Bivariate probit regression
Message/Author
 James W. Shaw posted on Wednesday, November 12, 2008 - 3:57 pm
A student of mine is analyzing the association of weight loss (Y1) and dietary supplement use (Y2) using data from NHANES. She is estimating the following system of equations using bivariate probit regression:

Y1 = X1 + ... + Xk + e1
Y2 = X1 + ... + Xk + e2

The errors, e1 and e2, are presumed to be correlated. The correlation (rho) is estimated.

We hypothesize that the magnitude of rho is related to a categorical regressor (X1). To test this, we are fitting a separate model in each stratum defined by X1 and then testing for differences among the estimated rhos.

The above-described analyses are being performed using another statistical application. However, I believe we could use M-plus to do the work. That is, fit probit regressions to Y1 and Y2 using the WITH command to account for the correlation between the errors.

Would there be any way in M-plus to formulate a test as to whether the correlation between the regressions' errors varies with X1? It would be desirable to do this in a single estimation. However, a multi-group approach similar to what I describe above would also be reasonable.

I do not believe that we could use XWITH since Y1 and Y2 are not being regressed on one another.
 Bengt O. Muthen posted on Thursday, November 13, 2008 - 10:43 am
There are 2 alternatives, using ML or using WLSMV.

With ML you would use probit and capture the residual covariance by a factor that influences both y's (fix one loading and the factor mean and variance, free the other loading). The multiple-group approach would be handled via "Knownclass" using Type=Mixture (see UG).

With WLSMV you could specify a residual covariance using y1 WITH y2 and use multiple-group analysis.

Both approaches can use Model Test to get a Wald test of any assumed form for the size of the correlation as a function of x1 group membership. Neither approach, unfortunately can use the VARIABLE command "Constraint=x1;" option (because it is not available with ML in connection with numerical integration or with WLSMV).
 JEP posted on Wednesday, October 24, 2012 - 5:50 pm
Is there a way to compute predicted probabilities after running a bivariate probit regression? I found a formula on this page (http://www.ats.ucla.edu/stat/mplus/seminars/intromplus_cfa/default.htm#3.%20Confirmatory%20factor%20analysis%20with%20categorical%20outcomes) - scroll to after it presents R-square output - but I would prefer not to have to calculate them manually. Thank you.
 Bengt O. Muthen posted on Thursday, October 25, 2012 - 10:57 am
There is not an automatic way to compute them. But the formulas you see there can be expressed in terms of model parameters in Model Constraint.