st: What is the rationale of the covariance estimator using by Brant test?

[<kirin_guess@yahoo.com.tw>]

Dear all,
.
I would like to know the rationale of the asymptotic covariance matrix of 
the Brant test.
Consider that there is a 3-category dependent variable z. To do Brant test, 
we dichotomise z as two variable, zj, and zl. There two variables are 
separately modelled by two binary logit models, j and l. Let bj is the MLEs 
of the model j, and bj the MLEs of the model l. Brant presented an estimator 
for the covariance between bj and bl (X is the matrix of independent 
variables with a constant vector. Wjl = diag(πl-πjπl), where πj is the 
predicted probability of zj=1 estimated by the model j, and πl is Pr(yl=1) 
estimated by the model l.)
.
COV(bj,bl) = [(X’ Wjj X)^-1] [X’ Wjl X] [(X’ Wll X)^-1]
.
Where is this estimator derived from? I know that, if we use this estimator 
to compute cov(bj, bj), then the equation above shrinks back to
.
COV(bj,bj) = (X’ Wjj X)^-1
.
which is just the covariance matrix of binary logit model j. However, when 
we compute COV(bj,bl), the estimator looks like a “sandwich”:
.
COV(bj,bl) = COV(bj,bj) [X’ Wjl X] COV(bl,bl)
.
The first and the third term in the right-hand side are the covariance 
matrices of the model j and l respectively. But, my questions are: (1) where 
does the middle term in the right-hand side come from? (2) why should the 
three terms in the right-hand side be put together in that way?
.
I have compared the covariance computed by Brant’s estimator with that 
computed by the command “suest”. The results were different. So Brant’s 
estimator cannot be explained by the theory of the seemingly unrelated 
estimation. (If they are indeed theoretically different, then which 
estimator is better?) Moreover, I have compared the results computed by 
Brant’s estimator and by gologit2. They are also different. (Specifically, 
I restricted the coefficients of gologit2 to b1 and b2, and then compared 
the e(V) of the gologits with the covariance computed by Brant’s 
estimator.) So Brant’s estimator cannot be directly explained by the way 
that the generalized ordered logit estimates the covariance.
.
I’d appreciate it if someone can explain the rationale of Brant’s 
covariance estimator.
(PS: I know how to compute it.  I want to know how it comes from.) Thank you 
very much.
.
The paper of the Brant test is:
Brant, Rollin. 1990. "Assessing Proportionality in the Proportional Odds 
Model for Ordinal Logistic Regression." Biometrics 46(4): 1171-1178.
https://docs.google.com/open?id=0B984NoKuZv46WUdlZlZHbmliV2s
.
.
.
Chi-lin Tsai 

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