title: Meta-analysis and the reversed Theorem of the Means.
abstract: Conventional meta-analysis estimators are weighted means of study measures, meant to estimate an overall population measure. For measures such as means, mean differences and risk differences, a weighted arithmetic mean is the conventional estimator. When the measures are ratios, such as odds ratios, logarithms of the study measures are most frequently used, and the back-transform is a weighted geometric mean, rather than the arithmetic mean. For numbers needed to treat, a weighted harmonic mean is the back-transform. The Theorem of the Means effectively states that unless all of the studies have an equal result, the arithmetic mean must be greater than the geometric mean, which must be greater than the harmonic mean. When the weights are fixed sampling weights, the inequalities are in the expected direction. However, when the weights are the usual reciprocal variance estimates, the inequalities go in the opposite direction. The use of reciprocal variance weights is therefore questioned as perhaps having a fundamental flaw. An example is shown of a meta-analysis of frequencies of two classes of drug-resistant HIV-1 mutations.
authors.affiliation: Everest Clinical Research, Markham, ON, Canada, LR3 0B8.
authors.forename: Michael D deB
authors.initials: MD
authors.lastname: Edwardes
mesh_terms: D000465:Algorithms; D015982:Bias; D002986:Clinical Trials as Topic / Q000706:statistics & numerical data*; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D016013:Likelihood Functions; D015201:Meta-Analysis as Topic; D015233:Models, Statistical; D016017:Odds Ratio; D017063:Outcome Assessment, Health Care / Q000379:methods*; D012044:Regression Analysis; D015203:Reproducibility of Results; D018401:Sample Size; D012680:Sensitivity and Specificity
keywords: correlated proportions; geometric mean; harmonic mean; numbers needed to treat; odds ratio; reciprocal variance; risk ratio; transformations; weighted means