Simulating a matrix of variables with predefined correlation structure

Question

For a simulation study I am working on, we are trying to test an algorithm that aims to identify specific culprit factors that predict a binary outcome of interest from a large mixture of possible exposures that are mostly unrelated to the outcome. To test this algorithm, I am trying to simulate the following data:

1. A binary dependent variable
2. A set of, say, 1000 variables, most binary and some continuous, that are not associated with the outcome (that is, are completely independent from the binary dependent variable, but that can still be correlated with one another).
3. A group of 10 or so binary variables which will be associated with the dependent variable. I will a-priori determine the magnitude of the correlation with the binary dependent variable, as well as their frequency in the data.

Generating a random set of binary variables is easy. But is there a way of doing this while ensuring that none of these variables are correlated with the dependent outcome?

Thank you!

Answer 1

"But is there a way of doing this while ensuring that none of these variables are correlated with the dependent outcome?"

With statistical sampling you can't ensure anything, you can only adjust the acceptable risk. Finding an acceptable level of risk may be harder than many people think.

Spurious correlations are a very real phenomenon. Real independent observations will often contain correlations, and if you want to actually test your algorithm to see how it will perform in reality then your tests should produce such phenomena in a manner similar to the real world—you should be generating independent candidate factors and allowing spurious correlations to occur.

If you are performing ~1000 independent tests of candidate factors, and you're targeting a risk level of α = 0.05, you can expect 50 non-significant terms to leak through into your analysis. To avoid this, you need to adjust your testing threshold using something along the lines of a Bonferroni correction. Recall that statistical discriminating power is based on standard error, which is inversely proportional to the square root of the sample size. Bonferroni says that 1000 simultaneous tests need their individual test threshold to be adjusted by a factor of 1000, which in turn means the sample size needs to be a million times larger than when performing a single test for significance.

So in summary I'd say that you shouldn't attempt to ensure lack of correlation, it's going to occur in the real world. You can mitigate the risk of non-predictive factors being included due to spurious correlation by generating massive amounts of data. In practice there will be non-predictors that leak through unless you can obtain enough data, so I'd suggest that your testing should address the rates of occurrence as a function of number of candidate factors and the sample size.