Congeneric reliability
In statistical models applied to psychometrics, congeneric reliability
("rho C")[1] 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.
is a structural equation model (SEM)-based reliability coefficients and is obtained from on a unidimensional model.
; also known as Cronbach's alpha), and is often recommended as its alternative.
A quantity similar (but not mathematically equivalent) to congeneric reliability first appears in the appendix to McDonald's 1970 paper on factor analysis, labeled
[2] In McDonald's work, the new quantity is primarily a mathematical convenience: a well-behaved intermediate that separates two values.
[3][4] Seemingly unaware of McDonald's work, Jöreskog first analyzed a quantity equivalent to congeneric reliability in a paper the following year.
[4][5] Jöreskog defined congeneric reliability (now labeled ρ) with coordinate-free notation,[5] and three years later, Werts gave the modern, coordinatized formula for the same.
[6] Both of the latter two papers named the new quantity simply "reliability".
[1][7][8] Applied statisticians have subsequently coined many names for
measures the statistical reliability of composite scores.
[1][9] As psychology calls "constructs" any latent characteristics only measurable through composite scores,[10]
[11] Following McDonald's more recent expository work on testing theory, some SEM-based reliability coefficients, including congeneric reliability, are referred to as "reliability coefficient
[1][12][13] Congeneric reliability applies to datasets of vectors: each row X in the dataset is a list Xi 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 Xi 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
where Ei is a statistically independent noise term.
[5] In this context, λi 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 Xi by F is then simply
More generally, given any covector w, the proportion of variance in wX explained by F is
which is maximized when w ∝ 𝔼[EE*]λ.
[5] ρC is this proportion of explained variance in the case where w ∝ [1 1 ... 1] (all components of X equally important):
These are the estimates of the factor loadings and errors: Compare this value with the value of applying tau-equivalent reliability to the same data.
", assumes that all factor loadings are equal (i.e.
In reality, this is rarely the case and, thus, it systematically underestimates the reliability.
) explicitly acknowledges the existence of different factor loadings.
However, such values should not be misunderstood as strict cutoff boundaries between "good" and "bad".
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.
