1 1 Analysis Of Preventive Intervention Data Using Mixture Modeling In Mplus Bengt Muthén UCLA [email protected]Society for Prevention Research pre-conference workshop, Washington DC, May 29, 2007 2 • Inefficient dissemination of statistical methods: – Many good methods contributions from biostatistics, psychometrics, etc are underutilized in practice • Fragmented presentation of methods: – Technical descriptions in many different journals – Many different pieces of limited software • Mplus: Integration of methods in one framework – Easy to use: Simple, non-technical language, graphics – Powerful: General modeling capabilities Mplus Background • Mplus versions – V1: November 1998 – V3: March 2004 – V2: February 2001 – V4: February 2006 • Mplus team: Linda & Bengt Muthén, Thuy Nguyen, Tihomir Asparouhov, Michelle Conn
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Analysis Of Preventive Intervention DataUsing Mixture Modeling In Mplus
• Latent class models• Mixture models• Discrete-time survival models• Missing data models
Models That Use Latent Variables
Mplus integrates the statistical concepts captured by latent variables into a general modeling framework that includes not only all of the models listed above but also combinations and extensions of these models.
• Observed variablesx background variables (no model structure)y continuous and censored outcome variablesu categorical (dichotomous, ordinal, nominal) and
count outcome variables• Latent variables
f continuous variables– interactions among f’s
c categorical variables– multiple c’s
6
Mplus
Several programs in one
• Structural equation modeling
• Item response theory analysis
• Latent class analysis
• Latent transition analysis
• Survival analysis
• Multilevel analysis
• Complex survey data analysis
• Monte Carlo simulation
Fully integrated in the general latent variable framework
Further Studies• Mplus web site: www.statmodel.com
• Short courses
• Johns Hopkins University, August 20-22, 2007 (twice a year): Multilevel modeling
• University of Florence, Italy, September 10-12, 2007: Mixture, growth, and multilevel modeling
• Web videos of courses: 10 weeks, 2 days, 1 day, 2 hours
• Reference section
• Paper section (pdf’s)
• Free demo and Mplus User's Guide
• Mplus Discussion
• Syllabus, handouts and suggested readings from the UCLA course Statistical Methods for School-Based Intervention Studies, http://www.gseis.ucla.edu/faculty/muthen/courses_Ed255C.htm
Finding Subgroups By Mixture Modeling• Baseline data analysis: Latent class analysis
• Analysis of developmental trajectory classes in the absence of intervention– Latent transition analysis– Growth mixture analysis
• Analysis of developmental trajectory classes in the presence of intervention - for whom is an intervention effective?
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Latent Class Analysis
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13
c
x
inatt1 inatt2 hyper1 hyper21.0
0.9
0.80.70.60.50.40.30.20.1
Latent Class Analysis
inat
t1Class 2
Class 3
Class 4
Class 1
Item Probability
Item
inat
t2
hype
r1
hype
r2
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Latent Class Analysis (Continued)Introduced by Lazarsfeld & Henry, Goodman, Clogg, Dayton & Mcready
• Setting– Cross-sectional data– Multiple items measuring a construct– Hypothesized construct represented as latent class variable (categorical
latent variable
• Aim– Identify items that indicate classes well– Estimate class probabilities– Relate class probabilities to covariates– Classify individuals into classes (posterior probabilities)
• Applications– Diagnostic criteria for alcohol dependence. National sample, n = 8313– Antisocial behavior items measured in the NLSY. National sample,
n = 7326
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15
Latent Class Analysis ModelDichotomous (0/1) indicators u: u1, u2, … , ur
Categorical latent variable c: c = k ; k = 1, 2, … , K.
Marginal probability for item uj = 1,
Joint probability of all u’s, assuming conditional independence
P(u1, u2, … , ur) =
P(c = k) P(u1 | c = k) P(u2 | c = k) … P(ur | c = k)
Note analogies with the case of continuous outcomes and continuousfactors
ΣK
k = 1
ΣK
k = 1
u1 u2 u3
c
P(uj = 1) = P(c = k) P(uj = 1 | c = k).
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LCA Estimation
Posterior Probabilities:
P(c = k | u1, u2, … , ur) =
Maximum-likelihood estimation via the EM algorithm:c seen as missing data. EM: maximize E(complete-data log likelihood |ui1, ui2 ,…, uir) wrt parameters.
