Mplus input H1: (cf. ex. p. 150 of Topic 7 short course) VARIABLE: CLUSTER=centre WITHIN = x_1-x_6 CATEGORICAL = u_1-u_15; ANALYSIS: ESTIMATOR = MLR MODEL: %WITHIN% eta BY u_1-u_15@1; ! 1-PL: loadings=1 eta on x_1-x_6; %BETWEEN% etab BY u_1-u_15@1;
H0: no CLUSTER, WITHIN, %WITHIN% keyword, %BETWEEN% section.
1) Is the syntax correct?
2) Mplus output gives on level 2 the interceptís variance and the 15 thresholds, not the interceptís mean beta_0. Why?
Thank you very much for answering so quickly to my inquiry.
A. Regarding 2), I have followed the example of your short course, but it is unclear to me why the unique random intercept (beta_Oj) on level 2 is modeled as 15 thresholds (etab BY u1-u15). Since the measurement part is considered to be invariant across clusters, why should the thresholds (tau_m) be modeled on level 2? Curiously, under H1, the estimated 15 thresholds on level 2 are (almost) identical to the thresholds under H0, with an (almost) constant shift (0.363 to 0.367). Has the cluster-specific random intercept been somehow integrated into the cluster-invariant thresholds?
B. Regarding 1), yes, the results are consistent with what I expected, and suggest a clustering effect. Am I right to use a loglikelihood difference testing with the scaling correction factor as described in your technical appendix?
C. However, I get a warning (NON-POSITIVE DEFINITE FIRST-ORDER DERIVATIVE PRODUCT MATRIX... THE NONIDENTIFICATION IS MOST LIKELY DUE TO HAVING MORE PARAMETERS THAN THE NUMBER OF CLUSTERS...). There are only 9 clusters, totaling 800 individuals, and 22 parameters. A multiple group analysis with the KNOWNCLASS option ran into other numerical problems (message ONE OR MORE MULTINOMIAL LOGIT PARAMETERS WERE FIXED TO AVOID SINGULARITY OF THE INFORMATION MATRIX). How should I address this?
does not mean that the random intercept is modeled as 15 thresholds. It means that he covariation among the u1-u15 random intercepts is explained by etab. The etab parameter is its variance. The thresholds end up on level 2 in line with multilevel modeling where all means appear on level 2, not level 1. The thresholds are indeed cluster invariant. Means/thresholds are often not different in single- and two-level analyses.
B. This is a tricky topic - see
"Likelihood ratio tests in linear mixed models with one variance component" Crainiceanu-Ruppert (2004) JRSSB 66, Part 1, pp. 165-185.
To avoid this, I would simply report your two-level results.
C. 9 clusters is smaller than we recommend. At least 20 are typically needed for good SEs and variance estimation. You should also check that you don't have more between-level parameters than clusters. An alternative is to create 8 dummy variables and use these as covariates in a single-level analysis. So changing from random to fixed mode for the clusters.
Thank you very much for your prompt and helpful answer. It's really great to have such a support from both of you and your team!
A(2). This clears up the subject to me -> C(2a). Maybe it is a very basic question: what are then the free parameters at level 2? In the modeling equations of my initial post, the only unknown parameter at level 2 is the variance (+/- mean) of the cluster-specific intercept beta_0j. However, in the Mplus input, the level 2 parameters also include the 15 thresholds (and exceed nb of clusters). Was there a mistake in my equations?
B(2). Thank you for the reference. The technical details are a bit hard for me, but I got it that LRT may suffer a serious bias in this multilevel setting.
C(2b). Thank you for suggesting this alternative of having centre as a nominal dummy coded covariate --- well suited to our problem, since cross-group variability is on the intercept only. It works fine (results are similar to the ones I obtained in the multilevel approach). Am I right using loglikelihood difference testing in this setting to infer a centre effect?
A linked question: for this latent regression with binary outcomes, should ML or MLR estimation be preferred? If I use MLR, does this setting lend itself to the scaling correction factor technique?
Many thanks again for your much invaluable time and help.