An email came in today asking whether three different short (4- and 5-item) questionnaires could be expected to provide reasonable quality measurement. Here’s my response.

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Thanks for raising this question. The questionnaire plainly was not designed to provide data suitable for measurement. Though much can be learned about making constructs measurable from data produced by this kind of questionnaire, “Rasch analysis” cannot magically create a silk purse from a sow’s ear (as the old expression goes). Use Linacre’s (1993) generalizability theory nomograph to see what reliabilities are expected for each subscale, given the numbers of items and rating categories, and applying a conservative estimate of the adjusted standard deviations (1.0 logit, for instance). Convert the reliability coefficients into strata (Fisher, 1992, 2008; Wright & Masters, 1982, pp. 92, 105-106) to make the practical meaning of the precision obtained obvious.

So if you have data, analyze it and compare the expected and observed reliabilities. If the uncertainties are quite different, is that because of targeting issues? But before you do that, ask experts in the area to rank order:

- the courses by relevance to the job;
- the evaluation criteria from easy to hard; and
- the skills/competencies in order of importance to job performance.

Then study the correspondence between the rankings and the calibration results. Where do they converge and diverge? Why? What’s unexpected? What can be learned?

Analyze all of the items in each area (student, employer, instructor) together in Winsteps and study each of the three tables 23.x, setting PRCOMP=S. Remember that the total variance explained is not interpreted simply in terms of “more is better” and that the total variance explained is not as important as the ratio of that variance to the variance in the first contrast (see Linacre, 2006, 2008). If the ratio is greater than 3, the scale is essentially unidimensional (though significant problems may remain to be diagnosed and corrected).

Common practice holds that unexplained variance eigenvalues should be less than 1.5, but this overly simplistic rule of thumb (Chou & Wang, 2010; Raîche, 2005) has been contradicted in practice many times, since, even if one or more eigenvalues are over 1.5, theory may say the items belong to the same construct, and the disattenuated correlations of the measures implied by the separate groups of items (provided in tables 23.x) may still approach 1.00, indicating that the same measures are produced across subscales. See Green (1996) and Smith (1996), among others, for more on this.

If subscales within each of the three groups of items are markedly different in the measures they produce, then separate them in different analyses. If these further analyses reveal still more multidimensionalities, it’s time to go back to the drawing board, given how short these scales are. If you define a plausible scale, study the item difficulty orders closely with one or more experts in the area. If there is serious interest in precision measurement and its application to improved management, and not just a bureaucratic need for data to satisfy empty demands for a mere appearance of quality assessment, then trace the evolution of the construct as it changes from less to more across the items.

What, for instance, is the common theme addressed across the courses that makes them all relevant to job performance? The courses were each created with an intention and they were brought together into a curriculum for a purpose. These intentions and purposes are the raw material of a construct theory. Spell out the details of how the courses build competency in translation.

Furthermore, I imagine that this curriculum, by definition, was set up to be effective in training students no matter who is in the courses (within the constraints of the admission criteria), and no matter which particular challenges relevant to job performance are sampled from the universe of all possible challenges. You will recognize these unexamined and unarticulated assumptions as what need to be explicitly stated as hypotheses informing a model of the educational enterprise. This model transforms implicit assumptions into requirements that are never fully satisfied but can be very usefully approximated.

As I’ve been saying for a long time (Fisher, 1989), please do not accept the shorthand language of references to “the Rasch model”, “Rasch scaling”, “Rasch analysis”, etc. Rasch did not invent the form of these models, which are at least as old as Plato. And measurement is not a function of data analysis. Data provide experimental evidence testing model-based hypotheses concerning construct theories. When explanatory theory corroborates and validates data in calibrated instrumentation, the instrument can be applied at the point of use with no need for data analysis, to produce measures, uncertainty (error) estimates, and graphical fit assessments (Connolly, Nachtman, & Pritchett, 1971; Davis, et al., 2008; Fisher, 2006; Fisher, Kilgore, & Harvey, 1995; Linacre, 1997; many others).

So instead of using those common shorthand phrases, please speak directly to the problem of modeling the situation in order to produce a practical tool for managing it.

