Congeneric reliability

In statistical models applied to psychometrics, congeneric reliability $\rho _{C}$ ("rho C") is the reliability of a unidimensional congeneric measurement model. It can be interpreted as an indicator of how reliably the items of such a measurement model reflect the same underlying variable. Synonymous terms are composite reliability, unidimensional omega and Raju (1977) coefficient).^[1]

Relevance

Congeneric measurement model

Tau-equivalent reliability ( $\rho_T$ ), which has traditionally been called "Cronbach's $\alpha$ ", assumes that all factor loadings are equal (i.e. $\lambda _{1}=\lambda _{2}=...=\lambda _{k}$ ). In reality, this is rarely the case and, thus, it systematically underestimates the reliability. In contrast, congeneric reliability ( $\rho _{C}$ ) explicitly acknowledges the existence of different factor loadings. According to Bagozzi & Yi (1988), $\rho _{C}$ should have a value of at least around 0.6.^[2] Often, higher values are desirable. However, such values should not be misunderstood as strict cutoff boundaries between "good" and "bad".^[3] Moreover, $\rho _{C}$ values close to 1 might indicate that items are too similar. Another property of a "good" measurement model besides reliability is construct validity.

History

Congeneric reliability was first introduced by Jöreskog (1971), when he used the term "reliability" but implicitly referred to congeneric measurement models.^[4] Werts et al. (1978) also used the more general term "reliability" but for the first time also used the term "composite reliability" to distinguish it from "single-item reliability".^[5] As no other term was available, the term "composite reliability" was subsequently in use but has also been criticized since.^[1] More recently, Cho (2016) suggested to use the term "congeneric reliability" instead.^[1]

Calculation

There are different ways to calculate congeneric reliability. These ways are equivalent, i.e. they will lead to the same result. Traditionally, $\rho _{C}$ is calculated as follows:

\rho _{C}={\frac {\left(\sum _{i=1}^{k}\lambda _{i}\right)^{2}}{\left(\sum _{i=1}^{k}\lambda _{i}\right)^{2}+\sum _{i=1}^{k}\sigma _{e_{i}}^{2}}}

Here, $k$ is the number of items of the measurement model, $\lambda _{i}$ the factor loading of item $i$ and $\sigma _{e_{i}}^{2}$ the observed variance of the error $e_{i}$ . An alternative way to calculate congeneric reliability suggested by Cho (2016) is realized as follows, where $\sigma _{X}^{2}$ is the variance of the test result:

\rho _{C}={\frac {\left(\sum _{i=1}^{k}\lambda _{i}\right)^{2}}{\sigma _{X}^{2}}}

The advantage of this alternative formula is that it is embedded in Cho's (2016) system of formulas, which makes it easier to be compared to other coefficients, e.g. tau-equivalent reliability (= "Cronbach's $\alpha$ "). This system is also the reason why the term "composite reliability" should be dismissed for the benefit of the term "congeneric reliability".

Example

The following illustrative example to calculate $\rho _{C}$ using the aforementioned formulas is based on Cho (2016).^[1] These are the raw data of the covariance matrix:

Fitted/implied covariance matrix
	$X_{1}$	$X_{2}$	$X_3$	$X_4$
$X_{1}$	$10.00$
$X_{2}$	$4.42$	$11.00$
$X_3$	$4.98$	$5.71$	$12.00$
$X_4$	$6.98$	$7.99$	$9.01$	$13.00$
$\Sigma$	$124.23=2\times \left(\Sigma _{subdiagonal}+\Sigma _{diagonal}\right)-\Sigma _{diagonal}$

These are the raw data of the factor loadings and errors:

Factor loadings and errors
	${\hat {\lambda }}_{i}$	${\hat {\sigma }}_{e_{i}}^{2}$
$X_{1}$	$1.96$	$6.13$
$X_{2}$	$2.25$	$5.92$
$X_3$	$2.53$	$5.56$
$X_4$	$3.55$	$.37$
$\Sigma$	$10.30$	$18.01$
$\Sigma ^{2}$	$106.22$

Now the traditional formula from above can be used to calculate $\rho _{C}$ , whereby a hat above a variable indicates that the calculation is based on sample data:

{\hat {\rho }}_{C}={\frac {\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}}{\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}+\sum _{i=1}^{k}{\hat {\sigma }}_{e_{i}}^{2}}}={\frac {106.22}{106.22+18.01}}=.8550

Accordingly with the alternative formula of Cho (2016):

{\hat {\rho }}_{C}={\frac {\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}}{{\hat {\sigma }}_{X}^{2}}}={\frac {106.22}{124.23}}=.8550

Related coefficients

A related coefficient is average variance extracted.

References

1 2 3 4 Cho (2016), https://dx.doi.org/10.1177/1094428116656239
↑ Bagozzi & Yi (1988), https://dx.doi.org/10.1177/009207038801600107
↑ Guide & Ketokivi (2015), https://dx.doi.org/10.1016/S0272-6963(15)00056-X
↑ Jöreskog (1971), https://dx.doi.org/10.1007/BF02291393
↑ Werts et al. (1978), https://dx.doi.org/10.1177/001316447803800412

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.

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[Cho2016-1] 1 2 3 4 Cho (2016), https://dx.doi.org/10.1177/1094428116656239

[2] Bagozzi & Yi (1988), https://dx.doi.org/10.1177/009207038801600107

[3] Guide & Ketokivi (2015), https://dx.doi.org/10.1016/S0272-6963(15)00056-X

[Joreskog1971-4] Jöreskog (1971), https://dx.doi.org/10.1007/BF02291393

[Werts1978-5] Werts et al. (1978), https://dx.doi.org/10.1177/001316447803800412