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Meta‐analysis and the reversed Theorem of the Means
Author(s) -
Edwardes Michael D. deB.
Publication year - 2014
Publication title -
research synthesis methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.376
H-Index - 35
eISSN - 1759-2887
pISSN - 1759-2879
DOI - 10.1002/jrsm.1118
Subject(s) - harmonic mean , geometric mean , mathematics , weighted geometric mean , weighted arithmetic mean , estimator , reciprocal , statistics , logarithmic mean , variance (accounting) , logarithm , inequality of arithmetic and geometric means , population mean , inequality , mathematical analysis , philosophy , linguistics , accounting , rearrangement inequality , business , linear inequality
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. Copyright © 2014 John Wiley & Sons, Ltd.