Archive for the ‘Models’ Category

2011 IMEKO Conference Papers Published Online

January 13, 2012

Papers from the Joint International IMEKO TC1+ TC7+ TC13 Symposium held August 31st to September 2nd,  2011, in Jena, Germany are now available online at http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24575/IMEKO2011_TOC.pdf. The following will be of particular interest to those interested in measurement applications in the social sciences, education, health care, and psychology:

Nikolaus Bezruczko
Foundational Imperatives for Measurement with Mathematical Models
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24419/ilm1-2011imeko-030.pdf

Nikolaus Bezruczko, Shu-Pi C. Chen, Connie Hill, Joyce M. Chesniak
A Clinical Scale for Measuring Functional Caregiving of Children Assisted with Medical Technologies
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24507/ilm1-2011imeko-032.pdf

Stefan Cano, Anne F. Klassen, Andrea L. Pusic, Andrea
From Breast-Q © to Q-Score ©: Using Rasch Measurement to Better Capture Breast Surgery Outcomes
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24429/ilm1-2011imeko-039.pdf

Gordon A. Cooper, William P. Fisher, Jr.
Continuous Quantity and Unit; Their Centrality to Measurement
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24494/ilm1-2011imeko-019.pdf

William P. Fisher, Jr.
Measurement, Metrology and the Coordination of Sociotechnical Networks
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24491/ilm1-2011imeko-017.pdf

William .P Fisher, Jr., A. Jackson Stenner
A Technology Roadmap for Intangible Assets Metrology
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24493/ilm1-2011imeko-018.pdf

Carl V. Granger, Nikolaus Bezruczko
Body, Mind, and Spirit are Instrumental to Functional Health: A Case Study
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24494/ilm1-2011imeko-019.pdf

Thomas Salzberger
The Quantification of Latent Variables in the Social Sciences: Requirements for Scientific Measurement and Shortcomings of Current Procedures
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24417/ilm1-2011imeko-029.pdf

A. Jackson Stenner, Mark Stone, Donald Burdick
How to Model and Test for the Mechanisms that Make Measurement Systems Tick
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24416/ilm1-2011imeko-027.pdf

Mark Wilson
The Role of Mathematical Models in Measurement: A Perspective from Psychometrics
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24178/ilm1-2011imeko-005.pdf

Also of interest will be Karl Ruhm’s plenary lecture and papers from the Fundamentals of Measurement Science session and the Special Session on the Role of Mathematical Models in Measurement:

Karl H. Ruhm
From Verbal Models to Mathematical Models – A Didactical Concept not just in Metrology
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24167/ilm1-2011imeko-002.pdf

Alessandro Giordani, Luca Mari
Quantity and Quantity Value
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24414/ilm1-2011imeko-025.pdf

Eric Benoit
Uncertainty in Fuzzy Scales Based Measurements
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24415/ilm1-2011imeko-020.pdf

Susanne C.N. Töpfer
Application of Mathematical Models in Optical Coordinate Metrology
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24445/ilm1-2011imeko-008.pdf

Giovanni Battista Rossi
Measurement Modelling: Foundations and Probabilistic Approach
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24446/ilm1-2011imeko-009.pdf

Sanowar H. Khan, Ludwik Finkelstein
The Role of Mathematical Modelling in the Analysis and Design of Measurement Systems
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24448/ilm1-2011imeko-010.pdf

Roman Z. Morawski
Application-Oriented Approach to Mathematical Modelling of Measurement Processes
http://www.db-thueringen.de/servlets/DerivateServlet/Derivate-24449/ilm1-2011imeko-011.pdf

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

A Second Simple Example of Measurement’s Role in Reducing Transaction Costs, Enhancing Market Efficiency, and Enables the Pricing of Intangible Assets

March 9, 2011

The prior post here showed why we should not confuse counts of things with measures of amounts, though counts are the natural starting place to begin constructing measures. That first simple example focused on an analogy between counting oranges and measuring the weight of oranges, versus counting correct answers on tests and measuring amounts of ability. This second example extends the first by, in effect, showing what happens when we want to aggregate value not just across different counts of some one thing but across different counts of different things. The point will be, in effect, to show how the relative values of apples, oranges, grapes, and bananas can be put into a common frame of reference and compared in a practical and convenient way.

For instance, you may go into a grocery store to buy raspberries and blackberries, and I go in to buy cantaloupe and watermelon. Your cost per individual fruit will be very low, and mine will be very high, but neither of us will find this annoying, confusing, or inconvenient because your fruits are very small, and mine, very large. Conversely, your cost per kilogram will be much higher than mine, but this won’t cause either of us any distress because we both recognize the differences in the labor, handling, nutritional, and culinary value of our purchases.

But what happens when we try to purchase something as complex as a unit of socioeconomic development? The eight UN Millennium Development Goals (MDGs) represent a start at a systematic effort to bring human, social, and natural capital together into the same economic and accountability framework as liquid and manufactured capital, and property. But that effort is stymied by the inefficiency and cost of making and using measures of the goals achieved. The existing MDG databases (http://data.un.org/Browse.aspx?d=MDG), and summary reports present overwhelming numbers of numbers. Individual indicators are presented for each year, each country, each region, and each program, goal by goal, target by target, indicator by indicator, and series by series, in an indigestible volume of data.

Though there are no doubt complex mathematical methods by which a philanthropic, governmental, or NGO investor might determine how much development is gained per million dollars invested, the cost of obtaining impact measures is so high that most funding decisions are made with little information concerning expected returns (Goldberg, 2009). Further, the percentages of various needs met by leading social enterprises typically range from 0.07% to 3.30%, and needs are growing, not diminishing. Progress at current rates means that it would take thousands of years to solve today’s problems of human suffering, social disparity, and environmental quality. The inefficiency of human, social, and natural capital markets is so overwhelming that there is little hope for significant improvements without the introduction of fundamental infrastructural supports, such as an Intangible Assets Metric System.

A basic question that needs to be asked of the MDG system is, how can anyone make any sense out of so much data? Most of the indicators are evaluated in terms of counts of the number of times something happens, the number of people affected, or the number of things observed to be present. These counts are usually then divided by the maximum possible (the count of the total population) and are expressed as percentages or rates.

As previously explained in various posts in this blog, counts and percentages are not measures in any meaningful sense. They are notoriously difficult to interpret, since the quantitative meaning of any given unit difference varies depending on the size of what is counted, or where the percentage falls in the 0-100 continuum. And because counts and percentages are interpreted one at a time, it is very difficult to know if and when any number included in the sheer mass of data is reasonable, all else considered, or if it is inconsistent with other available facts.

A study of the MDG data must focus on these three potential areas of data quality improvement: consistency evaluation, volume reduction, and interpretability. Each builds on the others. With consistent data lending themselves to summarization in sufficient statistics, data volume can be drastically reduced with no loss of information (Andersen, 1977, 1999; Wright, 1977, 1997), data quality can be readily assessed in terms of sufficiency violations (Smith, 2000; Smith & Plackner, 2009), and quantitative measures can be made interpretable in terms of a calibrated ruler’s repeatedly reproducible hierarchy of indicators (Bond & Fox, 2007; Masters, Lokan, & Doig, 1994).

The primary data quality criteria are qualitative relevance and meaningfulness, on the one hand, and mathematical rigor, on the other. The point here is one of following through on the maxim that we manage what we measure, with the goal of measuring in such a way that management is better focused on the program mission and not distracted by accounting irrelevancies.

Method

As written and deployed, each of the MDG indicators has the face and content validity of providing information on each respective substantive area of interest. But, as has been the focus of repeated emphases in this blog, counting something is not the same thing as measuring it.

Counts or rates of literacy or unemployment are not, in and of themselves, measures of development. Their capacity to serve as contributing indications of developmental progress is an empirical question that must be evaluated experimentally against the observable evidence. The measurement of progress toward an overarching developmental goal requires inferences made from a conceptual order of magnitude above and beyond that provided in the individual indicators. The calibration of an instrument for assessing progress toward the realization of the Millennium Development Goals requires, first, a reorganization of the existing data, and then an analysis that tests explicitly the relevant hypotheses as to the potential for quantification, before inferences supporting the comparison of measures can be scientifically supported.

A subset of the MDG data was selected from the MDG database available at http://data.un.org/Browse.aspx?d=MDG, recoded, and analyzed using Winsteps (Linacre, 2011). At least one indicator was selected from each of the eight goals, with 22 in total. All available data from these 22 indicators were recorded for each of 64 countries.

The reorganization of the data is nothing but a way of making the interpretation of the percentages explicit. The meaning of any one country’s percentage or rate of youth unemployment, cell phone users, or literacy has to be kept in context relative to expectations formed from other countries’ experiences. It would be nonsense to interpret any single indicator as good or bad in isolation. Sometimes 30% represents an excellent state of affairs, other times, a terrible one.

Therefore, the distributions of each indicator’s percentages across the 64 countries were divided into ranges and converted to ratings. A lower rating uniformly indicates a status further away from the goal than a higher rating. The ratings were devised by dividing the frequency distribution of each indicator roughly into thirds.

For instance, the youth unemployment rate was found to vary such that the countries furthest from the desired goal had rates of 25% and more(rated 1), and those closest to or exceeding the goal had rates of 0-10% (rated 3), leaving the middle range (10-25%) rated 2. In contrast, percentages of the population that are undernourished were rated 1 for 35% or more, 2 for 15-35%, and 3 for less than 15%.

Thirds of the distributions were decided upon only on the basis of the investigator’s prior experience with data of this kind. A more thorough approach to the data would begin from a finer-grained rating system, like that structuring the MDG table at http://mdgs.un.org/unsd/mdg/Resources/Static/Products/Progress2008/MDG_Report_2008_Progress_Chart_En.pdf. This greater detail would be sought in order to determine empirically just how many distinctions each indicator can support and contribute to the overall measurement system.

Sixty-four of the available 336 data points were selected for their representativeness, with no duplications of values and with a proportionate distribution along the entire continuum of observed values.

Data from the same 64 countries and the same years were then sought for the subsequent indicators. It turned out that the years in which data were available varied across data sets. Data within one or two years of the target year were sometimes substituted for missing data.

The data were analyzed twice, first with each indicator allowed its own rating scale, parameterizing each of the category difficulties separately for each item, and then with the full rating scale model, as the results of the first analysis showed all indicators shared strong consistency in the rating structure.

Results

Data were 65.2% complete. Countries were assessed on an average of 14.3 of the 22 indicators, and each indicator was applied on average to 41.7 of the 64 country cases. Measurement reliability was .89-.90, depending on how measurement error is estimated. Cronbach’s alpha for the by-country scores was .94. Calibration reliability was .93-.95. The rating scale worked well (see Linacre, 2002, for criteria). The data fit the measurement model reasonably well, with satisfactory data consistency, meaning that the hypothesis of a measurable developmental construct was not falsified.

The main result for our purposes here concerns how satisfactory data consistency makes it possible to dramatically reduce data volume and improve data interpretability. The figure below illustrates how. What does it mean for data volume to be drastically reduced with no loss of information? Let’s see exactly how much the data volume is reduced for the ten item data subset shown in the figure below.

The horizontal continuum from -100 to 1300 in the figure is the metric, the ruler or yardstick. The number of countries at various locations along that ruler is shown across the bottom of the figure. The mean (M), first standard deviation (S), and second standard deviation (T) are shown beneath the numbers of countries. There are ten countries with a measure of just below 400, just to the left of the mean (M).

The MDG indicators are listed on the right of the figure, with the indicator most often found being achieved relative to the goals at the bottom, and the indicator least often being achieved at the top. The ratings in the middle of the figure increase from 1 to 3 left to right as the probability of goal achievement increases as the measures go from low to high. The position of the ratings in the middle of the figure shifts from left to right as one reads up the list of indicators because the difficulty of achieving the goals is increasing.

Because the ratings of the 64 countries relative to these ten goals are internally consistent, nothing but the developmental level of the country and the developmental challenge of the indicator affects the probability that a given rating will be attained. It is this relation that defines fit to a measurement model, the sufficiency of the summed ratings, and the interpretability of the scores. Given sufficient fit and consistency, any country’s measure implies a given rating on each of the ten indicators.

For instance, imagine a vertical line drawn through the figure at a measure of 500, just above the mean (M). This measure is interpreted relative to the places at which the vertical line crosses the ratings in each row associated with each of the ten items. A measure of 500 is read as implying, within a given range of error, uncertainty, or confidence, a rating of

  • 3 on debt service and female-to-male parity in literacy,
  • 2 or 3 on how much of the population is undernourished and how many children under five years of age are moderately or severely underweight,
  • 2 on infant mortality, the percent of the population aged 15 to 49 with HIV, and the youth unemployment rate,
  • 1 or 2 the poor’s share of the national income, and
  • 1 on CO2 emissions and the rate of personal computers per 100 inhabitants.

