# 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 = (θ12/θ23)/( 1 + (θ12/θ23)) Eq. (2) DC = 1- Item loading SD Eq. (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_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

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