# Calculation of the individual standard error of the mean for meta analysis bias estimation

I am trying to run the egger's test to check my meta analysis for publication bias. My data consists of a correlation effect size and a sample size from each study. I understand that the egger's regression is based on the individual standard error of the mean for each observation. I need help to calculate that based on the effect size and sample size which are the only variables I have.

Thanks

• I don't think you can calculate the SE from only the effect size & N. Do you also have raw means? You can assess publication bias with a forest plot, though. – gung Mar 14 '16 at 18:55
• Thanks, Correlations and sample size are all I have because as this is what the studies in my field generally report – Hossein Mahdavi Mar 14 '16 at 19:53

If these are standard correlation coefficients which you meta-analysed using Fisher's hyperbolic arctangent transformation (also known as z) then you can calculate the standard error for z using the sample size. Then you have what you want. Otherwise you can use a version of Egger's method but not using standard error. See

@article{moreno09,
author = {Moreno, S G and Sutton, A J and Ades, A E and Stanley, T D
and Abrams, K R and Peters, J L and Cooper, N J},
title = {Assessment of regression--based methods to adjust for
publication bias through a comprehensive simulation study},
journal = {BMC Medical Research Methodology},
year = {2009},
volume = {9},
number = {2},
keywords = {meta-analysis, publication bias, meta-regression}
}


for various options or

@article{peters06,
author = {Peters, J L and Sutton, A J and Jones, D R and Abrams, K R and
Rushton, L},
title = {Comparison of two methods to detect publication bias in
meta--analysis},
journal = {Journal of the American Medical Association},
year = {2006},
volume = {295},
pages = {676--680},
keywords = {meta-analysis, publication bias, meta-regression}
}

• Just to add that if you are using R then the metafor package lets you do the test with a whole range of alternatives to precision with its regtest function. – mdewey Mar 15 '16 at 10:10

I do not favor the term 'publication bias', as the broader concept of small study effects is more appropriate and meaningfulfor most meta-analyses or meta-epidemiologic studies. See for instance this work by Rucker et al, authors of 'Meta-Analysis with R':

http://onlinelibrary.wiley.com/doi/10.1002/bimj.201000151/abstract

In any case, there are several tests appraising small study effects, and, for instance, as also suggested by mdewey, the Peters test is suitable for your case (being based on point estimate and inverse of sample size):

http://jama.jamanetwork.com/article.aspx?articleid=202337

In this article, Peters et al clearly spell out that they '...recommend a simple weighted linear regression with lnOR as the dependent variable and the inverse of the total sample size as the independent variable. This is a minor modification of Macaskill's test, with the inverse of the total sample size as the independent variable rather than total sample size.' They go on elaborating that the 'use of sample size reduces the correlation between the lnOR and its SE'.

• Thanks for the elaborate response. I am just wondering about the term "total sample size" .Does not that mean regressing the LnOR over a constant? – Hossein Mahdavi Mar 15 '16 at 2:03
• No, your model will be a linear regression model as y = a + b * x, with y being the ln (OR), and x the inverse of the sample size. The test is significant if the p values/confidence interval for the b coefficient rejects the null hypothesis (at odds with Egger test which focuses on the intercept). Basically, the dataset will contain, per each study, a row, with the corresponding unique ln (OR), and the corresponding unique study sample size. – Joe_74 Mar 15 '16 at 8:55
• Thanks. The term "Total sample size" was confusing and alarming. – Hossein Mahdavi Mar 15 '16 at 18:05