Investigating differential changes using Mixture Latent Change Scores (MLCS) modeling SESSION 2.5: Advanced Longitudinal Modeling Techniques Emil Coman1, Judith Fifield1, Jack McArdle2, Monique Davis-Smith3 1Ethel Donahue TRIPP Center, U. of Connecticut Health Center, 2 University of Southern California, 3 Medical Center of Central Georgia The 1st author thanks David Kenny for introducing him to the causal modeling world through the path analysis backdoor, for his constant mentoring and extensive generous discussions and advice, and to SEMNET mentors who so generously shared their time and expertise and taught Latent Variable modeling to many students of applied statistics across the globe for many years. Modern Modeling Methods Conference, Storrs, CT, May 20-21, 2014 Origin of structural models – Sewall Wright “Accounting for correlations” using the Method of Path Coefficients "Senior Animal Husbandman in Animal Genetics, Bureau of Animal Industry, United States Department of Agriculture“ Wright, S. (1921). Correlation and causation. Part I Method of path coefficients. Journal of agricultural research, 20(7), 557-585. [CITED BY 2119, GOOGLE.SCOLAR.COM ] MMM 2014 LCS Mixture 2 Some works on analyzing changes I should have read Kenny, D. A. (1975). A quasi-experimental approach to assessing treatment effects in the nonequivalent control group design. Psychological Bulletin, 82(3), 345. CITED BY 206, GOOGLE.SCOLAR.COM ] MMM 2014 LCS Mixture 3 Some works on analyzing changes I should have read Kenny, D. A., & Cohen, S. H. (1980). A reexamination of selection and growth processes in the nonequivalent control group design. Sociological methodology, 290-313. CITED BY 15, GOOGLE.SCOLAR.COM ] MMM 2014 LCS Mixture 4 Comparative Effectiveness of Treatments Main old issue: how to best compare changes. TX Comparative Effectiveness Outcome The common questions in CE are: 1. Are the 2 treatments benefiting patients equally? 2. Is one much better than the other? - statistically vs. clinically vs. patient-wise MMM 2014 LCS Mixture 5 Models of change MMM 2014 LCS Mixture 6 AR1 model σ2Y1= est. Y1 αY2 = est. Y2 β µY1 = est. αe @ 0 error σ2e= est. Notes: All un-labeled regression coefficients are equal to 1 (unity). MMM 2014 LCS Mixture 7 Models of change MMM 2014 LCS Mixture 8 LCS: model MMM 2014 LCS Mixture 9 LCS: Latent Change Score model σ2e= 0 Set=1 FG1 FG2 Set=1 Change depends Set=1 on initial state 1-to-2 FG Latent Change Inter-individual differences in intra-individual changes can depend on prior state, but also on other factors, like a constant change cumulative effect, similar to LGM slope. Notes: All un-labeled regression coefficients are equal to 1 (unity). MMM 2014 LCS Mixture 10 LCS: Latent Change Score model αe@0 e Y2 σ2Y1= est. αY2@0 Y2 Y1 µY1 = est. σ2e@0 γ = β-1 LCS(1->2) αLCS = est. σ2LCS= est. Notes: All un-labeled regression coefficients are equal to 1 (unity). URL for a brief visual history. MMM 2014 LCS Mixture 11 LGM: Latent Growth Model Intercept Y1 Y2 Y3 @0 @1 @2 Slope Notes: All un-labeled regression coefficients are equal to 1 (unity). MMM 2014 LCS Mixture 12 LCS: Latent Change Score model Y1 Y3 Y2 est. LCS21 est. LCS32 Notes: All un-labeled regression coefficients are equal to 1 (unity). MMM 2014 LCS Mixture 13 LCS model replicating LGM 0,0 Initial *,* level 0,0 0,0 @0 @0 Y1 0,= @0 * Y3 Y2 0,= @0 @0 LCS21 Constant Change LCS32 *,* Notes: All un-labeled regression coefficients are equal to 1 (unity); 1st digit is mean, second variance, if alone, covariance; hexagons are intercepts. MMM 