Saddle Points

To compute the maximum likelihood estimates the log-likelihood function L is maximized with respect to all model parameters. To check that the maximization has been achieved two things have to be satisfied: 1. The vector of first derivatives with respect to the model parameter L′ should be equal to 0. 2. The negative of the matrix of the second derivatives −L′′ should be a positive definite matrix. Most maximization algorithms are based on following the direction of the derivative until maximization and thus continue iterating through the parameter spaces until condition 1 is met. At that point the maximization algorithm has no direction to move on and stops hoping that condition 2 is also satisfied. After the iterations are completed condition 2 is checked and if it is satisfied then Mplus will conclude that indeed the maximization is complete. If the ML estimator is used then the information matrix (−L′′)−1 is used as the estimator of the asymptotic variance covariance for the parameter estimates. If the MLF estimator is used the estimator for the asymptotic variance covariance is given by L′(L′)T where the bar symbol here means averaging over the independent units in the data (observations in single level models or clusters in multilevel models). If the MLR estimator is used the estimator for the asymptotic variance covariance is given by (−L′′)−1L′(L′)T (−L′′)−1. All three estimators are asymptotically equivalent when the model is correctly specified, however for small or medium sample size the MLF estimator may overestimate the standard errors. This is particularly the case if the ratio between the number of independent units