A Note on Estimating Unreported Sample Statistics for Meta-Analysis

Abstract

A major challenge confronting meta-analysts seeking to synthesize existing empirical research on a given topic is the frequent failure of primary studies to fully report their sample statistics. Because such research cannot be included in a meta-analysis unless the unreported statistics can somehow be recovered, a number of methods have been devised to estimate the sample mean and standard deviation from other quantities. This note compares several recently proposed sets of estimators that rely on extrema and/or quartiles to estimate unreported statistics for any given sample. The simplest method relies on an underlying model of normality, while the more complex methods are explicitly designed to accommodate non-normality. Our empirical comparison uses a previously developed data set containing 58 samples, ranging in size from 48 to 2,528 observations, from a standard depression screening instrument, the nine-item Patient Health Questionnaire (PHQ-9). When only the median and extrema are known, we find that the estimation method based on normality yields the most accurate estimates of both the mean and standard deviation, despite the existence of asymmetry throughout the data set; and when other information is given, the normality-based estimators have accuracy comparable to that of the other estimators reviewed here. Additionally, if the sample size is unknown, the method based on normality is the only feasible approach. The simplicity of the normality-based approach provides an added convenience for practitioners.

Unified approach to optimal estimation of mean and standard deviation from sample summaries

“…While the work on this problem can be traced back all the way to Tippett, 5 interest in the systematic research of the estimators resurfaced in Hozo et al 6 and a significant body of the literature soon followed. [7][8][9][10][11][12][13][14][15][16][17][18][19] Typically, three scenarios are investigated, depending on the summaries being reported in an addition to a sample size: Scenario 1: { min , median, max}, Scenario 2: {first quartile, median, third quartile}, or Scenario 3: { min , first quartile, median, third quartile, max}.…”

“…While the work on this problem can be traced back all the way to Tippett, 5 interest in the systematic research of the estimators resurfaced in Hozo et al 6 and a significant body of the literature soon followed. 7–19 Typically, three scenarios are investigated, depending on the summaries being reported in an addition to a sample size: 0.25em falsefalse{ prefixtruemin , median , prefixtruemax falsefalse}, 0.25em falsefalse{ first quartile , median , third quartile falsefalse}, or 0.25em falsefalse{ prefixtruemin , first quartile , median , third quartile , prefixtruemax falsefalse}. However, other summaries are also possible; for example, Bowley’s “seven-figure summary,” including the minimum, first deciles, first quartile, median, third quartile, last decile, and maximum. 20 In general, estimators for the sample mean and sample SD are developed separately and independently of each other.…”