For any one country with a measure of 500 on this scale, ten percentages or rates that appear completely incommensurable and incomparable are found to contribute consistently to a single valued function, developmental goal achievement. Instead of managing each separate indicator as a universe unto itself, this scale makes it possible to manage development itself at its own level of complexity. This ten-to-one ratio of reduced data volume is more than doubled when the total of 22 items included in the scale is taken into account.

This reduction is conceptually and practically important because it focuses attention on the actual object of management, development. When the individual indicators are the focus of attention, the forest is lost for the trees. Those who disparage the validity of the maxim, you manage what you measure, are often discouraged by the the feeling of being pulled in too many directions at once. But a measure of the HIV infection rate is not in itself a measure of anything but the HIV infection rate. Interpreting it in terms of broader developmental goals requires evidence that it in fact takes a place in that larger context.

And once a connection with that larger context is established, the consistency of individual data points remains a matter of interest. As the world turns, the order of things may change, but, more likely, data entry errors, temporary data blips, and other factors will alter data quality. Such changes cannot be detected outside of the context defined by an explicit interpretive framework that requires consistent observations.

-100  100     300     500     700     900    1100    1300
|-------+-------+-------+-------+-------+-------+-------|  NUM   INDCTR
1                                 1  :    2    :  3     3    9  PcsPer100
1                         1   :   2    :   3            3    8  CO2Emissions
1                    1  :    2    :   3                 3   10  PoorShareNatInc
1                 1  :    2    :  3                     3   19  YouthUnempRatMF
1              1   :    2   :   3                       3    1  %HIV15-49
1            1   :   2    :   3                         3    7  InfantMortality
1          1  :    2    :  3                            3    4  ChildrenUnder5ModSevUndWgt
1         1   :    2    :  3                            3   12  PopUndernourished
1    1   :    2   :   3                                 3    6  F2MParityLit
1   :    2    :  3                                      3    5  DebtServExpInc
|-------+-------+-------+-------+-------+-------+-------|  NUM   INDCTR
-100  100     300     500     700     900    1100    1300
                   1
       1   1 13445403312323 41 221    2   1   1            COUNTRIES
       T      S       M      S       T

Discussion

A key element in the results obtained here concerns the fact that the data were about 35% missing. Whether or not any given indicator was actually rated for any given country, the measure can still be interpreted as implying the expected rating. This capacity to take missing data into account can be taken advantage of systematically by calibrating a large bank of indicators. With this in hand, it becomes possible to gather only the amount of data needed to make a specific determination, or to adaptively administer the indicators so as to obtain the lowest-error (most reliable) measure at the lowest cost (with the fewest indicators administered). Perhaps most importantly, different collections of indicators can then be equated to measure in the same unit, so that impacts may be compared more efficiently.

Instead of an international developmental aid market that is so inefficient as to preclude any expectation of measured returns on investment, setting up a calibrated bank of indicators to which all measures are traceable opens up numerous desirable possibilities. The cost of assessing and interpreting the data informing aid transactions could be reduced to negligible amounts, and the management of the processes and outcomes in which that aid is invested would be made much more efficient by reduced data volume and enhanced information content. Because capital would flow more efficiently to where supply is meeting demand, nonproducers would be cut out of the market, and the effectiveness of the aid provided would be multiplied many times over.

The capacity to harmonize counts of different but related events into a single measurement system presents the possibility that there may be a bright future for outcomes-based budgeting in education, health care, human resource management, environmental management, housing, corrections, social services, philanthropy, and international development. It may seem wildly unrealistic to imagine such a thing, but the return on the investment would be so monumental that not checking it out would be even crazier.

A full report on the MDG data, with the other references cited, is available on my SSRN page at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1739386.

Goldberg, S. H. (2009). Billions of drops in millions of buckets: Why philanthropy doesn’t advance social progress. New York: Wiley.

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Reasoning by analogy in social science education: On the need for a new curriculum

April 12, 2010

I’d like to revisit the distinction between measurement models and statistical models. Rasch was well known for joking about burning all books containing the words “normal distribution” (Andersen, 1995, p. 385). Rasch’s book and 1961 article both start on their first pages with a distinction between statistical models describing intervariable relations at the group level and measurement models prescribing intravariable relations at the individual level. I think confusion between these kinds of models has caused huge problems.

We typically assume all statistical analyses are quantitative. We refer to any research that uses numbers as quantitative even when nothing is done to map a substantive and invariant unit on a number line. We distinguish between qualitative and quantitative data and methods as though quantification has ever been achieved in the history of science without substantive qualitative understandings of the constructs.

Quantification in fact predates the emergence of statistics by millennia. It seems to me that there is a great deal to be gained from maintaining a careful distinction between statistics and measurement. Measurement is not primarily performed by someone sitting at a computer analyzing data. Measurement is done by individuals using calibrated instruments to obtain immediately useful quantitative information expressed in a universally uniform unit.

Rasch was correct in his assertion that we can measure the reading ability of a child with the same kind of objectivity with which we measure his or her weight or height. But we don’t commonly express individual height and weight measures in statistical terms. 

Information overload is one of the big topics of the day. Which will contribute more to reducing that overload in efficient and meaningful ways: calibrated instruments measuring in common units giving individual users immediate feedback that summarizes responses to dozens of questions, or ordinal group-level item-by-item statistics reported six months too late to do anything about them?

Instrument calibration certainly makes use of statistics, and statistical models usually assume measurement has taken place, but much stands to be gained from a clear distinction between inter- and intra-variable models. And so I respectfully disagree with those who assert that “the Rasch model is first of all a statistical model.” Maxwell’s method of making analogies from well known physical laws (Nersessian, 2002; Turner, 1955) was adopted by Rasch (1960, pp. 110-115) so that his model would have the same structure as the laws of physics.

Statistical models are a different class of models from the laws of physics (Meehl, 1967), since they allow cross-variable interactions in ways that compromise and defeat the possibility of testing the hypotheses of constant unit size, parameter separation, sufficiency, etc.

I’d like to suggest a paraphrase of the first sentence of the abstract from a recent paper (Silva, 2007) on using analogies in science education: Despite its great importance, many students and even their teachers still cannot recognize the relevance of measurement models to build up psychosocial knowledge and are unable to develop qualitative explanations for mathematical expressions of the lawful structural invariances that exist within the social sciences.

And so, here’s a challenge: we need to make an analogy from Silva’s (2007) work in physics science education and develop a curriculum for social science education that follows a parallel track. We could trace the development of reading measurement from Rasch (1960) through the Anchor Test Study (Jaeger, 1973; Rentz & Bashaw, 1977) to the introduction of the Lexile Framework for Reading (Stenner, 2001) and its explicit continuity with Rasch’s use of Maxwell’s method of analogy (Burdick, Stone, & Stenner, 2006) and full blown predictive theory (Stenner & Stone, 2003).

With the example of the Rasch Reading Law in hand, we could then train students and teachers to think about structural invariance in the context of psychosocial constructs. It may be that, without the development and dissemination of at least a college-level curriculum of this kind, we will never overcome the confusion between statistical and measurement models.

References

Andersen, E. B. (1995). What George Rasch would have thought about this book. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 383-390). New York: Springer-Verlag.

Burdick, D. S., Stone, M. H., & Stenner, A. J. (2006). The Combined Gas Law and a Rasch Reading Law. Rasch Measurement Transactions, 20(2), 1059-60 [http://www.rasch.org/rmt/rmt202.pdf].

Jaeger, R. M. (1973). The national test equating study in reading (The Anchor Test Study). Measurement in Education, 4, 1-8.

Meehl, P. E. (1967). Theory-testing in psychology and physics: A methodological paradox. Philosophy of Science, 34(2), 103-115.

Nersessian, N. J. (2002). Maxwell and “the Method of Physical Analogy”: Model-based reasoning, generic abstraction, and conceptual change. In D. Malament (Ed.), Essays in the history and philosophy of science and mathematics (pp. 129-166). Lasalle, Illinois: Open Court.

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.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 321-333 [http://www.rasch.org/memo1960.pdf]). Berkeley, California: University of California Press.

Rentz, R. R., & Bashaw, W. L. (1977, Summer). The National Reference Scale for Reading: An application of the Rasch model. Journal of Educational Measurement, 14(2), 161-179.

Silva, C. C. (2007, August). The role of models and analogies in the electromagnetic theory: A historical case study. Science & Education, 16(7-8), 835-848.

Stenner, A. J. (2001). The Lexile Framework: A common metric for matching readers and texts. California School Library Journal, 25(1), 41-2.

Stenner, A. J., & Stone, M. (2003). Item specification vs. item banking. Rasch Measurement Transactions, 17(3), 929-30 [http://www.rasch.org/rmt/rmt173a.htm].

Turner, J. (1955, November). Maxwell on the method of physical analogy. British Journal for the Philosophy of Science, 6, 226-238.

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Reliability Revisited: Distinguishing Consistency from Error

August 28, 2009

When something is meaningful to us, and we understand it, then we can successfully restate it in our own words and predictably reproduce approximately the same representation across situations  as was obtained in the original formulation. When data fit a Rasch model, the implications are (1) that different subsets of items (that is, different ways of composing a series of observations summarized in a sufficient statistic) will all converge on the same pattern of person measures, and (2) that different samples of respondents or examinees will all converge on the same pattern of item calibrations. The meaningfulness of propositions based in these patterns will then not depend on which collection of items (instrument) or sample of persons is obtained, and all instruments might be equated relative to a single universal, uniform metric so that the same symbols reliably represent the same amount of the same thing.

Statistics and research methods textbooks in psychology and the social sciences commonly make statements like the following about reliability: “Reliability is consistency in measurement. The reliability of individual scale items increases with the number of points in the item. The reliability of the complete scale increases with the number of items.” (These sentences are found at the top of p. 371 in Experimental Methods in Psychology, by Gustav Levine and Stanley Parkinson (Lawrence Erlbaum Associates, 1994).) The unproven, perhaps unintended, and likely unfounded implication of these statements is that consistency increases as items are added.

Despite the popularity of doing so, Green, Lissitz, and Mulaik (1977) argue that reliability coefficients are misused when they are interpreted as indicating the extent to which data are internally consistent. “Green et al. (1977) observed that though high ‘internal consistency’ as indexed by a high alpha results when a general factor runs through the items, this does not rule out obtaining high alpha when there is no general factor running through the test items…. They concluded that the chief defect of alpha as an index of dimensionality is its tendency to increase as the number of items increase” (Hattie, 1985, p. 144).

In addressing the internal consistency of data, the implicit but incompletely realized purpose of estimating scale reliability is to evaluate the extent to which sum scores function as sufficient statistics. How limited is reliability as a tool for this purpose? To answer this question, five dichotomous data sets of 23 items and 22 persons were simulated. The first one was constructed so as to be highly likely to fit a Rasch model, with a deliberately orchestrated probabilistic Guttman pattern. The second one was made nearly completely random. The third, fourth, and fifth data sets were modifications of the first one in which increasing numbers of increasingly inconsistent responses were introduced. (The inconsistencies were not introduced in any systematic way apart from inserting contrary responses in the ordered matrix.) The data sets are shown in the Appendix. Tables 1 and 2 summarize the results.

Table 1 shows that the reliability coefficients do in fact decrease, along with the global model fit log-likelihood chi-squares, as the amount of randomness and inconsistency is increased. Contrary to what is implied in Levine and Parkinson’s statements, however, reliability can vary within a given number of items, as it might across different data sets produced from the same test, survey, or assessment, depending on how much structural invariance is present within them.

Two other points about the tables are worthy of note. First, the Rasch-based person separation reliability coefficients drop at a faster rate than Cronbach’s alpha does. This is probably an effect of the individualized error estimates in the Rasch context, which makes its reliability coefficients more conservative than correlation-based, group-level error estimates. (It is worth noting, as well, that the Winsteps and SPSS estimates of Cronbach’s alpha match. They are reported to one fewer decimal places by Winsteps, but the third decimal place is shown for the SPSS values for contrast.)

Second, the fit statistics are most affected by the initial and most glaring introduction of inconsistencies, in data set three. As the randomness in the data increases, the reliabilities continue to drop, but the fit statistics improve, culminating in the case of data set two, where complete randomness results in near-perfect model fit. This is, of course, the situation in which both the instrument and the sample are as well targeted as they can be, since all respondents have about the same measure and all the items about the same calibration; see Wood (1978) for a commentary on this situation, where coin tosses fit a Rasch model.

Table 2 shows the results of the Winsteps Principal Components Analysis of the standardized residuals for all five data sets. Again, the results conform with and support the pattern shown in the reliability coefficients. It is, however, interesting to note that, for data sets 4 and 5, with their Cronbach’s alphas of about .89 and .80, respectively, which are typically deemed quite good, the PCA shows more variance left unexplained than is explained by the Rasch dimension. The PCA is suggesting that two or more constructs might be represented in the data, but this would never be known from Cronbach’s alpha alone.

