Fixed Effect Model

A fixed effect model in meta analysis is a statistical approach that assumes all studies in the analysis are estimating a single common (fixed) treatment effect, and that any variation between study results is due solely to random sampling error. The pooled estimate from a fixed effect model represents this assumed common effect, weighted by the precision (inverse of the variance) of each study.

The choice between a fixed effect model and a random effects model is one of the most consequential decisions in meta analysis, because it directly affects the pooled estimate, the confidence interval, and the interpretation of results. A fixed effect model is appropriate when the studies are functionally identical in terms of populations, interventions, comparators, and outcomes, and when the analyst believes there is one true effect that all studies are estimating. In such cases, the fixed effect model provides the most precise (narrowest confidence interval) estimate of that common effect. Understanding when this assumption is reasonable, and when it is not, is essential for producing credible evidence syntheses.

In a fixed effect meta analysis, each study's effect estimate is weighted by the inverse of its variance (1 divided by the squared standard error). Larger, more precise studies receive more weight because they provide better estimates of the common effect. The pooled estimate is the weighted average of the individual study estimates.

The most common methods for fixed effect meta analysis are the inverse variance method (which can be applied to any type of effect estimate) and the Mantel Haenszel method (which is particularly suited to binary outcome data and performs well with sparse data). The Peto method is another fixed effect approach used specifically for odds ratios, especially when events are rare.

The fixed effect model does not incorporate a between study variance component. Consequently, the confidence interval for the pooled estimate reflects only within study sampling variability, so it is narrower than a random effects model's whenever the estimated between study variance is above zero, and identical to it when that estimate is zero. If the assumption of a common effect is correct, this is appropriate. If the assumption is wrong (that is, if the true effect actually varies across studies), the fixed effect model will produce confidence intervals that are too narrow and may give a false sense of precision.

Heterogeneity statistics (the Q test and the I squared statistic) show whether the results are compatible with a common effect, but the Cochrane Handbook states that the choice between a fixed effect and a random effects meta analysis should never be made on the basis of a statistical test for heterogeneity. The choice should rest on whether a single common effect is plausible, judged from the clinical and methodological similarity of the studies, and should be made in advance. A non significant Q test does not show that the assumption holds, because the test has low power, especially when the number of studies is small.

A fixed effect model is appropriate when the included studies are sufficiently similar in design, population, intervention, and setting that a single common effect is a reasonable assumption. It is also appropriate when the analyst's goal is to estimate the average effect for the specific set of studies included (rather than generalizing to a broader population of studies). With only a few studies, some analysts prefer it because the random effects model's estimate of between study variance is imprecise, but the fixed effect result still assumes a common effect and applies only to the included studies. Cochrane reviews sometimes present both fixed effect and random effects analyses to assess the sensitivity of results to the modeling choice.

A common error is using a fixed effect model when there is substantial heterogeneity among studies, which violates the core assumption and produces misleadingly narrow confidence intervals. Researchers sometimes choose the fixed effect model because it produces a more statistically significant result (with narrower confidence intervals), which constitutes model selection based on results rather than on assumptions. Another mistake is interpreting the fixed effect pooled estimate as applicable to future studies or new populations, when it strictly pertains only to the common effect assumed to underlie the included studies. Confusing the fixed effect model in meta analysis with fixed effects models in econometrics or panel data analysis (which address within unit variation across time) is another source of misunderstanding.

A meta analysis pools five clinical trials of a blood pressure medication, all conducted in similar populations (adults aged 45 to 65 with uncomplicated hypertension) using the same dose and duration. The individual study effects (mean reductions in systolic blood pressure) are 8.2, 7.9, 8.5, 8.1, and 8.3 mmHg, with standard errors of 0.40, 0.50, 0.45, 0.35, and 0.55 mmHg. The fixed effect pooled estimate is 8.2 mmHg (95% CI: 7.8 to 8.6), and the most precise trial (standard error 0.35) carries the most weight, about 31%. The fixed effect model was chosen in advance because the trials are so similar that a single common effect is a reasonable assumption. The heterogeneity statistics are consistent with that assumption (Q = 0.92 on 4 degrees of freedom, p = 0.92, I squared = 0%), although with only five trials they cannot rule out small differences in the true effects. Because the estimated between study variance is zero, a random effects analysis gives the same result here.