# Variance covariance matrix of regression coefficients with a probit link

Suppose we are performing ordinal regression using a probit link function. The data are doses (log transformed) and responses. The responses are ordinal and can be from 0 to 4. Suppose that some of the responses are not seen in the data (e.g. 1 and 3). The variance covariance matrix from R looks like:

                        0|2      2|4             log10(dataframe[, 1])
0|2                   3           4                     8
2|4                   57          59                    25.3434
log10(dataframe[, 1]) 37         36                     18


Note that there is no  0|1 ,  1|2 , or  2|3  since there were no ones and threes in the data set. I am interested in the variance of the intercept, the variance of the probit slope, and the covariance between the intercept and the probit slope.

In the above matrix, would these values be the $(1,1)$, $(3,3)$ and $(1,3)$ entries respectively?

An ordinal regression model has multiple intercepts describing the baseline frequency of each outcome. In your case, you only had values 0, 2, and 4, so you have two intercepts denoted 0|2 and 2|4. So your question about the variance of the intercept cannot be interpreted unless you tell the which intercept are you talking about. So element $(1,1)$ is the variance of the first intercept that separates 0 from above 0, and element $(2,2)$ is the variance of the second intercept which separates 2 and below from above 2.
There is only one slope per predictor, however, so its variance is element $(3,3)$ of the matrix.