Alpha alone would indicate the presence of a unidimensional construct for data sets 3, 4 and 5, despite large standard deviations in the fit statistics and even though more than half the variance cannot be explained by the primary dimension. Worse, for the fifth data set, more variance is captured in the first three contrasts than is explained by the Rasch dimension. But with Cronbach’s alpha at .80, most researchers would consider this scale quite satisfactorily unidimensional and internally consistent.

These results suggest that, first, in seeking high reliability, what is sought more fundamentally is fit to a Rasch model (Andrich & Douglas, 1977; Andrich, 1982; Wright, 1977). That is, in addressing the internal consistency of data, the popular conception of reliability is taking on the concerns of construct validity. A conceptually clearer sense of reliability focuses on the extent to which an instrument works as expected every time it is used, in the sense of the way a car can be reliable. For instance, with an alpha of .70, a screening tool would be able to reliably distinguish measures into two statistically distinct groups (Fisher, 1992; Wright, 1996), problematic and typical. Within the limits of this purpose, the tool would meet the need for the repeated production of information capable of meeting the needs of the situation. Applications in research, accountability, licensure/certification, or diagnosis, however, might demand alphas of .95 and the kind of precision that allows for statistically distinct divisions into six or more groups. In these kinds of applications, where experimental designs or practical demands require more statistical power, measurement precision articulates finer degrees of differences. Finely calibrated instruments provide sensitivity over the entire length of the measurement continuum, which is needed for repeated reproductions of the small amounts of change that might accrue from hard to detect treatment effects.

Separating the construct, internal consistency, and unidimensionality issues  from the repeatability and reproducibility of a given degree of measurement precision provides a much-needed conceptual and methodological clarification of reliability. This clarification is routinely made in Rasch measurement applications (Andrich, 1982; Andrich & Douglas, 1977; Fisher, 1992; Linacre, 1993, 1996, 1997). It is reasonable to want to account for inconsistencies in the data in the error estimates and in the reliability coefficients, and so errors and reliabilities are routinely reported in terms of both the modeled expectations and in a fit-inflated form (Wright, 1995). The fundamental value of proceeding from a basis in individual error and fit statistics (Wright, 1996), is that local imprecisions and failures of invariance can be isolated for further study and selective attention.

The results of the simulated data analyses suggest, second, that, used in isolation, reliability coefficients can be misleading. As Green, et al. say, reliability estimates tend to systematically increase as the number of items increases (Fisher, 2008). The simulated data show that reliability coefficients also systematically decrease as inconsistency increases.

The primary problem with relying on reliability coefficients alone as indications of data consistency hinges on their inability to reveal the location of departures from modeled expectations. Most uses of reliability coefficients take place in contexts in which the model remains unstated and expectations are not formulated or compared with observations. The best that can be done in the absence of a model statement and test of data fit to it is to compare the reliability obtained against that expected on the basis of the number of items and response categories, relative to the observed standard deviation in the scores, expressed in logits (Linacre, 1993). One might then raise questions as to targeting, data consistency, etc. in order to explain larger than expected differences.

A more methodical way, however, would be to employ multiple avenues of approach to the evaluation of the data, including the use of model fit statistics and Principal Components Analysis in the evaluation of differential item and person functioning. Being able to see which individual observations depart the furthest from modeled expectation can provide illuminating qualitative information on the meaningfulness of the data, the measures, and the calibrations, or the lack thereof.  This information is crucial to correcting data entry errors, identifying sources of differential item or person functioning, separating constructs and populations, and improving the instrument. The power of the reliability-coefficient-only approach to data quality evaluation is multiplied many times over when the researcher sets up a nested series of iterative dialectics in which repeated data analyses explore various hypotheses as to what the construct is, and in which these analyses feed into revisions to the instrument, its administration, and/or the population sampled.

For instance, following the point made by Smith (1996), it may be expected that the PCA results will illuminate the presence of multiple constructs in the data with greater clarity than the fit statistics, when there are nearly equal numbers of items representing each different measured dimension. But the PCA does not work as well as the fit statistics when there are only a few items and/or people exhibiting inconsistencies.

This work should result in a full circle return to the drawing board (Wright, 1994; Wright & Stone, 2003), such that a theory of the measured construct ultimately provides rigorously precise predictive control over item calibrations, in the manner of the Lexile Framework (Stenner, et al., 2006) or developmental theories of hierarchical complexity (Dawson, 2004). Given that the five data sets employed here were simulations with no associated item content, the invariant stability and meaningfulness of the construct cannot be illustrated or annotated. But such illustration also is implicit in the quest for reliable instrumentation: the evidentiary basis for a delineation of meaningful expressions of amounts of the thing measured. The hope to be gleaned from the successes in theoretical prediction achieved to date is that we might arrive at practical applications of psychosocial measures that are as meaningful, useful, and economically productive as the theoretical applications of electromagnetism, thermodynamics, etc. that we take for granted in the technologies of everyday life.

Table 1

Reliability and Consistency Statistics

22 Persons, 23 Items, 506 Data Points

Data set Intended reliability Winsteps Real/Model Person Separation Reliability Winsteps/SPSS Cronbach’s alpha Winsteps Person Infit/Outfit Average Mn Sq Winsteps Person Infit/Outfit SD Winsteps Real/Model Item Separation Reliability Winsteps Item Infit/Outfit Average Mn Sq Winsteps Item Infit/Outfit SD Log-Likelihood Chi-Sq/d.f./p
First Best .96/.97 .96/.957 1.04/.35 .49/.25 .95/.96 1.08/0.35 .36/.19 185/462/1.00
Second Worst .00/.00 .00/-1.668 1.00/1.00 .05/.06 .00/.00 1.00/1.00 .05/.06 679/462/.0000
Third Good .90/.91 .93/.927 .92/2.21 .30/2.83 .85/.88 .90/2.13 .64/3.43 337/462/.9996
Fourth Fair .86/.87 .89/.891 .96/1.91 .25/2.18 .79/.83 .94/1.68 .53/2.27 444/462/.7226
Fifth Poor .76/.77 .80/.797 .98/1.15 .24/.67 .59/.65 .99/1.15 .41/.84 550/462/.0029
Table 2

Principal Components Analysis

Data set Intended reliability % Raw Variance Explained by Measures/Persons/Items % Raw Variance Captured in First Three Contrasts Total number of loadings > |.40| in first contrast
First Best 76/41/35 12 8
Second Worst 4.3/1.7/2.6 56 15
Third Good 59/34/25 20 14
Fourth Fair 47/27/20 26 13
Fifth Poor 29/17/11 41 15

References

Andrich, D. (1982, June). An index of person separation in Latent Trait Theory, the traditional KR-20 index, and the Guttman scale response pattern. Education Research and Perspectives, 9(1), http://www.rasch.org/erp7.htm.

Andrich, D. & G. A. Douglas. (1977). Reliability: Distinctions between item consistency and subject separation with the simple logistic model. Paper presented at the Annual Meeting of the American Educational Research Association, New York.

Dawson, T. L. (2004, April). Assessing intellectual development: Three approaches, one sequence. Journal of Adult Development, 11(2), 71-85.

Fisher, W. P., Jr. (1992). Reliability statistics. Rasch Measurement Transactions, 6(3), 238  [http://www.rasch.org/rmt/rmt63i.htm].

Fisher, W. P., Jr. (2008, Summer). The cash value of reliability. Rasch Measurement Transactions, 22(1), 1160-3.

Green, S. B., Lissitz, R. W., & Mulaik, S. A. (1977, Winter). Limitations of coefficient alpha as an index of test unidimensionality. Educational and Psychological Measurement, 37(4), 827-833.

Hattie, J. (1985, June). Methodology review: Assessing unidimensionality of tests and items. Applied Psychological Measurement, 9(2), 139-64.

Levine, G., & Parkinson, S. (1994). Experimental methods in psychology. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Linacre, J. M. (1993). Rasch-based generalizability theory. Rasch Measurement Transactions, 7(1), 283-284; [http://www.rasch.org/rmt/rmt71h.htm].

Linacre, J. M. (1996). True-score reliability or Rasch statistical validity? Rasch Measurement Transactions, 9(4), 455 [http://www.rasch.org/rmt/rmt94a.htm].

Linacre, J. M. (1997). KR-20 or Rasch reliability: Which tells the “Truth?”. Rasch Measurement Transactions, 11(3), 580-1 [http://www.rasch.org/rmt/rmt113l.htm].

Smith, R. M. (1996). A comparison of methods for determining dimensionality in Rasch measurement. Structural Equation Modeling, 3(1), 25-40.

Stenner, A. J., Burdick, H., Sanford, E. E., & Burdick, D. S. (2006). How accurate are Lexile text measures? Journal of Applied Measurement, 7(3), 307-22.

Wood, R. (1978). Fitting the Rasch model: A heady tale. British Journal of Mathematical and Statistical Psychology, 31, 27-32.

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. (1980). Foreword, Afterword. In Probabilistic models for some intelligence and attainment tests, by Georg Rasch (pp. ix-xix, 185-199. http://www.rasch.org/memo63.htm) [Reprint; original work published in 1960 by the Danish Institute for Educational Research]. Chicago, Illinois: University of Chicago Press.

Wright, B. D. (1994, Summer). Theory construction from empirical observations. Rasch Measurement Transactions, 8(2), 362 [http://www.rasch.org/rmt/rmt82h.htm].

Wright, B. D. (1995, Summer). Which standard error? Rasch Measurement Transactions, 9(2), 436-437 [http://www.rasch.org/rmt/rmt92n.htm].

Wright, B. D. (1996, Winter). Reliability and separation. Rasch Measurement Transactions, 9(4), 472 [http://www.rasch.org/rmt/rmt94n.htm].

Wright, B. D., & Stone, M. H. (2003). Five steps to science: Observing, scoring, measuring, analyzing, and applying. Rasch Measurement Transactions, 17(1), 912-913 [http://www.rasch.org/rmt/rmt171j.htm].

Appendix

Data Set 1

01100000000000000000000

10100000000000000000000

11000000000000000000000

11100000000000000000000

11101000000000000000000

11011000000000000000000

11100100000000000000000

11110100000000000000000

11111010100000000000000

11111101000000000000000

11111111010101000000000

11111111101010100000000

11111111111010101000000

11111111101101010010000

11111111111010101100000

11111111111111010101000

11111111111111101010100

11111111111111110101011

11111111111111111010110

11111111111111111111001

11111111111111111111101

11111111111111111111100

Data Set 2

01101010101010101001001

10100101010101010010010

11010010101010100100101

10101001010101001001000

01101010101010110010011

11011010010101100100101

01100101001001001001010

10110101000110010010100

01011010100100100101001

11101101001001001010010

11011010010101010100100

10110101101010101001001

01101011010000101010010

11010110101001010010100

10101101010000101101010

11011010101010010101010

10110101010101001010101

11101010101010110101011

11010101010101011010110

10101010101010110111001

01010101010101101111101

10101010101011011111100

Data Set 3

01100000000000100000010

10100000000000000010001

11000000000000100000010

11100000000000100000000

11101000000000100010000

11011000000000000000000

11100100000000100000000

11110100000000000000000

11111010100000100000000

11111101000000000000000

11111111010101000000000

11111111101010100000000

11111111111010001000000

11011111111111010010000

11011111111111101100000

11111111111111010101000

11011111111111101010100

11111111111111010101011

11011111111111111010110

11111111111111111111001

11011111111111111111101

10111111111111111111110

Data Set 4

01100000000000100010010

10100000000000000010001

11000000000000100000010

11100000000000100000001

11101000000000100010000

11011000000000000010000

11100100000000100010000

11110100000000000000000

11111010100000100010000

11111101000000000000000

11111011010101000010000

11011110111010100000000

11111111011010001000000

11011111101011110010000

11011111101101101100000

11111111110101010101000

11011111111011101010100

11111111111101110101011

01011111111111011010110

10111111111111111111001

11011111111111011111101

10111111111111011111110

Data Set 5

11100000010000100010011

10100000000000000011001

11000000010000100001010

11100000010000100000011

11101000000000100010010

11011000000000000010011

11100100000000100010000

11110100000000000000011

11111010100000100010000

00000000000011111111111

11111011010101000010000

11011110111010100000000

11111111011010001000000

11011111101011110010000

11011111101101101100000

11111111110101010101000

11011111101011101010100

11111111111101110101011

01011111111111011010110

10111111101111111111001

11011111101111011111101

00111111101111011111110

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Statistics and Measurement: Clarifying the Differences

August 26, 2009

Measurement is qualitatively and paradigmatically quite different from statistics, even though statistics obviously play important roles in measurement, and vice versa. The perception of measurement as conceptually difficult stems in part from its rearrangement of most of the concepts that we take for granted in the statistical paradigm as landmarks of quantitative thinking. When we recognize and accept the qualitative differences between statistics and measurement, they both become easier to understand.

