Introduction

The common meta-analysis as described by Hedges and Olkin combines multiple sets of effect size (ES) and their Standard Errors (SE) to produce a set of combined summary ES and SE. A major assumption underpinning the algorithm is that the data is normally distributed.

Meta-analysis of Correlation Coefficient (ρ) presents a difficulty because ρ is contrained by the range of -1 to +1. It is normally distributed when ρ=0, but the distribution becomes skewed in other values, with a long tail towards the 0 value, and short tail towards -1 or +1.

A comon method to overcme this constraint is to use the Fisher's Z transformation, where ρ is transformed to a normally distributed value of Z, and the Standard Error of Z is estimated according to the sample size. Meta-analysis can then be carried out on multiple sets of Z and its SE. The resulting Z value and its 95% confidence interval can then be reverse transformed into ρ values.

The second and third program on this page provides algorithm to convert n and ρ to Z and SE, and reverse transform Z to ρ after meta-analysis

Hunter and Schmidt proposed an alternative method, where ρ and its SE are weighted and averaged. The first program on this page provides these calculations.

Each of the two approaches have different advantages and disadvantages. When there are numerous studies with adequate sample size, the ρ values not near the extreme, and the data not heterogeneous, the results are approximately the same. Away from these ideal situations, there is a need to choose the method that is more appropriate to that individual set of data. StatsToDo presents both method as a tool kit, but suggests that the user consult the references provided or seek expert advice to decide which method should be used for rheir particular meta-analysis

Individual Programs

Meta-analysis using weighted means (Hunter and Schmidt)

This algorithm uses the commonly accepted idea that the Correlation Coefficient (ρ) is approximately normally distributed, with its Standard Error SE = sqrt((n-2) / (1 + ρ²))

The algorithm then combines the values to produce the combined summary ρ and its SE, placing weightings on the individual input ρ values according to its sample size

Fisher's Z Transformation of ρ for Meta-analysis (Hedges and Olkin)

Fisher's formula transforms sample size and the non-parametric Correlation Coefficient (ρ) to normally distributed Z and its Standard Error (se), so they can be used in meta-analysis. The formulae are

Z = 0.5 * log((1 + ρ) / (1 - ρ)), and
SE = 1 / sqrt(n - 3), where n is the sample size

Reverse Transformation from Fisher's Z to Correlation Ciefficient ρ

This is a short utility program to allow the reverse transformation of Fisher's Z back to Correlation Coefficient ρ. the Formula is

ρ = exp(2 * Z) - 1) / (exp(2 * Z) + 1)

This is useful to transform the combined summary Z and its 95% confidence intervals back into Correlation Coefficient (ρ) values after meta-analysis

References

SAS paper discussing the need for Fisher's Transformation http://www2.sas.com/proceedings/sugi31/170-31.pdf

For introduction to Meta-analysis of Correlation Coefficients, and description of the algorithms, http://www.statisticshell.com/docs/meta.pdf

For a comparison between the Hunter Schmidt and Hedges Olkin algorithms, http://www.statsdirect.com/help/default.htm#meta_analysis/correlation.htm.

For references to original descriptions of the algorithm

• Hunter, J. E., & Schmidt, F. L. (2004). Methods of meta-analysis: Correcting error and bias in research findings (Second edition). Newbury Park, CA: Sage
• Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando, FL: Academic Press

For calculations of Combined Summary Effect Size using Fixed and Random Effect Models :

Sutton AJ, Abrams KR, Jones DR, Sheldon TA, and Song F (2000) Methods for Meta-Analysis in Medical Research. John Wiley & Sons Ltd., Chichester UK. ISBN 0-471-49066-0.p 58-63, p 80-82

Data for Meta-analysis: All rows have 2 columns     - Each row from a separate study     - Column 1 = Sample Size (n)     - Column 2 = Correlation Coefficient (ρ) Meta-analysis using weighted means (Hunter and Schmidt) Fisher's Z Transformation of ρ for Meta-analysis (Hedges and Olkin) Reverse Transformation from Fisher's Z to Correlation Ciefficient ρ     - Data a single column of Z values