## 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 $$\eta$$. Let $$x_{kj}$$ $$(k = 1,\dots, K_j)$$ be an indicator (observable) belonging to construct $$\eta_j$$ $$(j = 1\dots, J)$$ and $$w_{kj}$$ be a weight. A composite is definied as: $\hat{\eta}_j = \sum^{K_j}_{k = 1} w_{kj} x_{kj}$ Again, $$\hat{\eta}_j$$ may represent a latent variable $$\eta_j$$ but may also serve as composite in its own right in which case we would essentially say that
$$\hat{\eta}_j = \eta_j$$ and refer to $$\eta_j$$ as a construct instead of a latent variable. Since $$\hat{\eta}_j$$ generally does not have a natural scale, weights are usually chosen such that $$\hat{\eta}_j$$ is standardized. Therefore, unless otherwise stated:

$E(\hat\eta_j) = 0\quad\quad \text{and}\quad\quad Var(\hat\eta_j) = E(\hat\eta^2_j) = 1$

Since the relations between concepts(or its statistical sibbling 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 $$j$$ and $$i$$ to zero. If all constructs of the reserchers model are connected by 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: $x_{kj} = \eta_{kj} + \varepsilon_{kj} = \lambda_{kj}\eta_j + \varepsilon_{kj}\quad\text{for}\quad k = 1, \dots, K_j\quad\text{and}\quad j = 1, \dots, J$

Call $$\eta_{kj} = \lambda_{kj}\eta_j$$ the (indicator) true/population score and $$\eta_j$$ the underlying latent variable supposed to be the common factor or cause of the $$K_j$$ indicators connected to latent variable $$\eta_j$$. Call $$\lambda_{kj}$$ the loading or direct effect of the latent variable on its indicator. Let $$x_{kj}$$ be an indicator (observable), $$\varepsilon_{kj}$$ 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 $$\eta_j$$ based on a weighted sum of observables, where $$w_{kj}$$ is a weight to be determined and $$\bar\eta_j$$ 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 $$\eta_{kj}$$ and the proxy true score which is the true score for $$\hat\eta_j$$ (i.e, the true score of a score that is in fact a linear combination of (indicator) scores!).

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

Assume that $$E(\varepsilon_{kj}) = E(\eta_j) = Cov(\eta_j, \varepsilon_{kj}) = 0$$. Further assume that $$Var(\eta_j) = E(\eta^2_j) = 1$$ 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 $$\hat\eta$$ is a in fact a simple sum score which implies that all weighs are set to one. Treatment is more general here since $$\hat{\eta}$$ 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:

\begin{align} Cov(x_k, \eta) &= \lambda_k \\ Var(\eta_k) &= \lambda^2_k \\ Var(x_k) &= \lambda^2_k + Var(\varepsilon_k) \\ Cor(x_k, \eta) &= \rho_{x_k, \eta} = \frac{\lambda_k}{\sqrt{Var(x_k)}} \\ Cov(\eta_k, \eta_l) &= Cor(\eta_k, \eta_l) = E(\eta_k\eta_l) = \lambda_k\lambda_lE(\eta^2) = \lambda_k\lambda_l \\ Cov(x_k, x_l) &= \lambda_k\lambda_lE(\eta^2) + \lambda_kE(\eta\varepsilon_k) + \lambda_lE(\eta\varepsilon_l) + E(\varepsilon_k\varepsilon_l) = \lambda_k\lambda_l + \delta_{kl} \\ Cor(x_k, x_l) &= \frac{\lambda_k\lambda_l + \delta_{kl}}{\sqrt{Var(x_k)Var(x_l)}} \\ Var(\bar\eta) &= E(\bar\eta^2) = \sum w_k^2\lambda^2_k + 2\sum_{k < l} w_k w_l \lambda_k\lambda_l = \left(\sum w_k\lambda_k \right)^2 = (\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2 \\ Var(\bar\varepsilon) &= E(\bar\varepsilon^2) = \sum w_k^2E(\varepsilon_k^2) + 2\sum_{k < l} w_k w_lE(\varepsilon_k\varepsilon_l)\\ Var(\hat\eta) &= E(\hat\eta^2) = \sum w_k^2(\lambda^2_k + Var(\varepsilon_k)) + 2\sum_{k < l} w_k w_l (\lambda_k\lambda_l + \delta_{kl}) \\ &= \sum w_k^2\lambda^2_k + 2\sum_{k < l} w_k w_l \lambda_k\lambda_l + \sum w_k^2Var(\varepsilon_k) + 2\sum_{k < l} w_k w_l \delta_{kl} \\ &=Var(\bar\eta) + Var(\bar\varepsilon) = (\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2 + Var(\bar\varepsilon) = \boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\Sigma}}\boldsymbol{\mathbf{w}} \\ Cov(\eta, \hat\eta) &= E\left(\sum w_k \lambda_k \eta^2\right) = \sum w_k\lambda_k = \boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}}= \sqrt{Var(\bar\eta)} \end{align}

where $$\delta_{kl} = Cov(\varepsilon_{k}, \varepsilon_{l})$$ for $$k \neq l$$ is the measurement error covariance and $$\boldsymbol{\mathbf{\Sigma}}$$ 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 $$\hat\eta$$ 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

Symbol Dimension Description
$$x_{kj}$$ $$(1 \times 1)$$ The $$k$$’th indicator of construct $$j$$
$$\eta_{kj}$$ $$(1 \times 1)$$ The $$k$$’th (indicator) true score related to construct $$j$$
$$\eta_j$$ $$(1 \times 1)$$ The $$j$$’th common factor/latent variable
$$\lambda_{kj}$$ $$(1 \times 1)$$ The $$k$$’th (standardized) loading or direct effect of $$\eta_j$$ on $$x_{kj}$$
$$\varepsilon_{kj}$$ $$(1 \times 1)$$ The $$k$$’th measurement error or error score
$$\hat\eta_j$$ $$(1 \times 1)$$ The $$j$$’th test score/composite/proxy for $$\eta_j$$
$$w_{kj}$$ $$(1 \times 1)$$ The $$k$$’th weight
$$\bar\eta_j$$ $$(1 \times 1)$$ The $$j$$’th (proxy) true score, i.e. the weighted sum of (indicator) true scores
$$\delta_{kl}$$ $$(1 \times 1)$$ The covariance between the $$k$$’th and the $$l$$’th measurement error
$$\boldsymbol{\mathbf{w}}$$ $$(K \times 1)$$ A vector of weights
$$\boldsymbol{\mathbf{\lambda}}$$ $$(K \times 1)$$ A vector of loadings