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How to interpret meta-analysis models: fixed effect and random effects meta-analyses
  1. Adriani Nikolakopoulou1,
  2. Dimitris Mavridis1,2,
  3. Georgia Salanti1
  1. 1Department of Hygiene and Epidemiology, University of Ioannina School of Medicine, Ioannina, Greece;
  2. 2Department of Primary Education, University of Ioannina, Ioannina, Greece
  1. Correspondence to Dr Dimitris Mavridis, Department of Hygiene and Epidemiology, University of Ioannina School of Medicine, Ioannina, 45110, Greece; dmavridi{at}cc.uoi.gr

Abstract

This section of the journal is aimed at providing the essential information readers should know about the topics that are addressed in the ‘Statistics in practice’ paper published in the same issue of the journal. This stand-alone section has to be seen as an articulated summary of the main notions clinicians have to know about some basic concepts in statistics, which may be useful for their evidence-based practice. After going through these notes, readers are encouraged to read the ‘Statistics in practice’ articles. Of course, we welcome any feedback from you (via email or Twitter) about this!

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Relative treatment effects studied in trials are typically measured using an effect size. The observed effect sizes are synthesised to obtain a summary treatment effect via meta-analysis.

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Fixed effect meta-analysis

Fixed effect meta-analysis assumes there is a common treatment effect across all study settings. Any differences between observed effect sizes are due to sampling error.

  • The summary treatment effect in a fixed effect model is a weighted average of study-specific effect sizes.

  • The weight assigned to each study is equal to the inverse of the variance of the study effect size.

  • The more precise the result of the study, the larger its weight in the meta-analysis and its contribution to the summary estimate.

  • The inferences drawn from a fixed effect meta-analysis should be restricted to the population included in the analysis.

Random effects meta-analysis

Under the random effects assumption, treatment effects are supposed to vary from study to study. The differences in observed effect sizes are attributed not only to random error but also to variation in true treatment effects (this is called heterogeneity).

  • As in the fixed effect model the summary treatment effect from a random effects model is a weighted average of study-specific effect sizes.

  • The weights assigned to each study equal the inverse of the variance of the study effect size plus an additional variance term that represents heterogeneity (which is the between-study variance).

  • Random effects meta-analysis is more reliable when making predictions about treatment effects in future trials.

  • Prediction intervals express the dispersion of the true effect sizes and can be interpreted as the predicted range for the true treatment effect in an individual study setting. Prediction intervals should be employed to aid interpretation of a random effects meta-analysis.

Comparison of fixed and random-effects meta-analysis

  • In the presence of small heterogeneity the two approaches give similar results.

  • Random effects meta-analysis gives more weight to imprecise (or small) studies compared to a fixed effect meta-analysis.

  • Random effects meta-analysis gives more conservative results unless there are small study effects (ie, small studies providing systematically different results from large studies).

  • The summary effect from a fixed effect model is an estimate of the assumed common underlying treatment effect; by contrast, for the random effects model is an estimate of the average of the distribution of treatment effects across various study settings.

Selection of the appropriate model

Selection of the appropriate synthesis method should be prespecified in the study protocol and based on researcher's beliefs about the model's underlying assumptions.

  • If there are important reasons to believe that the relative treatment effect is common in all included studies, then a fixed effect meta-analysis is a reasonable option.

  • When researchers expect that the treatment effects will be similar but not identical then random effects model is the appropriate one to use.

  • The choice of the synthesis method should not be based on the test of heterogeneity (the χ2 test).

  • Differences in the weights attributed to studies between the two models should be considered when choosing the meta-analysis model. Researchers should consider that small studies are assigned larger weights in a random effects model compared to a fixed effect model.

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Footnotes

  • Competing interests AN, DM and GS received research funding from the European Research Council (IMMA 260559).

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