Values from separate chains don't, so if you wanted it to look like one long chain, just concatenating them wouldn't look. The original article by gelman & rubin [1], as well as the bayesian data analysis textbook of gelman et al. Now i analyzed convergence 471 variables (parameters for each site).
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[2], recommends calculating the potential scale reduction factor (psrf). I have fit a bayesian sem using the r package blavaan. In convergence diagnosis in winbugs/jags/stan, there are different statistics reported for each variable.
Number of observations 58 number of missing patterns 1 statistic.
Note that individual chains have serial dependence; In jags with the runjags package,. I was going through the stan documentation which can be downloaded from here. I am using multivariate psrf statistics from gelman.diag() function to analyze my mcmc chains.
Arviz cites in which the authors discuss more robust psrf variants and also ess calculations. Common packages (like coda and boa in r) does not include a test for the multivariate version of the psrf, and in their paper, brooks and gelman (1998) do not include. In winbugs/stan, rhat ($\\hat{r}$) is reported. They also discuss a variant in case you're more interested in rare events;
