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The SURVEYLOGISTIC Procedure |
Two iterative maximum likelihood algorithms are available in PROC SURVEYLOGISTIC to obtain the pseudo-estimate of the model parameter
. The default is the Fisher scoring method, which is equivalent to fitting by iteratively reweighted least squares. The alternative algorithm is the Newton-Raphson method. Both algorithms give the same parameter estimates; the covariance matrix of
is estimated in the section Variance Estimation. For a generalized logit model, only the Newton-Raphson technique is available. You can use the TECHNIQUE= option in the MODEL statement to select a fitting algorithm.
Let be the response variable that takes values
. Let
index all observations and
be the value of response for the
th observation. Consider the multinomial variable
such that
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and . With
denoting the probability that the jth observation has response value i, the expected value of
is
, and
. The covariance matrix of
is
, which is the covariance matrix of a multinomial random variable for one trial with parameter vector
. Let
be the vector of regression parameters—for example,
for cumulative logit model. Let
be the matrix of partial derivatives of
with respect to
. The estimating equation for the regression parameters is
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where , and
and
are the WEIGHT and FREQ values of the
th observation.
With a starting value of , the pseudo-estimate of
is obtained iteratively as
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where ,
, and
are evaluated at the
th iteration
. The expression after the plus sign is the step size. If the log likelihood evaluated at
is less than that evaluated at
, then
is recomputed by step-halving or ridging. The iterative scheme continues until convergence is obtained—that is, until
is sufficiently close to
. Then the maximum likelihood estimate of
is
.
By default, starting values are zero for the slope parameters, and starting values are the observed cumulative logits (that is, logits of the observed cumulative proportions of response) for the intercept parameters. Alternatively, the starting values can be specified with the INEST= option in the PROC SURVEYLOGISTIC statement.
Let
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be the gradient vector and the Hessian matrix, where is the log likelihood for the
th observation. With a starting value of
, the pseudo-estimate
of
is obtained iteratively until convergence is obtained:
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where and
are evaluated at the
th iteration
. If the log likelihood evaluated at
is less than that evaluated at
, then
is recomputed by step-halving or ridging. The iterative scheme continues until convergence is obtained—that is, until
is sufficiently close to
. Then the maximum likelihood estimate of
is
.
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