In statistical models applied to psychometrics, congeneric reliability ("rho C")Cho, E. (2016). Making reliability reliable: A systematic approach to reliability coefficients. Organizational Research Methods, 19(4), 651–682. https://doi.org/10.1177/1094428116656239 a single-administration test score reliability (i.e., the reliability of persons over items holding occasion fixed) coefficient, commonly referred to as composite reliability, construct reliability, and coefficient omega. is a structural equation model (SEM)-based reliability coefficients and is obtained from a unidimensional model. is the second most commonly used reliability factor after Cronbach's alpha), and is often recommended as its alternative.
History and names A quantity similar (but not mathematically equivalent) to congeneric reliability first appears in the appendix to McDonald's 1970 paper on factor analysis, labeled .Although McDonald, R. P. (1985). Factor analysis and related methods. Hillsdale, NJ: Lawrence Erlbaum and (1999). Test theory. Mahwah, NJ: Lawrence Erlbaum claim that invented congeneric reliability, there is a subtle difference between the formula given there and the modern one. As discussed in , McDonald's denominator totals observed covariances, but the modern definition divides by the sum of fitted covariances. In McDonald's work, the new quantity is primarily a mathematical convenience: a well-behaved intermediate that separates two values. Seemingly unaware of McDonald's work, Jöreskog first analyzed a quantity equivalent to congeneric reliability in a paper the following year.Jöreskog, K. G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36(2), 109–133. https://doi.org/10.1007/BF02291393 Jöreskog defined congeneric reliability (now labeled ρ) with coordinate-free notation, and three years later, Werts gave the modern, coordinatized formula for the same.Werts, C. E., Linn, R. L., & Jöreskog, K. G. (1974). Intraclass reliability estimates: Testing structural assumptions. Educational and Psychological Measurement, 34, 25–33. Both of the latter two papers named the new quantity simply "reliability". The modern name originates with Jöreskog's name for the model whence he derived : a "congeneric model".Graham, J. M. (2006). Congeneric and (Essentially) Tau-Equivalent Estimates of Score Reliability What They Are and How to Use Them. Educational and Psychological Measurement, 66(6), 930–944. https://doi.org/10.1177/0013164406288165Lucke, J. F. (2005). “Rassling the Hog”: The Influence of Correlated Item Error on Internal Consistency, Classical Reliability, and Congeneric Reliability. Applied Psychological Measurement, 29(2), 106–125. https://doi.org/10.1177/0146621604272739
Applied statisticians have subsequently coined many names for . "Composite reliability" emphasizes that measures the statistical reliability of composite scores.Werts, C. E., Rock, D. R., Linn, R. L., & Jöreskog, K. G. (1978). A general method of estimating the reliability of a composite. Educational and Psychological Measurement, 38(4), 933–938. https://doi.org/10.1177/001316447803800412 As psychology calls "constructs" any latent characteristics only measurable through composite scores,Cronbach, L. J., & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52(4), 281–302. https://doi.org/10.1037/h0040957 has also been called "construct reliability".Hair, J. F., Babin, B. J., Anderson, R. E., & Black, W. C. (2018). Multivariate data analysis (8th ed.). Cengage. Following McDonald's more recent expository work on testing theory, some SEM-based reliability coefficients, including congeneric reliability, are referred to as "reliability coefficient ", often without a definition.Padilla, M. (2019). A Primer on Reliability via Coefficient Alpha and Omega. Archives of Psychology, 3(8), Article 8. https://doi.org/10.31296/aop.v3i8.125Revelle, W., & Zinbarg, R. E. (2009). Coefficients alpha, beta, omega, and the glb: Comments on Sijtsma. Psychometrika, 74(1), 145–154. https://doi.org/10.1007/s11336-008-9102-z
Formula and calculation
Congeneric reliability applies to datasets of vectors: each row in the dataset is a list of numerical scores corresponding to one individual. The congeneric model supposes that there is a single underlying property ("factor") of the individual , such that each numerical score is a noisy measurement of . Moreover, that the relationship between and is approximately linear: there exist (non-random) vectors and such that where is a statistically independent noise term.
In this context, is often referred to as the factor loading on item .
Because and are free parameters, the model exhibits affine invariance, and may be normalized to mean and variance without loss of generality. The fraction of variance explained in item by is then simply More generally, given any covector , the proportion of variance in explained by is which is maximized when .
is this proportion of explained variance in the case where (all components of equally important):
Example + Fitted/implied covariance matrix
These are the estimates of the factor loadings and errors: + Factor loadings and errors
Compare this value with the value of applying Cronbach's alpha to the same data.
Related coefficients Cronbach's alpha () assumes that all factor loadings are equal (i.e. ). In reality, this is rarely the case and, thus, it systematically underestimates the reliability. In contrast, congeneric reliability () explicitly acknowledges the existence of different factor loadings. According to Bagozzi & Yi (1988), should have a value of at least around 0.6.Bagozzi & Yi (1988), https://dx.doi.org/10.1177/009207038801600107 Often, higher values are desirable. However, such values should not be misunderstood as strict cutoff boundaries between "good" and "bad".Guide & Ketokivi (2015), https://dx.doi.org/10.1016/S0272-6963(15)00056-X Moreover, values close to 1 might indicate that items are too similar. Another property of a "good" measurement model besides reliability is construct validity.
A related coefficient is average variance extracted.
References
External links
RelCalc, tools to calculate congeneric reliability and other coefficients. Handbook of Management Scales, Wikibook that contains management related measurement models, their indicators and often congeneric reliability.