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.

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Tags: individual vs population, measurement, metrics, metrology, Models, ordinal, science, standards, Statistics, theory

August 28, 2009 at 06:57 |

This cogent comparison clarifies the differences between measurement and statistics in a way I haven’t seen before. I would like to make a chart that summarizes what you’ve said here to share with colleagues working in the field of mindfulness survey research. I think such a chart would be helpful as they first begin to form an understanding of these differences in reference to the value of the Rasch rating scale model for scale development and subsequently the measurement of mindfulness practice. In order to give credit where credit is due, shall I reference the chart to this blog site, or is there another citation of your work that you would prefer I reference?

August 28, 2009 at 09:39 |

Thanks, Sharon. Glad to provide something useful. I’ve submitted the piece for publication in Rasch Measurement Transactions (http://www.rasch.org/rmt), so you might cite it as submitted or in review there. Perhaps you’d consider posting the table here as a comment for everyone’s benefit? You should certainly include your own source to cite for it. Sounds like a good idea!

September 1, 2009 at 18:48 |

Hi Sharon,

I think Williams discussion is very important and is often overlooked not only in CAMS but in most areas where measurement is being used whether it be marketing research or clinical trials. Using badly crafted measures to try and capture difficult and nebulous concepts is as erroneous as having a design that is underpowered or badly executed, but it is often the least considered aspect of many clinical and non-clinical studies. Therefore I hope your colleagues will carefully considered measurement in any mindfulness surveys for the future. It is an area that I believe contributes substantially to type II errors in many areas of research.

I look forward to seeing your table.

Anita

September 27, 2009 at 17:32 |

I took statements directly from the blog above and constructed a table that compares a Measurement Models to Statistical Analysis Models. But, I don’t see any way to attach it and pasting it in causes it to lose its formatting…sharon

September 28, 2009 at 10:46 |

I have made the table taking direct quotes from the blog. But, I don’t see a way to attach it here and pasting it in loses the formatting…sharon

September 28, 2009 at 11:54 |

Thanks, Sharon. I’m not sure, but it might be that there is not a button for posting comments that does a paste from Word and retains the formatting (as there is in the blog composition window). If not, please just paste from the table’s columns one at a time, the second under the first, etc. I think that will convey the idea, and thanks for the effort!

William