• E (Expectation) step: compute E(ci | ui1,ui2,…,uir) = posterior probability for each class and E(ci uij | ui1, ui2,…,uir) for each class and uj
• M (Maximization) step: estimate P(uj | ck) and P(ck) parameters by regression and summation over posterior probabilities
P(c = k) P(u1 | c = k) P(u2 | c = k)…P(ur | c = k)P(u1, u2, … , ur)
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Number of H0 parameters in the (exploratory) LCA model with Kclasses and r binary u’s: K – 1 + K × r (H1 has 2r – 1 parameters).
H1 H0• 2 classes, 3 u: df = 0 computed as (8 – 1) – (1 + 6)• 2 classes, 4 u: df = 6 computed as (16 – 1) – (1 + 8)• 3 classes, 4 u: df = 1, but not identified because of 1 indeterminacy• 3 classes, 5 u: df = 14 computed as (32 – 1) – (2 + 15)
Confirmatory LCA modeling applies restrictions to the parameters.
Logit versus Probability Scale. The u-c relation is a logit regression (binary u),
P(u = 1 | c) = , (81)
Logit = log [P/(1 – P)]. (82) For example:Logit = 0: P = 0.5Logit = -1: P = 0.27Logit = 1: P = 0.73
LCA Parameters
11 + exp (–Logit)
Logit = -3: P = 0.05Logit = -10: P = 0.00005
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• Model fit to frequency tables. Overall test against data– When the model contains only u, summing over the cells,
χP = , (82)
χLR = 2 oi log oi / ei . (83)
LCA Testing Against Data
A cell that has non-zero observed frequency and expectedfrequency less than .01 is not included in the χ2 computation asthe default. With missing data on u, the EM algorithmdescribed in Little and Rubin (1987; chapter 9.3, pp. 181-185)is used to compute the estimated frequencies in the unrestrictedmultinomial model. In this case, a test of MCAR for theunrestricted model is also provided (Little & Rubin, 1987, pp.192-193).
• Model fit to univariate and bivariate frequency tables. MplusTECH10
Σi
2 (oi – ei)2
ei
Σi
2
10
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Latent Class AnalysisAlcohol Dependence Criteria, NLSY 1989 (n = 8313)
The multinomial logistic regression model expresses the probabilitythat individual i falls in class k of the latent class variable c as afunction of the covariate x,
P(ci = k | xi) = , (90)
where ακ = 0, γκ = 0 so that = 1.
This implies that the log odds comparing class k to the last class K is
log[P(ci = k | xi)/P(ci = K | xi)] = αk + γk xi . (91)
Multinomial Logistic Regression Of c ON x
ΣKs=1 eαs + γs xi
αk + γk xie
αK + γK xie
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27
White Males White Females
Black Males Black Females
CLASS 1CLASS 2
CLASS 3CLASS 4
CLASS 1CLASS 2
CLASS 3CLASS 4
CLASS 1CLASS 2
CLASS 3CLASS 4
CLASS 1CLASS 2
CLASS 3CLASS 4
Pro
b
0.016 17 18 19 20 21 22 23AGE
Pro
b
16 17 18 19 20 21 22 23AGE
Pro
b
16 17 18 19 20 21 22 23AGE
Pro
b
16 17 18 19 20 21 22 23AGE
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0.1
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1.0
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ASB Classes Regressed On Age,Male, Black In The NLSY (n=7326)
28
Clogg, C.C. (1995). Latent class models. In G. Arminger, C.C. Clogg& M.E. Sobel (eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 311-359). New York: Plenum Press.
Goodman, L.A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215-231.
Hagenaars, J.A & McCutcheon, A. (2002). Applied latent class analysis. Cambridge: Cambridge University Press.
Nestadt, G., Hanfelt, J., Liang, K.Y., Lamacz, M., Wolyniec, P., & Pulver, A.E. (1994). An evaluation of the structure of schizophrenia spectrum personality disorders. Journal of Personality Disorders, 8, 288-298.
Rindskopf, D., & Rindskopf, W. (1986). The value of latent class analysis in medical diagnosis. Statistics in Medicine, 5, 21-27.
Uebersax, J.S., & Grove, W.M. (1990). Latent class analysis of diagnostic agreement. Statistics in Medicine, 9, 559-572.
