Questions about standard error

Question

We are trying to create a table of standardised effects and standardised errors to compute for a meta-analysis. And I had a few questions around this

1. Can you get a standard error for pearson correlation, and if so can it be back calculated knowing the sample size, and r value. Or what other information do you need.

2. In a regression output I see people report unstandardised coefficients, standard error, and standardised coefficient. Is this standard error the standard error of the standardised coefficient or the unstandardised?

3. How do you back calculate a standardised error if you have a standardised coefficient, p value, and n?

Answer 1

Q1 The usual suggestion for meta-analysis of correlation coefficients is to transform them first. Although it is possible to calculate a standard error for the Pearson product moment statistic it is only really useful for very large sample sizes and population values of $r$ close to zero. The question linked to by @Glen_b in a comment Expected value and variance of sample correlation gives more details.

The usual transformation suggested first by Fisher is the hyperbolic arctangent $$ z = \frac{1}{2} \log \left(\frac{1 + r}{1 - r}\right) $$ with standard error $\frac{1}{\sqrt{N - 3}}$ where $N$ is the sample size. Note this is strictly only correct for bivariate normals but it is widely used nonetheless. It does of course have the advantage of mapping from [-1,1] to [$-\infty,\infty$]

Q2 I would assume the standard error was for the unstandardised coefficient but that is a wild assumption. You really need to ask the authors.

Q3 first back-calculate the value ot $t$ or $z$ corresponding to that $p$ and $n$. Then since you know that $t$ = coeff / se you can work out se.