Archive for the ‘precision’ Category

Just posted on www.economist.com in response to Sept 26 Schumpeter article

September 29, 2009

Let’s cut through the Gordian Knot to the real issue. That we manage what we measure is as close to an absolute truth as there ever was. What got us into this mess was the inadequacy of the vast majority of our measures. So-called “measures” that only get in the way of management are a sign that new standards, criteria, and methods of measurement are needed. The core issue we face is how to transform socialized externalities into capitalized internalities. Transaction costs are the most important and largest costs in any economic exchange. We reduce and control these via measurement. Human, social, and natural capital transaction costs are virtually uncontrolled and unmeasured. We need a metric system for universally uniform measures of abilities and skills, health, motivation, loyalty and trust, and environmental quality. And we needed it yesterday. But who is working on it? Who is talking about it? Most importantly, who is taking advantage of the huge strides that have been made in measurement science over the last 50 years, strides that have made measurement far more rigorous, practical, and flexible than anyone in business seems to know. As to business being an art, so is music, but music is played on and reproduced by some of the highest technology and finest precision instrumentation around. What we need to do is tune the instruments of the management arts and sciences so that we can harmonize our relationships, get with the beat, and sing the melodies we feel in our hearts and souls. For more information, see http://www.livingcapitalmetrics.com, or my blog at https://livingcapitalmetrics.wordpress.com.

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

September 28, 2009

The core issue in measurement is meaningfulness. There are three major aspects of meaningfulness to take into account in measurement. These have to do with the constancy of the unit, interpreting the size of differences in measures, and evaluating the coherence of the units and differences.

First, raw scores (counts of right answers or other events, sums of ratings, or rankings) do not stand for anything that adds up the way they do (see previous blogs for more on this). Any given raw score unit can be 4-5 times larger than another, depending on where they fall in the range. Meaningful measurement demands a constant unit. Instrument scaling methods provide it.

Second, meaningful measurement requires that we be able to say just what any quantitative amount of difference is supposed to represent. What does a difference between two measures stand for in the way of what is and isn’t done at those two levels? Is the difference within the range of error, and so random? Is the difference many times more than the error, and so repeatedly reproducible and constant? Meaningful measurement demands that we be able to make reliable distinctions.

Third, meaningful measurement demands that the items work together to measure the same thing. If reliable distinctions can be made between measures, what is the one thing that all of the items tap into? If the data exhibit a consistency that is shared across items and across persons, what is the nature of that consistency? Meaningful measurement posits a model of what data must look like to be interpretable and coherent, and then it evaluates data in light of that model.

When a constant unit is in hand, when the limits of randomness relative to stable differences are known, and when individual responses are consistent with one another, then, and only then, is measurement meaningful. Inconstant units, unknown amounts of random variation, and inconsistent data can never amount to the science we need for understanding and managing skills, abilities, health, motivations, social bonds, and environmental quality.

Managing our investments in human, social, and natural capital for positive returns demands that meaningful measurement be universalized in uniformly calibrated and accessible metrics. Scientifically rigorous, practical, and convenient methods for setting reference standards and making instruments traceable to them are readily available.

We have the means in hand for effecting order-of-magnitude improvements in the meaningfulness of the measures used in education, health care, human and environmental resource management, etc. It’s time we got to work on it.

We are what we measure. It’s time we measured what we want to be.

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LivingCapitalMetrics Blog by William P. Fisher, Jr., Ph.D. is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.
Based on a work at livingcapitalmetrics.wordpress.com.
Permissions beyond the scope of this license may be available at http://www.livingcapitalmetrics.com.

Reliability Coefficients: Starting from the Beginning

August 31, 2009

[This posting was prompted by questions concerning a previous blog entry, Reliability Revisited, and provides background on reliability that only Rasch measurement practitioners are likely to possess.] Most measurement applications based in ordinal data do not implement rigorous checks of the internal consistency of the observations, nor do they typically use the log-odds transformation to convert the nonlinear scores into linear measures. Measurement is usually defined in statistical terms, applying population-level models to obtain group-level summary scores, means, and percentages. Measurement, however, ought to involve individual-level models and case-specific location estimates. (See one of my earlier blogs for more on this distinction between statistics and measurement.)

Given the appropriate measurement focus on the individual, the instrument is initially calibrated and measures are estimated in a simultaneous conjoint process. Once the instrument is calibrated, the item estimates can be anchored, measures can be routinely produced from them, and new items can be calibrated into the system, and others dropped, over time. This method has been the norm in admissions, certification, licensure, and high stakes testing for decades (Fisher & Wright, 1994; Bezruczko, 2005).