Statistical analyses are commonly referred to as quantitative, even though the numbers analyzed most usually have not been derived from the mapping of an invariant substantive unit onto a number line. Measurement takes such mapping as its primary concern, focusing on the quantitative meaningfulness of numbers (Falmagne & Narens, 1983; Luce, 1978; ,  Marcus-Roberts & Roberts, 1987; Mundy, 1986; Narens, 2002; Roberts, 1999). Statistical models focus on group processes and relations among variables, while measurement models focus on individual processes and relations within variables (Duncan, 1992; Duncan & Stenbeck, 1988; Rogosa, 1987). Statistics makes assumptions about factors beyond its control, while measurement sets requirements for objective inference (Andrich, 1989). Statistics primarily involves data analysis, while measurement primarily calibrates instruments in common metrics for interpretation at the point of use (Cohen, 1994; Fisher, 2000; Guttman, 1985; Goodman, 1999a-c; Rasch, 1960).

Statistics focuses on making the most of the data in hand, while measurement focuses on using the data in hand to inform (a) instrument calibration and improvement, and (b) the prediction and efficient gathering of meaningful new data on individuals in practical applications. Where statistical “measures” are defined inherently by a particular analytic method, measures read from calibrated instruments—and the raw observations informing these measures—need not be computerized for further analysis.

Because statistical “measures” are usually derived from ordinal raw scores, changes to the instrument change their meaning, resulting in a strong inclination to avoid improving the instrument. Measures, in contrast, take missing data into account, so their meaning remains invariant over instrument configurations, resulting in a firm basis for the emergence of a measurement quality improvement culture. So statistical “measurement” begins and ends with data analysis, where measurement from calibrated instruments is in a constant cycle of application, new item calibrations, and critical recalibrations that require only intermittent resampling.

The vast majority of statistical methods and models make strong assumptions about the nature of the unit of measurement, but provide either very limited ways of checking those assumptions, or no checks at all. Statistical models are descriptive in nature, meaning that models are fit to data, that the validity of the data is beyond the immediate scope of interest, and that the model accounting for the most variation is regarded as best. Finally, and perhaps most importantly, statistical models are inherently oriented toward the relations among variables at the level of samples and populations.

Measurement models, however, impose strong requirements on data quality in order to achieve the unit of measurement that is easiest to think with, one that stays constant and remains invariant across the local particulars of instrument and sample. Measurement methods and models, then, provide extensive and varied ways of checking the quality of the unit, and so must be prescriptive rather than descriptive. That is, measurement models define the data quality that must be obtained for objective inference. In the measurement paradigm, data are fit to models, data quality is of paramount interest, and data quality evaluation must be informed as much by qualitative criteria as by quantitative.

To repeat the most fundamental point, measurement models are oriented toward individual-level response processes, not group-level aggregate processes. Herbert Blumer pointed out as early as 1930 that quantitative method is not equivalent to statistical method, and that the natural sciences had conspicuous degrees of success long before the emergence of statistical techniques (Hammersly, 1989, pp. 113-4). Both the initial scientific revolution in the 16th-17th centuries and the second scientific revolution of the 19th century found a basis in measurement for publicly objective and reproducible results, but statistics played little or no role in the major discoveries of the times.

The scientific value of statistics resides largely in the reproducibility of cross-variable data relations, and statisticians widely agree that statistical analyses should depend only on sufficient statistics (Arnold, 1982, p. 79). Measurement theoreticians and practitioners also agree, but the sufficiency of the mean and standard deviation relative to a normal distribution is one thing, and the sufficiency of individual responses relative to an invariant construct is quite another (Andersen, 1977; Arnold, 1985; Dynkin, 1951; Fischer, 1981; Hall, Wijsman, & Ghosh, 1965; van der Linden, 1992).

It is of historical interest, though, to point out that Rasch, foremost proponent of the latter, attributes credit for the general value of the concept of sufficiency to Ronald Fisher, foremost proponent of the former. Rasch’s strong statements concerning the fundamental inferential value of sufficiency (Andrich, 1997; Rasch, 1977; Wright, 1980) would seem to contradict his repeated joke about burning all the statistics texts making use of the normal distribution (Andersen, 1995, p. 385) were it not for the paradigmatic distinction between statistical models of group-level relations among variables, and measurement models of individual processes. Indeed, this distinction is made on the first page of Rasch’s (1980) book.

Now we are in a position to appreciate a comment by Ernst Rutherford, the winner of the 1908 Nobel Prize in Chemistry, who held that, if you need statistics to understand the results of your experiment, then you should have designed a better experiment (Wise, 1995, p. 11). A similar point was made by Feinstein (1995) concerning meta-analysis. The rarely appreciated point is that the generalizable replication and application of results depends heavily on the existence of a portable and universally uniform observational framework. The inferences, judgments, and adjustments that can be made at the point of use by clinicians, teachers, managers, etc. provided with additive measures expressed in a substantively meaningful common metric far outstrip those that can be made using ordinal measures expressed in instrument- and sample-dependent scores. See Andrich (1989, 2002, 2004), Cohen (1994), Davidoff (1999), Duncan (1992), Embretson (1996), Goodman (1999a, 1999b, 1999c), Guttman (1981, 1985), Meehl (1967), Michell (1986), Rogosa (1987), Romanowski and Douglas (2002), and others for more on this distinction between statistics and measurement.

These contrasts show that the confounding of statistics and measurement is a problem of vast significance that persists in spite of repeated efforts to clarify the distinction. For a wide variety of reasons ranging from cultural presuppositions about the nature of number to the popular notion that quantification is as easy as assigning numbers to observations, measurement is not generally well understood by the public (or even by statisticians!). And so statistics textbooks rarely, if ever, include even passing mention of instrument calibration methods, metric equating processes, the evaluation of data quality relative to the requirements of objective inference, traceability to metrological reference standards, or the integration of qualitative and quantitative methods in the interpretation of measures.

Similarly, in business, marketing, health care, and quality improvement circles, we find near-universal repetition of the mantra, “You manage what you measure,” with very little or no attention paid to the quality of the numbers treated as measures. And so, we find ourselves stuck with so-called measurement systems where,

• instead of linear measures defined by a unit that remains constant across samples and instruments we saddle ourselves with nonlinear scores and percentages defined by units that vary in unknown ways across samples and instruments;
• instead of availing ourselves of the capacity to take missing data into account, we hobble ourselves with the need for complete data;
• instead of dramatically reducing data volume with no loss of information, we insist on constantly re-enacting the meaningless ritual of poring over undigestible masses of numbers;
• instead of adjusting measures for the severity or leniency of judges assigning ratings, we allow measures to depend unfairly on which rater happens to make the observations;
• instead of using methods that give the same result across different distributions, we restrict ourselves to ones that give different results when assumptions of normality are not met and/or standard deviations differ;
• instead of calibrating instruments in an experimental test of the hypothesis that the intended construct is in fact structured in such a way as to make its mapping onto a number line meaningful, we assign numbers and make quantitative inferences with no idea as to whether they relate at all to anything real;
• instead of checking to see whether rating scales work as intended, with higher ratings consistently representing more of the variable, we make assumptions that may be contradicted by the order and spacing of the way rating scales actually work in practice;
• instead of defining a comprehensive framework for interpreting measures relative to a construct, we accept the narrow limits of frameworks defined by the local sample and items;
• instead of capitalizing on the practicality and convenience of theories capable of accurately predicting item calibrations and measures apart from data, we counterproductively define measurement empirically in terms of data analysis;
• instead of putting calibrated tools into the hands of front-line managers, service representatives, teachers and clinicians, we require them to submit to cumbersome data entry, analysis, and reporting processes that defeat the purpose of measurement by ensuring the information provided is obsolete by the time it gets back to the person who could act on it; and
• instead of setting up efficient systems for communicating meaningful measures in common languages with shared points of reference, we settle for inefficient systems for communicating meaningless scores in local incommensurable languages.

Because measurement is simultaneously ubiquitous and rarely well understood, we find ourselves in a world that gives near-constant lip service to the importance of measurement while it does almost nothing to provide measures that behave the way we assume they do. This state of affairs seems to have emerged in large part due to our failure to distinguish between the group-level orientation of statistics and the individual-level orientation of measurement. We seem to have been seduced by a variation on what Whitehead (1925, pp. 52-8) called the fallacy of misplaced concreteness. That is, we have assumed that the power of lawful regularities in thought and behavior would be best revealed and acted on via statistical analyses of data that themselves embody the aggregate mass of the patterns involved.

It now appears, however, in light of studies in the history of science (Latour, 1987, 2005; Wise, 1995), that an alternative and likely more successful approach will be to capitalize on the “wisdom of crowds” (Surowiecki, 2004) phenomenon of collective, distributed cognition (Akkerman, et al., 2007; Douglas, 1986; Hutchins, 1995; Magnus, 2007). This will be done by embodying lawful regularities in instruments calibrated in ideal, abstract, and portable metrics put to work by front-line actors on mass scales (Fisher, 2000, 2005, 2009a, 2009b). In this way, we will inform individual decision processes and structure communicative transactions with efficiencies, meaningfulness, substantive effectiveness, and power that go far beyond anything that could be accomplished by trying to make inferences about individuals from group-level statistics.

We ought not accept the factuality of data as the sole criterion of objectivity, with all theory and instruments constrained by and focused on the passing ephemera of individual sets of local particularities. Properly defined and operationalized via a balanced interrelation of theory, data, and instrument, advanced measurement is not a mere mathematical exercise but offers a wealth of advantages and conveniences that cannot otherwise be obtained. We ignore its potentials at our peril.

References
Akkerman, S., Van den Bossche, P., Admiraal, W., Gijselaers, W., Segers, M., Simons, R.-J., et al. (2007, February). Reconsidering group cognition: From conceptual confusion to a boundary area between cognitive and socio-cultural perspectives? Educational Research Review, 2, 39-63.

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Andersen, E. B. (1995). What George Rasch would have thought about this book. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 383-390). New York: Springer-Verlag.

Andrich, D. (1989). Distinctions between assumptions and requirements in measurement in the social sciences. In J. A. Keats, R. Taft, R. A. Heath & S. H. Lovibond (Eds.), Mathematical and Theoretical Systems: Proceedings of the 24th International Congress of Psychology of the International Union of Psychological Science, Vol. 4 (pp. 7-16). North-Holland: Elsevier Science Publishers.

Andrich, D. (1997). Georg Rasch in his own words [excerpt from a 1979 interview]. Rasch Measurement Transactions, 11(1), 542-3. [http://www.rasch.org/rmt/rmt111.htm#Georg].

Andrich, D. (2002). Understanding resistance to the data-model relationship in Rasch’s paradigm: A reflection for the next generation. Journal of Applied Measurement, 3(3), 325-59.

Andrich, D. (2004, January). Controversy and the Rasch model: A characteristic of incompatible paradigms? Medical Care, 42(1), I-7–I-16.

Arnold, S. F. (1982-1988). Sufficient statistics. In S. Kotz, N. L. Johnson & C. B. Read (Eds.), Encyclopedia of Statistical Sciences (pp. 72-80). New York: John Wiley & Sons.

Arnold, S. F. (1985, September). Sufficiency and invariance. Statistics & Probability Letters, 3, 275-279.

Cohen, J. (1994). The earth is round (p < 0.05). American Psychologist, 49, 997-1003.

Davidoff, F. (1999, 15 June). Standing statistics right side up (Editorial). Annals of Internal Medicine, 130(12), 1019-1021.

Douglas, M. (1986). How institutions think. Syracuse, New York: Syracuse University Press.

Dynkin, E. B. (1951). Necessary and sufficient statistics for a family of probability distributions. Selected Translations in Mathematical Statistics and Probability, 1, 23-41.

Duncan, O. D. (1992, September). What if? Contemporary Sociology, 21(5), 667-668.

Duncan, O. D., & Stenbeck, M. (1988). Panels and cohorts: Design and model in the study of voting turnout. In C. C. Clogg (Ed.), Sociological Methodology 1988 (pp. 1-35). Washington, DC: American Sociological Association.

Embretson, S. E. (1996, September). Item Response Theory models and spurious interaction effects in factorial ANOVA designs. Applied Psychological Measurement, 20(3), 201-212.

Falmagne, J.-C., & Narens, L. (1983). Scales and meaningfulness of quantitative laws. Synthese, 55, 287-325.

Feinstein, A. R. (1995, January). Meta-analysis: Statistical alchemy for the 21st century. Journal of Clinical Epidemiology, 48(1), 71-79.

Fischer, G. H. (1981, March). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59-77.

Fisher, W. P., Jr. (2000). Objectivity in psychosocial measurement: What, why, how. Journal of Outcome Measurement, 4(2), 527-563.

Fisher, W. P., Jr. (2005). Daredevil barnstorming to the tipping point: New aspirations for the human sciences. Journal of Applied Measurement, 6(3), 173-9.

Fisher, W. P., Jr. (2009a). Bringing human, social, and natural capital to life: Practical consequences and opportunities. In M. Wilson, K. Draney, N. Brown & B. Duckor (Eds.), Advances in Rasch Measurement, Vol. Two (p. in press). Maple Grove, MN: JAM Press.

Fisher, W. P., Jr. (2009b, July). Invariance and traceability for measures of human, social, and natural capital: Theory and application. Measurement (Elsevier), in press.

