Suppose there are n observations from the model , where X is
an n ×k matrix of covariate values (including the
intercept), y is
a vector of responses, and is a vector of
errors with survival distribution function S, cumulative
distribution function F, and probability density function f.
That is, ,
, and f(t)=dF(t)/dt,
where is a component of the error vector.
Then, if all the responses are observed,
the log likelihood, L, can be written as
If some of the responses are left, right, or
interval censored, the log likelihood can be written as
with the first sum over uncensored observations, the second sum
over right-censored observations, the third sum over left-censored
observations, the last sum over interval-censored observations, and
where zi is the lower end of a censoring interval.
If the response is specified in the binomial format,
events/trials, then the log-likelihood function is
where ri is the number of events and ni is
the number of trials for the ith observation.
In this case, .
For the symmetric distributions, logistic and normal,
this is the same as .
Additional information on censored and limited
dependent variable models can be found in
Kalbfleisch and Prentice (1980) and Maddala (1983).
The estimated covariance matrix of the parameter estimates is
computed as the negative inverse of I, which is
matrix of second derivatives of L with respect to
the parameters evaluated at the final parameter estimates.
If I is not positive definite, a positive
definite submatrix of I is inverted, and the
remaining rows and columns of the inverse are set to zero.
If some of the parameters, such as the scale and
intercept, are restricted, the corresponding elements
of the estimated covariance matrix are set to zero.
The standard error estimates for the parameter estimates are
taken as the square roots of the corresponding diagonal elements.
For restrictions placed on the intercept, scale,
and shape parameters, one-degree-of-freedom
Lagrange multiplier test statistics are computed.
These statistics are computed as
where g is the derivative of the log likelihood with respect
to the restricted parameter at the restricted maximum and
where the 1 subscripts refer to the restricted parameter
and the 2 subscripts refer to the unrestricted parameters.
The information matrix is evaluated at the restricted maximum.
These statistics are asymptotically distributed as
chi-squares with one degree of freedom under the null
hypothesis that the restrictions are valid, provided
that some regularity conditions are satisfied.
See Rao (1973, p. 418) for a more complete discussion.
It is possible for these statistics to be missing if the
observed information matrix is not positive definite.
Higher degree-of-freedom tests for multiple
restrictions are not currently computed.
V = I11 - I12I-122I21
A Lagrange multiplier test statistic
is computed to test this constraint.
Notice that this test statistic is comparable to the
Wald test statistic for testing that the scale is one.
The Wald statistic is the result of squaring the difference
of the estimate of the scale parameter from one and
dividing this by the square of its estimated standard error.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.