Shrinkage Estimator
Message/Author
 Max Nachbauer posted on Thursday, April 25, 2019 - 3:52 am
Dear Mplus-Team,

I am trying to understand the Mplus estimation procedures within multilevel modeling. Especially I am interested in dealing with level 2 sampling error, because in my study in most classes only a part of the students participated. To my understanding, a shrinkage estimator (or empirical bayes estimate) for estimating random intercepts and slopes (Raudenbush & Bryk 2002) should reduce the level 2 sampling error. On the other hand, Lüdtke et al (2008) state that latent modeling of cluster means is way to deal with level 2 sampling error. So I would like to raise two questions:

Q1: Are the Mplus multilevel estimates of random intercepts and slopes by default based on a shrinkage estimator?

Q2: Are the Mplus multilevel random intercepts and slopes by default modelled as latent variables? If yes: is this the Lüdtke approach or does the Lüdtke approach only apply to latent modelling of predictors?
 Tihomir Asparouhov posted on Friday, April 26, 2019 - 1:16 pm
Lüdtke approach applies to all of Mplus, i.e., all intercept and slopes are latent variables including intercepts for covariates.

Raudenbush & Bryk 2002 also use Lüdtke approach, i.e., all intercept and slopes are latent variables, except for the intercepts for the covariates, which they usually center with what we call observed centering (as in Mplus define command prior to model estimation).

The shrinkage estimator is not technically used during the model estimation in the way you are thinking of. It is used say in the EM algorithm or the MCMC in Bayes, however, not as a point estimate but as the mean of the posterior distribution, i.e., that is just the mean - the error of the estimate is still accounted for.

In the Mplus world the way we think of the shrinkage estimator is that this is your factor score for the latent variable (a point estimate for the cluster specific parameter).

One could say that instead of using the sample mean to center covariates you can use the shrinkage estimator. I don't think this will help much. That is because even with this estimator you are not accounting for the uncertainty of the mean - you would still be treating that as a fixed quantity rather than something that is measured with error and that is where all the problems with observed centering come from.
 Max Nachbauer posted on Thursday, May 02, 2019 - 4:19 am
Thank you Tihomir for these clarifications.

Let me raise one follow up question. There are two types of shrinkage estimation (Raudenbush & Bryk 2002): unconditional shrinkage estimaton applies to multilevel models with no level 2 predictors (shrinkage towards the grand mean), whereas conditional shrinkage estimation applies to multilevel models with level 2 predictors (shrinkage towards the predicted mean).

Is the principle of conditional shrinkage estimation also used within Mplus? That is, does the inclusion of level 2 predictors affect the estimation of random intercepts/slopes (of dependent variables)?
 Bengt O. Muthen posted on Thursday, May 02, 2019 - 11:25 am
Q1: Yes.

Q2: Yes, it does - as it should. The factor scores depend on the whole model.