## 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 |