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Philostratus the Younger, Imagines 10 (trans. Fairbanks) (Greek rhetorician C3rd A.D.) :
"[From the description of a painting :] A troup of dancers here, like the chorus which Daidalos (Daedalus) is aid to have given to Ariadne, the daughter of Minos. What does the art represent? Young men and maidens with joined hands are dancing."
Pausanias, Description of Greece 1. 20. 3 (trans. Jones) (Greek travelogue C2nd A.D.) :
"Beside this picture [in the temple of Dionysos at Athens] there are also represented . . . Ariadne asleep, Theseus putting out to sea, and Dionysos on his arrival to carry off Ariadne."
Pausanias, Description of Greece 10. 28. 3 :
"Ariadne was taken away from Theseus by Dionysos, who sailed against him with superior forces, and either fell in with Ariadne by chance or else set an ambush to catch her."
Pausanias, Description of Greece 1. 23. 7 - 8 (trans. Jones) (Greek travelogue C2nd A.D.) :
"They say that the god [Dionysos], having made war on Perseus, afterwards laid aside his enmity, and received great honors at the hands of the Argives, including this precinct set specially apart for himself. It was afterwards called the precinct of Kres (Cres) "the Kretan", because, when Ariadne died, Dionysos buried her here. But Lykeas (Lyceas) says that when the [new] temple [of Dionysos] was being rebuilt an earthenware coffin was found, and that it was Ariadne's. He also said that both he himself and other Argives had seen it."
Pausanias, Description of Greece 10. 28. 3 (trans. Jones) (Greek travelogue C2nd A.D.) :
"[In a painting of the Underworld by Polygnotos at Delphoi (Delphi) :] Ariadne, seated on a rock, is looking at her sister Phaidra (Phaedra)."
Pausanias, Description of Greece 5. 19. 1 (trans. Jones) (Greek travelogue C2nd A.D.) :
"[Amongst the scenes depicted on the chest of Kypselos (Cypselus) dedicated at Olympia :] There is Theseus holding a lyre, and by his side is Ariadne gripping a crown."
Based on recent psychometric developments, this paper presents a conceptual and practical guide for estimating internal consistency reliability of measures obtained as item sum or mean. The internal consistency reliability coefficient is presented as a by-product of the measurement model underlying the item responses. A three-step procedure is proposed for its estimation, including descriptive data analysis, test of relevant measurement models, and computation of internal consistency coefficient and its confidence interval. Provided formulas include: (a) Cronbach's alpha and omega coefficients for unidimensional measures with quantitative item response scales, (b) coefficients ordinal omega, ordinal alpha and nonlinear reliability for unidimensional measures with dichotomic and ordinal items, (c) coefficients omega and omega hierarchical for essentially unidimensional scales presenting method effects. The procedure is generalized to weighted sum measures, multidimensional scales, complex designs with multilevel and/or missing data and to scale development. Four illustrative numerical examples are fully explained and the data and the R syntax are provided.
In Case 1 an essentially tau-equivalent measurement model with high factor loadings close to .65 underlay the data, and the response distributions were symmetric. Accordingly, it was expected that descriptive statistics would suggest analyzing data as quantitative, that the essentially tau-equivalent measurement model would be the best fitting model, and that the omega value would be equal to alpha value. In Case 2, the underlying model was the congeneric measurement model with homogeneously high factor loadings and symmetric response distributions. Consequently, it was expected that item responses could be treated as quantitative, the best fitting model would be congeneric measurement model, and alpha would be expected to be close to omega due to the homogeneously high factor loadings. In Case 3, the underlying model had highly variable factor loadings plus three items with correlated errors, response distributions still being symmetric. Therefore, there were three expectations, descriptive statistics to support a quantitative subsequent analysis, the best fitting model to be that of measures with correlated errors, and alpha value to be unduly greater than omega mainly due to the fact that omega corrects for the correlation between errors whereas alpha treats this correlation as true variance. Finally, in Case 4 the underlying model was congeneric measurement model with highly variable factor loadings and with strong ceiling effects in the response distributions. In this case, it was expected that descriptive statistics would suggest treating data as ordinal and that the congeneric measurement model would show the best fit. Regarding the two reliability coefficients, they are expected to show a sizeable difference as ordinal alpha estimates the reliability of essentially tau-equivalent latent responses, whereas the non-linear SEM reliability coefficient estimates the reliability of congeneric observed responses.
