Can you include effect sizes in meta-analysis if the independent and dependent variable are the opposite way around to the other effect sizes? I am conducting a meta-analysis on the relationship between an exposure and outcome. I am wondering if it is ok to include effect sizes on this relationship which have been derived from models in which the exposure was modelled as the dependent variable (as opposed to independent variable) and the outcome modelled as the independent variable (as opposed to dependent variable). The other effect sizes included in the meta-analysis are all derived from models in which the exposure was the independent variable and the outcome was the dependent variable.
I assume that including these effect sizes would only be appropriate in instances in which the effect size was a correlation coefficient or standardised beta, because these effect sizes don't differ depending on whether the variable is modelled as independent or dependent. But other effect sizes won't be comparable?
Any advice would be greatly appreciated.
 A: So there are two types of studies:

*

*The variable that is considered the 'exposure' (child maltreatment) is dichotomized to create two groups (exposed vs not exposed) for which the mean levels of the 'outcome' variable (depression level) are reported and based on this a standardized mean difference can be computed.


*The variable that is considered the 'outcome' (depression) is dichtomized to create two groups (cases with depression versus controls without) for which the mean levels of the 'exposure' variable (child maltreatment severity) are reported and based on this a standardized mean difference can be computed.
Given that both variables are in principle continuous, I would compute the biserial correlation coefficient for both types of studies. The biserial correlation is an estimate of the underlying correlation between the two continuous variables based on data where one of two variables has been artificially dichotomized. And it is applicable regardless of which of the two variables has been dichotomized.
The nice thing is that you don't need anything beyond what you need to compute the standardized mean differences (means, SDs, and group sizes). Moreover, you could even combine these two types of studies with a third type of study:


*Neither the exposure nor the outcome are dichotomized and the authors are directly reporting the Pearson product-moment correlation between the two variables of interest (i.e., without any artificial dichotomization).

And there could even be a fourth type of study:


*Both the exposure and the outcome are artificially dichotomized and the authors are reporting the 2x2 table that is created based on the two dichotomized variables. In that case, you can compute a tetrachoric correlation coefficient, which again is an estimate of the underlying correlation between the two continuous variables.

See Jacobs and Viechtbauer (2017) for a paper that describes the computation of the biserial correlation coefficient and its sampling variance for meta-analytic purposes. Most relevant is the illustrative example that shows how to combine results from these different study types. The supplemental materials include the data and R code for the example.
If you have a sufficient number of studies, I would code the study type (i.e., which of the two variables was dichotomized vs neither dichotomized vs both dichotomized -- so 4 levels) and use this as part of a moderator analysis to examine whether there are differences in the strength of the association depending on the study type. While in theory (and under certain assumptions) all four cases provide an estimate of the correlation between the underlying continuous variables, studies that choose a particular design may differ from studies with other designs in other ways. If so, you should report separate estimates. If not, then you can combine the estimates across all types/designs.
Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180.
