I'd like to aggregate statistical findings from heterogeneous studies that report regression models with different functional designs. All regressions have the same dependent variable (e.g. Y as financial growth rate), but they do not use the same set of independent variables. For instance, four studies report the following models: Y(x1,x2), Y(x1,x3), Y(x2,x4), and Y(x5). The other issue is that these four models don’t have a similar statistical structure; it’s a mix of linear, logistic, and non-linear regressions!
I believe that as far as those models contribute to understand a same problem (Y is the same across the models), I should be able to aggregate them with all of their independent variables (x1 to x5) into one general model; however, I have difficulty finding a statistical technique (or a meta-analysis or meta-regression method) that is capable of doing so.
Do you know of any related articles, meta-analysis cases, or statistical techniques? How common is this challenge?
Mohammad,
My immediate thought is to treat this is as a missing-by-design problem.
If -- and this is a big assumption -- you are willing to argue that all study samples are representative of the same population, you could combine samples and model Y regressed on X1 through X4, where X3 and X4 are missing for sample 1, X2 and X4 missing for sample 2, etc.
With the assumption, these data are MAR because they are missing by design -- you know the mechanism perfectly. Study 4, with just X5, might present problems as it has a disjoint predictor set with all of the others -- in other words, there is no covariation information between X5 and any other X.
If the assumption is too strong, you *might* be able to get somewhere with methods for non-random missingness, but I don't have enough experience with those methods to say much more than that.
Pat
What do you really want to demonstrate with your analysis ?
If you want to demonstrate that the different regressors are somewhat 'unified' by their ability to predict a given Y, you could use a partial correlation strategy demonstrating that partialling out the correlation between regressors by Y the correlation disappears, if you want to demonstarte the global equivalence between models you could compute the 'estimated Y's by models from 1 to n (you can call them Y1, Y2,..Yn) and make a PCA in the Y1-Yn looking at the emerging of one major component (they are all Y) and then looking ay minor 'shape' components mirroring the different models relative similarities in terms of loading patterns on shape components, this will give us an appreciation of the different kind of errors made by the different models and thus the similarities and differences of their 'views' on Y.
Thank you for your suggestions. Given the small sizes of study populations, my main motivation is to pool the results of all those available studies (regression models) and increase the power of detecting real effects in a larger population. What makes my case a little more challenging than typical meta-analysis projects is that I’m trying to aggregate all those models into one general model including all independent variables (not focusing on only one effect size). The hope is also to develop a more complete picture of important causal mechanisms and improve precision and accuracy of findings.
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