged in, and for products of all kinds to be purchased with confidence. Life is safer and more convenient, and science and industry are more innovative and profitable, because of standards.

The point of my third-place paper is that life could be even safer and more convenient, and science and industry could be yet more innovative and profitable, if standards and conformity assessment procedures for outcomes in education, health care, social services, and environmental resource management were developed and implemented. Rasch measurement demonstrates the consistent reproducibility of meaningful measures across samples and different collections of construct-relevant items. Within any specific area of interest, then, Rasch measures have the potential of serving as the kind of mediating instruments or objects recognized as essential to the process of linking science with the economy (Fisher & Stenner, 2011b; Hussenot & Missonier, 2010; Miller & O’Leary, 2007). Recent white papers published by NIST and NSF document the challenges and benefits likely to be encountered and produced by initiatives moving in this direction (Fisher, 2009; Fisher & Stenner, 2011a).

A diverse array of Rasch measurement presentations were made at the recent International Measurement Confederation (IMEKO) meeting of metrology engineers in Jena, Germany (see RMT 25 (1), p. 1318). With that start at a new dialogue between the natural and social sciences, the NIST and NSF white papers, and with the award in the World Standards Day paper competition, the U.S. and international standards development communities have shown their interest in exploring possibilities for a new array of standard units of measurement, standardized outcome product definitions, standard conformity assessment procedures, and outcome product quality standards. The increasing acceptance and recognition of the viability of such standards is a logical consequence of observations like these:

  • “Where this law [relating reading ability and text difficulty to comprehension rate] can be applied it provides a principle of measurement on a ratio scale of both stimulus parameters and object parameters, the conceptual status of which is comparable to that of measuring mass and force. Thus…the reading accuracy of a child…can be measured with the same kind of objectivity as we may tell its weight” (Rasch, 1960, p. 115).
  • “Today there is no methodological reason why social science cannot become as stable, as reproducible, and hence as useful as physics” (Wright, 1997, p. 44).
  • “…when the key features of a statistical model relevant to the analysis of social science data are the same as those of the laws of physics, then those features are difficult to ignore” (Andrich, 1988, p. 22).

Rasch’s work has been wrongly assimilated in social science research practice as just another example of the “standard model” of statistical analysis. Rasch measurement rightly ought instead to be treated as a general articulation of the three-variable structure of natural law useful in framing the context of scientific practice. That is, Rasch’s models ought to be employed primarily in calibrating instruments quantitatively interpretable at the point of use in a mathematical language shared by a community of research and practice. To be shared in this way as a universally uniform coin of the realm, that language must be embodied in a consensus standard defining universally uniform units of comparison.

Rasch measurement offers the potential of shifting the focus of quantitative psychosocial research away from data analysis to integrated qualitative and quantitative methods enabling the definition of standard units and the calibration of instruments measuring in that unit. An intangible assets metric system will, in turn, support the emergence of new product- and performance-based standards, management system standards, and personnel certification standards. Reiterating once again Rasch’s (1960, p. xx) insight, we can acknowledge with him that “this is a huge challenge, but once the problem has been formulated it does seem possible to meet it.”

 References

Andrich, D. (1988). Rasch models for measurement. (Vols. series no. 07-068). Sage University Paper Series on Quantitative Applications in the Social Sciences. Beverly Hills, California: Sage Publications.

Fisher, W. P.. Jr. (2009). Metrological infrastructure for human, social, and natural capital (NIST Critical National Need Idea White Paper Series, Retrieved 25 October 2011 from http://www.nist.gov/tip/wp/pswp/upload/202_metrological_infrastructure_for_human_social_natural.pdf). Washington, DC: National Institute for Standards and Technology.

Fisher, W. P., Jr., & Stenner, A. J. (2011a, January). Metrology for the social, behavioral, and economic sciences (Social, Behavioral, and Economic Sciences White Paper Series). Retrieved 25 October 2011 from http://www.nsf.gov/sbe/sbe_2020/submission_detail.cfm?upld_id=36. Washington, DC: National Science Foundation.

