Congeneric reliability

Formula and calculation

Congeneric reliability applies to datasets of vectors: each row X in the dataset is a list X_i of numerical scores corresponding to one individual. The congeneric model supposes that there is a single underlying property ("factor") of the individual F, such that each numerical score X_i is a noisy measurement of F. Moreover, that the relationship between X and F is approximately linear: there exist (non-random) vectors λ and μ such that $${\displaystyle X_i={\lambda }_{i}F+{\mu }_i+E_i\text{,}}$$ where E_i is a statistically independent noise term.

In this context, λ_i is often referred to as the factor loading on item i.

Because λ and μ are free parameters, the model exhibits affine invariance, and F may be normalized to mean 0 and variance 1 without loss of generality. The fraction of variance explained in item X_i by F is then simply $${\displaystyle {\rho }_i=\frac{{\lambda }_i^2}{{\lambda }_i^2+\mathbb{V}[E_i]}\text{.}}$$ More generally, given any covector w, the proportion of variance in wX explained by F is $${\displaystyle \rho =\frac{(w\lambda {)}^2}{(w\lambda {)}^2+\mathbb{E}[(wE{)}^2]}\text{,}}$$ which is maximized when w ∝ 𝔼[EE^*]λ.

ρ_C is this proportion of explained variance in the case where w ∝ [1 1 ... 1] (all components of X equally important): $${\displaystyle {\rho }_C=\frac{{\left(\sum _{i=1}^k{\lambda }_i\right)}^2}{{\left(\sum _{i=1}^k{\lambda }_i\right)}^2+\sum _{i=1}^k{\sigma }_{E_i}^2}}$$

Example

These are the estimates of the factor loadings and errors:

${\displaystyle {\hat{\rho }}_C=\frac{{\left(\sum _{i=1}^k{\hat{\lambda }}_i\right)}^2}{{\hat{\sigma }}_X^2}=\frac{106.22}{124.23}=.8550}$

${\displaystyle {\hat{\rho }}_C=\frac{{\left(\sum _{i=1}^k{\hat{\lambda }}_i\right)}^2}{{\left(\sum _{i=1}^k{\hat{\lambda }}_i\right)}^2+\sum _{i=1}^k{\hat{\sigma }}_{e_i}^2}=\frac{106.22}{106.22+18.01}=.8550}$

Compare this value with the value of applying tau-equivalent reliability to the same data.

Related coefficients

Tau-equivalent reliability (${\displaystyle {\rho }_T}$), which has traditionally been called "Cronbach's ${\displaystyle \alpha }$", assumes that all factor loadings are equal (i.e. ${\displaystyle {\lambda }_1={\lambda }_2=...={\lambda }_k}$). In reality, this is rarely the case and, thus, it systematically underestimates the reliability. In contrast, congeneric reliability (${\displaystyle {\rho }_C}$) explicitly acknowledges the existence of different factor loadings. According to Bagozzi & Yi (1988), ${\displaystyle {\rho }_C}$ should have a value of at least around 0.6. Often, higher values are desirable. However, such values should not be misunderstood as strict cutoff boundaries between "good" and "bad". Moreover, ${\displaystyle {\rho }_C}$ values close to 1 might indicate that items are too similar. Another property of a "good" measurement model besides reliability is construct validity.

A related coefficient is average variance extracted.