Measurement modelling of individual response processes has to be stochastic, or else we run into the attenuation paradox (Engelhard, 1993, 1994). This is the situation in which a deterministic progression of observations from one end of the instrument to the other produces apparently error-free data strings that look like this (1 being a correct answer, a higher rating, or the presence of an attribute, and 0 being incorrect, a lower rating, or the absence of the attribute):

00000000000

10000000000

11000000000

11100000000

11110000000

11111000000

11111100000

11111111000

11111111100

11111111110

11111111111

In this situation, strings with all 0s and all 1s give no information useful for estimating measures (rows) or calibrations (columns). It is as though some of the people are shorter than the first unit on the ruler, and others are taller than the top unit. We don’t really have any way of knowing how short or tall they are, so their rows drop out. But eliminating the top and bottom rows makes the leftmost and rightmost columns all 0s and 1s, and eliminating them then gives new rows with all 0s and 1s, etc., until there’s no data left. (See my Revisiting Reliability blog for evaluations of five different probabilistically-structured data sets of this kind simulated to contrast various approaches to assessing reliability and internal consistency.)

The problem for estimation (Linacre, 1991, 1999, 2000) in data like those shown above is that the lack of informational overlaps between the columns, on the one hand, and between the rows, on the other, gives us no basis for knowing how much more of the variable is represented by any one item relative to any other, or by any one person measured relative to any other. In addition, whenever we actually construct measures of abilities, attitudes, or behaviors that conform with this kind of Guttman (1950) structure (Andrich, 1985; Douglas & Wright, 1989; Engelhard, 2008), the items have to be of such markedly different difficulties or agreeabilities that the results tend to involve large numbers of indistinguishable groups of respondents. But when that information is present in a probabilistically consistent way, we have an example of the phenomenon of stochastic resonance (Fisher, 1992b), so called because of the way noise amplifies weak deterministic signals (Andò & Graziani, 2000; Benzi, Sutera, & Vulpiani, 1981; Bulsara & Gammaitoni, 1996; Dykman & McClintock, 1998; Schimansky-Geier, Freund, Neiman, & Shulgin, 1998).

We need the noise, but we can’t let it overwhelm the system. We have to be able to know how much error there is relative to actual signal. Reliability is traditionally defined (Guilford 1965, pp. 439-40) as an estimate of this relation of signal and noise:

“The reliability of any set of measurements is logically defined as the proportion of their variance that is true variance…. We think of the total variance of a set of measures as being made up of two sources of variance: true variance and error variance… The true measure is assumed to be the genuine value of whatever is being measured… The error components occur independently and at random.”

Traditional reliability coefficients, like Cronbach’s alpha, are correlational, implementing a statistical model of group-level information. Error is taken to be the unexplained portion of the variance:

“In his description of alpha Cronbach (1951) proved (1) that alpha is the mean of all possible split-half coefficients, (2) that alpha is the value expected when two random samples of items from a pool like those in the given test are correlated, and (3) that alpha is a lower bound to the proportion of test variance attributable to common factors among the items” (Hattie, 1985, pp. 143-4).

But measurement models of individual-level response processes (Rasch, 1960; Andrich, 1988; Wright, 1977; Fisher & Wright, 1994; Bond & Fox, 2007; Wilson, 2005; Bezruczko, 2005) employ individual-level error estimates (Wright, 1977; Wright & Stone, 1979; Wright & Masters, 1982), not correlational group-level variance estimates. The individual measurement errors are statistically equivalent to sampling confidence intervals, as is evident in both Wright’s equations and in plots of errors and confidence intervals (see Figure 4 in Fisher, 2008). That is, error and confidence intervals both decline at the same rate with larger numbers of item responses per person, or larger numbers of person responses per item.

This phenomenon has a constructive application in instrument design. If a reasonable expectation for the measurement standard deviation can be formulated and related to the error expected on the basis of the number of items and response categories, a good estimate of the measurement reliability can be read off a nomograph (Linacre, 1993).

Wright (Wright & Masters, 1982, pp. 92, 106; Wright, 1996) introduced several vitally important measurement precision concepts and tools that follow from access to individual person and item error estimates. They improve on the traditional KR-20 or Cronbach reliability coefficients because the individualized error estimates better account for the imprecisions of mistargeted instruments, and for missing data, and so more accurately and conservatively estimate reliability.