Fisher, W. P., Jr., & Stenner, A. J. (2011b). A technology roadmap for intangible assets metrology. In Fundamentals of measurement science. International Measurement Confederation (IMEKO), Jena, Germany, August 31 to September 2.

Hussenot, A., & Missonier, S. (2010). A deeper understanding of evolution of the role of the object in organizational process. The concept of ‘mediation object.’ Journal of Organizational Change Management, 23(3), 269-286.

Miller, P., & O’Leary, T. (2007, October/November). Mediating instruments and making markets: Capital budgeting, science and the economy. Accounting, Organizations, and Society, 32(7-8), 701-34.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests (Reprint, with Foreword and Afterword by B. D. Wright, Chicago: University of Chicago Press, 1980). Copenhagen, Denmark: Danmarks Paedogogiske Institut.

Wright, B. D. (1997, Winter). A history of social science measurement. Educational Measurement: Issues and Practice, 16(4), 33-45, 52 [http://www.rasch.org/memo62.htm].

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LivingCapitalMetrics Blog by William P. Fisher, Jr., Ph.D. is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.
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Simple ideas, complex possibilities, elegant and beautiful results

February 11, 2011

Possibilities of great subtlety, elegance, and power can follow from the simplest ideas. Leonardo da Vinci is often credited with offering a variation on this theme, but the basic idea is much older. Philosophy, for instance, began with Plato’s distinction between name and concept. This realization that words are not the things they stand for has informed and structured each of several scientific revolutions.

How so? It all begins from the reasons why Plato required his students to have studied geometry. He knew that those familiar with the Pythagorean theorem would understand the difference between any given triangle and the mathematical relationships it represents. No right triangle ever definitively embodies a perfect realization of the assertion that the square of the hypotenuse equals the sum of the squares of the other two sides. The mathematical definition or concept of a triangle is not the same thing as any actual triangle.

The subtlety and power of this distinction became apparent in its repeated application throughout the history of science. In a sense, astronomy is a geometry of the heavens, Newton’s laws are a geometry of gravity, Ohm’s law is a geometry of electromagnetism, and relativity is a geometry of the invariance of mass and energy in relation to the speed of light. Rasch models present a means to geometries of literacy, numeracy, health, trust, and environmental quality.

We are still witnessing the truth, however partial, of Whitehead’s assertion that the entire history of Western culture is a footnote to Plato. As Husserl put it, we’re still struggling with the possibility of creating a geometry of experience, a phenomenology that is not a mere description of data but that achieves a science of living meaning. The work presented in other posts here attests to a basis for optimism that this quest will be fruitful.

Geometrical and algebraic expressions of scientific laws

April 12, 2010

Geometry provides a model of scientific understanding that has repeatedly proven itself over the course of history. Einstein (1922) considered geometry to be “the most ancient branch of physics” (p. 14). He accorded “special importance” to his view that “all linear measurement in physics is practical geometry,” “because without it I should have been unable to formulate the theory of relativity” (p. 14).

Burtt (1954) concurs, pointing out that the essential question for Copernicus was not “Does the earth move?” but, rather, “…what motions should we attribute to the earth in order to obtain the simplest and most harmonious geometry of the heavens that will accord with the facts?” (p. 39). Maxwell similarly employed a geometrical analogy in working out his electromagnetic theory, saying

“By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena. If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solution of the questions which the mathematical theory suggests.” (Maxwell, 1965/1890, p. 159).

Maxwell was known for thinking visually, once as a student offering a concise geometrical solution to a problem that resisted a lecturer’s lengthy algebraic efforts (Forfar, 2002, p. 8). His approach seemed to be one of playing with images with the aim of arriving at simple mathematical representations, instead of thinking linearly through a train of analysis. A similar method is said to have been used by Einstein (Holton, 1988, pp. 385-388).

Gadamer (1980) speaks of the mathematical transparency of geometric figures to convey Plato’s reasons for requiring mathematical training of the students in his Academy, saying:

“Geometry requires figures which we draw, but its object is the circle itself…. Even he who has not yet seen all the metaphysical implications of the concept of pure thinking but only grasps something of mathematics—and as we know, Plato assumed that such was the case with his listeners—even he knows that in a manner of speaking one looks right through the drawn circle and keeps the pure thought of the circle in mind.” (p. 101)

But exactly how do geometrical visualizations lend themselves to algebraic formulae? More specifically, is it possible to see the algebraic structure of scientific laws in geometry?

Yes, it is. Here’s how. Starting from the Pythagorean theorem, we know that the square of a right triangle’s hypotenuse is equal to the sum of the squares of the other two sides. For convenience, imagine that the lengths of the sides of the triangle, as shown in Figure 1, are 3, 4, and 5, for sides a, b, and c, respectively. We can count the unit squares within each side’s square and see that the 25 in the square of the hypotenuse equal the sum of the 9 in the square of side a and the 16 in the sum of side b.

That mathematical relationship can, of course, be written as

a2 + b2 = c2

which, for Figure 1, is

32 + 42 = 52 = 9 + 16 = 25

Now, most scientific laws are written in a multiplicative form, like this:

m = f / a

or

f = m * a

which, of course, is how Maxwell presented Newton’s Second Law. So how would the Pythagorean Theorem be written like a physical law?

Since the advent of small, cheap electronic calculators, slide rules have fallen out of fashion. But these eminently useful tools are built to take advantage of the way the natural logarithm and the number e (2.71828…) make division interchangeable with subtraction, and multiplication interchangeable with addition.

That means the Pythagorean Theorem could be written like Newton’s Second Law of Motion, or the Combined Gas Law. Here’s how it works. The Pythagorean Theorem is normally written as

a2 + b2 = c2

but does it make sense to write it as follows?

a2 * b2 = c2

Using the convenient values for a, b, and c from above

32 + 42 = 52

and

9 + 16 = 25

so, plainly, simply changing the plus sign to a multiplication sign will not work, since 9 * 16 is 144. This is where the number e comes in. What happens if e is taken as a base raised to the power of each of the parameters in the equation? Does this equation work?

e9 * e16 = e25

which, substituting a for e9, b for e16, and c for e25, could be represented by

a * b = c

and which could be solved as

8103 * 8,886,015 ≈ 72,003,378,611

Yes, it works, and so it is possible to divide through by e16 and arrive at the form of the law used by Maxwell and Rasch:

8103 ≈ 72,003,378,611 / 8,886,015

or

e9 = e25 / e16

or, again substituting a for e9, b for e16, and c for e25, could be represented by

a = c / b

which, when converted back to the additive form, looks like this:

a = c – b

and this

9 = 25 – 16 .

Rasch wrote his model in the multiplicative form of

εvi = θvσi

and it is often written in the form of

Pr {Xni = 1} = eβnδi / 1 + eβnδi

or

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

which is to say that the probability of a correct response from person n on item i is equal to e taken to the power of the difference between the estimate β (or B) of person n‘s ability and the estimate δ (or D) of item i‘s difficulty, divided by one plus e to that same power.

Logit estimates of Rasch model parameters taken straight from software output usually range between ­-3.0 or so and 3.0. So what happens if a couple of arbitrary values are plugged into these equations? If someone has a measure of 2 logits, what is their probability of a correct answer on an item that calibrates at 0.5 logits? The answer should be

e2-0.5 / (1 + e2-0.5).

Now,

e1.5 = 2.718281.5 = 4.481685….

and

4.481685 / (1 + 4.481685) ≈ 0.8176

For a table of the relationships between logit differences, odds, and probabilities, see Table 1.4.1 in Wright & Stone (1979, p. 16), or Table 1 in Wright (1977).

This form of the model

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

can be rewritten in an equivalent form as

[Pni / (1 – Pni)] = exp(Bn – Di) .