2014 LCS Mixture 14 Mixture modeling (MM) – Jeff Harring MM is old. Karl Pearson showed in 1894 that if data are the result of a mixture of probability distributions (like two normal distributions with different means and variances), the resulting distribution will appear as one asymmetric and bimodal, when in fact it represents two homogenous normal distributions, or two subpopulations. Harring, J. R. (December 4, 5 & 6, 2013). Introduction to Finite Mixture Models. Details: http://www.cilvr.umd.edu/Workshops/CILVRworkshoppageFMM2013.html College Park, Maryland. Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 185, 71-110. MMM 2014 LCS Mixture 15 Mixture modeling (MM) – Jeff Harring The general mixture specification assumes a composite (mixed) distribution as the sum of K classes pdf’s (probability density functions) distributions, each with class-specific parameters θk, and weights or mixing proportions φi, which specify in fact the class proportions. The likelihood for one patient’s observations is then obtained as the probability of his/her data as a function of the parameters, and then a global likelihood function L is calculated as the product of individual patients’ likelihoods. Using Maximum Likelihood (ML) or Bayesian estimation, one can then obtain the parameter values that maximize the loglikelihood L. MM software use in practice is an Expectation-Maximization (EM) algorithm, which acts as an optimizer (not estimator) by generating starting values for the parameters θk and then, given posterior probabilities for φik, obtains new estimates for φik and θk, and continues in such steps until the change in the likelihood from successive iterations is sufficiently small. (Harring, 2013). Harring, J. R. (December 4, 5 & 6, 2013). Introduction to Finite Mixture Models. Details: http://www.cilvr.umd.edu/Workshops/CILVRworkshoppageFMM2013.html. College Park, Maryland. MMM 2014 LCS Mixture 16 Study background • A church and faith-based diabetes prevention intervention among African Americans and Blacks (AA&B), that implemented two versions of the DPP (Diabetes Prevention Program). • CBDPT-2 is a two-site (Macon GA and Hartford CT), church-randomized, controlled trial of N=570 prediabetics recruited from 42 African American and Black churches. • The preliminary CBDPT-2 data to date contains repeated measures collected over 4 waves, on FG and BMI, as well as data on physical activity, religious orientation, readiness to change, and the EQ-5D health status index Diabetes Prevention Program Research, G., 10-year follow-up of diabetes incidence and weight loss in the Diabetes Prevention Program Outcomes Study. The Lancet, 2009. 374(9702): p. 16771686. Boltri, J.M., et al., Diabetes prevention in a faith-based setting: results of translational research. Journal of Public Health Management and Practice, 2008. 14(1): p. 29-32. MMM 2014 LCS Mixture 17 Sample GMM syntax Mplus GMM : CLASSES = whohow (3); ANALYSIS: TYPE = MIXTURE; MODEL: %OVERALL% i s | FG_B@0 FG_6@1 FG_12@2 FG_24@4; %whohow#2% i ; s ; i with s; !can add also s on interv ; MMM 2014 LCS Mixture 18 Sample LCS syntax Mplus MODEL: !LCS model LFG1 by FG_B; LFG2 by FG_6 ; !setup true scores with placeholder LFG3 by FG_12 ; LFG4 by FG_12@0 ;! LFG5 by FG_24 ; FG_B-FG_24 (MeasErr); ! equal measurement errors LFG2 on LFG1@1; LFG3 on LFG2@1; LFG4 on LFG3@1; LFG5 on LFG4@1; dLX21 by LFG2@1; dLX32 by LFG3@1; !define LCS by the 2nd variable dLX43 by LFG4@1; dLX54 by LFG5@1; [LFG1-LFG5@0 ] ; !intercepts of latents set = 0 [dLX21-dLX54@0 ]; ! residual LCSs set =0 LFG1-LFG5@0; ! residual latents set =0 initial BY LFG1@1; !initial level linked to 1st latent only ctchange BY dLX21-dLX54@1; !constant change pointing to all LCS [initial] (MInit); [ctchange](AvCtCh); initial (VInit); ctchange (VCtCh) ;initial WITH ctchange (CvInCtCh); dLX21 on LFG1 (DLXonX); ! proportional changes dLX32 on LFG2 (DLXonX); dLX43 on LFG3 (DLXonX); dLX54 on LFG4 (DLXonX); DLX32 WITH DLX21 @0; !Unwated correlations DLX43 WITH DLX21 @0; etc. MMM 2014 LCS Mixture 19 LCS benefits LCS models are better equipped to explore dynamic inter-connected processes of mutual changes, e.g.: ILC0 LIX LIY γYX βX ΔLILC10 ILC1 γYX βX ILC2 ΔLILC21 ξYX ξXY γXY ΔLIUS10 γXY βY βY IUS0 ΔLIUS21 IUS1 IUS2 Notes: X=ILC, Y=IUS; only three waves are shown for simplicity; parameters represent: β - the proportional growth (dotted arrows); γ – coupling (interrupted double lines); ξ - changes-to-changes (double line arrows); the constant change slope factors feed into all their respective repeated measures (not shown for clarity); all unlabeled paths are set to unity. Coman, E., Lin, C. A., Suggs, L. S., Iordache, E., McArdle, J. J., & Barbour, R. Altering dynamic pathways to reduce substance use among youth: changes achieved by dynamic coupling. Addiction Research & Theory. doi: 10.3109/16066359.2014.892932 20 MMM 2014 LCS Mixture GMM vs. LCSM 3 class solutions GMM LCSM c1LCSM c2LCSM c3LCSM c1GMM 482 390 66 26 c2GMM 35 22 10 3 c3GMM 3 0 412 1 77 2 31 520 The 520 patients were classified like GMM classes (left gray shaded column) 482c1GMM, 35c2GMM, and 3c3GMM . LCSM finds 3 classes of patients (bottom green row) 412c1LCSM, 77c2LCSM and 31c3LCSM MMM 2014 LCS Mixture 21 GMM brief results Comparative Effectiveness of M (More) vs. L (Less) DPP in Fasting Glucose changes for 2 classes - GMM Lc1 GMM3 Mc1 GMM3 Lc2 GMM3 Mc2 GMM3 Single lines - Less DPP, double lines More DPP; interrupted lines Class 1 GMM, continuous lines Class 2 GMM means . Mc2 vs. Lc2 100.0 Mc1 vs. Lc1 Note: the third class had no cases with all four wave FC valid values. 70.0 Mc1 vs. Lc1: p = .048 (favors L; linear time-by-condition interaction effect) Mc2 vs. Lc2: p = .128, Quadratic p=.095. MMM 2014 LCS Mixture 22 LCSM brief results Comparative Effectiveness of M (More) vs. L (Less) DPP in Fasting Glucose changes for 3 classes - LCSM Mc1 vs. Lc1 Mc2 vs. Lc2 Lc1 LCSM3 Lc2 LCSM3 Lc3 LCSM3 Mc1 LCSM3 Mc2 LCSM3 Mc3 LCSM3 100.0 70.0 Mc3 vs. Lc3 Single lines - Less DPP, double lines More DPP (larger marker); similar patterns: means of LCSM classes 1 , 2 and 3 . Mc1 vs. Lc1: p = .900; Mc2 vs. Lc2: p = .033 (favors L; Quadratic time-bycondition p=.002); Mc3 vs. Lc3: p = .969 (Quadratic p = .013; Cubic p = .025). MMM 2014 LCS Mixture 23 LCS Mixture modeling 1. The Latent Change Score Mixture (LCSM) modeling promises more nuance and fine-tuning of classes of patients who are bound to develop differently, but following similar complex dynamic patterns of change. LCS mixture models allow for endless extensions and opportunities. 2. Practitioners and researchers can use early change information to predict class membership, and hence re-assign patients to more beneficial treatments. 3. The meaningfulness of classes needs explorations; reasons for adherence may be a factor. 1. Unique combinations of known moderators of effects are likely predicting classes. 2. Patient input in assigning meaning is key. MMM 2014 LCS Mixture 24 DONE! Thanks [email protected] MMM 2014 LCS Mixture 25
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