﻿ Random effects model
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# Random effects model

random effects model, random effects model vs fixed effects model
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 the analysis of hierarchical or panel data when one assumes no 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, 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

## Contents

• 1 Qualitative description
• 2 Simple example
• 3 Variance components
• 4 Unbiasedness
• 6 Notes

## Qualitative description

Random effect models assist in controlling for unobserved heterogeneity when the 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[citation needed]

Two common assumptions are made about the individual specific effect: the random effects assumption and the fixed effects assumption The random effects assumption 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 efficient than the fixed effects model However, if this assumption does not hold, the random effects model is not consistent[citation needed]

## Simple example

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 Yij be the score of the jth pupil at the ith school A simple way to model the relationships of these quantities is

Y i j = μ + U i + W i j , =\mu +U_+W_,\,}

where μ is the average test score for the entire population In this model Ui 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 The term Wij is the individual-specific random effect, ie, it's the deviation of the j-th pupil’s score from the average for the i-th school

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

Y i j = μ + β 1 S e x i j + β 2 R a c e i j + β 3 P a r e n t s E d u c i j + U i + W i j , =\mu +\beta _\mathrm _+\beta _\mathrm _+\beta _\mathrm _+U_+W_,\,}

where Sexij is the dummy variable for boys/girls, Raceij is the dummy variable for white/black pupils, and ParentsEducij 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, Race, and Parents' Education

## Variance components

The variance of Yij is the sum of the variances τ2 and σ2 of Ui and Wij respectively

Let

Y ¯ i ∙ = 1 n ∑ j = 1 n Y i j }_=}\sum _^Y_}

be the average, not of all scores at the ith school, but of those at the ith school that are included in the random sample Let

Y ¯ ∙ ∙ = 1 m n ∑ i = 1 m ∑ j = 1 n Y i j }_=}\sum _^\sum _^Y_}

be the grand average

Let

S S W = ∑ i = 1 m ∑ j = 1 n Y i j − Y ¯ i ∙ 2 ^\sum _^Y_-}_^\,} S S B = n ∑ i = 1 m Y ¯ i ∙ − Y ¯ ∙ ∙ 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[citation needed] that

1 m n − 1 E S S W = σ 2 }ESSW=\sigma ^}

and

1 m − 1 n E S S B = σ 2 n + τ 2 }ESSB=}}+\tau ^}

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

## Unbiasedness

In general, random effects are efficient, and should be used over fixed effects if the assumptions underlying them are believed to be satisfied For random effects to work in the school example it is necessary that the school-specific effects be uncorrelated 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

• Bühlmann model
• Hierarchical linear modeling
• Fixed effects
• MINQUE
• Covariance estimation
• Conditional variance

## Notes

1. ^ 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
2. ^ 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
3. ^ Laird, Nan M; Ware, James H 1982 "Random-Effects Models for Longitudinal Data" Biometrics 38 4: 963–974 JSTOR 2529876
4. ^ Gardiner, Joseph C; Luo, Zhehui; Roman, Lee Anne 2009 "Fixed effects, random effects and GEE: What are the differences" Statistics in Medicine 28: 221–239 doi:101002/sim3478

• Christensen, Ronald 2002 Plane Answers to Complex Questions: The Theory of Linear Models Third ed New York: Springer ISBN 0-387-95361-2
• Gujarati, Damodar N; Porter, Dawn C 2009 "Panel Data Regression Models" Basic Econometrics Fifth international ed Boston: McGraw-Hill pp 591–616 ISBN 978-007-127625-2
• Hsiao, Cheng 2003 Analysis of Panel Data Second ed New York: Cambridge University Press pp 73–92 ISBN 0-521-52271-4
• Wooldridge, Jeffrey M 2013 "Random Effects Estimation" Introductory Econometrics: A Modern Approach Fifth international ed Mason, OH: South-Western pp 474–478 ISBN 978-1-111-53439-4

• Fixed and random effects models
• How to Conduct a Meta-Analysis: Fixed and Random Effect Models

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