# Simulate missing data for multivariate distribution?

(Following the private request from a more senior CV member, I am editing this question to make it more readable and comply with CV standards).

I am looking for a method to simulate multivariate, non-normal data with a pre-specified pattern of missingness while making sure that the missingness on does not invalidate my non-normality.

I thought about doing this as a two-step process: (1) Generate multivariate, non-normal data following the method described in Vale & Maurelli 1983 (the multivariate extension of Fleishman 1978). The Vale & Maurelli process allows me to specify the population values of skewness/kurtosis in all 1-dimensional marginals. (2) Once a sample is taken from said distribution with given population values of skewness/kurtosis on the one-dimensional marginals , generate some missing data pattern on the sample data.

I immediatley recognized that by inducing missingness, my non-normality could disappear (e.g. if I sample from a skewed distribution and then have missing data on the higher values, the skewness could be severely diminished or disappear all togehter). My attempt at a solution was to introduce a 3rd step where (3) Once the data were sampled and the missingness introduced, I would calculate the sample estimates of skewness/kurtosis and only retain those datasets for further analysis that fell within certain "acceptable" (<--- this would be set by me) range of skewness/kurtosis.

A professor commented on the fact that my step (3) would be invalid because #1 the definitions of skewness/kurtosis we have only exist for complete datasets and #2 if I only kept certain datasets and threw away others I could no longer claim I was sampling randomly from my distribution and I could only do this under MCAR (Missing Compeletely at Random) but not under MAR (Missing At Random) and NMAR (Not Missing At Random).

So I am stuck now and would very much appreciate any insights on anyone who could have some idea about how to implement said algorithm. I believe maybe going about with a different sampling scheme than the one I suggested?

• I would like to suggest that making a consistent distinction between estimates of skewness/kurtosis and their population values would clarify this question (and perhaps even answer it for you). It seems clear that this distinction completely resolves question (1), because missingness has no effect on the population moments. What I cannot figure out from your text is whether you are specifying skewness/kurtosis/correlation for the population or for the sample. (And if it's the sample, then why?) – whuber Jan 5 '14 at 16:27
• Sorry about that. The values of skewness/kurtosis are specified on the population. From said population (which has all the skewness/kurtosis of each one-dimensional marginal specified by me) I take a sample and then I induce a missing data pattern on the sample. – S. Punky Jan 5 '14 at 19:01