Cronbach's Alpha with the Dimension Coefficient to Jointly Assess a Scale’s Quality

Reliability is a necessary, but not sufficient, component of validity (Downing, 2003; Feldt, Brennan, 1989). The dimension coefficient (DC) is, therefore, necessarily incorporated with Cronbach’s α to completely and fully describe a scale’s characteristics (van der et al., 2003), because not all reliable scales are valid (Cook, Beckman, 2006).

We manipulated data sets containing two types of item length (12 and 20). Each, with 5-point polytomous responses, was uniformly distributed across a ± 2 logit range. This was done for 6 kinds of normally distributed sample sizes (n = 12, 30, 50, 100, 300, and 500) with trait standard deviations (SDs) uniformly distributed from 0.5 to 9.5 logits across numbers of misfit items from 0 to 2, all of which misfit items are related to the true score with a zero correlation under Rasch model conditions. A total of 720 (= 2 item lengths x 6 sample sizes x 20 SDs x 3 numbers of misfit items) simulation datasets were administered in this study. True-score reliability and dimension coefficients were simultaneously calculated for each simulation data set.

In this case, DCs were temporarily defined by 5 respective approaches, such as Cronbach a, EGA_ratio as Eq.1 that applies the logic of scree plots to propose a ratio by computing the first and second eigenvalues (R12 = λ1/λ2) with that of the second and third ones (R23 = λ2/λ3)( Lord, 1980; Divgi, 1980), EGA_angle_ratio as Eq.2 that computes a ratio on angles at the second and third eigenvalues, Rasch loading SD as Eq.3 and Rasch_EGA_ratio as Eq.(4) derived from Rasch PCA on standardized residuals.

DC = (R12/R23)/(1 + (R12/R23))Eq. (1)
DC = (θ1223)/( 1 + (θ1223))Eq. (2)
DC = 1- Item loading SDEq. (3)
DC = (RR12/RR23)/(1 + (RR12/RR23))Eq. (4)
TypeSensitivitySpecificityROC95% CICut-off
EGA_ratio92.4697.030.970.94 to 0.98>0.67
EGA_angle_ratio94.5075.200.870.83 to 0.91>0.62
Cronbach α62.3199.010.820.77 to 0.86>0.95
Rasch item loading73.8776.240.820.77 to 0.86>0.54
Rasch_EGA_ratio74.8754.460.670.61 to 0.73≤0.55

The results were shown in Table 1 using the receiver operating characteristic (ROC) (Fawcett ,2006), in which the area under the curve, sensitivity and specificity for a binary classifier of one and multiple dimensions determined by parallel analysis(Horn, 1965). We found that the EGA_ratio with high sensitivity and specificity can be an approach to compute DC with a cut-off point (>0.67) determining the dimension strength. In our simulation study, the median of DC in Rasch unidimensionality scales without misfit items is 0.94, the highest DC can reach to 0.98.

If an instrument is valid, particularly if the unidimensionality is acceptable, we expect it to be reliable as well. However, an instrument can be both valid and reliable and still not acceptably unidimensional (DC < 0.70). It is also possible to have an instrument with low reliability and low unidimensionality.

This is why we proposed to incorporate Cronbach’s α with the DC to jointly assess a scale’s quality, and responded to the argument (Sijtsma, 2009) that using Cronbach’s α often goes hand-in-hand with the PCA approach in practical test construction, especially when validity is not easily obtained because the true score is unknown.

Tsair-Wei Chien
Chi Mei Medical Center, Taiwan

References:

Cook, D.A., & Beckman, T.J. (2006). Current Concepts in Validity and Reliability for Psychometric Instruments: Theory and Application. Am J Med., 119, 166.e7-166.

Divgi, D.R. (1980). Dimensionality of binary items: Use of a mixed model. Paper presented at the annual meeting of the National Council on Measurement in Education. Boston, MA.