Input For FMA Of 9 Alcohol ItemsIn The NLSY 1989 (Continued)
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35
Latent Transition Analysis
36
• Setting– Cross-sectional or longitudinal data– Multiple items measuring several different constructs– Hypothesized simple structure for measurements– Hypothesized constructs represented as latent class variables
(categorical latent variables)
Latent Transition Analysis
• Aim– Identify items that indicate classes well– Test simple measurement structure– Study relationships between latent class variables– Estimate class probabilities– Relate class probabilities to covariates– Classify individuals into classes (posterior probabilities)
• Application– Latent transition analysis with four latent class indicators at two
time points and a covariate
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Transition Probabilities Time Point 1 Time Point 2
0.8 0.2
0.4 0.6
1 2c2
2
c11
Latent Transition Analysis
u11 u12 u13 u14 u21 u22 u23 u24
c1 c2
x
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TYPE = MIXTURE;ANALYSIS:
%OVERALL%c2#1 ON c1#1 x;c1#1 ON x;
MODEL:
NAMES ARE u11-u14 u21-u24 x xc1 xc2;
USEV = u11-u14 u21-u24 x;
CATEGORICAL = u11-u24;
CLASSES = c1(2) c2(2);
VARIABLE:
FILE = mc2tx.dat;DATA:
Latent transition analysis for two time points and a covariate
TITLE:
Input For LTA WithTwo Time Points And A Covariate
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Input For LTA WithTwo Time Points And A Covariate (Continued)
MODEL c1:%c1#1%[u11$1-u14$1] (1-4);%c1#2%[u11$1-u14$1] (5-8);
MODEL c2:%c2#1%[u21$1-u24$1] (1-4);%c2#2%[u21$1-u24$1] (5-8);
OUTPUT: TECH1 TECH8;
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Tests Of Model Fit
LoglikelihoodH0 Value -3926.187
Information CriteriaNumber of Free Parameters 13Akaike (AIC) 7878.374Bayesian (BIC) 7942.175Sample-Size Adjusted BIC 7900.886
(n* = (n + 2) / 24)Entropy 0.902
Output Excerpts LTA WithTwo Time Points And A Covariate
21
41
Chi-Square Test of Model Fit for the Latent Class Indicator Model Part
Pearson Chi-Square
Value 250.298Degrees of Freedom 244P-Value 0.3772
Likelihood Ratio Chi-Square
Value 240.811Degrees of Freedom 244P-Value 0.5457
Final Class CountsFINAL CLASS COUNTS AND PROPORTIONS OF TOTAL SAMPLE SIZE BASEDON ESTIMATED POSTERIOR PROBABILITIES
Output Excerpts LTA WithTwo Time Points And A Covariate (Continued)
Estimates S.E. Est./S.E.
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Chung, H., Park, Y., & Lanza, S.T. (2005). Latent transition analysis with covariates: pubertal timing and substance use behaviors in adolescent females. Statistics in Medicine, 24, 2895 - 2910.
Collins, L.M. & Wugalter, S.E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 27, 131-157.
Collins, L.M., Graham, J.W., Rousculp, S.S., & Hansen, W.B. (1997). Heavy caffeine use and the beginning of the substance use onset process: An illustration of latent transition analysis. In K. Bryant, M. Windle, & S. West (Eds.), The science of prevention: Methodological advances from alcohol and substance use research. Washington DC: American Psychological Association. pp. 79-99.
Kaplan, D. (2006). An overview of Markov chain methods for the study of stage-sequential developmental processes. Submitted for publication.
Nylund, K.L., Muthén, B., Nishina, A., Bellmore, A. & Graham, S. (2006). Stability and instability of peer victimization during middle school: Using latent transition analysis with covariates, distal outcomes, and modeling extensions. Submitted for publication.
