The Combined Gas Law and a Rasch Reading Law

Many physical laws are expressed as universal conditionals among variable triplets. Newton's second law, for example, formalizes the relationship between mass and acceleration when holding force constant (i.e., conditioning on) as F = MA. Similarly, the combined gas law specifies the relationship between volume and temperature conditioning on pressure. After transformation (e.g., loge pressure + loge volume - loge temperature = constant, given a frame of reference specified by the number of molecules), each of these laws can be abstracted to a common form (a + b - c = constant). Note that these laws permit causal claims expressible as counterfactual conditionals. If we have 20 Liters of a gas at 2000°K under 20 atmospheres of pressure and we cool the gas to 1000°K, we will observe a decrease in the pressure to 10 atmospheres.

The value of such laws may lie more in the explicit causal organization of key constructs than in accuracy of prediction in the real world. Cartwright (1983) made a useful distinction between "the tidy and simple mathematical equations of abstract theory, and the intricate and messy descriptions, in either words or formulae, which express our knowledge of what happens in real systems made of real materials" (p. 128). This distinction led Cartwright to the view that "fundamental equations do not govern objects in reality; they govern only objects in models [i.e., idealizations]" (p. 129).

The human sciences, for the most part, lack laws such as those stated above and consequently lack causal stories that are universal in application: "Lacking a 'complete (causal) theory' of what influences what, and how much, we simply cannot compute expected numerical changes in stochastic dependencies when moving from one population or setting to another" (Meehl, 1978, p. 814, emphasis in original). In this note we build on the abstracted formalism derived above and imagine the form of a Rasch Reading Law.

Table 1
Comprehension Rates for Readers of Different Ability with Texts of the Same Readability
or How Reader Ability and Comprehension Rate Relate Under Constant Text Readability
ReaderAbilitySports Illustrated ReadabilityComprehension Rate

Contemporary reading theory recognizes three related constructs: reader ability (a stable attribute of persons), text readability (a stable attribute of text), and comprehension (the rate at which a particular reader makes meaning from a particular text). As a result of 25 years of ongoing research, we know that comprehension is a function of the difference between reader ability and text readability (Stenner & Burdick, 1997). Table 1 illustrates the relationship between reader ability and comprehension rate with text readability held constant. With increasing reader ability, the model forecasts increasing comprehension rate conditioning on text readability. This description of the relationship between reader, text, and comprehension echoes the description of the combined gas law (Table 2).

Table 2
How Temperature and Pressure Relate Under Constant Volume
Temperature Volume Pressure
20 Liters
20 Liters
20 Liters
20 Liters
20 Liters
20.0 atm
10.0 atm
5.0 atm
2.5 atm
1.25 atm

In fact, logit transformed comprehension rate + text measure - reader measure = the constant 1.1 (given a frame of reference that specifies 75% comprehension whenever text measure = reader measure). Therefore, a + b - c = constant holds as the common abstracted form of both the combined gas law and the Rasch Reading Law as well as many other physical laws. Below are several causal corollaries of the Rasch Reading Law.

(1) For any reader (and thus for all readers), an increase in text measure causes a decrease in comprehension.

(2) For any reader (and thus for all readers), a decrease in text measure causes an increase in comprehension.

(3) For any text (and thus for all texts), an increase in reader ability causes an increase in comprehension.

(4) For any text (and thus for all texts), a decrease in reader ability causes a decrease in comprehension.

Corollaries such as those above are consequences of the highly abstracted a + b - c = constant, holding in a domain of enquiry. Tables 1 and 2 concretize this abstraction for the gas law and reading law. The Rasch model, in concert with a substantive theory, is a powerful tool for discovering and testing the adequacy of such formulations. Note, however, that the fact that data fit a Rasch model says nothing about causality. Rasch models are associational rather than causal. Substantive theory provides the causal story for the variation detected by a measurement procedure. Specification equations formalize these causal stories and allow precise predictions.

These causal explanations have truth built into them. When I infer from an effect to a cause, I am asking what made the effect occur, what brought it about. No explanation of that sort explains at all unless it does present a cause, and in accepting such an explanation, I am accepting not only that it explains in the sense of organizing and making plain, but also that it presents me with a cause. (Cartwright, 1983, p. 91)

If one of our children cannot summarize what he just read in his fifth grade science text, we explain this by pointing out that he is a 580L
reader and the text book is at 830L
. The equation that models comprehension rate as a function of the difference between reader measure and text measure produces an expected comprehension rate below 50%. We hypothesize that the child's failure to produce a good summary has a cause: low comprehension. Suppose that we go to the Web and find a 600L
article on the same science topic, and the child reads the article and produces a coherent summary of the text. We conclude that, indeed, low comprehension was the cause of poor summarization. Manipulating the reader-text match caused an increase in comprehension, which in turn caused a change in summary performance. Clearly I am inferring from effect to probable cause. Note that this explanation is unintelligible "without the direct implication that there are [readers, texts and comprehension rates]" (Cartwright, 1983, p. 92).

We wonder how many other variable triplets in the human sciences can be abstracted to the form a + b - c = constant. The implications of this kind of law-making for construct validity should be evident (see Borsboom, 2005).

Donald S. Burdick
Mark H. Stone
A. Jackson Stenner


Borsboom, D. (2005). Measuring the mind. Cambridge: Cambridge University Press.

Cartwright, N. (1983). How the laws of physics lie. New York: Oxford Press.

Meehl, P. E. (1978). Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald, and the slow progress of soft psychology. Journal of Consulting and Clinical Psychology, 46, 806-834.

Stenner, A. J., & Burdick, D. S. (1997). The objective measurement of reading comprehension: In response to technical questions raised by the California Department of Education technical study group. Unpublished manuscript.

The Combined Gas Law and a Rasch Reading Law, Burdick, D.S., Stone, M.H., Stenner, A.J. … Rasch Measurement Transactions, 2006, 20:2 p. 1059-60

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets),
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps),


The URL of this page is