Notation

The structural model

The structural model specifies the relationships between constructs (i.e., the statistical representation of a concept) via paths (arrows) and associated path coefficients. The path coefficients - sometimes also called structural coefficients - express the magnitude of the influence exerted by the construct at the start of the arrow on the variable at the arrow’s end. In composite-based SEM constructs are always operationalized (not modeled!!) as composites, i.e., weighted linear combinations of its respective indicators. Consequently, depending on how a given construct is modeled, such a composite may either serve as a proxy for an underlying latent variable (common factor) or as a composite in its own right. Despite this crucial difference, we stick with the common - although somewhat ambivalent - notation and represent both the construct and the latent variable (which is only a possible construct) by η. Let x_kj (k = 1, …, K_j) be an indicator (observable) belonging to construct η_j (j = 1…, J) and w_kj be a weight. A composite is definied as: $$\hat{\eta}_j = \sum^{K_j}_{k = 1} w_{kj} x_{kj} $$ Again, η̂_j may represent a latent variable η_j but may also serve as composite in its own right in which case we would essentially say that
η̂_j = η_j and refer to η_j as a construct instead of a latent variable. Since η̂_j generally does not have a natural scale, weights are usually chosen such that η̂_j is standardized. Therefore, unless otherwise stated:

E(η̂_j) = 0    and    Var(η̂_j) = E(η̂_j²) = 1

Since the relations between concepts(or its statistical sibling the constructs) are a product of the researcher’s theory and assumptions to be analyzed, some constructs are typically not directly connected by a path. Technically this implies a restriction of the path between construct a path we call the structural model saturated. If at least one path is restricted to zero, the structural model is called non-saturated.

The reflective measurement model

Define the general reflective (congeneric) measurement model as: x_kj = η_kj + ε_kj = λ_kjη_j + ε_kj  for  k = 1, …, K_j  and  j = 1, …, J

Call η_kj = λ_kjη_j the (indicator) true/population score and η_j the underlying latent variable supposed to be the common factor or cause of the K_j indicators connected to latent variable η_j. Call λ_kj the loading or direct effect of the latent variable on its indicator. Let x_kj be an indicator (observable), ε_kj be a measurement error and
$$\hat{\eta}_j = \sum^{K_j}_{k = 1} w_{kj} x_{kj} = \sum^{K_j}_{k = 1} w_{kj} \eta_{kj} + \sum^{K_j}_{k = 1} w_{kj} \varepsilon_{kj} = \bar\eta_{j} + \bar\varepsilon_{j} = \eta_j\sum_{k=1}^{K_J}w_{kj}\lambda_{kj} + \bar\varepsilon_{kj}, $$ be a proxy/test score/composite/stand-in for/of η_j based on a weighted sum of observables, where w_kj is a weight to be determined and η̄_j the proxy true score, i.e., a weighted sum of (indicator) true scores. Note the distinction between what we refer to as the indicator true score η_kj and the proxy true score which is the true score for η̂_j (i.e, the true score of a score that is in fact a linear combination of (indicator) scores!).

We will usually refer to η̂_j as a proxy for η_j as it stresses the fact that η̂_j is generally not the same as η_j unless ε̄_j = 0 and $\sum_{k=1}^{K_J}w_{kj}\lambda_{kj} = 1$.

Assume that E(ε_kj) = E(η_j) = Cov(η_j, ε_kj) = 0. Further assume that Var(η_j) = E(η_j²) = 1 to determine the scale.

It often suffices to look at a generic test score/latent variable. For the sake of clarity the index j is therefore dropped unless it is necessary to avoid confusion.

Note that most of the classical literature on quality criteria such as reliability is centered around the idea that the proxy η̂ is a in fact a simple sum score which implies that all weighs are set to one. Treatment is more general here since η̂ is allowed to be any weighted sum of related indicators. Readers familiar with the “classical treatment” may simply set weights to one (unit weights) to “translate” results to known formulae.

Based on the assumptions and definitions above the following quantities necessarily follow:

$$ $$

where δ_kl = Cov(ε_k, ε_l) for k ≠ l is the measurement error covariance and Σ is the indicator variance-covariance matrix implied by the measurement model:

$$ \boldsymbol{\mathbf{\Sigma }}= \begin{pmatrix} \lambda^2_1 + Var(\varepsilon_1) & \lambda_1\lambda_2 + \delta_{12} & \dots & \lambda_1\lambda_K + \delta_{1K} \\ \lambda_2\lambda_ 1 + \delta_{21} & \lambda^2_2 + Var(\varepsilon_2) & \dots & \lambda_2\lambda_K +\delta_{1K} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_{K}\lambda_1 + \delta_{K1} & \lambda_K\lambda_2 + \delta_{K2} &\dots &\lambda^2_K + Var(\varepsilon_K) \end{pmatrix} $$

In cSEM indicators are always standardized and weights are always appropriately scaled such that the variance of η̂ is equal to one. Furthermore, unless explicitly specified measurement error covariance is restricted to zero. As a consequence, it necessarily follows that:

$$ \begin{align} Var(x_k) &= 1 \\ Cov(x_k, \eta) &= Cor(x_k, \eta) \\ Cov(x_k, x_l) &= Cor(x_k, x_l) \\ Var(\hat\eta) &= \boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\Sigma}}\boldsymbol{\mathbf{w}} = 1 \\ Var(\varepsilon_k) &= 1 - Var(\eta_k) = 1 - \lambda^2_k \\ Cov(\varepsilon_k, \varepsilon_l) &= 0 \\ Var(\bar\varepsilon) &= \sum w_k^2 (1 - \lambda_k^2) \end{align} $$ For most formulae this implies a significant simplification, however, for ease of comparison to extant literature formulae we stick with the “general form” here but mention the “simplified form” or “cSEM form” in the Methods and Formula sections.

Notation table