Congeneric Reliability
   HOME

TheInfoList



OR:

In statistical models applied to
psychometrics Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...
, congeneric reliability \rho_C ("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. \rho_C is a structural equation model(SEM)-based reliability coefficients and is obtained from on a unidimensional model. \rho_C is the second most commonly used reliability factor after
tau-equivalent reliability Cronbach's alpha (Cronbach's \alpha), also known as tau-equivalent reliability (\rho_T) or coefficient alpha (coefficient \alpha), is a reliability coefficient that provides a method of measuring internal consistency of tests and measures. Numero ...
(\rho_T), and is often recommended as its alternative.


Formula and calculation


Systematic and conventional formula

Let X_i denote the observed score of item i and X(=X_1 + X_2 + \cdots + X_k) denote the sum of all items in a test consisting of k items. It is assumed that each item's (observation) score consists of the item's (unobserved) true score and the item's error (i.e., X_i=T_i+e_i). The congeneric model assumes that each item's true score is a linear combination of a common factor (F) (i.e., T_i=\mu_i+\lambda_i F). \lambda_i is often referred to as a
factor loading Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
of item i. \sigma^2_X is the sum of all the elements of the fitted/implied covariance matrix of X obtained from estimates of \lambda_i 's and \sigma_'s. \rho_C's "systematic formula" is: : \rho_C = \frac Its conventional (i.e., more often used) formula is: : \rho_C = \frac


Example

These are the estimates of the factor loadings and errors: :\hat_ = \frac = \frac = .8550 :\hat_ = \frac = \frac = .8550 Compare this value with the value of applying
tau-equivalent reliability Cronbach's alpha (Cronbach's \alpha), also known as tau-equivalent reliability (\rho_T) or coefficient alpha (coefficient \alpha), is a reliability coefficient that provides a method of measuring internal consistency of tests and measures. Numero ...
to the same data.


History

_'s formula was first introduced by Jöreskog (1971) in a matrix notation.Jöreskog, K. G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36(2), 109–133. https://doi.org/10.1007/BF02291393 Its conventional formula first appeared in Werts et al. (1974).Werts, C. E., Linn, R. L., & Jöreskog, K. G. (1974). Intraclass reliability estimates: Testing structural assumptions. Educational and Psychological Measurement, 34, 25–33. https://doi.org/10.1177/001316447403400104 They didn't give the formula a special name and just referred to it as "reliability." In other words, this formula has no official name, and this absence causes various versions of the name to be created.


Names of congeneric reliability

_ has been referred to by various names between applied researchers and between reliability researchers. In addition, the names used by applied researchers differ from the names used by reliability researchers. This diversity and difference create confusion and inaccuracies in communication.


Composite reliability

The term composite reliability is short for the 'reliability of composite scores'. Unless measured by a single item, all reliability coefficients are composite reliability. Therefore, this name is not suitable as a specific formula name. The name composite reliability gives the impression that this reliability coefficient is complex, or that it has been synthesized from other reliability coefficients. Werts et al. (1978)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 also called this formula "reliability." However, they used the expression "the composite reliability" once as an abbreviation of the reliability of a composite score to distinguish the reliability of a single item. Since then, this unintended name has been used as the name of this formula. Applied researchers most often use this name when referring to \rho_. Researchers who publish papers on reliability rarely use this name.


Construct reliability

Construct reliability is short for the 'reliability of a construct'. Construct is synonymous with concept. A construct is a theoretical and abstract entity and is embodied through measurement. We can estimate the reliability of a measurement, but not the reliability of a construct. For example, you can say the reliability of "a measure of height," but not the reliability of the concept of "height." Construct reliability is a term that is not logically established. Let's say that this term makes sense. All other reliability coefficients also originate from the measurement of a construct and should be called construct reliability. Construct reliability is not suitable as a term referring to a specific reliability coefficient. The term has been used in the books of Hair and his colleagues, the world's best sellers for practical statistical analysis. Applied researchers use the term construct reliability at a frequency of 1/3 of composite reliability. Researchers who publish papers on reliability rarely use this name.


Coefficient omegaCho, E. and Chun, S. (2018), Fixing a broken clock: A historical review of the originators of reliability coefficients including Cronbach’s alpha. Survey Research, 19(2), 23–54.

Various SEM-based reliability coefficients are referred to as \omega, typically without a definition. Therefore, it is difficult for readers to know exactly what the name \omega refers to. This practice reduces the accuracy of communication. If we need a generic name to refer to a variety of reliability coefficients, using \rho rather than \omega is more traditional. The name coefficient \omega is based on McDonald's (1985, 1999)McDonald, R. P. (1985).
Factor analysis Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
and related methods. Hillsdale, NJ: Lawrence Erlbaum.
McDonald, R. P. (1999). Test theory: A unified treatment. Mahwah, NJ: Lawrence Erlbaum. claim that McDonald (1970)McDonald, R. P. (1970). Theoretical canonical foundations of principal factor analysis, canonical factor analysis, and alpha factor analysis. British Journal of Mathematical and Statistical Psychology, 23, 1-21. doi:10.1111/j.2044-8317.1970.tb00432.x. first developed \rho_. In his paper on exploratory factor analysis (EFA), McDonald (1970) presents a reliability formula using the \theta symbol. This formula was included in the footnote of the article without any explanation. McDonald (1985) refers to a formula algebraically equivalent to \rho_ as \omega in his book. He also says that the \theta presented by McDonald (1970) is renamed \omega. McDonald (1999) describes various types of reliability coefficients (e.g., unidimensional and multi-dimensional models) as \omega. He explicitly declares that he first proposed \rho_. McDonald (1985, 1999)) does not cite Jöreskog (1971) or Werts et al. (1974). The following objections were made. First, the formula proposed by McDonald (1970) was not new. If this formula were of high academic value at that time, it would not have been presented without explanation in the footnotes. In the context of EFA, there are studies suggesting similar reliability formulas. Second, McDonald (1970) 's \theta differs from \rho_. The denominator of the formula given by McDonald (1970) is observed covariances, and the denominator of \rho_ is fitted covariances. Third, McDonald (1970) did not discuss how to actually obtain this coefficient. While it is easy to derive a reliability formula, the more important barrier at that time was how to obtain estimates of each parameter. Jöreskog has addressed this issue across studies. Fourth, it was Jöreskog (1971) that actually influenced users. McDonald (1970) was occasionally cited in EFA literature, but rarely cited in reliability literature. The expression coefficients \omega was rarely used before 2009. Applied researchers rarely use this name. Researchers who publish papers on reliability often use this name recently.


Congeneric reliability

Unlike other names that give no information about the characteristics of the coefficients, the name congeneric reliability contains information about when this coefficient should be used. Jöreskog (1971) did not propose a name for \rho_, but referred to the measurement model from which \rho_ was derived as a congeneric model. The name congeneric reliability has been used occasionally in reliability literature since then.Lucke, 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 Cho (2016) proposed that this coefficient be referred to as \rho_{C} for a consistent system with other reliability coefficients.


Related coefficients

A related coefficient is
average variance extracted In statistics (classical test theory), average variance extracted (AVE) is a measure of the amount of variance that is captured by a construct in relation to the amount of variance due to measurement error.Fornell & Larcker (1981), https://www.jst ...
.


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. Comparison of assessments Psychometrics Statistical reliability