ANALYSIS: TYPE = MIXTURE MISSING;PROCESS = 2;STARTS = 100 20;
MODEL:%OVERALL%c2#1 ON c1#1@0
cg#1 (p0);[c2#1] (p1);
MODEL cg:%cg#1%c2#1 ON c1#1 (p2);%cg#2%c2#1 ON c1#1 (p3);
MODEL c1:%c1#1%[u11$1-u15$1*1] (1-5);%c1#2%[u11$1-u15$1*-1] (11-15);
Input For LTA With An Intervention
50
MODEL c2:%c2#1%[u21$1-u25$1*1] (1-5);
%c2#2%[u21$1-u25$1*-1] (11-15);
MODEL CONSTRAINT:NEW(p011 p012 p021 p022 p111 p112 p121 p122 lowlow highlow);!p0*, p1* contain probabilities for the 4 cells for control and !intervention groups !lowlow is the probability effect of intervention on staying in !the low class!highlow is the probability effect of intervention on moving from !high to low class!the effect is calculated as P(intervention)-P(control)p011 = exp(p0+p1+p2)/(exp(p0+p1+p2)+1);p012 = 1/(exp(p0+p1+p2)+1);p021 = exp(p0+p1)/(exp(p0+p1)+1);p022 = 1/(exp(p0+p1)+1);p111 = exp(p1+p3)/(exp(p1+p3)+1);p112 = 1/(exp(p1+p3)+1);p121 = exp(p1)/(exp(p1)+1);p122 = 1/(exp(p1)+1);lowlow = p111-p011;highlow = p121-p021;
Schizophrenia DataGrowth Model for Binary OutcomesWith a Treatment Variable and Scaling Factors
TITLE:
Alternative language:
i BY illness1-illness6@1;s BY illness1@0 illness2@1 illness4@3 illness6@5;[illness1$1 illness2$1 illness4$1 illness6$1] (1);[s];i s ON drug;!{illness1@1 illness2-illness6};
MODEL:
64
Tests Of Model Fit
LoglikelihoodHO Value -486.337
Information CriteriaNumber of Free Parameters 7Akaike (AIC) 986.674Bayesian (BIC) 1012.898Sample-Size Adjusted BIC 990.696
(n* = (n + 2) / 24)
Output Excerpts Schizophrenia Data Growth ModelFor Binary Outcomes With A Treatment Variable
Gibbons, R.D. & Hedeker, D. (1997). Random effects probit and logistic regression models for three-level data. Biometrics, 53, 1527-1537.
Hedeker, D. & Gibbons, R.D. (1994). A random-effects ordinal regression model for multilevel analysis. Biometrics, 50, 933-944.
Muthén, B. (1996). Growth modeling with binary responses. In A. V. Eye, & C. Clogg (Eds.), Categorical variables in developmental research: methods of analysis (pp. 37-54). San Diego, CA: Academic Press. (#64)
Muthén, B. & Asparouhov, T. (2002). Latent variable analysis with categorical outcomes: Multiple-group and growth modeling in Mplus. Mplus Web Note #4 (www.statmodel.com).
Further Readings On Growth Analysis With Categorical Outcomes
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iu
iy sy
su
maleblackhispesfh123hsdrpcoll
qy
qu
y18 y19 y20 y24 y25
u18 u19 u20 u24 u25
neve
r
once
2 or
3 ti
mes
4 or
5 ti
mes
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7 ti
mes
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9 ti
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10 o
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es
HD83
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Two-Part Growth Modeling:Frequency Of Heavy Drinking Ages 18 – 25
• Latent classes for binary and continuous parts may be incorrectly picked up as additional factors in conventionalanalysis
• Multilevel
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Growth Mixture Modeling
36
71
Individual Development Over Time
y1
w
y2 y3 y4
η0 η1
ε1 ε2 ε3 ε4
t = 1 t = 2 t = 3 t = 4
(1) yti = η0i + η1i xt + εti
(2a) η0i = α0 + γ0 wi + ζ0i
(2b) η1i = α1 + γ1 wi + ζ1i
i = 1i = 2
i = 3
y
x
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Mixtures And Latent Trajectory Classes
Modeling motivated by substantive theories of:
• Multiple Disease Processes: Prostate cancer (Pearson et al.)
• Multiple Pathways of Development: Adolescent-limited versus life-course persistent antisocial behavior (Moffitt), crime curves (Nagin), alcohol development (Zucker, Schulenberg)
• Subtypes: Subtypes of alcoholism (Cloninger, Zucker)
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73
Example: Mixed-Effects Regression Models ForStudying The Natural History Of Prostate Disease
Source: Pearson, Morrell, Landis and Carter (1994), Statistics in Medicine
MIXED-EFFECT REGRESSION MODELS
Years Before Diagnosis Years Before Diagnosis
Figure 2. Longitudinal PSA curves estimated from the linear mixed-effects model for the group average (thick solid line) and for each individual in the study (thin solid lines)
Poor Development: 20% Moderate Development: 28% Good Development: 52%
69% 8% 1%Dropout:
7 8 9 10
4060
8010
0
Grades 7-107 8 9 10
4060
8010
0
Grades 7-107 8 9 10
4060
8010
0
Grades 7-10
Mplus Graphics For LSAY MathAchievement Trajectory Classes
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Female
Hispanic
Black
Mother’s Ed.
Home Res.