Downing, S.M. (2003). Validity: on the meaningful interpretation of assessment data. Med Educ., 37, 830-837.

Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27, 861-874.

Feldt, L.S., & Brennan, R.L. (1989). Reliability. In: Linn RL, editor. Educational Measurement, 3rd Ed. New York: American Council on Education and Macmillan.

Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.

Lord, F.M. (1980). Applications of item response theory to practical testing problems. Hillside, NJ: Erlbaum.

Sijtsma, K. (2009). On the Use, the Misuse, and the Very Limited Usefulness of Cronbach’s Alpha. Psychometrika, 74, 107-120.

van der Heijden, P.G., van Buuren, S., Fekkes, M., Radder, J., & Verrips, E. (2003). Unidimensionality and reliability under Mokken scaling of the Dutch language version of the SF-36. Qual Life Res., 12(2), 189-98.


Cronbach’s Alpha with the Dimension Coefficient to Jointly Assess a Scale’s Quality. Tsair-Wei Chien … Rasch Measurement Transactions, 2012, 26:3 p. 1379




Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 2nd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
Oct. 17 - Nov. 14, 2014, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Oct. 31 - Nov. 28, 2014, Fri.-Fri. On-line workshop: Rasch Applications, Part 2: Clinical Assessment, Survey Research, and Educational Measurement (W. Fisher), www.statistics.com
Nov. 14, 2014, Fri. In-person workshop: IX Workshop on Rasch Models in Business Administration, Tenerife, Canary Islands, Spain, www.institutos.ull.es/viewcontent/institutos/iude/46416/es
Nov. 30, 2014, Sun. Submission deadline: 6th Rasch Conference: Sixth International Conference on Probabilistic Models for Measurement in Education, Psychology, Social Science and Health, Cape Town, South Africa www.rasch.co.za/conference.php
Dec. 3-5, 2014, Wed.-Fri. In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
Jan. 2-30, 2015, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 12-14, 2015, Mon.-Wed. 6th Rasch Conference: Sixth International Conference on Probabilistic Models for Measurement in Education, Psychology, Social Science and Health, Cape Town, South Africa www.rasch.co.za/conference.php
March 11-13, 2015, Wed.-Fri. In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
March 20, 2015, Fri. UK Rasch User Group Meeting, London, United Kingdom, www.rasch.org.uk
March 26-27, 2015, Thur.-Fri. In-person workshop: Introduction to Rasch Measurement with Winsteps (W. Boone), Cincinnati, raschmeasurementanalysis.com
April 16-20, 2015, Thurs.-Mon. AERA Annual Meeting, Chicago IL www.aera.net
April 21-22, 2015, Tues.-Wed. IOMC 2015: International Outcomes Measurement Conference, Chicago IL www.jampress.org
May 13-15, 2015, Wed.-Fri. In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
May 18-20, 2015, Mon.-Wed. In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
May 29 - June 26, 2015, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
July 3-31, 2015, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 14 - Sept. 11, 2015, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Sept. 9-11, 2015, Wed.-Fri. In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
Sept. 14-16, 2015, Mon.-Wed. In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
Sept. 17-18, 2015, Thur.-Fri. In-person workshop: Advanced Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
Oct. 16 - Nov. 13, 2015, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Sept. 4 - Oct. 16, 2015, Fri.-Fri. On-line workshop: Rasch Applications, Part 1: How to Construct a Rasch Scale (W. Fisher), www.statistics.com
Oct. 23 - Nov. 20, 2015, Fri.-Fri. On-line workshop: Rasch Applications, Part 2: Clinical Assessment, Survey Research, and Educational Measurement (W. Fisher), www.statistics.com
Dec. 2-4, 2015, Wed.-Fri. In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric
Aug. 12 - Sept. 9, 2016, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src="http://www.rasch.org/events.txt"></script>

The URL of this page is www.rasch.org/rmt/rmt263c.htm,

Website: www.rasch.org/rmt/contents.htm,