Hi Linda - My fuzzy memory is that earlier, Mplus gave this warning unless the LL was replicated at least to three decimals for mixture models with random starts: WARNING: THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED.
In any case, I'm getting the warning currently for LL values that differ in the integers, but if the LL is different only in the decimals, no warning appears. For example, the two LL -1413.455 and -1413.458 do not produce the warning, but the values -4478.303 and -4482.279 do produce the warning. This makes me wonder what the criterion is for worrying about when the LL is not replicated with random starts. Do only the integers in the LL matter?
The question extends to comparing the LL from the K-1 Model, to the LL estimated with the VLMR or BLRT for the K-1 model when running a K model. In many recent models, I'm not getting exactly the same LL to 3 decimals, but the LL are the same to the integers. So, how close is close enough!?!
And this probably means that the estimates will get better/closer with an increase in the number of integration points (up to a point) -- slowing things down of course as the points are increased. There's always a trade-off!
But, now the OCD part of me can relax about those decimals when the warning isn't given!