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 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} = \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 |