A recently posted issue requested the implementation of Mx1's
\allint functions, which perform integrations between thresholds on a multivariate normal distribution. This functionality is already being used in FIML Objectives if the data are ordinal (using code supplied by Alan Genz), so the back-end implementation is not difficult.
The implementation does raise some front-end questions, however. The Mx1 version of these functions use an augmented matrix structure (described below) as input. I'm interested in feedback about how it should work in OpenMx.
The Mx1 Implementation:
\mnor() calculates the integral between upper and lower limits on each variable (that is, the volume of a hypercube between the specified limits). For
\mnor() the input matrix is the vertical adhesion of:
\mnor() returns a 1x1 matrix containing the integral.
\allint() calculates the probabilities of all cells in a multivariate normal distribution that has been "sliced" by a varying number of hyperplanes along each dimension. For
\allint(), the input is:
\allint() in Mx1 returns a vector of values representing the integrals of each cell in the space, with the dimension represented by the rightmost covariance column cycling most rapidly.
I would rather avoid the use of augmented matrices and the use of special codes if there's an easier way. I'm inclined to say that our implementation of
\mnorm() could simply take a matrix and vectors for the upper and lower limits, and just use the special values for infinity (or use NA) as the limits when needed.
The catch is that avoiding augmented matrices means that each argument passed in is functionally different from the others. That is, the first argument is assumed to be a covariance matrix, the second a means vector, etc. We've avoided this kind of thing in OpenMx so far, but it seems like more Rish way of doing it than augmented matrices.
\allint provides a bigger problem, since there might be different numbers of thresholds for each column of the covariance matrix, so a matrix representation becomes less appropriate. I'd propose that
\allint() take a covariance matrix, a vector of means, and then a vector of thresholds for each column in the covariance matrix. While this makes conformance checking a little strange, it allows the user to specify any number of thresholds for a given column. What
\allint() outputs is also unclear, since the result should theoretically have as many dimensions as the distribution does. We'll have to flatten it, which might be ugly.
We've been keeping to the associated R functions so far--we could use the format of function
pmvnorm(lower, upper, mean, correlation/covariance) from the mvtnorm package to sub in for
\mnor(), but there's no associated function for
So, the questions: