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The LOGISTIC Procedure |
With pij denoting the probability
that the jth observation has response value i,
the expected value of Zj is
pj = (p1j, ... ,p(k+1)j)'. The
covariance matrix of
Zj is Vj, which
is the covariance matrix of a multinomial random variable
for one trial
with parameter vector pj.
Let be the
vector of regression parameters; in other words,
. And let
Dj be the matrix of partial derivatives of
pj with
respect to
.The estimating equation for the regression parameters is
where Wj = wj fj Vj-, wj and fj are the WEIGHT and FREQ values of the jth observation, and Vj- is a generalized inverse of Vj. PROC LOGISTIC chooses Vj- as the inverse of the diagonal matrix with pj as the diagonal.
With a starting value of , the maximum likelihood
estimate of
is obtained
iteratively as
The covariance matrix of
is estimated by
By default, starting values are zero for the slope parameters, and for the intercept parameters, starting values are the observed cumulative logits (that is, logits of the observed cumulative proportions of response). Alternatively, the starting values may be specified with the INEST= option.
With a starting value of , the maximum likelihood estimate
of
is obtained
iteratively until convergence is obtained:
The covariance matrix of
is estimated by
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