Latent variable model
Encyclopedia
A latent variable model is a statistical model
that relates a set of variables
(so-called manifest variables) to a set of latent variable
s.
It is assumed that 1) the responses on the indicators or manifest variables are the result of
an individual's position on the latent variable(s), and 2) that the manifest variables have nothing
in common after controlling for the latent variable (local independence
).
Different types of the latent variable model can be grouped according to whether the manifest and
latent variables are categorical or continuous:
Another name for latent trait analysis is item response theory
(IRT).
The most simple IRT model is the Rasch model
.
An important part of the latent profile analysis is the mixture model
.
In factor analysis
and latent trait analysis the latent variables are treated
as continuous normally distributed variables, and in latent profile analysis
and latent class analysis as from a multinomial distribution.
The manifest variables in factor analysis and latent profile analysis
are continuous and in most cases, their conditional distribution given the latent variables
is assumed to be normal. In latent trait analysis and latent class analysis,
the manifest variables are discrete. These variables could be dichotomous, ordinal or nominal variables.
Their conditional distributions are assumed to be binomial or multinomial.
Because the distribution of a continuous latent variable can be approximated by a discrete distribution,
the distinction between continuous and discrete variables turns out not to be fundamental at all.
Therefore there may be a psychometrical latent variable, but not a psychological
psychometric variable.
Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
that relates a set of variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
(so-called manifest variables) to a set of latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...
s.
It is assumed that 1) the responses on the indicators or manifest variables are the result of
an individual's position on the latent variable(s), and 2) that the manifest variables have nothing
in common after controlling for the latent variable (local independence
Local independence
Local independence is the underlying assumption of latent variable models.The observed items are conditionally independent of each other given an individual score on the latent variable. This means that the latent variable explains why the observed items are related to another...
).
Different types of the latent variable model can be grouped according to whether the manifest and
latent variables are categorical or continuous:
| Manifest variables | ||
|---|---|---|
| Latent variables | Continuous | Categorical |
| Continuous | Factor analysis Factor analysis Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables... |
Latent trait analysis |
| Categorical | Latent profile analysis | Latent class analysis |
Another name for latent trait analysis is item response theory
Item response theory
In psychometrics, item response theory also known as latent trait theory, strong true score theory, or modern mental test theory, is a paradigm for the design, analysis, and scoring of tests, questionnaires, and similar instruments measuring abilities, attitudes, or other variables. It is based...
(IRT).
The most simple IRT model is the Rasch model
Rasch model
Rasch models are used for analysing data from assessments to measure variables such as abilities, attitudes, and personality traits. For example, they may be used to estimate a student's reading ability from answers to questions on a reading assessment, or the extremity of a person's attitude to...
.
An important part of the latent profile analysis is the mixture model
Mixture model
In statistics, a mixture model is a probabilistic model for representing the presence of sub-populations within an overall population, without requiring that an observed data-set should identify the sub-population to which an individual observation belongs...
.
In factor analysis
Factor analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...
and latent trait analysis the latent variables are treated
as continuous normally distributed variables, and in latent profile analysis
and latent class analysis as from a multinomial distribution.
The manifest variables in factor analysis and latent profile analysis
are continuous and in most cases, their conditional distribution given the latent variables
is assumed to be normal. In latent trait analysis and latent class analysis,
the manifest variables are discrete. These variables could be dichotomous, ordinal or nominal variables.
Their conditional distributions are assumed to be binomial or multinomial.
Because the distribution of a continuous latent variable can be approximated by a discrete distribution,
the distinction between continuous and discrete variables turns out not to be fundamental at all.
Therefore there may be a psychometrical latent variable, but not a psychological
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
psychometric variable.