The analyses were carried out using R. Phase 1, descriptive analysis, was conducted using the reshape2 (Wickham, 2007) and psych (Revelle, 2016) packages to calculate the response percentages, other descriptive statistics, and the Pearson or polychoric correlation coefficients when appropriate. In Phase 2, the nested measurement models were analyzed using the cfa function from the lavaan package (Rosseel, 2012) choosing the ML estimate in the first three cases, as per quantitative data, and the WLSMV estimate in Case 4, as per ordinal data. In order to facilitate comparison, in Phase 3 both a and omega coefficients were obtained for the best fitting parsimonious measurement models using the reliability function of the semTools package (semTools Contributors, 2016). When available, the 95% confidence intervals were calculated using the ci.reliability function of the MBESS package (Kelley & Pornprasertmanit, 2016). All decision making was based on the criteria described in the previous sections. The data for the examples are available at and the syntax used can be found in Appendix A and Appendix B of this paper.
Table 1 presents univariate and bivariate descriptive statistics for all scenarios. In Case 1, the central categories showed the highest percentage of responses and no ceiling or floor effects were observed. The values of skewness ranged between -0.11 and 0.10, and those of kurtosis between -0.29 and -0.64, so that the data were treated as quantitative although they proceed from the responses to a five-point Likert scale. All Pearson correlation coefficients were positive and homogeneous ranging from .31 to .47. Therefore, we decided to test the two plausible measurement models, congeneric versus essentially tau-equivalent, using the ML estimator. The results are presented in the first two lines in Table 2. The most constrained model tested, the essentially tau-equivalent measurement model, showed good fit to the data, χ2 (14) = 22.02, p = .078, CFI = .992, TLI = .991, RMSEA = .031. As the χ2 difference with the more flexible congeneric measurement model was not statistically significant, χ2 (5) = .09, p = .999, we chose the essentially tau-equivalent measurement model in application of the parsimony principle. Thus, all assumptions were met for α (see Equation 4) to be a good estimator of internal consistency reliability. As expected, the α estimate of .809 was the same as the omega estimate. The internal consistency of the sum or average of the items in Case 1 was within the usual standards with 95%CI values between .784 and .831.
The exploration of the data in Case 2 also led us to treat them as quantitative. Indeed, descriptive statistics in Table 1 showed the frequencies on a five-point scale without ceiling or floor effects, with skewness not higher than 0.19 in absolute value, kurtosis not higher than 0.85 in absolute value, and homogeneous correlation coefficients between items in a range between .26 and .53. In consequence, the congeneric and essentially tau-equivalent measurement models were tested using the ML estimator. As seen in Table 2, unacceptable fit was obtained when the constraint of equal factor loadings was imposed (essentially tau-equivalent measures), χ2(14) = 46.78, p < .001, CFI = .969, TLI = .967, RMSEA = .062. A considerable improvement in fit was observed when factor loadings were allowed to be different across items in the more flexible congeneric measurement model, χ2 (9) = 20.46, p = .015, CFI = .989, TLI = .982, RMSEA = .046. Moreover, the χ2 difference between both models was statistically significant, χ2 (5) = 26.32, p
Again, in Case 3 all descriptive statistics suggested analyzing data as quantitative. The response distributions in five categories did not show extreme responses and the skewness and kurtosis indices were not higher than the absolute values of 0.17 and 0.51 respectively (see Table 1) and therefore the ML estimator was deemed appropriate. However, as expected, the correlation coefficients were not homogeneous since very high correlations, greater than .78, between three items (Y4, Y5, Y6) were observed, while the remaining correlations ranged between low and moderate from .05 to .43. These three items showed a special clustering that would be modeled as correlated errors.
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