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In statistics, a **random effects model**, also called a **variance components model**, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. In econometrics, random effects models are used in panel analysis of hierarchical or panel data when one assumes no fixed effects (it allows for individual effects). A random effects model is a special case of a mixed model.

- Qualitative description
- Simple example
- Variance components
- Applications
- See also
- Further reading
- References
- External links

Contrast this to the biostatistics definitions,^{ [1] }^{ [2] }^{ [3] }^{ [4] }^{ [5] } as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables).

Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the model.^{ [6] }

Two common assumptions can be made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual unobserved heterogeneity is uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables.^{ [6] }

If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects model.

Suppose *m* large elementary schools are chosen randomly from among thousands in a large country. Suppose also that *n* pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let *Y*_{ij} be the score of the *j*th pupil at the *i*th school. A simple way to model this variable is

where *μ* is the average test score for the entire population. In this model *U _{i}* is the school-specific

The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:

where Sex_{ij} is the dummy variable for boys/girls and ParentsEduc_{ij} records, say, the average education level of a child’s parents. This is a mixed model, not a purely random effects model, as it introduces fixed-effects terms for Sex and Parents' Education.

The variance of *Y*_{ij} is the sum of the variances τ^{2} and σ^{2} of *U*_{i} and *W*_{ij} respectively.

Let

be the average, not of all scores at the *i*th school, but of those at the *i*th school that are included in the random sample. Let

be the grand average.

Let

be respectively the sum of squares due to differences *within* groups and the sum of squares due to difference *between* groups. Then it can be shown ^{[ citation needed ]} that

and

These "expected mean squares" can be used as the basis for estimation of the "variance components" *σ*^{2} and *τ*^{2}.

The *τ*^{2} parameter is also called the intraclass correlation coefficient.

Random effects models used in practice include the Bühlmann model of insurance contracts and the Fay-Herriot model used for small area estimation.

- Baltagi, Badi H. (2008).
*Econometric Analysis of Panel Data*(4th ed.). New York, NY: Wiley. pp. 17–22. ISBN 978-0-470-51886-1. - Hsiao, Cheng (2003).
*Analysis of Panel Data*(2nd ed.). New York, NY: Cambridge University Press. pp. 73–92. ISBN 0-521-52271-4. - Wooldridge, Jeffrey M. (2002).
*Econometric Analysis of Cross Section and Panel Data*. Cambridge, MA: MIT Press. pp. 257–265. ISBN 0-262-23219-7. - Gomes, Dylan G.E. (20 January 2022). "Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?".
*PeerJ*.**10**: e12794. doi:10.7717/peerj.12794.

**Analysis of variance** (**ANOVA**) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the *t*-test beyond two means.

**Autocorrelation**, sometimes known as **serial correlation** in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

The method of **least squares** is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals made in the results of each individual equation.

An ** F-test** is any statistical test in which the test statistic has an

**Analysis of covariance** (**ANCOVA**) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

In signal processing, **cross-correlation** is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a *sliding dot product* or *sliding inner-product*. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.

In statistical modeling, **regression analysis** is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

In statistics, **ordinary least squares** (**OLS**) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

**Panel** (**data**) **analysis** is a statistical method, widely used in social science, epidemiology, and econometrics to analyze two-dimensional panel data. The data are usually collected over time and over the same individuals and then a regression is run over these two dimensions. Multidimensional analysis is an econometric method in which data are collected over more than two dimensions.

In statistics, **simple linear regression** is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective *simple* refers to the fact that the outcome variable is related to a single predictor.

**Difference in differences** is a statistical technique used in econometrics and quantitative research in the social sciences that attempts to mimic an experimental research design using observational study data, by studying the differential effect of a treatment on a 'treatment group' versus a 'control group' in a natural experiment. It calculates the effect of a treatment on an outcome by comparing the average change over time in the outcome variable for the treatment group to the average change over time for the control group. Although it is intended to mitigate the effects of extraneous factors and selection bias, depending on how the treatment group is chosen, this method may still be subject to certain biases.

In statistics, a **fixed effects model** is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity.

**Multilevel models** are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

In statistics, **one-way analysis of variance** is a technique that can be used to compare whether two samples means are significantly different or not. This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usually) categorical input data, the "X", always one variable, hence "one-way".

In statistics, a **sum of squares due to lack of fit**, or more tersely a **lack-of-fit sum of squares**, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the **pure-error sum of squares**.

In statistics, **jackknife variance estimates for random forest** are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects.

**Nonlinear mixed-effects models** constitute a class of statistical models generalizing linear mixed-effects models. Like linear mixed-effects models, they are particularly useful in settings where there are multiple measurements within the same statistical units or when there are dependencies between measurements on related statistical units. Nonlinear mixed-effects models are applied in many fields including medicine, public health, pharmacology, and ecology.

In statistics, **expected mean squares (EMS)** are the expected values of certain statistics arising in partitions of sums of squares in the analysis of variance (ANOVA). They can be used for ascertaining which statistic should appear in the denominator in an F-test for testing a null hypothesis that a particular effect is absent.

- ↑ Diggle, Peter J.; Heagerty, Patrick; Liang, Kung-Yee; Zeger, Scott L. (2002).
*Analysis of Longitudinal Data*(2nd ed.). Oxford University Press. pp. 169–171. ISBN 0-19-852484-6. - ↑ Fitzmaurice, Garrett M.; Laird, Nan M.; Ware, James H. (2004).
*Applied Longitudinal Analysis*. Hoboken: John Wiley & Sons. pp. 326–328. ISBN 0-471-21487-6. - ↑ Laird, Nan M.; Ware, James H. (1982). "Random-Effects Models for Longitudinal Data".
*Biometrics*.**38**(4): 963–974. doi:10.2307/2529876. JSTOR 2529876. - ↑ Gardiner, Joseph C.; Luo, Zhehui; Roman, Lee Anne (2009). "Fixed effects, random effects and GEE: What are the differences?".
*Statistics in Medicine*.**28**(2): 221–239. doi:10.1002/sim.3478. PMID 19012297. - ↑ Gomes, Dylan G.E. (20 January 2022). "Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?".
*PeerJ*.**10**: e12794. doi:10.7717/peerj.12794. - 1 2 Wooldridge, Jeffrey (2010).
*Econometric analysis of cross section and panel data*(2nd ed.). Cambridge, Mass.: MIT Press. p. 252. ISBN 9780262232586. OCLC 627701062.

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