correlation
I have a list of records with fields that can be seen as two sets of variables: dependent and independent. There are say 100 fields in each record and we consider 10 of them as dependent variables and 90 as independent.
What are the methods to find all the correlations that exist between the two sets?
(At introductory statistics course that I've taken we have been dealing with simple correlations between two variables but the task like this was not covered.)
There's no clear answer. The idea is that if you have a correlation that approaches 1 then you essentially have one variable and not multiple variables. So you could test against the hypotheses that r=1.00. With that said, the idea of MANOVA is to give you something more than a series of ANOVA tests. It helps you find a relationship with one test because you're able to lower your mean square error when combining dependent variables. It just won't help if you have highly correlated dependent variables.
Simply do a t-test of the transformed correlations, exactly as you would test any two sets of data to compare their means. The test technically is a comparison of the mean transformed correlations, but for most purposes that's not a problem. (How meaningful would an arithmetic mean of correlations be in the first place? Arguably, the transformed correlation coefficients are the meaningful quantities!)
The whole point to the Fisher Z transformation $$\rho\to (\log(1+\rho)-\log(1-\rho))/2$$ is to make comparisons legitimate. When $n$ bivariate data are independently sampled from a near-bivariate Normal distribution with given correlation $\rho,$ the Fisher Z- transformed sample correlation coefficient will have close to a Normal distribution, with mean equal to the transformed value of $\rho$ and variance $1/(n-3)$--regardless of the value of $\rho.$ This is just what is needed to justify applying the Student t test (with equal variances in each group) or Analysis of Variance.
To demonstrate, I simulated samples of size $n=50$ from various bivariate Normal distributions having a range of correlations $\rho,$ repeating this $50,000$ times to obtain $50,000$ sample correlation coefficients for each $\rho$. To make these results comparable, I subtracted the Fisher Z transformation of $\rho$ from each transformed sample correlation coefficient, calling the result "$Z,$" so as to produce distributions that ought to be approximately Normal, all of zero mean, and all with the same standard deviation of $\sqrt{1/(50-3)} \approx 0.15.$ For comparison I have overplotted the density function of that Normal distribution on each histogram.
You can see that across this wide range of underlying correlations (as extreme as $-0.95$), the Fisher-transformed sample correlations indeed look like they have nearly Normal distributions, as promised.
For those who might be worried about extreme cases, I extended the simulations out to $\rho=0.9999$ (with $\rho=0$ shown as a reference at the left). The transformed distributions are still Normal and still have the promised variances:
Finally, the picture doesn't change much with small sample sizes. Here's the same simulation with samples of just $n=8$ bivariate Normal values:
A tiny bit of skewness towards less extreme values is apparent, and the standard deviations seem a little smaller than expected, but these variations are so small as to be of no concern.
Best Answer
I agree with Guillaume that a CCA could be useful. CCA is symmetric in the X and Y variables - so neither is presumed to be the cause of the other. If you truly believe that the "ten" are dependent on the "90", you could do a MANOVA. It's in R.
The MANOVA uses a principal components analysis to find the linear combination of the dependent variables, that is best explained by the independent variables.