Iterative Algorithms for Model Fitting |
Two iterative maximum likelihood algorithms are available in PROC LOGISTIC. 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; however, the estimated covariance matrix of the parameter estimators can differ slightly. This is due to the fact that Fisher scoring is based on the expected information matrix while the Newton-Raphson method is based on the observed information matrix. In the case of a binary logit model, the observed and expected information matrices are identical, resulting in identical estimated covariance matrices for both algorithms. You can specify the TECHNIQUE= option to select a fitting algorithm, and specify the FIRTH option to perform a bias-reducing penalized maximum likelihood fit. Note for generalized logit models that only the Newton-Raphson technique is available. For conditional logistic regression, see the section Conditional Logistic Regression for a list of methods used.
Consider the multinomial variable such that
With denoting the probability that the jth observation has response value i, the expected value of is where . 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; in other words, . Let be the matrix of partial derivatives of with respect to . The estimating equation for the regression parameters is
where , and are the weight and frequency of the th observation, and is a generalized inverse of . PROC LOGISTIC chooses as the inverse of the diagonal matrix with as the diagonal.
With a starting value of , the maximum likelihood estimate of is obtained iteratively as
where , , and are evaluated at . The expression after the plus sign is the step size. If the likelihood evaluated at is less than that evaluated at , then is recomputed by step-halving or ridging as determined by the value of the RIDGING= option. The iterative scheme continues until convergence is obtained—that is, until is sufficiently close to . Then the maximum likelihood estimate of is .
The covariance matrix of is estimated by
where and are, respectively, and evaluated at . is the information matrix, or the negative expected Hessian matrix, evaluated at .
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 can be specified with the INEST= option.
For cumulative models, let the parameter vector be , and for the generalized logit model let . The gradient vector and the Hessian matrix are given, respectively, by
where is the log likelihood for the th observation. With a starting value of , the maximum likelihood estimate of is obtained iteratively until convergence is obtained:
where and are evaluated at . If the likelihood evaluated at is less than that evaluated at , then is recomputed by step-halving or ridging.
The covariance matrix of is estimated by
where the observed information matrix is computed by evaluating at .
Firth’s method is currently available only for binary logistic models. It replaces the usual score (gradient) equation
where is the number of parameters in the model, with the modified score equation
where the s are the th diagonal elements of the hat matrix and . The Hessian matrix is not modified by this penalty, and the optimization method is performed in the usual manner.