Goodman, S. N. (1999a, 6 April). Probability at the bedside: The knowing of chances or the chances of knowing? (Editorial). Annals of Internal Medicine, 130(7), 604-6.

Goodman, S. N. (1999b, 15 June). Toward evidence-based medical statistics. 1: The p-value fallacy. Annals of Internal Medicine, 130(12), 995-1004.

Goodman, S. N. (1999c, 15 June). Toward evidence-based medical statistics. 2: The Bayes factor. Annals of Internal Medicine, 130(12), 1005-1013.

Guttman, L. (1981). What is not what in theory construction. In I. Borg (Ed.), Multidimensional data representations: When & why. Ann Arbor, MI: Mathesis Press.

Guttman, L. (1985). The illogic of statistical inference for cumulative science. Applied Stochastic Models and Data Analysis, 1, 3-10.

Hall, W. J., Wijsman, R. A., & Ghosh, J. K. (1965). The relationship between sufficiency and invariance with applications in sequential analysis. Annals of Mathematical Statistics, 36, 575-614.

Hammersley, M. (1989). The dilemma of qualitative method: Herbert Blumer and the Chicago Tradition. New York: Routledge.

Hutchins, E. (1995). Cognition in the wild. Cambridge, Massachusetts: MIT Press.

Latour, B. (1987). Science in action: How to follow scientists and engineers through society. New York: Cambridge University Press.

Latour, B. (1995). Cogito ergo sumus! Or psychology swept inside out by the fresh air of the upper deck: Review of Hutchins’ Cognition in the Wild, MIT Press, 1995. Mind, Culture, and Activity: An International Journal, 3(192), 54-63.

Latour, B. (2005). Reassembling the social: An introduction to Actor-Network-Theory. (Clarendon Lectures in Management Studies). Oxford, England: Oxford University Press.

Luce, R. D. (1978, March). Dimensionally invariant numerical laws correspond to meaningful qualitative relations. Philosophy of Science, 45, 1-16.

Magnus, P. D. (2007). Distributed cognition and the task of science. Social Studies of Science, 37(2), 297-310.

Marcus-Roberts, H., & Roberts, F. S. (1987). Meaningless statistics. Journal of Educational Statistics, 12(4), 383-394.

Meehl, P. E. (1967). Theory-testing in psychology and physics: A methodological paradox. Philosophy of Science, 34(2), 103-115.

Michell, J. (1986). Measurement scales and statistics: A clash of paradigms. Psychological Bulletin, 100, 398-407.

Mundy, B. (1986, June). On the general theory of meaningful representation. Synthese, 67(3), 391-437.

Narens, L. (2002, December). A meaningful justification for the representational theory of measurement. Journal of Mathematical Psychology, 46(6), 746-68.

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.

Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy, 14, 58-94.

Roberts, F. S. (1999). Meaningless statements. In R. Graham, J. Kratochvil, J. Nesetril & F. Roberts (Eds.), Contemporary trends in discrete mathematics, DIMACS Series, Volume 49 (pp. 257-274). Providence, RI: American Mathematical Society.

Rogosa, D. (1987). Casual [sic] models do not support scientific conclusions: A comment in support of Freedman. Journal of Educational Statistics, 12(2), 185-95.

Romanoski, J. T., & Douglas, G. (2002). Rasch-transformed raw scores and two-way ANOVA: A simulation analysis. Journal of Applied Measurement, 3(4), 421-430.

Stevens, S. S. (1951). Mathematics, measurement, and psychophysics. In S. S. Stevens (Ed.), Handbook of experimental psychology (pp. 1-49). New York: John Wiley & Sons.

Surowiecki, J. (2004). The wisdom of crowds: Why the many are smarter than the few and how collective wisdom shapes business, economies, societies and nations. New York: Doubleday.

van der Linden, W. J. (1992). Sufficient and necessary statistics. Rasch Measurement Transactions, 6(3), 231 [http://www.rasch.org/rmt/rmt63d.htm].

Whitehead, A. N. (1925). Science and the modern world. New York: Macmillan.

Wise, M. N. (Ed.). (1995). The values of precision. Princeton, New Jersey: Princeton University Press.

Wright, B. D. (1980). Foreword, Afterword. In Probabilistic models for some intelligence and attainment tests, by Georg Rasch (pp. ix-xix, 185-199. http://www.rasch.org/memo63.htm) [Reprint; original work published in 1960 by the Danish Institute for Educational Research]. Chicago, Illinois: University of Chicago Press.

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Contesting the Claim, Part III: References

July 24, 2009

References

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Andersen, E. B. (1995). What George Rasch would have thought about this book. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 383-390). New York: Springer-Verlag.

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.

Andrich, D. (1998). Thresholds, steps and rating scale conceptualization. Rasch Measurement Transactions, 12(3), 648-9 [http://209.238.26.90/rmt/rmt1239.htm].

Arnold, S. F. (1985, September). Sufficiency and invariance. Statistics & Probability Letters, 3, 275-279.

Bond, T., & Fox, C. (2001). Applying the Rasch model: Fundamental measurement in the human sciences. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Burdick, D. S., Stone, M. H., & Stenner, A. J. (2006). The Combined Gas Law and a Rasch Reading Law. Rasch Measurement Transactions, 20(2), 1059-60 [http://www.rasch.org/rmt/rmt202.pdf].

Burdick, H., & Stenner, A. J. (1996). Theoretical prediction of test items. Rasch Measurement Transactions, 10(1), 475 [http://www.rasch.org/rmt/rmt101b.htm].

Choi, E. (1998, Spring). Rasch invents “Ounces.” Popular Measurement, 1(1), 29 [http://www.rasch.org/pm/pm1-29.pdf].

Cohen, J. (1994). The earth is round (p < 0.05). American Psychologist, 49, 997-1003.

DeBoeck, P., & Wilson, M. (Eds.). (2004). Explanatory item response models: A generalized linear and nonlinear approach. (Statistics for Social and Behavioral Sciences). New York: Springer-Verlag.

Dynkin, E. B. (1951). Necessary and sufficient statistics for a family of probability distributions. Selected Translations in Mathematical Statistics and Probability, 1, 23-41.

Embretson, S. E. (1996, September). Item Response Theory models and spurious interaction effects in factorial ANOVA designs. Applied Psychological Measurement, pp. 201-212.

Falmagne, J.-C., & Narens, L. (1983). Scales and meaningfulness of quantitative laws. Synthese, 55, 287-325.

Fischer, G. H. (1981, March). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59-77.

Fischer, G. H. (1995). Derivations of the Rasch model. In G. Fischer & I. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 15-38). New York: Springer-Verlag.

Fisher, W. P., Jr. (1988). Truth, method, and measurement: The hermeneutic of instrumentation and the Rasch model [diss]. Dissertation Abstracts International, 49, 0778A, Dept. of Education, Division of the Social Sciences: University of Chicago (376 pages, 23 figures, 31 tables).

Fisher, W. P., Jr. (1997). Physical disability construct convergence across instruments: Towards a universal metric. Journal of Outcome Measurement, 1(2), 87-113.

Fisher, W. P., Jr. (1997, June). What scale-free measurement means to health outcomes research. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 357-373.

Fisher, W. P., Jr. (1999). Foundations for health status metrology: The stability of MOS SF-36 PF-10 calibrations across samples. Journal of the Louisiana State Medical Society, 151(11), 566-578.

Fisher, W. P., Jr. (2000). Objectivity in psychosocial measurement: What, why, how. Journal of Outcome Measurement, 4(2), 527-563.

Fisher, W. P., Jr. (2004, October). Meaning and method in the social sciences. Human Studies: A Journal for Philosophy and the Social Sciences, 27(4), 429-54.

Fisher, W. P., Jr. (2008, Summer). The cash value of reliability. Rasch Measurement Transactions, 22(1), 1160-3 [http://www.rasch.org/rmt/rmt221.pdf].

Fisher, W. P., Jr. (2009, July). Invariance and traceability for measures of human, social, and natural capital: Theory and application. Measurement (Elsevier), in press.

Goodman, S. N. (1999, 15 June). Toward evidence-based medical statistics. 1: The p-value fallacy. Annals of Internal Medicine, 130(12), 995-1004.

Guttman, L. (1985). The illogic of statistical inference for cumulative science. Applied Stochastic Models and Data Analysis, 1, 3-10.

Hall, W. J., Wijsman, R. A., & Ghosh, J. K. (1965). The relationship between sufficiency and invariance with applications in sequential analysis. Annals of Mathematical Statistics, 36, 575-614.

Linacre, J. M. (1993). Rasch-based generalizability theory. Rasch Measurement Transactions, 7(1), 283-284 [http://www.rasch.org/rmt/rmt71h.htm].

Luce, R. D., & Tukey, J. W. (1964). Simultaneous conjoint measurement: A new kind of fundamental measurement. Journal of Mathematical Psychology, 1(1), 1-27.

Meehl, P. E. (1967). Theory-testing in psychology and physics: A methodological paradox. Philosophy of Science, 34(2), 103-115.

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, 806-34.

Michell, J. (1999). Measurement in psychology: A critical history of a methodological concept. Cambridge: Cambridge University Press.

Moulton, M. (1993). Probabilistic mapping. Rasch Measurement Transactions, 7(1), 268 [http://www.rasch.org/rmt/rmt71b.htm].

Mundy, B. (1986, June). On the general theory of meaningful representation. Synthese, 67(3), 391-437.

Narens, L. (2002). Theories of meaningfulness (S. W. Link & J. T. Townsend, Eds.). Scientific Psychology Series. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Newby, V. A., Conner, G. R., Grant, C. P., & Bunderson, C. V. (2009). The Rasch model and additive conjoint measurement. Journal of Applied Measurement, 10(4), 348-354.

Pelton, T., & Bunderson, V. (2003). The recovery of the density scale using a stochastic quasi-realization of additive conjoint measurement. Journal of Applied Measurement, 4(3), 269-81.

Ramsay, J. O., Bloxom, B., & Cramer, E. M. (1975, June). Review of Foundations of Measurement, Vol. 1, by D. H. Krantz et al. Psychometrika, 40(2), 257-262.

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.

Roberts, F. S., & Rosenbaum, Z. (1986). Scale type, meaningfulness, and the possible psychophysical laws. Mathematical Social Sciences, 12, 77-95.

Romanoski, J. T., & Douglas, G. (2002). Rasch-transformed raw scores and two-way ANOVA: A simulation analysis. Journal of Applied Measurement, 3(4), 421-430.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test. Psychological Bulletin, 57(5), 416-428.

Smith, R. M., & Taylor, P. (2004). Equating rehabilitation outcome scales: Developing common metrics. Journal of Applied Measurement, 5(3), 229-42.

Thurstone, L. L. (1928). Attitudes can be measured. American Journal of Sociology, XXXIII, 529-544. Reprinted in L. L. Thurstone, The Measurement of Values. Midway Reprint Series. Chicago, Illinois: University of Chicago Press, 1959, pp. 215-233.

van der Linden, W. J. (1992). Sufficient and necessary statistics. Rasch Measurement Transactions, 6(3), 231 [http://www.rasch.org/rmt/rmt63d.htm].

Velleman, P. F., & Wilkinson, L. (1993). Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician, 47(1), 65-72.

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

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Contesting the Claim, Part II: Are Rasch Measures Really as Objective as Physical Measures?

July 22, 2009

When a raw score is sufficient to the task of measurement, the model is the Rasch model, we can estimate the parameters consistently, and we can evaluate the fit of the data to the model. The invariance properties that follow from a sufficient statistic include virtually the entire class of invariant rules (Hall, Wijsman, & Ghosh, 1965; Arnold, 1985), and similar relationships with other key measurement properties follow from there (Fischer, 1981, 1995; Newby, Conner, Grant, & Bunderson, 2009; Wright, 1977, 1997).

What does this all actually mean? Imagine we were able to ask an infinite number of people an infinite number of questions that all work together to measure the same thing. Because (1) the scores are sufficient statistics, (2) the ruler is not affected by what is measured, (3) the parameters separate, and (4) the data fit the model, any subset of the questions asked would give the same measure. This means that any subscore for any person measured would be a function of any and all other subscores. When a sufficient statistic is a function of all other sufficient statistics, it is not only sufficient, it is necessary, and is referred to as a minimally sufficient statistic. Thus, if separable, independent model parameters can be estimated, the model must be the Rasch model, and the raw score is both sufficient and necessary (Andersen, 1977; Dynkin, 1951; van der Linden, 1992).

This means that scores, ratings, and percentages actually stand for something measurable only when they fit a Rasch model.  After all, what actually would be the point of using data that do not support the estimation of independent parameters? If the meaning of the results is tied in unknown ways to the specific particulars of a given situation, then those results are meaningless, by definition (Roberts & Rosenbaum, 1986; Falmagne & Narens, 1983; Mundy, 1986; Narens, 2002; also see Embretson, 1996; Romanoski and Douglas, 2002). There would be no point in trying to learn anything from them, as whatever happened was a one-time unique event that tells us nothing we can use in any future event (Wright, 1977, 1997).