Expectations
Drop Thoughts
Arrested
Expelled
Math7 Math8 Math9 Math10
High SchoolDropout
i s
c
LSAY Math Achievement Trajectory Classes
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Growth Mixture Modeling Of Developmental Pathways
Outcome
Escalating
Early Onset
Normative
Agex
i
u
s
c
y1 y2 y3 y4
q
18 37
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• New setting:
– Sequential, linked processes
• New aims:
– Using an earlier process to predict a later process– Early prediction of failing class
Application: General growth mixture modeling of first- andsecond-grade reading skills and their Kindergarten precursors;prediction of reading failure (Muthén, Khoo, Francis, Boscardin,1999). Suburban sample, n = 410.
General Growth Mixture Modeling With Sequential Processes
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Assessment Of Reading Skills Development
• Longitudinal multiple-cohort design involving approximately 1000 children with measurements taken four times a year from Kindergarten through grade two (October, December, February, April)
2), letters/names/sounds (Kindergarten only), rapid naming• Standardized reading comprehension tests at the end of Grade 1 and
Grade 2 (May).Three research hypotheses (EARS study; Francis, 1996):• Kindergarten children will differ in their growth and development in
precursor skills• The rate of development of the precursor skills will relate to the rate of
development and the level of attainment of reading and spelling skills – and the individual growth rates in reading and spelling skills will predict performance on standardized tests of reading and spelling
• The use of growth rates for skills and precursors will allow for earlier identification of children at risk for poor academic outcomes and lead to more stable predictions regarding future academic performance
NAMES ARE gender eth wc pa1-pa4 wr1-wr8 l1-l4 s1 r1 s2 r2 rnaming1 rnaming2 rnaming3 rnaming4;USEVAR = pa1-wr8 rnaming4;MISSING ARE ALL (999);CLASSES = c(5);
VARIABLE:
TECH8;OUTPUT:
Growth mixture model for reading skills developmentTITLE:
Input For Growth Mixture ModelFor Reading Skills Development
84
Five Classes Of Reading Skills Development
21
0-1
-2-3
1 2 3 4 5 6 7 8 9 10 11 12
21
0-1
-2-3
Time Point Time Point
Kindergarten Growth (Five Classes)
Phonemic Awareness
Grades 1 and 2 Growth (Five Classes)
Word Recognition
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Muthén, B. (2001). Second-generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class/latent growth modeling. In Collins, L.M. & Sayer, A. (Eds.), New methods for the analysis of change (pp. 291-322). Washington, D.C.: APA. (#82)
Muthén, B. (2001). Latent variable mixture modeling. In G. A. Marcoulides & R. E. Schumacker (eds.), New developments and techniques in structural equation modeling (pp. 1-33). Lawrence Erlbaum Associates. (#86)
Muthén, B. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81-117. (#96)
Muthén, B. (2004). Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data. In D. Kaplan (ed.), Handbook of quantitative methodology for the social sciences (pp. 345-368). Newbury Park, CA: Sage Publications. (#100)
Further Readings On Growth Mixture Modeling
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Further Readings On Growth Mixture Modeling (Continued)
Muthén, B. & Asparouhov, T. (2006). Growth mixture analysis: Models with non-Gaussian random effects. Forthcoming in Fitzmaurice, G., Davidian, M., Verbeke, G. & Molenberghs, G. (eds.), Advances in Longitudinal Data Analysis. Chapman & Hall/CRC Press.
Muthén, B. & Muthén, L. (2000). Integrating person-centered and variable-centered analysis: growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882-891. (#85)
Muthén, B. & Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463-469. (#78)
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Different treatment effects in different trajectory classes
Muthén, B., Brown, C.H., Masyn, K., Jo, B., Khoo, S.T., Yang, C.C.,Wang, C.P. Kellam, S., Carlin, J., & Liao, J. (2002). General growthmixture modeling for randomized preventive interventions. Biostatistics,3, 459-475.
