# Different assumptions between independent normal and multivariate normal distributions

I have been dealing with multivariate meta analysis lately. In the literature the various authors say that the main assumption of the MVMA is that the different variables are Multivariately distributed. I understand that. However I don't understand how that assumption is stronger than the assumption that the different variables are independently normally distributed. Could somebody explain that?

• Marginal normal independence is simply a special case of multivariate normal with zero covariance. L Feb 21 '17 at 0:26

Basically, if you assume that your data is distributed as independent normals, then you also assume that the variables are uncorrelated. On another hand, if you assume that they are distributed as multivariate normal, then it also means that you consider that they may be some non-zero correlation between the variables (recall that multivariate normal is parametrized by covariance matrix $\Sigma$).

• To me it seems that the assumption of independence is a much stronger assumption than allowing the data to infer some correlation, if any. By that logic you would say that MVMA imposes less strong assumptions since it allows for more flexibility. Is that wrong?
– Nay
Feb 20 '17 at 21:08
• @GNik I don't want to argue with that since I do not really know what exactly you are referring to (you did not provide any reference or quote). Assuming independent normals may be seen as stronger assumption in the sense that you assume that the correlation is zero, while in the case of multivariate normal you do not assume that but allow flexibility, so your model may adapt to data. But rather discussing "strongness" or not, it is much more important to focus on the consequences (i.e. assuming no correlation and treating data as uncorrelated).
– Tim
Feb 20 '17 at 21:16
• Ok that makes sense. (I am working on the application of multivariate meta analysis in the context of Health Technology Assessments, where usually multiple outcomes are simultaneously meta analysed. Until recently people where ok with assuming independence but the truth is that allowing for correlation among outcomes can yield great advantages [increased precision due to the borrowing of strength]. I am not working on a specific project. Instead, just trying to understand at what costs [additional assumptions] do these advantages come.)
– Nay
Feb 20 '17 at 21:31
• @GNik the cost is estimating the covariance matrix. Independence is a common assumption, that makes things much easier in many cases (see also stats.stackexchange.com/questions/213464/… ).
– Tim
Feb 20 '17 at 21:41