Someone recently asked me about how to incorporate results from Structural Equation Models into a meta-analysis. The short answer is 'with great difficulty', although that's not terribly helpful.

One approach it to do the meta-analysis at the correlation level, which is good if the SEM studies report the zero-order correlations (which hopefully they do). That is, you use meta-analysis to estimate pooled values of correlations between variables. Imagine you had three variables: anxiety, parenting, age and a bunch of papers that report relationships between some of these papers. Let's say you have 53 papers in total, and 52 report the correlation between age and anxiety, then you use these 52 studies to get a pooled value of r for that relationship. If only 9 of the studies report the relationship between age and anxiety, then you use only these to get a pooled r for this association and so on. You stick these pooled values into a correlation matrix and then do an SEM on it to test whatever model you want to test. So, the meta-analysis part of the analysis informs the correlation matrix on which the resulting SEM is based. So, it's running an SEM on data that others have collected (and you have pooled together). This is called meta-analytic structural equation modeling (MASEM; Cheung & Chan, 2005, 2009). You can implement it with Mike Cheung's metaSEM package in R. Mike Cheung has some great resources here.

One approach it to do the meta-analysis at the correlation level, which is good if the SEM studies report the zero-order correlations (which hopefully they do). That is, you use meta-analysis to estimate pooled values of correlations between variables. Imagine you had three variables: anxiety, parenting, age and a bunch of papers that report relationships between some of these papers. Let's say you have 53 papers in total, and 52 report the correlation between age and anxiety, then you use these 52 studies to get a pooled value of r for that relationship. If only 9 of the studies report the relationship between age and anxiety, then you use only these to get a pooled r for this association and so on. You stick these pooled values into a correlation matrix and then do an SEM on it to test whatever model you want to test. So, the meta-analysis part of the analysis informs the correlation matrix on which the resulting SEM is based. So, it's running an SEM on data that others have collected (and you have pooled together). This is called meta-analytic structural equation modeling (MASEM; Cheung & Chan, 2005, 2009). You can implement it with Mike Cheung's metaSEM package in R. Mike Cheung has some great resources here.

## References

- Cheung, M. W. L., & Chan, W. (2009). A two-stage approach to synthesizing covariance matrices in meta-analytic structural equation modeling. Structural Equation Modeling, 16, 28-53
- Cheung, M. W. L., & Chan, W. (2005). Meta-analytic structural equation modeling: A two-stage approach. Psychological Methods, 10, 40-64.