# Chapter 25.1: Fixed-Effects and Random-Effects Models

In a *meta-analysis*, results from 2 or more *primary studies* are combined statistically. The meta-analyst seeking a method to combine primary study results can do so by using either a *fixed-effects model* or a *random-effects model*.^{1}

We explain the differences between the 2 models based on the underlying assumptions, statistical considerations, and how the choice of model affects the results (Table 25.1-1). Note, however, that this is a controversial area within the field of meta- analysis, and expert statisticians disagree even with the characterizations in Table 25.1-1. The approach we take is, however, largely consistent with that of the Cochrane Collaboration.

**Comparison of Fixed-Effects and Random-Effects Models**

**Comparison of Fixed-Effects and Random-Effects Models**

Fixed-Effects Models | Random-Effects Models | |

Conceptual considerations | Estimates effect in this sample of studies Assumes effects are the same in all studies | Estimates effect in a population of studies from which the available studies are a random sample Assumes effects differ across studies and the pooled estimate is the mean effect |

Statistical considerations | Variance is only derived from within-study variance | Variance is derived from both within-study and between-study variances |

Practical considerations | Narrow CI Large studies have much more weight than small studies | Wider CI Large studies have more weight than small studies, but the gradient is smaller than in fixed-effects models |

A fixed-effects model considers the set of studies included in the meta-analysis and assumes that there is a single true value underlying all of the study results.^{2} That is, the assumption is that if all studies that address the same question were infinitely large and completely free of *bias*, they would yield identical estimates of the effect. Thus, with the assumption that studies are free of *risk of bias*, the observed estimates of effect differ from one another only because of *random error*.^{3} This assumes that any differences in the patients enrolled, the way the intervention was administered, and the way the outcome was measured have no (or minimal) impact on the magnitude of effect. The error term for a fixed-effects model comes only from within-study variation (study *variance*); the model does not consider between-study variability in results (known as *heterogeneity*) (see Chapter 23, Understanding and Applying the Results of a Systematic Review and Meta-analysis). A fixed-effects model aims to estimate this common-truth effect and the uncertainty about it.

A random-effects model assumes that the studies included are a random sample of a population of studies that address the question posed in the meta-analysis.^{4} Because there are inevitably differences in the patients, interventions, and outcomes among studies that address a particular research question, each study estimates a different underlying true effect, and these effects will have a normal ...

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