IBM® SPSS® Amos™ 28

The Convergence Statistic, labeled C.S. in the output, is computed as for a single scalar estimand.

SD is the estimated standard deviation of the posterior distribution, calculated as, where is the i-th retained observation on the estimand and N is the number of retained observations.

SE is an estimate of the standard error of obtained by the method of batch means. SE is a measure of the variability in that is attributable to the fact that N is finite. By default, 20 batches are used to estimate SE. To change the number of batches, click View → Options → MCMC.

The formula for C.S. is similar to one by Gelman, et al. (2013).

The Convergence Statistic is based on the idea that there is no point in taking a very large number of observations in an attempt to make the MCMC error (SE) very close to zero. Even if you obtained an infinite number of MCMC observations so that SE became zero, there would still be uncertainty in your knowledge of the parameter value as measured by SD -- the standard deviation of its posterior distribution. The convergence statistic is a measure of how much you could reduce your uncertainty about an estimand by increasing the number of MCMC observations to infinity. C.S. should be close to 1. Gelman et al give the following rule of thumb. "...'near' 1 depends on the problem at hand; for most examples, values below 1.1 are acceptable, but for a final analysis in a critical problem, a higher level of precision may be required." (p. 297)

Gelman et al add the caution, "In addition, even if an iterative simulation appears to converge and has passed all tests of convergence, it still may actually be far from convergence if important areas of the target distribution were not captured by the starting distribution and are not easily reachable by the simulation algorithm." (p. 297)

The global C.S. value (the value that affects whether a happy face is displayed) is the maximum of the C.S. values for the individual parameters. By default, a happy face is displayed when the convergence statistic for each model parameter is less than the threshold 1.002. The default convergence criterion of 1.002 was chosen as a result of experience showing that it is quite conservative. You can change the threshold to another value by clicking View → Options → MCMC and changing the Convergence criterion. The default value of 1.002 might be too stringent if the number of parameters is very large.