What we’ve done here is akin to taking a narrative stroll through a garden of mathematical proofs. These conceptual analyses can be very convincing, but actual demonstrations of them are essential. Demonstrations would be especially persuasive if there would be some way of showing three things. First, shouldn’t there be some way of constructing ordinal ratings or scores for one or another physical variable that, when scaled, give us measures that are the same as the usual measures we are accustomed to?

This would show that we can use the type of instrument usually found in the social sciences to construct physical measures with the characteristics we expect. There are four available examples, in fact, involving paired comparisons of weights (Choi, 1998), measures of short lengths (Fisher, 1988), ratings of medium-range distances (Moulton, 1993), and a recovery of the density scale (Pelton & Bunderson, 2003). In each case, the Rasch-calibrated experimental instruments produced measures equivalent to the controls, as shown in linear plots of the pairs of measures.

A second thing to build out from the mathematical proofs are experiments in which we check the purported stability of measures and calibrations. We can do this by splitting large data sets, using different groups of items to produce two or more measures for each person, or using different groups of respondents/examinees to provide data for two or more sets of item calibrations. This is a routine experimental procedure in many psychometric labs, and results tend to conform with theory, with strong associations found between increasing sample sizes and increasing reliability coefficients for the respective measures or calibrations. These associations can be plotted (Fisher, 2008), as can the pairs of calibrations estimated from different samples (Fisher, 1999), and the pairs of measures estimated from different instruments (Fisher, Harvey, Kilgore, et al., 1995; Smith & Taylor, 2004). The theoretical expectation of tighter plots for better designed instruments, larger sample sizes, and longer tests is confirmed so regularly that it should itself have the status of a law of nature (Linacre, 1993).

A third convincing demonstration is to compare studies of the same thing conducted in different times and places by different researchers using different instruments on different samples. If the instruments really measure the same thing, there will not only be obvious similarities in their item contents, but similar items will calibrate in similar positions on the metric across samples. Results of this kind have been obtained in at least three published studies (Fisher, 1997a, 1997b; Belyukova, Stone, & Fox, 2004).

All of these arguments are spelled out in greater length and detail, with illustrations, in a forthcoming article (Fisher, 2009). I learned all of this from Benjamin Wright, who worked directly with Rasch himself, and who, perhaps more importantly, was prepared for what he could learn from Rasch in his previous career as a physicist. Before encountering Rasch in 1960, Wright had worked with Feynman at Cornell, Townes at Bell Labs, and Mulliken at the University of Chicago. Taught and influenced not just by three of the great minds of twentieth-century physics, but also by Townes’ philosophical perspectives on meaning and beauty, Wright had left physics in search of life. He was happy to transfer his experience with computers into his new field of educational research, but he was dissatisfied with the quality of the data and how it was treated.

Rasch’s ideas gave Wright the conceptual tools he needed to integrate his scientific values with the demands of the field he was in. Over the course of his 40-year career in measurement, Wright wrote the first software for estimating Rasch model parameters and continuously improved it; he adapted new estimation algorithms for Rasch’s models and was involved in the articulation of new models; he applied the models to hundreds of data sets using his software; he vigorously invested himself in students and colleagues; he founded new professional societies, meetings, and journals;  and he never stopped learning how to think anew about measurement and the meaning of numbers. Through it all, there was always a yardstick handy as a simple way of conveying the basic requirements of measurement as we intuitively understand it in physical terms.

Those of us who spend a lot of time working with these ideas and trying them out on lots of different kinds of data forget or never realize how skewed our experience is relative to everyone else’s. I guess a person lives in a different world when you have the sustained luxury of working with very large databases, as I have had, and you see the constancy and stability of well-designed measures and calibrations over time, across instruments, and over repeated samples ranging from 30 to several million.

When you have that experience, it becomes a basic description of reasonable expectation to read the work of a colleague and see him say that “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). After calibrating dozens of instruments over 25 years, some of them many times over, it just seems like the plainest statement of the obvious to see the same guy say “Our measurement principles should be the same for properties of rocks as for the properties of people. What we say has to be consistent with physical measurement” (Andrich, 1998, p. 3).

And I find myself wishing more people held the opinion expressed by two other colleagues, that “scientific measures in the social sciences must hold to the same standards as do measures in the physical sciences if they are going to lead to the same quality of generalizations” (Bond & Fox, 2001, p. 2). When these sentiments are taken to their logical conclusion in a practical application, the real value of “attempting for reading comprehension what Newtonian mechanics achieved for astronomy” (Burdick & Stenner, 1996) becomes apparent. Rasch’s analogy of the structure of his model for reading tests and Newton’s Second Law can be restated relative to any physical law expressed as universal conditionals among variable triplets; a theory of the variable measured capable of predicting item calibrations provides the causal story for the observed variation (Burdick, Stone, & Stenner, 2006; DeBoeck & Wilson, 2004).

Knowing what I know, from the mathematical principles I’ve been trained in and from the extensive experimental work I’ve done, it seems amazing that so little attention is actually paid to tools and concepts that receive daily lip service as to their central importance in every facet of life, from health care to education to economics to business. Measurement technology rose up decades ago in preparation for the demands of today’s challenges. It is just plain weird the way we’re not using it to anything anywhere near its potential.

I’m convinced, though, that the problem is not a matter of persuasive rhetoric applied to the minds of the right people. Rather, someone, hopefully me, has got to configure the right combination of players in the right situation at the right time and at the right place to create a new form of real value that can’t be created any other way. Like they say, money talks. Persuasion is all well and good, but things will really take off only when people see that better measurement can aid in removing inefficiencies from the management of human, social, and natural capital, that better measurement is essential to creating sustainable and socially responsible policies and practices, and that better measurement means new sources of profitability.  I’m convinced that advanced measurement techniques are really nothing more than a new form of IT or communications technology. They will fit right into the existing networks and multiply their efficiencies many times over.

And when they do, we may be in a position to finally

“confront the remarkable fact that throughout the gigantic range of physical knowledge numerical laws assume a remarkably simple form provided fundamental measurement has taken place. Although the authors cannot explain this fact to their own satisfaction, the extension to behavioral science is obvious: we may have to await fundamental measurement before we will see any real progress in quantitative laws of behavior. In short, ordinal scales (even continuous ordinal scales) are perhaps not good enough and it may not be possible to live forever with a dozen different procedures for quantifying the same piece of behavior, each making strong but untestable and basically unlikely assumptions which result in nonlinear plots of one scale against another. Progress in physics would have been impossibly difficult without fundamental measurement and the reader who believes that all that is at stake in the axiomatic treatment of measurement is a possible criterion for canonizing one scaling procedure at the expense of others is missing the point” (Ramsay, Bloxom, and Cramer, 1975, p. 262).

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

The “Standard Model,” Part II: Natural Law, Economics, Measurement, and Capital

July 15, 2009

At Tjalling Koopmans’ invitation, Rasch became involved with the Cowles Commission, working at the University of Chicago in the 1947 academic year, and giving presentations in the same seminar series as Milton Friedman, Kenneth Arrow, and Jimmie Savage (Linacre, 1998; Cowles Foundation, 1947, 1952; Rasch, 1953). Savage would later be instrumental in bringing Rasch back to Chicago in 1960.

Rasch was prompted to approach Savage about giving a course at Chicago after receiving a particularly strong response to some of his ideas from his old mentor, Frisch, when Frisch had come to Copenhagen to receive an honorary doctorate in 1959. Frisch shared the first Nobel Prize in economics with Tinbergen, was a co-founder, with Irving Fisher, of the Econometric Society,  invented words such as “econometrics” and “macro-economics,” and was the editor of Econometrica for many years. As recounted by Rasch (1977, pp. 63-66; also see Andrich, 1997; Wright, 1980, 1998), Frisch was struck by the disappearance of the person parameter from the comparisons of item calibrations in the series of equations he presented. In response to Frisch’s reaction, Rasch formalized his mathematical ideas in a Separability Theorem.

Why were the separable parameters  significant to Frisch? Because they addressed the problem that was at the center of Frisch’s network of concepts: autonomy, better known today as structural invariance (Aldrich, 1989, p. 15; Boumans, 2005, pp. 51 ff.; Haavelmo, 1948). Autonomy concerns the capacity of data to represent a pattern of relationships that holds up across the local particulars. It is, in effect, Frisch’s own particular way of extending the Standard Model. Irving Fisher (1930) had similarly stated what he termed a Separation Theorem, which, in the manner of previous work by Walras, Jevons, and others, was also presented in terms of a multiplicative relation between three variables. Frisch (1930) complemented Irving Fisher’s focus on an instrumental approach with a mathematical, axiomatic approach (Boumans, 2005) offering necessary and sufficient conditions for tests of Irving Fisher’s theorem.

When Rasch left Frisch, he went directly to London to work with Ronald Fisher, where he remained for a year. In the following decades, Rasch became known as the foremost advocate of Ronald Fisher’s ideas in Denmark. In particular, he stressed the value of statistical sufficiency, calling it the “high mark” of Fisher’s work (Fisher, 1922). Rasch’s student, Erling Andersen, later showed that when raw scores are both necessary and sufficient statistics for autonomous, separable parameters, the model employed is Rasch’s (Andersen, 1977; Fischer, 1981; van der Linden, 1992).

Whether or not Rasch’s conditions exactly reproduce Frisch’s, and whether or not his Separability Theorem is identical with Irving Fisher’s Separation Theorem, it would seem that time with Frisch exerted a significant degree of influence on Rasch, likely focusing his attention on statistical sufficiency, the autonomy implied by separable parameters, and the multiplicative relations of variable triples.

These developments, and those documented in previous of my blogs, suggest the existence of powerful and untapped potentials hidden within psychometrics and econometrics. The story told thus far remains incomplete. However compelling the logic and personal histories may be, central questions remain unanswered. To provide a more well-rounded assessment of the situation, we must take up several unresolved philosophical issues (Fisher, 2003a, 2003b, 2004).

It is my contention that, for better measurement to become more mainstream, a certain kind of cultural shift is going to have to happen. This shift has already been underway for decades, and has precedents that go back centuries. Its features are becoming more apparent as long term economic sustainability is understood to involve significant investments in humanly, socially and environmentally responsible practices.  For such practices to be more than just superficial expressions of intentions that might be less interested in the greater good than selfish gain, they have to emerge organically from cultural roots that are already alive and thriving.

It is not difficult to see how such an organic emergence might happen, though describing it appropriately requires an ability to keep the relationship of the local individual to the global universal always in mind. And even if and when that description might be provided, having it in hand in no way shows how it could be brought about. All we can do is to persist in preparing ourselves for the opportunities that arise, reading, thinking, discussing, and practicing. Then, and only then, might we start to plant the seeds, nurture them, and see them grow.

References

Aldrich, J. (1989). Autonomy. Oxford Economic Papers, 41, 15-34.

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Andrich, D. (1997). Georg Rasch in his own words [excerpt from a 1979 interview]. Rasch Measurement Transactions, 11(1), 542-3. [http://www.rasch.org/rmt/rmt111.htm#Georg].

Boumans, M. (2001). Fisher’s instrumental approach to index numbers. In M. S. Morgan & J. Klein (Eds.), The age of economic measurement (pp. 313-44). Durham, North Carolina: Duke University Press.

Bjerkholt, O. (2001). Tracing Haavelmo’s steps from Confluence Analysis to the Probability Approach (Tech. Rep. No. 25). Oslo, Norway: Department of Economics, University of Oslo, in cooperation with The Frisch Centre for Economic Research.

Boumans, M. (1993). Paul Ehrenfest and Jan Tinbergen: A case of limited physics transfer. In N. De Marchi (Ed.), Non-natural social science: Reflecting on the enterprise of “More Heat than Light” (pp. 131-156). Durham, NC: Duke University Press.

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

Burdick, D. S., Stone, M. H., & Stenner, A. J. (2006). The Combined Gas Law and a Rasch Reading Law. Rasch Measurement Transactions, 20(2), 1059-60 [http://www.rasch.org/rmt/rmt202.pdf].

Cowles Foundation for Research in Economics. (1947). Report for period 1947, Cowles Commission for Research in Economics. Retrieved 7 July 2009, from Yale University Dept. of Economics: http://cowles.econ.yale.edu/P/reports/1947.htm.

Cowles Foundation for Research in Economics. (1952). Biographies of Staff, Fellows, and Guests, 1932-1952. Retrieved 7 July 2009 from Yale University Dept. of Economics: http://cowles.econ.yale.edu/P/reports/1932-52d.htm#Biographies.

Fischer, G. H. (1981, March). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59-77.

Fisher, I. (1930). The theory of interest. New York: Macmillan.

Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, A, 222, 309-368.

Fisher, W. P., Jr. (1992). Objectivity in measurement: A philosophical history of Rasch’s separability theorem. In M. Wilson (Ed.), Objective measurement: Theory into practice. Vol. I (pp. 29-58). Norwood, New Jersey: Ablex Publishing Corporation.

