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Random effects model
Random effects model
Random effects model
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Background

In econometrics, a random effects model, also called a variance components model, is a statistical model where the model effects 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. A random effects model is a special case of a mixed model.

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).

Qualitative description

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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.

Simple example

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Suppose large elementary schools are chosen randomly from among thousands in a large country. Suppose also that pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let be the score of the -th pupil at the -th school.

A simple way to model this variable is

where is the average test score for the entire population.

In this model is the school-specific random effect: it measures the difference between the average score at school and the average score in the entire country. The term is the individual-specific random effect, i.e., it's the deviation of the -th pupil's score from the average for the -th school.

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

where is a binary dummy variable and 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.

Variance components

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The variance of is the sum of the variances and of and respectively.

Let

be the average, not of all scores at the -th school, but of those at the -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" and .

The parameter is also called the intraclass correlation coefficient.

Marginal likelihood

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For random effects models the marginal likelihoods are important.[7]

Applications

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

See also

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Further reading

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Random effects model

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In statistics, the random effects model is a framework for analyzing where certain factors or effects are treated as random variables drawn from a specific , typically normal, enabling inferences about a broader of possible levels rather than just the observed ones. This approach contrasts with fixed effects models by assuming that the random effects capture unexplained variability across groups or units, such as treatments sampled from a larger set, and is formally expressed in a basic one-way model as Yij=μ+τi+ϵijY_{ij} = \mu + \tau_i + \epsilon_{ij}, where τi∼N(0,στ2)\tau_i \sim N(0, \sigma_\tau^2) represents the random effect for the ii-th level and ϵij∼N(0,σ2)\epsilon_{ij} \sim N(0, \sigma^2) is the residual , with between τi\tau_i and ϵij\epsilon_{ij}. The total variance of observations is then στ2+σ2\sigma_\tau^2 + \sigma^2, and a key is the coefficient ρ=στ2/(στ2+σ2)\rho = \sigma_\tau^2 / (\sigma_\tau^2 + \sigma^2), which quantifies the proportion of total variance due to the random effects. Random effects models differ fundamentally from fixed effects models in their scope of inference and assumptions about the factors involved. In fixed effects models, the levels of a factor are considered exhaustive and specifically chosen by the researcher, limiting inferences to those particular levels and testing hypotheses about equality of means (e.g., H0:μi=μH_0: \mu_i = \mu); random effects, however, treat levels as a random sample from a larger , broadening inferences to that population and testing whether the variance of the effects is zero (e.g., H0:στ2=0H_0: \sigma_\tau^2 = 0). This distinction arises because random effects account for hierarchical or clustered data structures, where observations within groups are correlated, adjusting standard errors for tests on other effects and preventing underestimation of variability. Random effects models are integral to linear mixed models (LMMs), which combine fixed effects—representing all possible levels of interest for direct inference, such as specific treatments—with random effects for sampled factors like blocks, replications, or locations to model additional variability. In LMMs, random effects are specified separately (e.g., via a random statement in software), and they adjust the standard errors for fixed effect tests, making the model suitable for experimental designs like split-plot or repeated measures where repeated effects over time require structures. Applications extend to fields like for analyzing field trials across varying locations, and in for where individual-specific effects are assumed uncorrelated with regressors. In meta-analysis, the random effects model assumes true effects vary across studies due to heterogeneity, incorporating between-study variance to provide more conservative estimates than fixed-effect alternatives.

Fundamentals

Definition and Qualitative Description

A random effects model is a statistical framework in which certain model parameters, such as intercepts or slopes, are treated as random variables drawn from a specified , enabling the incorporation of unobserved heterogeneity or variation across groups, individuals, or units in the data. This approach allows the model to account for clustering or in observations that arise from shared unmeasured factors, such as repeated measures on the same subject, rather than assuming all observations are independent. Qualitatively, random effects models provide an intuitive way to model data where the levels of a factor are viewed as a random sample from a broader , rather than fixed and exhaustive categories of . For instance, in studying treatment effects across multiple , a random effects model treats school-specific deviations as draws from a distribution, capturing natural variation due to unmeasured school characteristics like culture or resources, which induces correlation among students within the same school. In contrast, fixed effects would assume uniform parameters across all units, ignoring such group-level variability. This perspective shifts the focus from estimating specific effects for each level to estimating the overall variance in those effects, facilitating generalizations beyond the observed sample. Key assumptions underlying random effects models include that the random effects follow a with mean zero and constant variance, reflecting their role as deviations from the population mean without systematic bias. In models including fixed effects and covariates, the random effects are assumed to be independent of the covariates, ensuring that unobserved heterogeneity does not correlate with observed predictors and supporting unbiased estimation of fixed effects. The origins of random effects models trace back to the development of variance components analysis in the 1940s and 1950s, building on R.A. Fisher's foundational work in the on analysis of variance for agricultural experiments at Rothamsted Experimental Station, where he introduced methods to partition variance into components attributable to different sources. Frank Yates, collaborating with Fisher, extended these ideas in the 1930s through studies on sampling designs and yield estimation, laying groundwork for handling random variation in experimental data. This evolved into modern mixed-effects models through seminal contributions like the 1982 framework by Laird and Ware, which formalized random effects for longitudinal .

