I have what is a very basic question about meta-analysis. If data from individual studies about the relationship of interest are presented in both bivariate and multivariate analyses, which should I use? Would it be appropriate to calculate effect sizes based on multivariate analyses (e.g., adjusted odds ratios), or should only data from bivariate analyses be used?
You can use both types of effect sizes, but you should make sure that you run seperate meta-analyses. That is, one synthesis should be based on bivariate effect sizes; a separate meta-analysis should focus on partial effects (e.g., regression coefficients; adjusted odds ratio etc.) sizes. Here is a recommendation from Aloe/Thompson (2013: 400) (you can download the paper free of charge):
"Partial effect sizes should not be combined with bivariate correlations, nor should different types of partial effect sizes be combined. Meta-analysts should present two sets of analyses: one for bivariate correlations and one for partial effects. "
Aloe, A. M., & Thompson, C. G. (2013). The synthesis of partial effect sizes. Journal of the Society for Social Work and Research, 4(4), 390 – 405.