Reliability Coefficients: Starting from the Beginning

[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):












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.


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