Viechtbauer 2010 - Fixed, Random and Mixed-Effects Models

Summary

Meta-analysis starts from $k$ independent effect estimates $y_i={\theta }_i+e_i$ with known sampling variances $v_i$. The random-effects model adds ${\theta }_i=\mu +u_i$, $u_i\sim N(0,{\tau }^2)$; the mixed-effects model replaces $\mu$ with a linear predictor in study-level moderators. Crucially, a fixed-effects model does not assume homogeneity; it simply restricts inference to the $k$ studies at hand, whereas random/mixed models infer to a hypothetical population of studies.

Overview

Section 2 of the paper states the models underlying the entire package. It is deliberately brief and points to Hedges and Olkin (1985), Berkey et al. (1995), van Houwelingen et al. (2002) and Raudenbush (2009).

Main Content

Sampling model

Observed effect model (Eq. 1)

For $i=1,\dots ,k$ independent studies,

$$y_i={\theta }_i+e_i,e_i\sim N(0,v_i),$$

where $y_i$ is the observed effect, ${\theta }_i$ the unknown true effect of study $i$, $e_i$ the sampling error and $v_i$ the sampling variance, assumed known. The $y_i$ are assumed unbiased, normally distributed estimates of ${\theta }_i$. A bias-correcting, normalizing or variance-stabilising transformation may be needed (log for odds ratios, Fisher’s $r$-to-$z$ for correlations; see Viechtbauer 2010 - Effect Size Measures and escalc).

Random-effects model

Random-effects model (Eq. 2)

$${\theta }_i=\mu +u_i,u_i\sim N(0,{\tau }^2),$$

so marginally $y_i\sim N(\mu ,v_i+{\tau }^2)$. $\mu$ is the average true effect and ${\tau }^2$ the total amount of heterogeneity among true effects. If ${\tau }^2=0$ then ${\theta }_1=\cdots ={\theta }_k\equiv \theta$ and $\mu =\theta$.

Mixed-effects model

Mixed-effects model / meta-regression (Eq. 3)

$${\theta }_i={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_{p^{\prime }}{x}_{i{p}^{\prime }}+u_i,u_i\sim N(0,{\tau }^2),$$

with $x_{ij}$ the value of moderator $j$ for study $i$ ($p^{\prime }$ moderators, $p=p^{\prime }+1$ coefficients with intercept). Now ${\tau }^2$ is the residual heterogeneity, the variability of true effects not accounted for by the moderators. Categorical moderators enter by dummy coding.

What the fixed-effects model estimates

Fixed-effects targets (Eq. 4-5)

With weighted least squares and typical weights $w_i=1/v_i$, the fixed-effects model estimates the weighted average of the true effects,

$${\bar{\theta }}_w=\frac{{\sum }_i{w}_i{\theta }_i}{{\sum }_i{w}_i}.$$

With unweighted least squares it estimates the simple average ${\bar{\theta }}_u={\sum }_i{\theta }_i/k$ (Laird and Mosteller 1990).

Fixed versus random inference (Sec. 2.3)

1. A fixed-effects model gives a conditional inference about the $k$ included studies (more precisely, about sets of $k$ studies whose true effects equal those of the included studies), answering “How large is the average true effect in this set of $k$ studies?” (Hedges and Vevea 1998).
2. Random/mixed-effects models give an unconditional inference about a larger population of studies (conducted, conductable or future) of which the $k$ are a random sample, answering “How large is $\mu$ in that population?”
3. Contrary to common statements, the fixed-effects model does not assume homogeneous true effects. It gives valid inference under heterogeneity provided conclusions are restricted to the included studies.
4. If the true effects are homogeneous, $\mu ={\bar{\theta }}_w={\bar{\theta }}_u=\theta$ and the distinction disappears.
5. Because there is no infallible homogeneity test, the type of inference should be chosen before examining the data.

Footnote 2 of the paper adds that weighted versus unweighted least squares is also a choice for random/mixed models but, since the parameters are the same either way, weighted estimation is preferred for efficiency.

Examples

Interpreting the two on the BCG data (values from Viechtbauer 2010 - BCG Vaccine Worked Example). The fixed-effects pooled log RR is $-0.4303$ (RR 0.65), dominated by the huge TPT Madras and Comstock 1974 trials, and answers “what is the precision-weighted average among these 13 trials?” The random-effects $\hat{\mu }=-0.7145$ (RR 0.49) answers “what is the average true effect across a population of BCG trials?”, with $\hat{\tau }=0.56$ describing spread across trials. The two differ because trials are heterogeneous ($I^2=92\%$) and the large trials sit near zero effect.

Connections

• Marketing measurement: a brand running geo tests in many markets and quarters must decide the estimand. A fixed-effect view reports the average lift of tests run; a random-effects view reports expected lift for the next market, which is what budget planning needs, and $\tau$ gives the prediction-interval width.
• Bayesian view: Roever 2020 - Normal-Normal Hierarchical Model and Likelihood writes the same model as a hierarchical normal-normal model; see Hierarchical Models and Hierarchical Linear Models.
• Viechtbauer 2010 - Fitting Models with rma, Moderators and Knapp-Hartung shows how each model is requested in code.

See Also

• Viechtbauer 2010 - Conducting Meta-Analyses in R with the metafor Package - Overview
• Viechtbauer 2010 - tau2 Estimators, Q-Test and Heterogeneity Statistics