In
structural equation modeling
Structural equation modeling (SEM) is a diverse set of methods used by scientists for both observational and experimental research. SEM is used mostly in the social and behavioral science fields, but it is also used in epidemiology, business, ...
, 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 measur ...
. 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 measur ...
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 bet ...
, 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 rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
.
Examples
There are several basic types of discrepancy functions, including
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
(ML),
generalized least squares
In statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a Linear regression, linear regression model. It is used when there is a non-zero amount of correlation between the Residual (statistics), resi ...
(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
In statistics, linear regression is a statistical model, model that estimates the relationship ...
(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, \ldots, x_N has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the ...
*
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, nam ...
*
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, \ldots, x_N has a low discrepancy of a sequence, discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the p ...
References
Structural equation models
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