Simulating multivariate ordinal data

[Leonardo Guizzetti]

This is a request to ask if there are any packages that implement an algorithm to simulate multivariate ordinal data with a specific rank (Spearman) correlation matrix? A cursory search didn't find anything.

The algorithm is described in Wicklin's book [1], but attributes the algorithms to [2,3].

If there is no such code then I will consider implementing it myself.

[1] Wicklin, R. (2013), Simulating Data with SAS, Cary, NC: SAS Institute Inc.

[2] Kaiser, S., Träger, D., and Leisch, F. (2011), Generating Correlated Ordinal Random Values, Technical report, University of Munich, Department of Statistics. URL http://epub.ub.uni-muenchen.de/12157/

[3] Demirtas, H. (2006), “A Method for Multivariate Ordinal Data Generation Given Marginal Distributions and Correlations,” Journal of Statistical Computation and Simulation, 76, 1017–1025.

[George Ford]

this might do it.

clear

matrix C = (1 , 0.5 \ 0.5 , 1)
corr2data x1 x2 , corr(C) n(1000)
xtile x1c = x1 , n(5)
xtile x2c = x2 , n(3)

corr x1 x2
polychoric x1c x2c

[Leonardo Guizzetti]

Thank you, George. I'm not familiar with polychoric correlation. It seems to be targeting an estimator of the Pearson correlation? I don't think this will work for my needs though, as I'm trying to generate ordinal variables with arbitrary distributions to follow a specified Spearman correlation distribution.

[George Ford]

maybe this then. seems to work with some different distributions, mainly because I'm converting to ranks (an order).

clear
set obs 1000
*g x1 = rnormal()
*g x2 = rnormal()
g x1 = rpoisson(10)
g x2 = rpoisson(5)
egen r1 = rank(x1)
egen r2 = rank(x2)
mkmat r1 r2 , mat(R) 
matrix C = (1 , 0.5 \ 0.5 , 1)
matrix Z = cholesky(C)
matrix y = R*Z'
svmat y
correl r1 r2
correl y1 y2

[George Ford]

Polychoric provides a correlation for categorical data (I was thinking categorical, not ordered as you indicated). Tetrachoric gets you the correlation between dummies.