Taking the natural logarithm of the response probabilities expresses the model in perhaps its most intuitive form, often written as

ln[Pni / (1 – Pni)] = Bn – Di .

Substituting a for ln[Pni / (1 – Pni)], b for Bn, and c for Di, we have the same equation as we had for the Pythagorean Theorem, above

a = c – b .

Plugging in the same values of 2.0 and 0.5 logits for Bn and Di,

ln[Pni / (1 – Pni)] = 2.0 – 0.5 = 1.5.

The logit value of 1.5 is obtained from response odds [Pni / (1 – Pni)] of about 4.5, making, again, Pni equal to about 0.82.

Rasch wrote the model in working from Maxwell like this:

Avj = Fj / Mv .

So when catapult j’s force F of 50 Newtons (361.65 poundals) is applied to object v’s mass M of 10 kilograms (22.046 pounds), the acceleration of this interaction is 5 meters (16.404 feet) per second, per second. Increases in force relative to the same mass result in proportionate increases in acceleration, etc.

The same consistent and invariant structural relationship is posited and often found in Rasch model applications, such that reasonable matches are found between the expected and observed response probabilities are found for various differences between ability, attitude, or performance measures Bn and the difficulty calibrations Di of the items on the scale, between different measures relative to any given item, and between different calibrations relative to any given person. Of course, any number of parameters may be added, as long as they are included in an initial calibration design in which they are linked together in a common frame of reference.

Model fit statistics, principal components analysis of the standardized residuals, statistical studies of differential item/person functioning, and graphical methods are all applied to the study of departures from the modeled expectations.

I’ve shown here how the additive expression of the Pythagorean theorem, the multiplicative expression of natural laws, and the additive and multiplicative forms of Rasch models all participate in the same simultaneous, conjoint relation of two parameters mediated by a third. For those who think geometrically, perhaps the connections drawn here will be helpful in visualizing the design of experiments testing hypotheses of converging yet separable parameters. For those who think algebraically, perhaps the structure of lawful regularity in question and answer processes will be helpful in focusing attention on how to proceed step by step from one definite idea to another, in the manner so well demonstrated by Maxwell (Forfar, 2002, p. 8). Either way, the geometrical and/or algebraic figures and symbols ought to work together to provide a transparent view on the abstract mathematical relationships that stand independent from whatever local particulars are used as the medium of their representation.

Just as Einstein held that it would have been impossible to formulate the theory of relativity without the concepts, relationships, and images of practical geometry, so, too, may it one day turn out that key advances in the social and human sciences depend on the invariance of measures related to one another in the simple and lawful regularities of geometry.

Figure 1. A geometrical proof of the Pythagorean Theorem

References

Burtt, E. A. (1954). The metaphysical foundations of modern physical science (Rev. ed.) [First edition published in 1924]. Garden City, New York: Doubleday Anchor.

Einstein, A. (1922). Geometry and experience (G. B. Jeffery, W. Perrett, Trans.). In Sidelights on relativity (pp. 12-23). London, England: Methuen & Co. LTD.

Forfar, J. (2002, June). James Clerk Maxwell: His qualities of mind and personality as judged by his contemporaries. Mathematics Today, 38(3), 83.

Gadamer, H.-G. (1980). Dialogue and dialectic: Eight hermeneutical studies on Plato (P. C. Smith, Trans.). New Haven: Yale University Press.

Holton, G. (1988). Thematic origins of scientific thought (Revised ed.). Cambridge, Massachusetts: Harvard University Press.

Maxwell, J. C. (1965/1890). The scientific papers of James Clerk Maxwell (W. D. Niven, Ed.). New York: Dover Publications.

Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14(2), 97-116 [http://www.rasch.org/memo42.htm].

Wright, B. D., & Stone, M. H. (1979). Best test design: Rasch measurement. Chicago, Illinois: MESA Press.

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LivingCapitalMetrics Blog by William P. Fisher, Jr., Ph.D. is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.
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