This directory contains a number of files concerning a simple ordinal
missing data problem.

Data were simulated using sas under a bivariate normal distribution of two
variables that correlate .5, using the sas script sim1.sas

Both variables are assigned to categories according to the following
table:

Normal score	Assigned Category 
x<0 		0
0<x<.5		1
.5<x<1.282	2
x>1.282		3

The sas program attrits the data so that if the first variable is in
category 0 then the second variable is missing.  This is a case of missing
at random, in Little & Rubin terminology.  The large output file of
100,000 cases is written to simord.rec

The Mx script sim.mx is a straightforward estimation of the correlation
and the threshold parameters.  The output sim.mxo shows that the missing
data have not created biases in the parameters.  The thresholds and the
correlation are accurately estimated at close to the population
parameters.

The much improved script simchol.mx shows a slightly different approach in
which a cholesky decomposition of the covariance matrix is estimated, and
non-linear constraints are used to constrain the diagonal of this matrix
to equal 1.  In addition, the thresholds parameters in M are multiplied by
a lower triangular matrix of 1's which has the effect of making rows 2 and
onwards of M the *deviation* of threshold i+1 from threshold i.  Together
with the constraint that these elements are positive, in the Bound
statement, this script ensures that no 'illegal' cases of threshold i+1 <
threshold i will be encountered during optimization. 


Mike Neale
March 22 1999