Congeneric reliability

The term congeneric reliability (pronunciation: “rho-C”; English congeneric reliability ) describes the reliability of one-dimensional congeneric measurement models . Such measurement models are characterized by the fact that the factor loadings of the indicators do not have to be homogeneous; H. can differ. is a key figure whose value shows the extent to which the indicators of a measurement model all measure something very similar, as intended: ideally the underlying latent variable . The congeneric reliability is thus u. a. important in psychometrics . There are numerous synonyms for the term (including in particular factor reliability and composite reliability , as well as construct reliability , unidimensional omega , Raju (1977) coefficient ). ${\displaystyle \rho _{C}}$ ${\displaystyle \rho _{C}}$

meaning

congeneric measurement model

The very widespread tau-equivalent reliability (= “Cronbach's ”) requires that the factor loadings of all indicators are the same (i.e. ). However, this is not the case in many measurement models, which systematically underestimates the reliability. The congeneric reliability provides a remedy here by explicitly taking different factor loadings into account. Similar to for , the values ​​are usually between 0 and 1, whereby according to Bagozzi & Yi (1988) values ​​of at least about 0.6 are desirable. In research practice, however, higher values ​​of at least 0.7 or even 0.8 are typically aimed for. Both for and , however, it should be noted that strict rules that automatically reject measurement models below a threshold value and automatically accept them above a threshold value are generally prohibited. In addition, a close to 1 can indicate that the indicators used are too similar in terms of redundancy. ${\displaystyle \rho _{T}}$ ${\displaystyle \alpha }$ ${\displaystyle \lambda _{1}=\lambda _{2}=\ldots =\lambda _{k}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$ ${\displaystyle \rho _{C}}$

history

For the first time kongenerische reliability by Jöreskog was introduced (1971), where this simply the term "reliability" (in the English original. Reliability ) was used. However, the author was referring to congeneric measurement models. Werts et al. (1978) generally used the term “reliability” for this, but also used the term “composite reliability” for the first time to distinguish between “single-item reliability”. As a result, in the absence of a conceptual alternative, “composite reliability” was often used, but the term was also criticized. Recently u. a. Cho (2016) the consistent use of the term "kongenerische reliability" (eng. congeneric reliability ) propagated.

calculation

There are several alternative ways of calculating congeneric reliability, but they are equivalent and thus lead to the same result. Traditionally it is calculated as follows: ${\displaystyle \rho _{C}}$

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

This is the number of indicators ( items ) of the measurement model , the factor loading of the indicator and the observed variance of the error . A calculation proposed by Cho (2016) is implemented as follows, where the variance of the test result stands: ${\displaystyle k}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle i}$ ${\displaystyle \sigma _{e_{i}}^{2}}$ ${\displaystyle e_{i}}$ ${\displaystyle \sigma _{X}^{2}}$

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

The advantage of the alternative formula is that it is embedded in the system of formulas presented by Cho (2016) and facilitates a comparison with other coefficients, for example for the tau-equivalent reliability (= "Cronbach's "). The previously missing systematic naming is also the reason why Cho does not use the term "composite reliability" and instead speaks of "congeneric reliability". ${\displaystyle \alpha }$

example

The following example for calculating using the two aforementioned formulas is taken from Cho (2016). First the raw data of the covariance matrix: ${\displaystyle \rho _{C}}$

Covariance matrix

	${\displaystyle X_{1}}$	${\displaystyle X_{2}}$	${\displaystyle X_{3}}$	${\displaystyle X_{4}}$
${\displaystyle X_{1}}$	${\displaystyle 10{,}00}$
${\displaystyle X_{2}}$	${\displaystyle 4{,}42}$	${\displaystyle 11{,}00}$
${\displaystyle X_{3}}$	${\displaystyle 4{,}98}$	${\displaystyle 5{,}71}$	${\displaystyle 12{,}00}$
${\displaystyle X_{4}}$	${\displaystyle 6{,}98}$	${\displaystyle 7{,}99}$	${\displaystyle 9{,}01}$	${\displaystyle 13{,}00}$
${\displaystyle \Sigma }$	${\displaystyle 124{,}23=2\times \left(\Sigma _{subdiagonal}+\Sigma _{diagonal}\right)-\Sigma _{diagonal}}$

The raw data for the factor loadings and errors are as follows:

Factor loadings and errors

	${\displaystyle {\hat {\lambda }}_{i}}$	${\displaystyle {\hat {\sigma }}_{e_{i}}^{2}}$
${\displaystyle X_{1}}$	${\displaystyle 1{,}96}$	${\displaystyle 6{,}13}$
${\displaystyle X_{2}}$	${\displaystyle 2{,}25}$	${\displaystyle 5{,}92}$
${\displaystyle X_{3}}$	${\displaystyle 2{,}53}$	${\displaystyle 5{,}56}$
${\displaystyle X_{4}}$	${\displaystyle 3{,}55}$	${\displaystyle 0{,}37}$
${\displaystyle \Sigma }$	${\displaystyle 10{,}30}$	${\displaystyle 18{,}01}$
${\displaystyle \Sigma ^{2}}$	${\displaystyle 106{,}22}$

Now you can calculate with the traditional formula from above; A roof over a variable indicates that the calculation is based on sample data: ${\displaystyle \rho _{C}}$

${\displaystyle {\hat {\rho }}_{C}={\frac {\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}}{\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}+\sum _{i=1}^{k}{\hat {\sigma }}_{e_{i}}^{2}}}={\frac {106{,}22}{106{,}22+18{,}01}}=0{,}8550}$

The alternative formula proposed by Cho (2016) results accordingly:

${\displaystyle {\hat {\rho }}_{C}={\frac {\left(\sum _{i=1}^{k}{\hat {\lambda }}_{i}\right)^{2}}{{\hat {\sigma }}_{X}^{2}}}={\frac {106{,}22}{124{,}23}}=0{,}8550}$

Further quality measures

A coefficient closely related to congeneric reliability is the average recorded variance (DEV, English AVE ). In addition to the reliability, other properties of a measurement model must be questioned, including e.g. B. the construct validity .

Web links

• RelCalc , tools for calculating congeneric reliability and other coefficients.
• The Handbook of Management Scales by Wikibooks collects business constructs, their indicators and often gives the congeneric reliability. (engl.)

credentials

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