Intended for healthcare professionals

Research Methods & Reporting Statistics Notes

Inverse probability weighting

BMJ 2016; 352 doi: (Published 15 January 2016) Cite this as: BMJ 2016;352:i189
  1. Mohammad Ali Mansournia, assistant professor of epidemiology1,
  2. Douglas G Altman, professor of statistics in medicine2
  1. 1Department of Epidemiology and Biostatistics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran
  2. 2Centre for Statistics in Medicine, Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, University of Oxford, Oxford OX3 7LD, UK
  1. Correspondence to: M A Mansournia mansournia_ma{at}
  • Accepted 5 January 2016

Statistical analysis usually treats all observations as equally important. In some circumstances, however, it is appropriate to vary the weight given to different observations. Well known examples are in meta-analysis, where the inverse variance (precision) weight given to each contributing study varies, and in the analysis of clustered data.1

Differential weighting is also used when different parts of the population are sampled with unequal probabilities of selection. Two examples of intentional unbalanced sampling are:

  1. Surveys with unequal probabilities of selection—In a national survey of hypertension prevalence, certain groups with relatively rare characteristics (such as people aged ≥65 years) were oversampled to improve the precision of estimates for those groups.2

  2. Two-phase prevalence studies—In the first phase of a two-phase prevalence study of mental health status, the sampled patients completed a short screening questionnaire. In the second phase, a subsample was selected for a definitive diagnostic test with oversampling of the screen-positive cases to ensure precise estimates for diagnostic prevalence.3

    In such cases the ordinary unweighted sample quantities, such as means or proportions, are likely to be biased estimates of their corresponding population quantities. This “selection bias” can be eliminated by performing a weighted estimation, giving each individual’s data a …

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