Help with meta-regression

I want to implement a meta-regression and require some assistance. Suppose that two univariate features ($$X$$ and $$Y$$) were measured from two samples $$A$$ and $$B$$ of size $$N_A$$ and $$N_B$$, respectively. Both $$A$$ and $$B$$ come from the same population, such that $$X$$ and $$Y$$ are Gaussian random variables which are identically distributed. For $$A$$, I have the regression coefficients associated with the equation $$Y_A=X_A\beta_{1,A} + \beta_{0,A}$$. Similarly, for $$B$$, I have the regression coefficients associated with the equation $$Y_B=X_B\beta_{1,B} + \beta_{0,B}$$. I also have the means $$\bar X_A$$, $$\bar X_B$$, $$\bar Y_A$$, $$\bar Y_B$$ and their estimated variances $$s^2 (X_A)$$, $$s^2 (X_B)$$, $$s^2 (Y_A)$$, $$s^2 (Y_B)$$. All regression coefficients have the same units. I want to calculate the regression coefficients for the combined sample $$C=A\cup B$$, i.e. for the equation $$Y_C=X_C\beta_{1,C} + \beta_{0,C}$$. How can I do this if I do not have access to the actual data?

• Not so simple. That depends if sample A is plausibly from the same population as sample B or not. So is it? – Carl Jul 7 at 4:50
• Yes, they are from the same population. – Neuroguy Jul 7 at 4:57
• How do you know? Did you test for significant difference? – Carl Jul 7 at 5:05
• I implemented a test of heterogeneity and obtained a p value close to 1. The two samples come from studies with the same inclusion/exclusion criteria, the measurements of $X$ and $Y$ were made using a very similar protocol. Eventually, I will likely run into studies where the null hypothesis of the heterogeneity test is rejected at a significant level, but for now I'd like to combine these two studies because they have a large sample size and they are the studies that interest me most. – Neuroguy Jul 7 at 5:12
• So combine the parameters. – Carl Jul 7 at 5:55