Random effects model


Random effects model

In statistics, a random effect(s) model, also called a variance components model is a kind of hierarchical linear model. It assumes that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. In econometrics, random effects models are used in analysis of hierarchical or panel data when one assumes no fixed effects (i.e. no individual effects). The fixed effects model is a special case of the random effects model.

Simple example

Suppose "m" large elementary schools are chosen randomly from among millions in a large country. Then "n" pupils 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. Then

:Y_{ij} = mu + U_i + W_{ij},,

where μ is the average of all scores in the whole population, "U""i" is the deviation of the average of all scores at the "i"th school from the average in the whole population, and "W""ij" is the deviation of the "j"th pupil's score from the average score at the "i"th school. It is assumed that W_{ij}sim N(0,sigma^2), that is, the deviations are normal with mean zero and variance sigma^2, the value of which is unknown.

Variance components

The variance of "Y""ij" is the sum of the variances τ2 and σ2 of "U""i" and "W""ij" respectively.

Let

:overline{Y}_{iullet} = frac{1}{n}sum_{j=1}^n Y_{ij}

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

:overline{Y}_{ulletullet} = frac{1}{mn}sum_{i=1}^msum_{j=1}^n Y_{ij}

be the "grand average".

Let

:SSW = sum_{i=1}^msum_{j=1}^n (Y_{ij} - overline{Y}_{iullet})^2 ,

:SSB = nsum_{i=1}^m (overline{Y}_{iullet} - overline{Y}_{ulletullet})^2 ,

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 that

: frac{1}{m(n - 1)}E(SSW) = sigma^2

and

: frac{1}{n}E(SSB) = frac{sigma^2}{n} + au^2.

These "expected mean squares" can be used as the basis for estimation of the "variance components" σ2 and τ2.

Random effects estimation

The estimation for the coefficients in multiple comparisons model in which the effects of different classes are random can be done via generalized least squares (GLS). If we assume random effects the error term in the model

:y_{it}=x_{it}eta+alpha_{i}+u_{it},,

where y_{it} is the dependent variable, x_{it} is the vector of regressors, eta is the vector of coefficients, alpha_{i}=alpha are the random effects, and u_{it} is the error term, then alpha_{i} should have a normal distribution with mean zero and a constant variance.

The coefficients can be estimated via

:widehat{eta}=(X'Omega^{-1} X)^{-1}(X'Omega^{-1}Y),:widehat{Omega}^{-1}=Iota otimes Sigma,

where "X" and "Y" are the matrix version of the regressor and independent variable, respectively, Iota is the identity matrix, Sigma is the variance of u_{it} and alpha, and Omega is the variance-covariance matrix.

ee also

*Bühlmann model
*Meta analysis
*Hierarchical linear modeling

References

* [http://www.jr2.ox.ac.uk/bandolier/booth/glossary/random.html Random effect model at Bandolier (Oxford EBM website)]
* [http://teaching.sociology.ul.ie/DCW/confront/node45.html Fixed and random effects models]
* [http://www.ioa.pdx.edu/newsom/mlrclass/ho_randfixd.doc Distinguishing Between Random and Fixed: Variables, Effects, and Coefficients]
* [http://www.pitt.edu/~SUPER1/lecture/lec1171/012.htm How to Conduct a Meta-Analysis: Fixed and Random Effect Models]
* [http://www.uwyo.edu/aadland/classes/econ5350/ch13.pdf ECON 5350 Class Notes: Chapter 13. Panel Data]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Random-Effects-Modell — Inhaltsverzeichnis 1 Abgrenzung statische und dynamische Modelle 2 Schätzverfahren in den statischen Modellen 3 Schätzverfahren in den dynamischen Modellen 4 Literatur // …   Deutsch Wikipedia

  • Fixed-effects- und Random-effects-Modell — Dieser Artikel wurde auf der Qualitätssicherungsseite des Portals Mathematik eingetragen. Dies geschieht, um die Qualität der Artikel aus dem Themengebiet Mathematik auf ein akzeptables Niveau zu bringen. Bitte hilf mit, die Mängel dieses… …   Deutsch Wikipedia

  • Random coil — A random coil is a polymer conformation where the monomer subunits are oriented randomly while still being bonded to adjacent units. It is not one specific shape, but a statistical distribution of shapes for all the chains in a population of… …   Wikipedia

  • Model selection — is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is …   Wikipedia

  • Fixed effects estimation — In econometrics and statistics the fixed effects estimator (also known as the within estimator) is an estimator for the coefficients in panel data analysis. If we assume fixed effects, we impose time independent effects for each entity.… …   Wikipedia

  • Mixed model — A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful… …   Wikipedia

  • Generalized linear model — In statistics, the generalized linear model (GLM) is a flexible generalization of ordinary least squares regression. It relates the random distribution of the measured variable of the experiment (the distribution function ) to the systematic (non …   Wikipedia

  • Generalized linear mixed model — In statistics, a generalized linear mixed model (GLMM) is a particular type of mixed model (multilevel model). It is an extension to the generalized linear model in which the linear predictor contains random effects in addition to the usual fixed …   Wikipedia

  • Fixed-Effects-Modell — Inhaltsverzeichnis 1 Abgrenzung statische und dynamische Modelle 2 Schätzverfahren in den statischen Modellen 3 Schätzverfahren in den dynamischen Modellen 4 Literatur // …   Deutsch Wikipedia

  • Rasch model — Rasch models are used for analysing data from assessments to measure things 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… …   Wikipedia