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| Level 2 random intercept in latent re... |
 
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Dear Mplus experts,
For the latent regression below, may I have some advice about random intercept usage?
A latent trait (eta) is measured by 15 binary responses (u_1-u_15). We test a centre effect on the latent trait, controlling for individual level covariates (x_1-x_6).
H0: SEM no clustering,
H1: centre has a random intercept in a 2-level model.
Measurement part = cluster-invariant 1-PL IRT. For item m, individual i, centre j:
P(u_{ijm} = 1) = F( eta_{ij} – tau_m )
tau_m=item threshold, F=logistic.
Structural part:
eta_ij = beta_0j + beta_1 x_{ij;1} + ... + beta_6 x_{ij;6} + epsilon_ij
H0: beta_0j = beta_0
H1: beta_0j = beta_0 + r_j
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 intercept’s variance and the 15 thresholds, not the intercept’s mean beta_0. Why?
Thanks very much for any help!
Mo |
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1. It looks correct. The best check is if you get the results you are looking for.
2. In a cross-sectional model, factor means are not identified when thresholds are free. |
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Dear Dr Muthen,
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? |
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A. Saying
etab BY u1-u15;
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. |
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Dear Professor Muthen,
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. |
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A(2). The between-level parameters are the between-level factor variance and the thresholds.
B(2). It is a tough topic. For an overview, you may also want to read Chapter 3 of the second edition of Joop Hox's Multilevel book.
C(2b). Center effect is simply seen as the Z test for each center dummy.
I don't think you see much difference between ML and MLR for this example, but I would use MLR and it works with the scaling correction. |
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