Comparison with Fixed Effects Models

Random effects models differ from fixed effects models primarily in the scope of and the treatment of factors. In fixed effects models, the levels of a factor are considered fixed and of specific interest to the researcher, with inferences limited to those levels; hypotheses typically test the equality of means across levels (e.g., H0:μi=μH_0: \mu_i = \mu). Random effects models, however, treat the levels as a random sample from a larger , allowing inferences about the variance of effects (e.g., H0:στ2=0H_0: \sigma_\tau^2 = 0) and enabling beyond the observed levels. This distinction is particularly relevant in hierarchical or clustered data, where random effects account for correlation within groups, adjusting estimates and standard errors accordingly. In applications like analysis in , fixed effects can control for unobserved time-invariant heterogeneity that may correlate with covariates, while random effects assume exogeneity (no such correlation) for efficiency and the ability to estimate time-invariant effects; the Hausman test can help select between them. However, in general statistical contexts such as ANOVA or linear mixed models, the focus remains on the inferential scope rather than bias correction.

Mathematical Formulation

Basic Model Structure

The random effects model, also known as a linear mixed-effects model, is formulated as a hierarchical that incorporates both fixed and random effects to account for variation across groups or clusters in the data. In its basic scalar form for the jj-th observation within the ii-th group, the model is expressed as yij=Xijβ+Zijbi+ϵij,y_{ij} = X_{ij} \beta + Z_{ij} b_i + \epsilon_{ij}, where yijy_{ij} is the response variable, XijβX_{ij} \beta represents the fixed effects contribution with β\beta as the vector of fixed-effect parameters, ZijbiZ_{ij} b_i captures the random effects for group ii with bi∼N(0,G)b_i \sim N(0, G) denoting the random effects vector assumed to follow a with mean zero and GG, and ϵij∼N(0,R)\epsilon_{ij} \sim N(0, R) is the residual error term with RR. This structure allows the model to handle clustered data by treating group-specific deviations bib_i as random draws from a distribution, thereby generalizing fixed effects approaches to induce dependence within groups. The model adopts a two-stage hierarchical interpretation. In the first stage, the conditional distribution of the response given the random effects is specified as yij∣bi∼N(Xijβ+Zijbi,R)y_{ij} | b_i \sim N(X_{ij} \beta + Z_{ij} b_i, R), modeling the within-group variability around the group-specific mean. The second stage then specifies the distribution of the random effects themselves as bi∼N(0,G)b_i \sim N(0, G), which introduces between-group variability and ensures that observations within the same group ii are correlated through the shared bib_i term, with the covariance between yijy_{ij} and yiky_{ik} (for j≠kj \neq k) arising from Cov(Zijbi,Zikbi)=ZijGZikT\text{Cov}(Z_{ij} b_i, Z_{ik} b_i) = Z_{ij} G Z_{ik}^T. This hierarchical setup facilitates inference about the fixed effects β\beta while borrowing strength across groups via the random effects distribution. In matrix notation, the full model for the stacked response vector y\mathbf{y} across all groups is y=Xβ+Zb+ϵ,\mathbf{y} = X \beta + Z \mathbf{b} + \epsilon, where y\mathbf{y} is the n×1n \times 1 response vector, XX is the n×pn \times p fixed-effects design matrix, ZZ is the n×qn \times q random-effects design matrix, b∼N(0,Σb)\mathbf{b} \sim N(0, \Sigma_b) is the stacked q×1q \times 1 random effects vector with covariance Σb=blockdiag(G1,…,Gm)\Sigma_b = \text{blockdiag}(G_1, \dots, G_m) for mm groups, and ϵ∼N(0,Σϵ)\epsilon \sim N(0, \Sigma_\epsilon) with Σϵ=blockdiag(R1,…,Rm)\Sigma_\epsilon = \text{blockdiag}(R_1, \dots, R_m). The resulting marginal covariance structure of y\mathbf{y} is V=ZΣbZT+ΣϵV = Z \Sigma_b Z^T + \Sigma_\epsilon, which captures the total variability as a combination of random effects and residual components, leading to a marginal normal distribution y∼N(Xβ,V)\mathbf{y} \sim N(X \beta, V). Key assumptions underlying this model include the normality of both the random effects b\mathbf{b} and the residuals ϵ\epsilon, independence between b\mathbf{b} and ϵ\epsilon, and, under the common conditional independence assumption, homoscedasticity within groups such that R=σ2IR = \sigma^2 I (implying equal residual variances and no additional autocorrelation beyond that induced by the random effects). These assumptions ensure that the induced correlations are solely attributable to the shared random effects within groups, enabling valid likelihood-based inference.

