In
structural equation modeling
Structural equation modeling (SEM) is a label for a diverse set of methods used by scientists in both experimental and observational research across the sciences, business, and other fields. It is used most in the social and behavioral scienc ...
, a discrepancy function is a mathematical function which describes how closely a structural model conforms to observed data; it is a measure of
goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
. Larger values of the discrepancy function indicate a poor fit of the model to data. In general, the parameter estimates for a given model are chosen so as to make the discrepancy function for that model as small as possible. Analogous concepts in statistics are known as
goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
or
statistical distance
In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be be ...
, and include
deviance and
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
.
Examples
There are several basic types of discrepancy functions, including
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
(ML),
generalized least squares
In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinar ...
(GLS), and
ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
(OLS), which are considered the "classical" discrepancy functions.
Discrepancy functions all meet the following basic criteria:
*They are non-negative, i.e., always greater than or equal to zero.
*They are zero only if the fit is perfect, i.e., if the model and parameter estimates perfectly reproduce the observed data.
*The discrepancy function is a continuous function of the elements of S, the sample covariance matrix, and Σ(θ), the "reproduced" estimate of S obtained by using the parameter estimates and the structural model.
In order for "maximum likelihood" to meet the first criterion, it is used in a revised form as the
deviance.
See also
*
Constructions of low-discrepancy sequences In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of ''N'', its subsequence ''x''1, ..., ''x'N'' has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of poi ...
*
Discrepancy theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, name ...
*
Low-discrepancy sequence In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of ''N'', its subsequence ''x''1, ..., ''x'N'' has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of poi ...
References
Structural equation models
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