# Tip for data extraction for meta-analysis – 6

April 22, 2019

## How do you calculate a standard error of a beta coefficient?

Kathy Taylor

I have shown how to rescale hazard ratios (HRs) to a common change in a predictor variable, and also beta coefficients which are HRs (natural) log-transformed. We need to enter beta coefficients and their standard errors (SEs) into meta-analysis software. In this post I’ll show how to calculate these SEs.

Beta coefficients may be reported with a SE or a t-value. HRs are usually reported with a confidence interval.

A bit of maths (see below if you’re interested) shows us:

and rescaling from a change of x units to y units in the predictor variable

and if the HR hasn’t already been rescaled to y units

Let me show you some examples. Returning to the studies mentioned in my previous post. I had already scaled them to a 5mmHg increase in systolic blood pressure variability:

study by Mancia et al has a beta coefficient of -0.092 and its

study by Hansen et al has a beta coefficient of 0.01 and its

study by Pringle et al has a beta coefficient of 0.16 with its

So the data are ready for meta-analysis, but all these data cannot be pooled together. The data of Mancia et al may be pooled with the data of Hansen et al, or the data of Pringle at al, but not the data from both studies, because the patient populations of these two studies overlap – double counting patients can inflate the effect estimates. I’ll return to the issue of overlapping patient populations in a later post.

Here’s a tip…

Standard errors of beta coefficients can be calculated from t values and confidence intervals

In my next post I will show how to pool data from studies reporting categorical risk data with different numbers of categories and different thresholds.

### Where did the equations come from?

(You can skip this if you are only interested in carrying out the calculations)

This can be rearranged to

A 95% confidence interval is

Therefore

upperCI = mean + 1.96 x SE

lowerCI mean – 1.96 x SE

This equation applies to any symmetric 95% confidence interval.

As HRx has a confidence interval (lowerCIto upperCIx)

As βx has a confidence interval (ln(lowerCIx) to ln(upperCIx))

Rescaling beta coefficients from a change of x units to y units in the predictor variable,

both follow on from the explanation I gave in my last post for

βbeta

The same applies for

which follows on from the explanation given for

Dr Kathy Taylor teaches data extraction in Meta-analysis. This is a short course that is also available as part of our MSc in Evidence-Based Health Care, MSc in EBHC Medical Statistics, and MSc in EBHC Systematic Reviews.