Variance Components

In the random effects model, the total variance of the observed response variable yy is decomposed into two primary components: the between-group variance attributable to the random effects, denoted σb2\sigma_b^2, and the within-group residual variance, denoted σϵ2\sigma_\epsilon^2. This partitioning is expressed as σy2=σb2+σϵ2,\sigma_y^2 = \sigma_b^2 + \sigma_\epsilon^2, where σy2\sigma_y^2 represents the marginal variance of yy. This decomposition highlights how unobserved heterogeneity across groups contributes to the overall variability in the data, separate from measurement error or other residuals. A key metric derived from this decomposition is the intraclass correlation coefficient (ICC), defined as ρ=σb2σb2+σϵ2.\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma_\epsilon^2}. The ICC quantifies the proportion of total variance explained by the random effects or grouping structure, ranging from 0 (no clustering) to 1 (complete clustering). Values of ρ\rho close to 0 suggest that observations within groups are nearly independent, while higher values indicate stronger dependence due to shared random effects. Conceptually, variance components are estimated by partitioning the into between-group and within-group portions, akin to analysis of variance (ANOVA) procedures, where expected mean squares inform the components under normality assumptions. This approach provides a foundation for interpreting heterogeneity, though detailed estimation techniques are addressed elsewhere. A large σb2\sigma_b^2 relative to σϵ2\sigma_\epsilon^2 signals substantial unobserved heterogeneity among groups, justifying the inclusion of random effects to account for clustering. In balanced designs with equal group sizes, components are readily identifiable from the ANOVA table; however, unbalanced designs introduce complexities, as varying group sizes affect the of sums of squares and can complicate the separation of variance sources.

Estimation and Inference

Maximum Likelihood Methods

In random effects models, (MLE) involves maximizing the with respect to the fixed effects parameters β\beta and the variance-covariance parameters Σ\Sigma, treating the random effects as parameters. To obtain a tractable form, the is constructed by integrating out the random effects bb, yielding L(β,Σ∣y)=∫L(y∣β,b,Σ)f(b∣Σ) dbL(\beta, \Sigma | y) = \int L(y | \beta, b, \Sigma) f(b | \Sigma) \, db, where L(y∣β,b,Σ)L(y | \beta, b, \Sigma) is the conditional likelihood of the observed yy given β\beta, bb, and Σ\Sigma, and f(b∣Σ)f(b | \Sigma) is the density of the random effects. Under normality assumptions for both the random effects and residuals, this integration results in a multivariate normal marginal distribution for the data: y∼N(Xβ,V)y \sim N(X\beta, V), where V=ZGZT+RV = Z G Z^T + R incorporates the variance components from the random effects design matrix ZZ, GG, and residual RR. The log-likelihood function is then ℓ(β,Σ)=−12[nlog⁡(2π)+log⁡∣V∣+(y−Xβ)TV−1(y−Xβ)]\ell(\beta, \Sigma) = -\frac{1}{2} \left[ n \log(2\pi) + \log|V| + (y - X\beta)^T V^{-1} (y - X\beta) \right], where nn is the sample size. Computing this requires evaluating the and inverse of VV, which poses numerical challenges for large datasets due to the high dimensionality and potential ill-conditioning of VV. Optimization proceeds iteratively, often using the expectation-maximization (EM) algorithm to handle the integration implicitly. The EM algorithm alternates between an expectation step, computing expected values of the random effects given current estimates, and a maximization step, updating β\beta via the estimator β^=(XTV−1X)−1XTV−1y\hat{\beta} = (X^T V^{-1} X)^{-1} X^T V^{-1} y and profiling the likelihood for Σ\Sigma to obtain variance component estimates. Under correct model specification and suitable regularity conditions, such as increasing sample size with fixed dimensionality of random effects, the MLEs are consistent and asymptotically normal, with n(β^−β)→N(0,(XTV−1X/n)−1)\sqrt{n} (\hat{\beta} - \beta) \to N(0, (X^T V^{-1} X / n)^{-1})
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