I have a model that estimates two separate LCA's (two 4-class models for two sets of behaviors comprised of ordinal and dichotmous indicators) and regresses the later class variable on the first. In addition, I include some covariates and test a covariate by latent class interaction.
To determine appropriate class sizes for the two LCA's I ran each separately, first, and used the adjusted BIC, LMR-LRT, and entropy as a guide (data is complex sample, so I did not use Tech14). Then I ran a combined model informed by these separate results.
The reviews were very positive, but one reviewer suggests the need to test the validity of the local independence assumption -- referencing Garrett & Zeger (2000) Log-Odds Ratio Check or evaluation of the bivariate residuals (Vermunt & Magidson, 2000).
My questions -- is this assumption necessary to test (wouldn't non-indepence be relaxing this assumption of the method); are the two recommended tests appropriate for my data type (i.e., ordinal categorical) and model; and are the readily implemented in Mplus (my sense from Karen Nylund's 2007 paper is "no").
With regard to Tech10 -- is there a particular standard I should use in evaluating local independence? With 6 variables (each having 2 to 4 levels), there are a number of standardized residual z-scores in excess of |1.96|. Does the presence of any significant residuals pose a problem for the local independence assumption, or is there a "rule of thumb" regarding how many (or what percent) are significant? Would a significant residual indicate that I need to have a direct relation between two given indicators in the model?
Also, is there a way to request (or assess) the expected information matrix to formally check model identification (another request from the reviewer)? I see the ratio of smallest to largest eigenvalue in my output (0.137E-05), would that indicate that none of my eigenvalues equal "0" -- and, thus, my model is identified?
The first thing I would do is try one more class and see if the significant standardized residuals are still significant. It may be you need one more class.
Significant standardized residuals indicate that the assumption of conditional independence is not met. There is no rule of thumb here as to how many are acceptable. Each residual covariance is one dimension of integration so adding more than 4 may not be feasible.
If the information matrix is singular, we give a message. The ratio of smallest to largest eigenvalue in your output (0.137E-05) indicates that your model is identified.
I had run additional models with more classes, but the Adjusted BIC, LMR Likelihood Ratio Test, and entropy did not suggest these models were appropriate. Significant bivariate residuals are still present (assuming that I'm interpreting the output correctly -- would the absolute value of the standardized residual (z-score) > 1.96 indicate significance?).
Do such findings invalidate the model results? In the case of conflicting evidence (i.e., if standardized residuals did decrease) -- which set of standards should guide model selection? Vermunt appears to indicate there is a trade-off between local independence and class size (i.e., by relaxing this assumption and allowing direct relations among indicators you may reduce classes). However, by allowing such relations you may also ignore potentially meaningful classes. I don't recall a clear statement as to how to weigh the decision.
Also, would one relax this assumption by modeling a covariance between indicators? In the current model, for example, I am modeling substance use classes based upon categorical indicators of frequency of use for various substances. Would I have to indicate that some of the substances (tobacco and alcohol use) are also indendently associated with one another apart from the class structure that I am reporting?
I am using the bivariate results in TECH10 to evaluate local independence of an IRT mixture model and would like to clarify the following:
1) How are the standardized residuals calculated?
2) Are the standardized residuals expected to follow a particular distribution (e.g., as in Yen's Q3 statistic which, under multivariate normality, is expected to be normally distributed)?
3) Is the chi-square for the bivariate association the same as the chi-square for local independence suggested by Chen (1997)?
4) How are the degrees of freedom for the chi-square statistics calculated? Are they simply (row-1) * (columns-1) (e.g., Df = 4 two variables with 3 ordinal categories each) or must the number of estimated parameters (thresholds) be taken into account (e.g., DF = 5 for two variables with three ordinal categories each)?
The standardized residuals given in tech10 are the standardized Pearson residuals. See Agresti's Categorical Data Analysis book, Sections 3.3.1 and 4.5.5. The original article on this topic is The Analysis of Residuals in Cross-Classified Tables, Shelby J. Haberman, Biometrics, Vol. 29, No. 1 (Mar., 1973), pp. 205-220. They are normally distributed z-scores.
For the bivariate tables the standardized residuals are computed by (O-E)/[sqrt(E)*sqrt(1-E/n)]. O and E are the Observed and Expected (model estimated) quantities for a pattern in the categorical data.
Rob Dvorak posted on Friday, November 27, 2009 - 8:55 am
Hi Drs. Muthen,
I was wondering if there is a way to evaluate the Condition Number for the Information Matrix. I have been told that > 0.xE-06 is a rule of thumb, but I'm wondering if there is a citation I am missing (perhaps in the Mplus manual that I've missed). In my analysis, mine is currently 0.133E-04.
I don't know about citations, although I assume the numerical analysis literature would have something on it. Depending on the algorithm, I think our epsilon limit for calling it singular, and most likely non-identified, is E-09 or E-10. I think different sized models with different sized parameter values can influence whether or not a small value should be seen as an indicator of non-identification. Then there is also the matter of which estimator of the information matrix one uses. Mplus works with MLF, ML, and MLR. MLF seems to be most sensitive to possible singularity/non-identification.
I am trying to evaluate local independence for a LCA with a four class solution. I have requested tech 10 from mplus, but, I wasnt sure what part of the output under the bivariate model information section to interpret. I see z scores for the different combinations of variables as well as two chi square tests.
Hi. I am doing a CFA with one factor and ordinal categorical outcome variables. I have used multiple imputation with wlsmv. I guess I can't get the bivariate correlation of the standardized residuals to evaluate local independence. Is there another way I can get this information?
Hi. I am doing a CFA with one factor and ordinal categorical outcome variables. In order to evaluate local independence, I am using Reeve's >.2 criterion. The ouput I get with the following syntax shows correlations across categories. Is there a way I can collpase this to just show the resdiual correlation averaged across the indicators?
This is the syntax I used:
TITLE: CFA safety tbi mlr with tech 10_CC DATA: FILE IS "C:\Users\kathy\Desktop\shepherd_safety_project\ ControlFIle_cg_cc3_3_2012.dat"; VARIABLE: NAMES ARE cc1 cc2 cc3 cc4 cc5 cc6 cc7 cc8 cc9 cc10 cc11 cc12 cc13 cc14 cc15 cc16 cc17 cc18 cc19 cc20 cc21; CATEGORICAL ARE cc1 cc2 cc3 cc4 cc5 cc6 cc7 cc8 cc9 cc10 cc11 cc12 cc13 cc14 cc15 cc16 cc17 cc18 cc19 cc20 cc21; MISSING ARE ALL (9); ANALYSIS: Estimator=mlr; Model: f BY cc1 cc2 cc3 cc4 cc5 cc6 cc7 cc8 cc9 cc10 cc11 cc12 cc13 cc14 cc15 cc16 cc17 cc18 cc19 cc20 cc21; f@1; OUTPUT: tech10 SAMPSTAT; STAND; RESIDUAL; PATTERNS; SAVEDATA: FILE IS COGCAP_03022012cfa.DAT; FORMAT IS F2.0;
Andy Daniel posted on Wednesday, December 12, 2012 - 6:59 am
I'm running a LCA with repeated measures of one nominal variable in a longitudinal dataset (6 categories in each wave). It is very plausible that the Local Independence Assumption isn't met in this case and that the measurement errors are associated.
I was wondering if there is a way to check the Local Independence Assumption in this model with mplus. Due to the fact that the variables are nominal TECH10 is not provided.
The following question would be if it is possible to model the assocation between the measurement errors to deal with the violation of the LI-Assumption.