Further information is available in the references below.

Aryadoust, S. V. (2009). Mapping Rasch-based measurement onto the argument-based validity framework. *Rasch Measurement Transactions, 23*(1), 1192-3 [http://www.rasch.org/rmt/rmt231.pdf].

Chang, C.-H. (1996). Finding two dimensions in MMPI-2 depression. *Structural Equation Modeling, 3*(1), 41-49.

Chou, Y. T., & Wang, W. C. (2010). Checking dimensionality in item response models with principal component analysis on standardized residuals. *Educational and Psychological Measurement, 70,* 717-731.

Connolly, A. J., Nachtman, W., & Pritchett, E. M. (1971). *Keymath: Diagnostic Arithmetic Test*. Circle Pines, Minnesota: American Guidance Service. Retrieved 23 June 2018 from https://images.pearsonclinical.com/images/pa/products/keymath3_da/km3-da-pub-summary.pdf

Davis, A. M., Perruccio, A. V., Canizares, M., Tennant, A., Hawker, G. A., Conaghan, P. G. et al. (2008, May). The development of a short measure of physical function for hip OA HOOS-Physical Function Shortform (HOOS-PS): An OARSI/OMERACT initiative. *Osteoarthritis Cartilage, 16*(5), 551-559.

Fisher, W. P., Jr. (1989). What we have to offer. *Rasch Measurement Transactions, 3*(3), 72 [http://www.rasch.org/rmt/rmt33d.htm].

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

Fisher, W. P., Jr. (2006). Survey design recommendations [expanded from Fisher, W. P. Jr. (2000) Popular Measurement, 3(1), pp. 58-59]. *Rasch Measurement Transactions, 20*(3), 1072-1074 [http://www.rasch.org/rmt/rmt203.pdf].

Fisher, W. P., Jr. (2008). The cash value of reliability. *Rasch Measurement Transactions, 22*(1), 1160-1163 [http://www.rasch.org/rmt/rmt221.pdf].

Fisher, W. P., Jr., Harvey, R. F., & Kilgore, K. M. (1995). New developments in functional assessment: Probabilistic models for gold standards. *NeuroRehabilitation, 5*(1), 3-25.

Green, K. E. (1996). Dimensional analyses of complex data. *Structural Equation Modeling, 3*(1), 50-61.

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. (1997). Instantaneous measurement and diagnosis. *Physical Medicine and Rehabilitation State of the Art Reviews, 11*(2), 315-324 [http://www.rasch.org/memo60.htm].

Linacre, J. M. (1998). Detecting multidimensionality: Which residual data-type works best? *Journal of Outcome Measurement, 2*(3), 266-83.

Linacre, J. M. (1998). Structure in Rasch residuals: Why principal components analysis? *Rasch Measurement Transactions, 12*(2), 636 [http://www.rasch.org/rmt/rmt122m.htm].

Linacre, J. M. (2003). PCA: Data variance: Explained, modeled and empirical. *Rasch Measurement Transactions, 17*(3), 942-943 [http://www.rasch.org/rmt/rmt173g.htm].

Linacre, J. M. (2006). Data variance explained by Rasch measures. *Rasch Measurement Transactions, 20*(1), 1045 [http://www.rasch.org/rmt/rmt201a.htm].

Linacre, J. M. (2008). PCA: Variance in data explained by Rasch measures. *Rasch Measurement Transactions, 22*(1), 1164 [http://www.rasch.org/rmt/rmt221j.htm].

Raîche, G. (2005). Critical eigenvalue sizes in standardized residual Principal Components Analysis. *Rasch Measurement Transactions, 19*(1), 1012 [http://www.rasch.org/rmt/rmt191h.htm].

Schumacker, R. E., & Linacre, J. M. (1996). Factor analysis and Rasch. *Rasch Measurement Transactions, 9*(4), 470 [http://www.rasch.org/rmt/rmt94k.htm].

Smith, E. V., Jr. (2002). Detecting and evaluating the impact of multidimensionality using item fit statistics and principal component analysis of residuals. *Journal of Applied Measurement, 3*(2), 205-31.

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

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

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