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| | Hi there. My name is Sophia Meagher although it is not the title on my beginning certificate. One of the things she enjoys most is canoeing and she's been performing it for quite a whilst. I've always cherished living in Mississippi. Invoicing is my occupation.<br><br>Feel free to surf to my weblog :: love psychic ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 your domain name]) |
| {{expert-subject|statistics||Talk=Lead|date=January 2011}}
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| In [[statistics]], a '''random effect(s) model''', also called a '''variance components model''', is a kind of [[hierarchical linear model]]. It assumes that the dataset being analysed consists of a hierarchy of different populations whose differences relate to that hierarchy. In [[econometrics]], random effects models are used in the analysis of hierarchical or [[panel data]] when one assumes no [[fixed effects estimator|fixed effects]] (it allows for individual effects). The random effects model is a special case of the [[fixed effects model]]. Contrast this to the [[biostatistics]] definitions,<ref>{{cite book |first=Peter J. |last=Diggle |first2=Patrick |last2=Heagerty |first3=Kung-Yee |last3=Liang |first4=Scott L. |last4=Zeger |year=2002 |title=Analysis of Longitudinal Data |edition=2nd |location= |publisher=Oxford University Press |pages=169–171 |isbn=0-19-852484-6 }}</ref><ref>{{cite book |first=Garrett M. |last=Fitzmaurice |first2=Nan M. |last2=Laird |first3=James H. |last3=Ware |year=2004 |title=Applied Longitudinal Analysis |location=Hoboken |publisher=John Wiley & Sons |pages=326–328 |isbn=0-471-21487-6 }}</ref><ref>{{cite journal |first=Nan M. |last=Laird |first2=James H. |last2=Ware |year=1982 |title=Random-Effects Models for Longitudinal Data |journal=[[Biometrics (journal)|Biometrics]] |volume=38 |issue=4 |pages=963–974 |jstor=2529876 }}</ref> 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]]).
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| ==Qualitative description==
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| Such models assist in controlling for [[unobserved heterogeneity]] when this heterogeneity is constant over time and correlated with independent variables. This constant can be removed from the data through differencing, for example by taking a first difference which will remove any time invariant components of the model.
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| There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a [[random effects model]]) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more [[Efficiency (statistics)|efficient]] than the fixed effects model. However, if this assumption does not hold (i.e., if the [[Durbin–Watson statistic|Durbin–Watson test]] fails), the random effects model is not [[Consistency (statistics)|consistent]].
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| ==Simple example==
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| 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''<sub>''ij''</sub> be the score of the ''j''th pupil at the ''i''th school. A simple way to model the relationships of these quantities is
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| : <math>
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| Y_{ij} = \mu + U_i + W_{ij},\,
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| </math>
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| where ''μ'' is the average test score for the entire population. In this model ''U<sub>i</sub>'' is the school-specific '''random effect''': it measures the difference between the average score at school ''i'' and the average score in the entire country and it is "random" because the school has been randomly selected from a larger population of schools. The term, ''W<sub>ij</sub>'' is the individual-specific error. That is, it is the deviation of the ''j''-th pupil’s score from the average for the ''i''-th school. Again this is regarded as random because of the random selection of pupils within the school, even though it is a fixed quantity for any given pupil.
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| The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:
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| : <math>
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| Y_{ij} = \mu + \beta_1 \mathrm{Sex}_{ij} + \beta_2 \mathrm{Race}_{ij} + \beta_3 \mathrm{ParentsEduc}_{ij} + U_i + W_{ij},\,
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| </math>
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| where Sex<sub>''ij''</sub> is the [[dummy variable]] for boys/girls, Race<sub>''ij''</sub> is the dummy variable for white/black pupils, and ParentsEduc<sub>''ij''</sub> records the average education level of child’s parents. This is a [[mixed model]], not a purely random effects model.{{Dubious|date=February 2011}}
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| ==Variance components==
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| The variance of ''Y''<sub>''ij''</sub> is the sum of the variances τ<sup>2</sup> and σ<sup>2</sup> of ''U''<sub>''i''</sub> and ''W''<sub>''ij''</sub> respectively.
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| Let
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| : <math>\overline{Y}_{i\bullet} = \frac{1}{n}\sum_{j=1}^n Y_{ij}</math>
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| 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
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| :<math>\overline{Y}_{\bullet\bullet} = \frac{1}{mn}\sum_{i=1}^m\sum_{j=1}^n Y_{ij}</math>
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| be the "grand average".
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| Let
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| :<math>SSW = \sum_{i=1}^m\sum_{j=1}^n (Y_{ij} - \overline{Y}_{i\bullet})^2 \, </math>
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| :<math>SSB = n\sum_{i=1}^m (\overline{Y}_{i\bullet} - \overline{Y}_{\bullet\bullet})^2 \,</math>
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| 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
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| :<math> \frac{1}{m(n - 1)}E(SSW) = \sigma^2</math> | |
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| and
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| :<math> \frac{1}{(m - 1)n}E(SSB) = \frac{\sigma^2}{n} + \tau^2.</math> | |
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| These "[[expected mean square]]s" can be used as the basis for [[estimation]] of the "variance components" σ<sup>2</sup> and τ<sup>2</sup>.
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| ==Unbiasedness==
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| In general, random effects is efficient, and should be used (over fixed effects) if the assumptions underlying it are believed to be satisfied. For RE to work in the school example it is necessary that the school-specific effects be orthogonal to the other covariates of the model. This can be tested by running fixed effects, then random effects, and doing a [[Hausman specification test]]. If the test rejects, then random effects is biased and fixed effects is the correct estimation procedure.
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| ==See also==
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| *[[Bühlmann model]]
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| *[[Hierarchical linear modeling]]
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| *[[Fixed effects]]
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| *[[MINQUE]]
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| *[[LS-VCE]] [http://engold.ui.ac.ir/~amiri/ThesisAmiri_Simkooei.pdf]
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| ==Notes==
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| {{reflist}}
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| ==Further reading==
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| * {{cite book
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| |title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald|location=New York|publisher=Springer|year=2002| edition=Third|isbn=0-387-95361-2}}
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| * {{cite book |last=Gujarati |first=Damodar N. |last2=Porter |first2=Dawn C. |chapter=Panel Data Regression Models |title=Basic Econometrics |location=Boston |publisher=McGraw-Hill |year=2009 |edition=Fifth international |isbn=978-007-127625-2 |pages=591–616 }}
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| * {{cite book |last=Wooldridge |first=Jeffrey M. |year=2013 |chapter=Random Effects Estimation |pages=474–478 |title=Introductory Econometrics: A Modern Approach |location=Mason, OH |publisher=South-Western |edition=Fifth international |isbn=978-1-111-53439-4 }}
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| ==External links==
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| *[http://teaching.sociology.ul.ie/DCW/confront/node45.html Fixed and random effects models]
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| *[http://www.upa.pdx.edu/IOA/newsom/mlrclass/ho_randfixd.pdf Distinguishing Between Random and Fixed: Variables, Effects, and Coefficients]
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| *[http://www.pitt.edu/~SUPER1/lecture/lec1171/012.htm How to Conduct a Meta-Analysis: Fixed and Random Effect Models]
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| {{DEFAULTSORT:Random Effects Model}}
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| [[Category:Statistical models]]
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| [[Category:Analysis of variance]]
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Hi there. My name is Sophia Meagher although it is not the title on my beginning certificate. One of the things she enjoys most is canoeing and she's been performing it for quite a whilst. I've always cherished living in Mississippi. Invoicing is my occupation.
Feel free to surf to my weblog :: love psychic (your domain name)