### Common factor

A common factor or latent variable is a type of construct. The name common factor is motivated by the theorized relationship to its indicators. This relation is commonly refered to as the measurement model. Two kinds of measurement models exist: the reflective and the (causal-)formative measurement model. The defining feature of a common factor in a reflective measurement model is the idea that the common factor is the common underlying (latent) cause of the realizations of a set of indicators that are said to measure the common factor. The idea of the reflective measurement model is closely related to the true score theory. Accordingly, indicators related to the common factor are modeled as measurement error-prone manifestations of a common variable, cause or factor. Although there are subtle conceptional differences, the terms common factor, true score, and latent variable are mostly used synommenoulsy in cSEM.

The common factor is also the central entity in the causal-formative measurment model. Here the indicators are modeled as causing the/one common factor. This type of measurement looks similar to the composite (measurement) model, however, both models are, in fact, quite different. While the causal-formative measurement model assumes that the common factor is imperfectly measured by its indicators (i.e., there is an error term with non-zero variance) the composite in a composite model is build/defined by its indicators, i.e., error-free by definition.

### Composite

A composite is a weighted sum of indicators. Composites may either serve as constructs in their own right or as proxies for a latent variable which, in turn, serves as a “statistical proxy” for a concept under study. The nature of the composite is therefore defined by the type of (measurement) model. If composites are error-free representations of a concept we refer to the measurement model as the composite (measurement) model. If composites are used as stand-ins for a latent variable, the measurement model is called causal-formative (Henseler 2017).

Note that, although we sometimes use it this way as well, the term “measurement” is acutally rather inadequate for the composite model since in a composite model the construct is build/formed by its related indicators. Hence no measurement in the acutal sense of the word takes place.

### Composite-based methods

Composite-based methods or composite-based SEM refers to the entirety of methods centered around the use as of composites (linear compounts of observables) as stand-ins or error-free representations for the concepts under investigation. Composite-based methods are often also called variance-based methods as focal parameters are usually retrived such that explained variance of the dependend constructs in a structural model is maximized.

### Concept

An entity defined by a conceptual/theoretical definition. In line with Rigdon (2016) variables representing or subsuming a concept are called conceptual variables. The precise nature/meaning of a conceptual variable depends on “differnt assumptions, philosophies or worldviews [of the researcher]” (Rigdon 2016, 2). Unless otherwise stated, in cSEM, it is sufficient to think of concepts as entities that exist simply because they have been defined. Hence, abstract terms such as “loyalty” or “depression” as well as designed entities (artifacts) such as the “OECD Better Life Index” are covered by the definition.

### Construct

Construct refers to a representation of a concept within a given statistical model. While a concept is defined by a conceptual (theoretical) definition, a construct for a concept is defined/created by the researcher’s operationalization of a concept within a statistical model. Concepts are either modeled as common factors/latent variables or as composites. Both operationalizations - the common factor and the composite - are called constructs in cSEM. As opposed to concepts, constructs therefore exist because they arise as the result of the act of modeling their relation to the observable variables (indicators) based on a specific set of assumptions. Constructs may therefore best be understood as stand-ins, i.e. statistical proxies for concepts. Consequently, constructs do not necessarily represent the concept they seek to represent, i.e., there may be a validity gap.

### Factor-based methods

Factor-based methods or factor-based SEM refers to the entirety of methods centered around the use of common factors as statistical proxies for the concepts under investigation. Factor-based methods are also called covariance-based methods as focal parameters are retrived such that the difference between the model-implied $$\boldsymbol{\mathbf{\Sigma}}(\theta)$$ and the empirical indicator covariance matrix $$\boldsymbol{\mathbf{S}}$$ is minimized.

### Indicator

An observable variable. In cSEM observable variables are generally refered to as indicators, however, terms such as item, manifest variable, or observable (variable) are sometimes used synommenoulsy.

### Latent variable

See: Common factor

### Measurement model

The measurement model is a statistical model relating indicators to constructs (the statistical representiation of a concept). If the concept under study is modeled as a common factor two measurement models exist:

• The reflective measurement model
• The causal-formative measurement model

If the concept under study is modeled as a composite we call the measurement model:

• composite (measurement) model

Note that, although we sometimes use it this way as well, the term “measurement” is acutally rather inadequate for the composite model since in a composite model the construct is build/formed by its related indicators. Hence no measurement in the acutal sense of the word takes place.

### Model

There are many ways to define the term “model.” In cSEM we use the term model to refer to both the theoretical and the statistical model.

Simply speaking theoretical models are a formalized set of hypotheses stating if and how entities (observable or unobservable) are related. Since theoretical models are by definition theoretical any statistical analysis inevitably mandates a statistical model.

A statistical model is typically defined by a set of (testable) restrictions. Statistical models are best understood as the operationalized version of the theoretical model. Note that the act of operationalizing a given theoretical model always entails the possibility for error. Using a construct modeled as a composite or a common factor, for instance, is an attempt to map a theoretical entity (the concept) from the theoretical space into the statistical space. If this mapping is not one-to-one the construct is only an imperfect representation of the concept. Consequently, there is a validity gap.

NOTE: Note that we try to refrain from using the term “model” when describing an estimation approach or algorithm such as partial least squares (PLS) as this helps to clearly distinguish between the model and the approach used to estimate a given model.

### Test score

A proxy for a true score. Usually, the test score is a simple (unweighted) sum score of observables/indicators, i.e. unit weights are assumed when building the test score. More generally, the test score can be any weighted sum of observables (i.e. a composite), however, the term “test score” is historically closely tied to the idea that it is indeed a simple sum score. Hence, whenever it is important to distinguish between a true score as representing a sum score and a true score as representing a weighted sum of indicators (where indicator weights a not necessarily one) we will explicitly state what kind of test score we mean.

### True score

The term true score derives from the true score theory which theorizes/models an indicator or outcome as the sum of a true score and an error score. The term is closely linked to the latent variable/common factor model in that the true score of a set of indicators are linear functions of some underlying common factor. Mathematically speaking, the correspondence is $$\eta_{jk} = \lambda_{jk}\eta_j$$
where $$\eta_{jk}$$ is the true score, $$\eta_j$$ the underlying latent variable and $$\lambda_{jk}$$ the loading. Despite some differences, the term true score can generally be used synommenoulsy to the terms common factor and latent variable in cSEM without risking a misunderstanding.

### Proxy

Any quantity that functions as a representation of some other quantity. Prominent proxies are the test score - which serves as a stand-in/proxy for the true score - and the composite - which serves as a stand-in/proxy for a common factor if not used as a composite in its own right. Proxies are usually - but not necessarily - error-prone representations of the quantity they seek to represent.

### Saturated and non-saturated models

A structural model is called “saturated” if all constructs of the model are allowed to freely covary. This is equivalent to saying that none of the path of the structural model are restricted to zero. A saturated model has zero degrees of freedom and hence carries no testable restrictions. If at least one path is restricted to zero, the structural model is called “non-saturated.”

See: Proxy

### Structural Equation Modeling (SEM)

The entirety of a set of related theories, mathematical models, methods, algorithms and terminologies related to analyzing the relationships between concepts and/or observables.

## Literatur

Henseler, Jörg. 2017. “Bridging Design and Behavioral Research with Variance-Based Structural Equation Modeling.” Journal of Advertising 46 (1): 178–92. https://doi.org/10.1080/00913367.2017.1281780.
Rigdon, Edward E. 2016. “Choosing PLS Path Modeling as Analytical Method in European Management Research: A Realist Perspective.” European Management Journal 34 (6). https://doi.org/10.1016/j.emj.2016.05.006.