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 X_i=\lambda_iF+\mu_i+E_i\text{,} 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 \rho_i=\frac{\lambda_i^2}{\lambda_i^2+\mathbb{V}[E_i]}\text{.} More generally, given any
covector , the proportion of variance in explained by is \rho=\frac{(w\lambda)^2}{(w\lambda)^2+\mathbb{E}[(wE)^2]}\text{,} which is maximized when . is this proportion of
explained variance in the case where (all components of equally important): \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^2_{E_i} }
Example These are the estimates of the factor loadings and errors: :\hat{\rho}_{C} = \frac{ \left( \sum_{i=1}^k \hat{\lambda}_i \right)^2 }{ \hat{\sigma}^{2}_{X} } = \frac{ 106.22 }{ 124.23 } = .8550 :\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}^{2}_{e_i} } = \frac{ 106.22 }{ 106.22 + 18.01 } = .8550 Compare this value with the value of applying
Cronbach's alpha to the same data. == Related coefficients ==