Fixed-Effect vs Random-Effects Models

Two methods for combining studies. A fixed-effect model assumes there is one true effect and that all variation between studies is sampling error; it weights studies by precision and gives narrower CIs — valid only when studies are homogeneous. A random-effects model assumes the true effect varies across studies (a distribution) and adds the between-study variance (τ²) on top of within-study error, giving wider, more conservative CIs — appropriate when heterogeneity exists.

One forest plot, the identical four studies, pooled two ways. The narrow gold diamond is the fixed-effect estimate; the wider pale diamond is the random-effects estimate. Both are centred near the same point, but the random-effects CI is broader because it carries the between-study variance (τ²) as well as each study’s own sampling error.

If the study CIs overlap well (low I²), the two diamonds are almost identical and either model is fine. The more the studies disagree (high I²), the more the random-effects diamond widens relative to the fixed one — sometimes enough to cross the null and lose significance. The width gap between the diamonds is a visual signal of how much heterogeneity is being accommodated.

Model choice can change the CI and even the conclusion. A fixed-effect model used despite real heterogeneity produces a falsely narrow, over-precise result. Random-effects gives an honest, wider interval — but it only accommodates heterogeneity, it does not explain it, and it up-weights small (often lower-quality) studies relative to fixed-effect.

• Fixed: one true effect; weight ∝ precision; narrow CI; needs homogeneity
• Random: effect varies; adds τ² → wider, more conservative CI
• High I² → random-effects, and investigate the heterogeneity

IV magnesium for acute MI — Cochrane Review (Li et al., 2007). Pooling early mortality across the trials showed marked heterogeneity, and the model choice flipped the answer: the fixed-effect analysis showed no benefit (OR 0.99, 95% CI 0.94–1.04), while the random-effects analysis showed an apparent large benefit (OR 0.66, 95% CI 0.53–0.82). The heterogeneity came from small early positive trials sitting against the huge neutral mega-trial (ISIS-4); the later definitive MAGIC trial confirmed no mortality benefit. Neither model was a substitute for explaining why the trials disagreed.

• Using a fixed-effect model when I² is high — the CI is falsely narrow and over-precise.
• Believing random-effects “corrects” heterogeneity — it only accommodates it; you must still explain it.
• Forgetting random-effects up-weights small studies, which can exaggerate the pooled effect (small-study/publication bias).