Terminology.Rmd

---
title: "Terminology"
date: "Last edited: `r Sys.Date()`"
author: "Manuel Rademaker"
output:
  prettydoc::html_pretty:
    theme: cayman
    highlight: github
    toc: true
    includes:
      before_body: preamble.mathjax.tex
vignette: >
  %\VignetteIndexEntry{csem-terminology}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: ../inst/REFERENCES.bib
---
<!-- used to print boldface greek symbols (\mathbf only works for latin symbols) -->
```{r child="preamble-tex.tex", echo=FALSE, include=FALSE}
```

### Common factor{#commonfactor}

A common factor or latent variable is a type of [construct](#construct). The name common factor
is motivated by the theorized relationship to its [indicators](#indicator). This relation is commonly refered to as
the [measurement model](#mm). 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](#truescore) 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](#composite), 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{#composite}

A composite is a weighted sum of [indicators](#indicator). Composites may 
either serve as [constructs](#construct) in their own right or as [proxies](#proxy) 
for a [latent variable](#latentvariable) which, in turn, serves as a 
"statistical proxy" for a [concept](#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](#standin) for a [latent variable](#latentvariable), 
the measurement model is called causal-formative [@Henseler2017]. 

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{#cbased}

Composite-based methods or composite-based SEM refers to the entirety of methods 
centered around the use as of [composites](#composite) (linear compounts of observables) 
as [stand-ins](#standin) or error-free representations for the [concepts](#concept) 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.

### Composite-based SEM{#cbasedsem}

See: [Composite-based methods](#cbased)

### Concept{#concept}

An entity defined by a conceptual/theoretical definition. In line with @Rigdon2016
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]" 
[@Rigdon2016, p. 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}

Construct refers to a representation of a [concept](#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](#standin), 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.

### Covariance-based SEM 

See: [Factor-based methods](#fbmethods)

### Factor-based methods{#fbmethods}

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

See: [Common factor](#commonfactor)

### Measurement model{#mm}

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

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

If the concept under study is modeled as a [composite](#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](#construct) modeled as a [composite](#composite) 
or a [common factor](#commonfactor), for instance, is an attempt to map a theoretical entity (the [concept](#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{#testscore}

A [proxy](#proxy) for a [true score](#truescore). Usually, the test score is a simple (unweighted) 
sum score of [observables/indicators](#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](#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{#truescore}

The term true score derives from the true score theory which theorizes/models
an [indicator](#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](#commonfactor) 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 {#proxy}

Any quantity that functions as a representation of some other quantity.
Prominent proxies are the [test score](#testscore) - which serves as a stand-in/proxy
for the [true score](#truescore) - and the [composite](#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](#construct) 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".

### Stand-in{#standin}

See: [Proxy](#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](#concept) and/or observables. 

### Variance-based methods {#vbmethods}

See: [Composite-based methods](#cbased)

## Literatur