Wright and Masters introduce a new reliability statistic, G, the measurement separation reliability index. The availability of individual error estimates makes it possible to estimate the true variance of the measures more directly, by subtracting the mean square error from the total variance. The standard deviation based on this estimate of true variance is then made the numerator of a ratio, G, having the root mean square error as its denominator.

Each unit increase in this G index then represents another multiple of the error unit in the amount of quantitative variation present in the measures. This multiple is nonlinearly represented in the traditional reliability coefficients expressed in the 0.00 – 1.00 range, such that the same separation index unit difference is found in the 0.00 to 0.50, 0.50 to 0.80, 0.80 to 0.90, 0.90 to 0.94, 0.94 to 0.96, and 0.96 to 0.97 reliability ranges (see Fisher, 1992a, for a table of values; available online: see references).

G can also be estimated as the square root of the reliability divided by one minus the reliability. Conversely, a reliability coefficient roughly equivalent to Cronbach’s alpha is estimated as G squared divided by G squared plus the error variance. Because individual error estimates are inflated in the presence of missing data and when an instrument is mistargeted and measures tend toward the extremes, the Rasch-based reliability coefficients tend to be more conservative than Cronbach’s alpha, as these sources of error are hidden within the variances and correlations. For a comparison of the G separation index, the G reliability coefficient, and Cronbach’s alpha over five simulated data sets, see the Reliability Revisited blog entry.

Error estimates can be made more conservative yet by multiplying each individual error term by the larger of either 1.0 or the square root of the associated individual mean square fit statistic for that case (Wright, 1995). (The mean square fit statistics are chi-squares divided by their degrees of freedom, and so have an expected value of 1.00; see Smith (1996) for more on fit, and see my recent blog, Revisiting Reliability, for more on the conceptualization and evaluation of reliability relative to fit.)

Wright and Masters (1982, pp. 92, 105-6) also introduce the concept of strata, ranges on the measurement continuum with centers separated by three errors. Strata are in effect a more forgiving expression of the separation reliability index, G, since the latter approximates strata with centers separated by four errors. An estimate of strata defined as having centers separated by four errors is very nearly identical with the separation index. If three errors define a 95% confidence interval, four are equivalent to 99% confidence.

There is a particular relevance in all of this for practical applications involving the combination or aggregation of physical, chemical, and other previously calibrated measures. This is illustrated in, for instance, the use of chemical indicators in assessing disease severity, environmental pollution, etc. Though any individual measure of the amount of a chemical or compound is valid within the limits of its intended purpose, to arrive at measures delineating disease severity, overall pollution levels, etc., the relevant instruments must be designed, tested, calibrated, and maintained, just as any instruments are (Alvarez, 2005; Cipriani, Fox, Khuder, et al., 2005; Fisher, Bernstein, et al., 2002; Fisher, Priest, Gilder, et al., 2008; Hughes, Perkins, Wright, et al., 2003; Perkins, Wright, & Dorsey, 2005; Wright, 2000).

The same methodology that is applied in this work, involving the rating or assessment of the quality of the outcomes or impacts counted, expressed as percentages, or given in an indicator’s native metric (parts per million, acres, number served, etc.), is needed in the management of all forms of human, social, and natural capital. (Watch this space for a forthcoming blog applying this methodology to the scaling of the UN Millennium Development Goals data.) The practical advantages of working from calibrated instrumentation in these contexts include data quality evaluations, the replacement of nonlinear percentages with linear measures, data volume reduction with no loss of information, and the integration of meaningful and substantive qualities with additive quantities on annotated metrics.

References

Alvarez, P. (2005). Several noncategorical measures define air pollution. In N. Bezruczko (Ed.), Rasch measurement in health sciences (pp. 277-93). Maple Grove, MN: JAM Press.

Andò, B., & Graziani, S. (2000). Stochastic resonance theory and applications. New York: Kluwer Academic Publishers.

Andrich, D. (1985). An elaboration of Guttman scaling with Rasch models for measurement. In N. B. Tuma (Ed.), Sociological methodology 1985 (pp. 33-80). San Francisco, California: Jossey-Bass.

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.

Benzi, R., Sutera, A., & Vulpiani, A. (1981). The mechanism of stochastic resonance. Journal of Physics. A. Mathematical and General, 14, L453-L457.

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.

Bulsara, A. R., & Gammaitoni, L. (1996, March). Tuning in to noise. Physics Today, 49, 39-45.

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.

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334.

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