# meta-analysis of linear regression coefficients

I have the summary results of a linear regression model with 26 covariates which has been fitted to two independent samples (with different sizes). Now I am planning to use the linear regression beta estimates of a covariate of interest within this model and its standard errors across the two studies to perform an inverse-variance-based meta-analysis of the betas and the p-value of its significance. Since I need to use the t-statistics in this process for tracking deviation of the meta-analysis beta from zero, I wonder what would be the correct degrees of freedom for converting the meta-t statistics to the final meta analysis p-value, as the two samples have different sizes, and therefore DOFs?

I already read this previous question How to perform a meta-analysis of regression coefficients? but seems like this does not answer my confusion with DOF in the meta-analysis stage.

• Why are you using the t-statistic for testing the statistical significance of the meta-analytic mean beta? Typically, inverse-variance weighted meta-analysis uses the z-distribution. There is some debate on this issue and some argue for using t. Give that this will be a mean of two beta's the DOF is 1. Jun 6 '18 at 15:14
• @dbwilson I'm a bit confused. In a simple linear regression model, wouldn't we use the t-statistic for testing significant deviation of a beta estimate from zero? Then how come this may change to z in the meta analysis step? Jun 7 '18 at 7:22
• If I understand your data, you have two beta's you want to average with meta-analysis. Each has a different standard error (and is based on a different sample size). To compute the average of these betas you are using inverse variance weighted meta-analysis. The significance of the mean beta is assessed with a z-test. In a fixed effects model, its standard error in the inverse of the sum of the weights (which are the inverse of the squared standard errors). Jun 7 '18 at 11:43
• What moves forward to the meta-analysis from the simple OLS regression models is the betas and standard errors, not the t-tests and p-values (the latter two have no unique information not contained in the estimate and its standard error). Jun 7 '18 at 11:44
• @dbwilson Now I'm starting to get it. Just one last question, if we can use z-transform (and the z-test) to assess deviation of the meta-beta estimate from zero using the inverse variance weighting, how does this approach reduce to the extreme case of only one sample meta-analysis (I know it is weird to do a meta test with only one sample). Maybe I am missing something here? Jun 7 '18 at 12:03