Dr Rick To proceed i believe both x and y have to be transformed in a way that the new intervals are all finite. I can then use trapezoidal or Simpson’s rules to approximate the integrals. Could you suggest the transformations? Haider ... View more

To explain my understanding you (Dr Rick) suggested to split an interval over several sub-intervals particularly nine subintervals which will give rise to nine regions for integration. The methods to my knowledge which can do this are Trapezoidal and Simpson's rules.  But the latter requires even number of subintervals but we have nine, so we are left with Trapezoidal rule. As this requires finite intervals we can't directly apply this method without variable transformations which will lead to finite intervals. I have tried x=rsin(theta) and y=rcos(theta) which gives a Jacobian of r. But I can't find the domains for theta and r. If we assume 0<theta<2*pi and 0<r<infinity, how can we split each interval into three subintervals as required? suggestion from anyone will allow me to move forward. Or will any other transformation   be needed and what are those? Haider ... View more

dear Dr Rick thanks for sending the Olssen paper through. I have read the paper and can easily program in SAS both the ML methods for estimating polychoric correlation. I prefer the two-step method because of its computational ease. However, the paper doesn't discuss MLE for polychoric correlation in complex surveys (weighted polychoric correlation).  in this context i have been striving to develop SAS codes when you suggested the four steps in your last post. while i can contemplate on the nine regions for finding area under a curve, i am unable to proceed without an example. Please send me an article with examples and ways to code them in SAS. Looking forward to your guidance. Haider ... View more

i am stuck with 1. you suggested nine regions which I can't understand. an example showing integration over several regions would help me to understand. Both x and y each have 3 domains for integration according to the cutoffs you indicated. Haider ... View more

Hi Dr Rick my statement is invalid when i suggested using trapezoidal or Simpsons one-third rule as the domains include infinity. please give example of how probbnrm function can be used over the 9 regions you mentioned. don't know how 9 regions come up. In steps 1 and 2 for calculating polychoric correlation there are three domains for integrating over y and another three domains for integrating over x. This gives us 6 regions of integration in total.  please let me know. Haider ... View more

so monte carlo integration isn't needed. Do the nine regions you mention are related to numerical approximation using trapezoid or simpson's one-third rule?  it would help if you can suggest an example to look at to solve the problem.  Haider ... View more

Dear Dr Rick in continuation it seems the multiple integration problem has no analytical solution or numerical approximation because of infinite domains for integration.  i have limited background of multivariate calculus now. what i think is that monte carlo integration can be used while quasi monte carlo integration would require fewer assumptions and can be applied. kindly help with this. SAS has features for these. Haider  ... View more

so the 3 integrals for y are over  (-Infinity, -0.842),  (-0.842, 0.524), (0.524, Infinity) let's say x has the same distribution as y, then the 3 integrals for x should be the same. How can they be expressed by probbnrm where probbnrm(a,b,rho)=integration of bivariate normal over (-Infinity, b) - integration of bivariate normal over (-Infinity, a) rho should be fixed say at 0.5 Please suggest. Haider ... View more

correction: the other integral for y is over (-.842,.524) and the same for x. This gives us 6 integrals-3 for y and 3 for x. How can they be expressed in terms of probbnrm? Haider ... View more

Dear Dr. Rick, thanks. for the example you provided for cutoffs, I have simplified them to (-infinity,.524) and (-infinity,-.524). will this be of any help? I can't write this in terms of probbnrm function as these are the limits of y only. Lets say the cutoffs for x are the same for simplicity. We would have 4 integrals in total. How can they be expressed in terms of probbnrm? we are doing polychoric correlation. Haider ... View more

Correction I meant how did you find the limits (-1,0) and (0,0.5) for the probability integral of a bivariate normal density? Haider ... View more

Dear Dr Rick Thanks. Please explain how you got the limits (-0.5,0) and (0,1). The paper stated two ways or is it just one method. Kindly explain. Haider ... View more

Dear Dr Rick, thanks very much. I am trying to understand the outputs. Does the graph mean that the ML estimate of rho is aroundr -0.9 where the probability is maximum?  another point is that the regions of integration you have assigned are (-1,0) and (0,0.5). According to the paper i sent, shouldn't these inner and outer limits of integration be non-negative fractions? please let me know. Haider ... View more

Yes the formulas you mention from the article are the ones to be used. I have just explained the steps as I understood. Thanks ... View more

Dear Dr Rick Using the article formulas I want to determine the weighted polychoric correlation (rho). The first step is to find the joint probability function which is a double integral of a bivariate density of two standard normal (latent) variables over finite boundaries as given which have to be determined comparing the observed ordinal variables against sum of survey weights in my data. The next step is to use this result of double integral to my data to maximise the log likelihood function with respect to rho when loglikelihood is sum(weight*log(double integral)). In the lit search I found Simpsons one third rule being suggested to find the double integral of a bivariate normal density. And there are other ways.thanks. Haider ... View more