Fisher, W. P., Jr. (2003a, December). Mathematics, measurement, metaphor, metaphysics: Part I. Implications for method in postmodern science. Theory & Psychology, 13(6), 753-90.

Fisher, W. P., Jr. (2003b, December). Mathematics, measurement, metaphor, metaphysics: Part II. Accounting for Galileo’s “fateful omission.” Theory & Psychology, 13(6), 791-828.

Fisher, W. P., Jr. (2004, October). Meaning and method in the social sciences. Human Studies: A Journal for Philosophy and the Social Sciences, 27(4), 429-54.

Fisher, W. P., Jr. (2007, Summer). Living capital metrics. Rasch Measurement Transactions, 21(1), 1092-3 [http://www.rasch.org/rmt/rmt211.pdf].

Fisher, W. P., Jr. (2008, March 28). Rasch, Frisch, two Fishers and the prehistory of the Separability Theorem. In Session 67.056. Reading Rasch Closely: The History and Future of Measurement. American Educational Research Association, Rasch Measurement SIG, New York University, New York City.

Frisch, R. (1930). Necessary and sufficient conditions regarding the form of an index number which shall meet certain of Fisher’s tests. Journal of the American Statistical Association, 25, 397-406.

Haavelmo, T. (1948). The autonomy of an economic relation. In R. Frisch &  et al. (Eds.), Autonomy of economic relations. Oslo, Norway: Memo DE-UO, 25-38.

Heilbron, J. L. (1993). Weighing imponderables and other quantitative science around 1800 Historical studies in the physical and biological sciences, 24 (Supplement), Part I, 1-337.

Jammer, M. (1999). Concepts of mass in contemporary physics and philosophy. Princeton, NJ: Princeton University Press.

Linacre, J. M. (1998). Rasch at the Cowles Commission. Rasch Measurement Transactions, 11(4), 603.

Maas, H. (2001). An instrument can make a science: Jevons’s balancing acts in economics. In M. S. Morgan & J. Klein (Eds.), The age of economic measurement (pp. 277-302). Durham, North Carolina: Duke University Press.

Mirowski, P. (1988). Against mechanism. Lanham, MD: Rowman & Littlefield.

Rasch, G. (1953, March 17-19). On simultaneous factor analysis in several populations. From the Uppsala Symposium on Psychological Factor Analysis. Nordisk Psykologi’s Monograph Series, 3, 65-71, 76-79, 82-88, 90.

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.

Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy,  14, 58-94.

van der Linden, W. J. (1992). Sufficient and necessary statistics. Rasch Measurement Transactions, 6(3), 231 [http://www.rasch.org/rmt/rmt63d.htm].

Wright, B. D. (1980). Foreword, Afterword. In Probabilistic models for some intelligence and attainment tests, by Georg Rasch (pp. ix-xix, 185-199. http://www.rasch.org/memo63.htm) [Reprint; original work published in 1960 by the Danish Institute for Educational Research]. Chicago, Illinois: University of Chicago Press.

Wright, B. D. (1994, Summer). Theory construction from empirical observations. Rasch Measurement Transactions, 8(2), 362 [http://www.rasch.org/rmt/rmt82h.htm].

Wright, B. D. (1998, Spring). Georg Rasch: The man behind the model. Popular Measurement, 1, 15-22 [http://www.rasch.org/pm/pm1-15.pdf].

Publications Documenting Score, Rating, Percentage Contrasts with Real Measures

July 7, 2009

A few brief and easy introductions to the contrast between scores, ratings, and percentages vs measures include:

Linacre, J. M. (1992, Autumn). Why fuss about statistical sufficiency? Rasch Measurement Transactions, 6(3), 230 [http://www.rasch.org/rmt/rmt63c.htm].

Linacre, J. M. (1994, Summer). Likert or Rasch? Rasch Measurement Transactions, 8(2), 356 [http://www.rasch.org/rmt/rmt82d.htm].

Wright, B. D. (1992, Summer). Scores are not measures. Rasch Measurement Transactions, 6(1), 208 [http://www.rasch.org/rmt/rmt61n.htm].

Wright, B. D. (1989). Rasch model from counting right answers: Raw scores as sufficient statistics. Rasch Measurement Transactions, 3(2), 62 [http://www.rasch.org/rmt/rmt32e.htm].

Wright, B. D. (1993). Thinking with raw scores. Rasch Measurement Transactions, 7(2), 299-300 [http://www.rasch.org/rmt/rmt72r.htm].

Wright, B. D. (1999). Common sense for measurement. Rasch Measurement Transactions, 13(3), 704-5  [http://www.rasch.org/rmt/rmt133h.htm].

Longer and more technical comparisons include:

Andrich, D. (1989). Distinctions between assumptions and requirements in measurement in the social sciences. In J. A. Keats, R. Taft, R. A. Heath & S. H. Lovibond (Eds.), Mathematical and Theoretical Systems: Proceedings of the 24th International Congress of Psychology of the International Union of Psychological Science, Vol. 4 (pp. 7-16). North-Holland: Elsevier Science Publishers.

van Alphen, A., Halfens, R., Hasman, A., & Imbos, T. (1994). Likert or Rasch? Nothing is more applicable than good theory. Journal of Advanced Nursing, 20, 196-201.

Wright, B. D., & Linacre, J. M. (1989). Observations are always ordinal; measurements, however, must be interval. Archives of Physical Medicine and Rehabilitation, 70(12), 857-867 [http://www.rasch.org/memo44.htm].

Zhu, W. (1996). Should total scores from a rating scale be used directly? Research Quarterly for Exercise and Sport, 67(3), 363-372.

The following lists provide some key resources. The lists are intended to be representative, not comprehensive.  There are many works in addition to these that document the claims in yesterday’s table. Many of these books and articles are highly technical.  Good introductions can be found in Bezruczko (2005), Bond and Fox (2007), Smith and Smith (2004), Wilson (2005), Wright and Stone (1979), Wright and Masters (1982), Wright and Linacre (1989), and elsewhere. The www.rasch.org web site has comprehensive and current information on seminars, consultants, software, full text articles, professional association meetings, etc.

Books and Journal Issues

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

Andrich, D., & Douglas, G. A. (Eds.). (1982). Rasch models for measurement in educational and psychological research [Special issue]. Education Research and Perspectives, 9(1), 5-118. [Full text available at www.rasch.org.]

Bezruczko, N. (Ed.). (2005). Rasch measurement in health sciences. Maple Grove, MN: JAM Press.

Bond, T., & Fox, C. (2007). Applying the Rasch model: Fundamental measurement in the human sciences, 2d edition. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Choppin, B. (1985). In Memoriam: Bruce Choppin (T. N. Postlethwaite ed.) [Special issue]. Evaluation in Education: An International Review Series, 9(1).

DeBoeck, P., & Wilson, M. (Eds.). (2004). Explanatory item response models: A generalized linear and nonlinear approach. Statistics for Social and Behavioral Sciences). New York: Springer-Verlag.

Embretson, S. E., & Hershberger, S. L. (Eds.). (1999). The new rules of measurement: What every psychologist and educator should know. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Engelhard, G., Jr., & Wilson, M. (1996). Objective measurement: Theory into practice, Vol. 3. Norwood, New Jersey: Ablex.

Fischer, G. H., & Molenaar, I. (1995). Rasch models: Foundations, recent developments, and applications. New York: Springer-Verlag.

Fisher, W. P., Jr., & Wright, B. D. (Eds.). (1994). Applications of Probabilistic Conjoint Measurement [Special Issue]. International Journal of Educational Research, 21(6), 557-664.

Garner, M., Draney, K., Wilson, M., Engelhard, G., Jr., & Fisher, W. P., Jr. (Eds.). (2009). Advances in Rasch measurement, Vol. One. Maple Grove, MN: JAM Press.

Granger, C. V., & Gresham, G. E. (Eds). (1993, August). New Developments in Functional Assessment [Special Issue]. Physical Medicine and Rehabilitation Clinics of North America, 4(3), 417-611.

Linacre, J. M. (1989). Many-facet Rasch measurement. Chicago, Illinois: MESA Press.

Liu, X., & Boone, W. (2006). Applications of Rasch measurement in science education. Maple Grove, MN: JAM Press.

Masters, G. N. (2007). Special issue: Programme for International Student Assessment (PISA). Journal of Applied Measurement, 8(3), 235-335.

Masters, G. N., & Keeves, J. P. (Eds.). (1999). Advances in measurement in educational research and assessment. New York: Pergamon.

Osborne, J. W. (Ed.). (2007). Best practices in quantitative methods. Thousand Oaks, CA: Sage.

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.

Smith, E. V., Jr., & Smith, R. M. (Eds.) (2004). Introduction to Rasch measurement. Maple Grove, MN: JAM Press.

Smith, E. V., Jr., & Smith, R. M. (2007). Rasch measurement: Advanced and specialized applications. Maple Grove, MN: JAM Press.

Smith, R. M. (Ed.). (1997, June). Outcome Measurement [Special Issue]. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 261-428.

Smith, R. M. (1999). Rasch measurement models. Maple Grove, MN: JAM Press.

von Davier, M. (2006). Multivariate and mixture distribution Rasch models. New York: Springer.

Wilson, M. (1992). Objective measurement: Theory into practice, Vol. 1. Norwood, New Jersey: Ablex.

Wilson, M. (1994). Objective measurement: Theory into practice, Vol. 2. Norwood, New Jersey: Ablex.

Wilson, M. (2005). Constructing measures: An item response modeling approach. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Wilson, M., Draney, K., Brown, N., & Duckor, B. (Eds.). (2009). Advances in Rasch measurement, Vol. Two (p. in press). Maple Grove, MN: JAM Press.

Wilson, M., & Engelhard, G. (2000). Objective measurement: Theory into practice, Vol. 5. Westport, Connecticut: Ablex Publishing.

Wilson, M., Engelhard, G., & Draney, K. (Eds.). (1997). Objective measurement: Theory into practice, Vol. 4. Norwood, New Jersey: Ablex.

Wright, B. D., & Masters, G. N. (1982). Rating scale analysis: Rasch measurement. Chicago, Illinois: MESA Press.

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

Wright, B. D., & Stone, M. H. (1999). Measurement essentials. Wilmington, DE: Wide Range, Inc. [http://www.rasch.org/memos.htm#measess].

Key Articles

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-73.

Andrich, D. (2002). Understanding resistance to the data-model relationship in Rasch’s paradigm: A reflection for the next generation. Journal of Applied Measurement, 3(3), 325-59.

Andrich, D. (2004, January). Controversy and the Rasch model: A characteristic of incompatible paradigms? Medical Care, 42(1), I-7–I-16.

Beltyukova, S. A., Stone, G. E., & Fox, C. M. (2008). Magnitude estimation and categorical rating scaling in social sciences: A theoretical and psychometric controversy. Journal of Applied Measurement, 9(2), 151-159.

Choppin, B. (1968). An item bank using sample-free calibration. Nature, 219, 870-872.

Embretson, S. E. (1996, September). Item Response Theory models and spurious interaction effects in factorial ANOVA designs. Applied Psychological Measurement, 20(3), 201-212.

Engelhard, G. (2008, July). Historical perspectives on invariant measurement: Guttman, Rasch, and Mokken. Measurement: Interdisciplinary Research & Perspectives, 6(3), 155-189.

Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359-374.

Fischer, G. H. (1981, March). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59-77.

Fischer, G. H. (1989). Applying the principles of specific objectivity and of generalizability to the measurement of change. Psychometrika, 52(4), 565-587.

Fisher, W. P., Jr. (1997). Physical disability construct convergence across instruments: Towards a universal metric. Journal of Outcome Measurement, 1(2), 87-113.

Fisher, W. P., Jr. (2004, October). Meaning and method in the social sciences. Human Studies: A Journal for Philosophy and the Social Sciences, 27(4), 429-54.

Fisher, W. P., Jr. (2009, July). Invariance and traceability for measures of human, social, and natural capital: Theory and application. Measurement (Elsevier), in press.

Grosse, M. E., & Wright, B. D. (1986, Sep). Setting, evaluating, and maintaining certification standards with the Rasch model. Evaluation & the Health Professions, 9(3), 267-285.

Hall, W. J., Wijsman, R. A., & Ghosh, J. K. (1965). The relationship between sufficiency and invariance with applications in sequential analysis. Annals of Mathematical Statistics, 36, 575-614.

Kamata, A. (2001, March). Item analysis by the Hierarchical Generalized Linear Model. Journal of Educational Measurement, 38(1), 79-93.

Karabatsos, G., & Ullrich, J. R. (2002). Enumerating and testing conjoint measurement models. Mathematical Social Sciences, 43, 487-505.

Linacre, J. M. (1997). Instantaneous measurement and diagnosis. Physical Medicine and Rehabilitation State of the Art Reviews, 11(2), 315-324.

Linacre, J. M. (2002). Optimizing rating scale category effectiveness. Journal of Applied Measurement, 3(1), 85-106.

Lunz, M. E., & Bergstrom, B. A. (1991). Comparability of decision for computer adaptive and written examinations. Journal of Allied Health, 20(1), 15-23.

Lunz, M. E., Wright, B. D., & Linacre, J. M. (1990). Measuring the impact of judge severity on examination scores. Applied Measurement in Education, 3/4, 331-345.

Masters, G. N. (1985, March). Common-person equating with the Rasch model. Applied Psychological Measurement, 9(1), 73-82.

Mislevy, R. J., Steinberg, L. S., & Almond, R. G. (2003). On the structure of educational assessments. Measurement: Interdisciplinary Research and Perspectives, 1(1), 3-62.

Pelton, T., & Bunderson, V. (2003). The recovery of the density scale using a stochastic quasi-realization of additive conjoint measurement. Journal of Applied Measurement, 4(3), 269-81.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 321-333 [http://www.rasch.org/memo1960.pdf]). Berkeley, California: University of California Press.

Rasch, G. (1966). An individualistic approach to item analysis. In P. F. Lazarsfeld & N. W. Henry (Eds.), Readings in mathematical social science (pp. 89-108). Chicago, Illinois: Science Research Associates.

Rasch, G. (1966, July). An informal report on the present state of a theory of objectivity in comparisons. Unpublished paper [http://www.rasch.org/memo1966.pdf].

Rasch, G. (1966). An item analysis which takes individual differences into account. British Journal of Mathematical and Statistical Psychology, 19, 49-57.

Rasch, G. (1968, September 6). A mathematical theory of objectivity and its consequences for model construction. [Unpublished paper [http://www.rasch.org/memo1968.pdf]], Amsterdam, the Netherlands: Institute of Mathematical Statistics, European Branch.

Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy, 14, 58-94.

Romanoski, J. T., & Douglas, G. (2002). Rasch-transformed raw scores and two-way ANOVA: A simulation analysis. Journal of Applied Measurement, 3(4), 421-430.

Smith, R. M. (1996). A comparison of methods for determining dimensionality in Rasch measurement. Structural Equation Modeling, 3(1), 25-40.

Smith, R. M. (2000). Fit analysis in latent trait measurement models. Journal of Applied Measurement, 1(2), 199-218.

Stenner, A. J., & Smith III, M. (1982). Testing construct theories. Perceptual and Motor Skills, 55, 415-426.

Stenner, A. J. (1994). Specific objectivity – local and general. Rasch Measurement Transactions, 8(3), 374 [http://www.rasch.org/rmt/rmt83e.htm].

Stone, G. E., Beltyukova, S. A., & Fox, C. M. (2008). Objective standard setting for judge-mediated examinations. International Journal of Testing, 8(2), 180-196.

Stone, M. H. (2003). Substantive scale construction. Journal of Applied Measurement, 4(3), 282-97.

Wilson, M., & Sloane, K. (2000). From principles to practice: An embedded assessment system. Applied Measurement in Education, 13(2), 181-208.

Wright, B. D. (1968). Sample-free test calibration and person measurement. In Proceedings of the 1967 invitational conference on testing problems (pp. 85-101 [http://www.rasch.org/memo1.htm]). Princeton, New Jersey: Educational Testing Service.

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. (1980). Foreword, Afterword. In Probabilistic models for some intelligence and attainment tests, by Georg Rasch (pp. ix-xix, 185-199. http://www.rasch.org/memo63.htm). Chicago, Illinois: University of Chicago Press.

Wright, B. D. (1984). Despair and hope for educational measurement. Contemporary Education Review, 3(1), 281-288 [http://www.rasch.org/memo41.htm].

Wright, B. D. (1985). Additivity in psychological measurement. In E. Roskam (Ed.), Measurement and personality assessment. North Holland: Elsevier Science Ltd.

Wright, B. D. (1996). Comparing Rasch measurement and factor analysis. Structural Equation Modeling, 3(1), 3-24.

Wright, B. D. (1997, June). Fundamental measurement for outcome evaluation. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 261-88.

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].

Wright, B. D. (1999). Fundamental measurement for psychology. In S. E. Embretson & S. L. Hershberger (Eds.), The new rules of measurement: What every educator and psychologist should know (pp. 65-104 [http://www.rasch.org/memo64.htm]). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Wright, B. D., & Bell, S. R. (1984, Winter). Item banks: What, why, how. Journal of Educational Measurement, 21(4), 331-345 [http://www.rasch.org/memo43.htm].

Wright, B. D., & Linacre, J. M. (1989). Observations are always ordinal; measurements, however, must be interval. Archives of Physical Medicine and Rehabilitation, 70(12), 857-867 [http://www.rasch.org/memo44.htm].

Wright, B. D., & Mok, M. (2000). Understanding Rasch measurement: Rasch models overview. Journal of Applied Measurement, 1(1), 83-106.

Model Applications

Adams, R. J., Wu, M. L., & Macaskill, G. (1997). Scaling methodology and procedures for the mathematics and science scales. In M. O. Martin & D. L. Kelly (Eds.), Third International Mathematics and Science Study Technical Report: Vol. 2: Implementation and Analysis – Primary and Middle School Years. Boston: Center for the Study of Testing, Evaluation, and Educational Policy.

Andrich, D., & Van Schoubroeck, L. (1989, May). The General Health Questionnaire: A psychometric analysis using latent trait theory. Psychological Medicine, 19(2), 469-485.

Beltyukova, S. A., Stone, G. E., & Fox, C. M. (2004). Equating student satisfaction measures. Journal of Applied Measurement, 5(1), 62-9.

Bergstrom, B. A., & Lunz, M. E. (1999). CAT for certification and licensure. In F. Drasgow & J. B. Olson-Buchanan (Eds.), Innovations in computerized assessment (pp. 67-91). Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc., Publishers.

Bond, T. G. (1994). Piaget and measurement II: Empirical validation of the Piagetian model. Archives de Psychologie, 63, 155-185.

Bunderson, C. V., & Newby, V. A. (2009). The relationships among design experiments, invariant measurement scales, and domain theories. Journal of Applied Measurement, 10(2), 117-137.

Cavanagh, R. F., & Romanoski, J. T. (2006, October). Rating scale instruments and measurement. Learning Environments Research, 9(3), 273-289.

Cipriani, D., Fox, C., Khuder, S., & Boudreau, N. (2005). Comparing Rasch analyses probability estimates to sensitivity, specificity and likelihood ratios when examining the utility of medical diagnostic tests. Journal of Applied Measurement, 6(2), 180-201.

Dawson, T. L. (2004, April). Assessing intellectual development: Three approaches, one sequence. Journal of Adult Development, 11(2), 71-85.

DeSalvo, K., Fisher, W. P. Jr., Tran, K., Bloser, N., Merrill, W., & Peabody, J. W. (2006, March). Assessing measurement properties of two single-item general health measures. Quality of Life Research, 15(2), 191-201.

Engelhard, G., Jr. (1992). The measurement of writing ability with a many-faceted Rasch model. Applied Measurement in Education, 5(3), 171-191.

Engelhard, G., Jr. (1997). Constructing rater and task banks for performance assessment. Journal of Outcome Measurement, 1(1), 19-33.

Fisher, W. P., Jr. (1998). A research program for accountable and patient-centered health status measures. Journal of Outcome Measurement, 2(3), 222-239.

Fisher, W. P., Jr., Harvey, R. F., Taylor, P., Kilgore, K. M., & Kelly, C. K. (1995, February). Rehabits: A common language of functional assessment. Archives of Physical Medicine and Rehabilitation, 76(2), 113-122.

Heinemann, A. W., Gershon, R., & Fisher, W. P., Jr. (2006). Development and application of the Orthotics and Prosthetics User Survey: Applications and opportunities for health care quality improvement. Journal of Prosthetics and Orthotics, 18(1), 80-85 [http://www.oandp.org/jpo/library/2006_01S_080.asp].

Heinemann, A. W., Linacre, J. M., Wright, B. D., Hamilton, B. B., & Granger, C. V. (1994). Prediction of rehabilitation outcomes with disability measures. Archives of Physical Medicine and Rehabilitation, 75(2), 133-143.

Hobart, J. C., Cano, S. J., O’Connor, R. J., Kinos, S., Heinzlef, O., Roullet, E. P., C., et al. (2003). Multiple Sclerosis Impact Scale-29 (MSIS-29):  Measurement stability across eight European countries. Multiple Sclerosis, 9, S23.

Hobart, J. C., Cano, S. J., Zajicek, J. P., & Thompson, A. J. (2007, December). Rating scales as outcome measures for clinical trials in neurology: Problems, solutions, and recommendations. Lancet Neurology, 6, 1094-1105.

Lai, J., Fisher, A., Magalhaes, L., & Bundy, A. C. (1996). Construct validity of the sensory integration and praxis tests. Occupational Therapy Journal of Research, 16(2), 75-97.

Lee, N. P., & Fisher, W. P., Jr. (2005). Evaluation of the Diabetes Self Care Scale. Journal of Applied Measurement, 6(4), 366-81.

Ludlow, L. H., & Haley, S. M. (1995, December). Rasch model logits: Interpretation, use, and transformation. Educational and Psychological Measurement, 55(6), 967-975.

Markward, N. J., & Fisher, W. P., Jr. (2004). Calibrating the genome. Journal of Applied Measurement, 5(2), 129-41.

Massof, R. W. (2007, August). An interval-scaled scoring algorithm for visual function questionnaires. Optometry & Vision Science, 84(8), E690-E705.

Massof, R. W. (2008, July-August). Editorial: Moving toward scientific measurements of quality of life. Ophthalmic Epidemiology, 15, 209-211.

Masters, G. N., Adams, R. J., & Lokan, J. (1994). Mapping student achievement. International Journal of Educational Research, 21(6), 595-610.

Mead, R. J. (2009). The ISR: Intelligent Student Reports. Journal of Applied Measurement, 10(2), 208-224.

Pelton, T., & Bunderson, V. (2003). The recovery of the density scale using a stochastic quasi-realization of additive conjoint measurement. Journal of Applied Measurement, 4(3), 269-81.

Smith, E. V., Jr. (2000). Metric development and score reporting in Rasch measurement. Journal of Applied Measurement, 1(3), 303-26.

Smith, R. M., & Taylor, P. (2004). Equating rehabilitation outcome scales: Developing common metrics. Journal of Applied Measurement, 5(3), 229-42.

Solloway, S., & Fisher, W. P., Jr. (2007). Mindfulness in measurement: Reconsidering the measurable in mindfulness. International Journal of Transpersonal Studies, 26, 58-81 [http://www.transpersonalstudies.org/volume_26_2007.html].

Stenner, A. J. (2001). The Lexile Framework: A common metric for matching readers and texts. California School Library Journal, 25(1), 41-2.

Wolfe, E. W., Ray, L. M., & Harris, D. C. (2004, October). A Rasch analysis of three measures of teacher perception generated from the School and Staffing Survey. Educational and Psychological Measurement, 64(5), 842-860.

Wolfe, F., Hawley, D., Goldenberg, D., Russell, I., Buskila, D., & Neumann, L. (2000, Aug). The assessment of functional impairment in fibromyalgia (FM): Rasch analyses of 5 functional scales and the development of the FM Health Assessment Questionnaire. Journal of Rheumatology, 27(8), 1989-99.

Creative Commons License
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.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

W

endt, A., & Tatum, D. S. (2005). Credentialing health care professionals. In N. Bezruczko (Ed.), Rasch measurement in health sciences (pp. 161-75). Maple Grove, MN: JAM Press.

What We Measure Matters

May 19, 2009

This comment was posted in reply to Chris Conley’s recent blog athttp://www.huffingtonpost.com/chip-conley/what-we-measure-matters_b_204676.html.

Right on, Chip!

The focus on meaning is essential. And unbeknownst to just about everyone but geeks like me, there is an extensive, longstanding, and mathematically rigorous scientific literature on meaningfulness in measurement.

We need to follow through from meaningful content to meaningful numbers, since survey and assessment ratings, scores, and response percentages are NOT measures in the everyday sense of what we mean when we deal with weight scales, clocks, thermometers, or rulers. That is, these numbers do not and cannot stand for something that adds up in the same way they do. The meaning of any given unit difference changes depending on where it falls in the measurement range, on who is measured, and/or on which item(s) are measuring.

For something we want to measure to be mapped onto a number line and to be truly and fully quantified, data have to have certain properties, like additivity, sufficiency, invariance, separable parameters, etc. When those properties are obtained, an instrument can be calibrated, data volume dramatically reduced, data quality assessed in terms of its internal consistency, and the measures made meaningfully interpretable.

Fortunately, scientific scaling methods have been applied in high stakes graduation, admissions, and professional certification/licensure testing for almost 40 years. Over the last 30 years, they have come to be applied in all kinds of survey research in health care and management consulting. Contact me for more information, see my web site at www.livingcapitalmetrics.com, or see www.rasch.org for full text articles.
More on Happiness
Read the Article at HuffingtonPost