Anonymous posted on Monday, August 22, 2005 - 10:03 am
hi Prof. Muthen
For two-level modeling with continuous dependent variables (only), unbalanced data and presence of missing data, Mplus allows ML, MLR, MLF for estimation in combination with integration or not. I can find from the tech. appendix the loglikelihood expression for balanced data. For unbalanced data, it says Mplus uses MUML. A few questions: 1) does it mean ML, MLR and MLF for unbalanced data are only still limited information method but the standard errors and chi-square test statistics are adjusted for unequal sample size differently? 2) from the expression (196) in the tech. appendix, there is no integration involved. when users specify: Algorithm=integration; (say in ex9.6) in mplus, what exactly the log-likelihood expression Mplus uses? When & Why we can choose do or not to do integration here? Which one give more reliable/better/robust results? 3) Any references you can recommend so I can read more or this.
Thanks a lot.
bmuthen posted on Monday, August 22, 2005 - 6:05 pm
1) ML, MLR, MLF give ML parameter estimates in both the balanced and unbalanced cases (they differ in terms of SEs, although all of them are correct). MUML gives ML parameter estimates only with balanced data (and no random slopes and no missingness).
2) Numerical integration is not needed with 2-level ML for continuous outcomes unless you have mixtures. It is needed for 2-level ML with categorical and other non-normal outcomes. In general, it is needed when the posterior distribution of the latent variables is not normal. If you use algo=int when it is not needed, you get the same results, but slower.
3) References related to numerical integration with ML and 2-level modeling include the Biometrics articles by Gibbons and Hedeker.