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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?

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  • $\begingroup$ Not so simple. That depends if sample A is plausibly from the same population as sample B or not. So is it? $\endgroup$ – Carl Jul 7 at 4:50
  • $\begingroup$ Yes, they are from the same population. $\endgroup$ – Neuroguy Jul 7 at 4:57
  • $\begingroup$ How do you know? Did you test for significant difference? $\endgroup$ – Carl Jul 7 at 5:05
  • $\begingroup$ 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. $\endgroup$ – Neuroguy Jul 7 at 5:12
  • $\begingroup$ So combine the parameters. $\endgroup$ – Carl Jul 7 at 5:55
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I realize now that I forgot to add an error term to my equation so that it matches the formula for linear regression. Thank you for that correction.

There is a lot of literature on the use of regression coefficients as measures of effect size in a meta-analysis. It’s certainly not the most common way to quantify effect size, but it it is not unheard of. Typically, standardized regression coefficients are used when this is done. For the application I’m interested in, the value of the regression coefficient is of greatest practical interest to begin with, and so the regression coefficient is what I decided to use.

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  • $\begingroup$ This is not an answer to the question. $\endgroup$ – Michael Chernick Jul 15 at 21:13

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