Fast multivariate normality test for large data sets in R I have a data set of about 260,000 observations of 50 variables. Although I highly suspect it's multivariate distribution to be non-normal, I still need a proof of it. I tried QuantPsyc package and it's mult.norm function but got a warning "Error: cannot allocate vector of size 794.5 Gb". I guess my data is too large for such tests. How can I test my data for multivariate normality or rather its absence?
 A: You can reduce the problem of refuting multivariate normality to the one-dimensional case. Just use the fact that a random vector $X\in\mathbb{R}^n$ is multivariate normal, if and only if $a^T X\in\mathbb{R}$ is a normal random variable for every vector $a\in\mathbb{R}^n$ (see first bullet of this section.).
Start with the margins, i.e. apply a standard univariate normality test to each $X_i=e_i ^TX.$ If all margins pass your test, either choose an $a$ according to some prior knowledge you have, or sample randomly uniform on the sphere.
If you want a proper test, i.e. with confidence level and such, you would need to account for multiple testing. And all caveats of hypothesis testing apply of course!
But I think actually much harder than performing tests, is thinking about what it is you like to achieve by those tests. Your data is not normal, this is clear, because it is discrete. Even more pertinent: If your data is real world data, even the underlying generating distribution will never be normal. It may be very close to normal. So how close to or far away from normal do you need for your application? And in what sense? There are many ways in which your data may deviate.
