Certain problems may occur during model estimation. These problems are:
1. Boundary solutions may be encountered
2. One or more model parameters may not be identified
3. Local solutions may be encountered
1) Boundary solutions may be encountered
Occasionally, maximizing the likelihood function yields a boundary solution; that is, a solution in which certain multinomial probabilities, Poisson rates or error variances in normal models turn out to be zero, or may converge to zero. Such problems are prevented in Latent GOLD by the imposition of prior distributions on the parameters through the use of Bayes constants as a default technical parameter setting.
By default, the technical parameters for Bayes constants in Latent GOLD are set to alpha = 1, which causes alpha (= 1) artificial observations to be added to the data for the purpose of parameter estimation. Following the model estimation, the artificial observations are ‘subtracted’ from the data, so that the number of cases reported in the summary output does not reflect the additional alpha observations. The artificial observation(s) are generated from a conservative null model.
Strictly speaking, when a non-zero Bayes constant is used, rather than maximum likelihood estimation, the estimation procedure utilized is referred to as ‘posterior mode’ estimation and the modified log likelihood function that is maximized is called the ‘log-posterior’. For further information on this and the prior distributions used by Latent GOLD constants see section 7.3 of the Latent GOLD Technical Guide (pages 54-56).
In exercise #B1 (below) we will change the Latent GOLD setting for Bayes constants from the default of 1 to 0, to illustrate a situation where the resulting maximum likelihood solution is a boundary solution, and interpret these results.
2) Identification issues
If insufficient information is available to obtain unique maximum likelihood estimates for
one or more model parameters, such parameters are said to be ‘not identified’. (For a more formal definition of parameter identification, see section 7.10 of the Latent GOLD Technical Guide.
The Latent GOLD program provides estimation warning messages when it encounters boundary or unidentified parameters. In exercise #B2 (below), we will examine some of these situations. One issue to be aware of is that the additional information provided by a non-zero Bayes constant may in some cases cause unidentified parameters to become identified.
3) Local solutions
Because the likelihood function is not guaranteed to be concave, the estimation algorithm for maximizing it may sometimes yield a solution that provides a maximum only within a local range of parameter values rather than globally over all possible combinations of parameter values. Depending upon the particular starting value used for the parameter estimates in the estimation algorithm, the resulting solution may be local. The best way to prevent ending up with a local solution is to use multiple sets of randomly generated starting values. By default, the Latent GOLD program uses 10 sets of random start values. See section 7.8 of the Latent GOLD Technical Guide (pages 61-62) for further information on this topic.