Growth Mixtures In Randomized Trials
See also Muthen & Curran, 1997 for monotonic treatment effects
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y2 y3 y4 y5y1 y6 y7
i s q
Txc
ANCOVA Growth Mixture Model
y1 y7
Tx
Modeling Treatment Effects
• GMM: treatment changes trajectory shape
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89Figure 1. Path Diagrams for Models 1 - 3
y
η
Ι
c
Model 1
y
η
Ι
c
u
Model 2
y1
η1
Ι
c1
Model 3
y2
η2
c2
90Figure 6. Estimated Mean Growth Curves and Observed Trajectories for
4-Class model 1 by Class and Intervention Status
High Class, Control Group
Grades 1-7
TOC
A-R
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1F 1S 2F 2S 3S 4S 5S 6S 7S
High Class, Intervention Group
Grades 1-7
TOC
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1F 1S 2F 2S 3S 4S 5S 6S 7S
Medium Class, Control Group
Grades 1-7
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A-R
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1F 1S 2F 2S 3S 4S 5S 6S 7S
Medium Class, Intervention Group
Grades 1-7
TOC
A-R
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1F 1S 2F 2S 3S 4S 5S 6S 7S
Low Class, Control Group
Grades 1-7
TOC
A-R
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12
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1F 1S 2F 2S 3S 4S 5S 6S 7S
Low Class, Intervention Group
Grades 1-7
TOC
A-R
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56
12
34
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1F 1S 2F 2S 3S 4S 5S 6S 7S
LS Class, Control Group
Grades 1-7
TOC
A-R
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34
56
12
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1F 1S 2F 2S 3S 4S 5S 6S 7S
LS Class, Intervention Group
Grades 1-7
TOC
A-R
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12
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1F 1S 2F 2S 3S 4S 5S 6S 7S
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TOC
A-R
3-Class Model 1
1F 1S 2F 2S 3S 4S 5S 6S 7SGrades 1 - 7
High Class, 15%Medium Class, 44%Low Class, 19%
ControlIntervention
LS Class, 22%
BIC=3394Entropy=0.80
4-Class Model 1
TOC
A-R
1F 1S 2F 2S 3S 4S 5S 6S 7SGrades 1 - 7
High Class, 14%Medium Class, 50%Low Class, 36%ControlIntervention
BIC=3421Entropy=0.83
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3-Class Model Estimated Mean Class TrajectoriesAnd Posterior Probability Weighted Means, Control Group
1F 1S 2F 2S 3S 4S 5S 6S 7SGrades 1 - 7
Class 1 = 14%Class 2 = 50%Class 3 = 36%
TOC
A-R
TOC
A-R
1F 1S 2F 2S 3S 4S 5S 6S 7SGrades 1 - 7
Class 1 = 14%Class 2 = 50%Class 3 = 36%
3-Class Model Estimated Mean Class TrajectoriesAnd Posterior Probability Weighted Means, Intervention Group
NAMES ARE sctaa11f sctaa11s sctaa12f sctaa12s sctaa13ssctaa14s sctaa15s sctaa16s sctaa17s intngrp;MISSING ARE ALL (999);USEVARIABLES ARE sctaa11f-sctaa17s tx;CLASSES = c(3);
VARIABLE:
growth mixtures in randomized trialsTITLE:
Input For Growth MixturesIn Randomized Trials
94
%c#1%[ac*3 bc qc]; bc qc ON tx;%c#2%[ac*2 bc qc]; bc qc ON tx;%c#3%[ac*1 bc qc]; bc qc ON tx;ac sctaa11f-sctaa17s;
Input For Growth MixturesIn Randomized Trials (Continued)
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95
Randomized Trials With Non-Compliance
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Randomized Trials With NonCompliance• Tx group (compliance status observed)
– Compliers– Noncompliers
• Control group (compliance status unobserved)– Compliers– NonCompliers
Compliers and Noncompliers are typically not randomly equivalentsubgroups.
Four approaches to estimating treatment effects:1. Tx versus Control (Intent-To-Treat; ITT)2. Tx Compliers versus Control (Per Protocol)3. Tx Compliers versus Tx NonCompliers + Control (As-Treated)4. Mixture analysis (Complier Average Causal Effect; CACE):
• Tx Compliers versus Control Compliers• Tx NonCompliers versus Control NonCompliers
CACE: Little & Yau (1998) in Psychological Methods
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CACE Estimation Via Mixture Modeling AndML Estimation In Mplus
The latent classes of people are principal strataStraightforward to add covariates for y and for c. Many extensions possible.
c
y
Z
u
98
Randomized Trials with NonCompliance: ComplierAverage Causal Effect (CACE) Estimation
c
y
Txx
UG Ex 7.23Ex 7.24
50
99
TRAINING DATATraining data can be used when latent class membership is known forcertain individuals in the sample.
Training data must include one variable for each latent class. Eachindividual receives a value of 0 or 1 for each class variable. A zeroindicates that the individual is not allowed to be in the class. A oneindicates that the individual is allowed to be in the class.
CACE Application
With CACE models, there are two classes, compliers and noncompliers.The treatment group has known class membership. The control groupdoes not. Therefore, the training data is as follows:
10Treatment Group NonCompliers01Treatment Group Compliers11Control Group
Class 2 Non-Compliers
Class 1Compliers
100
JOBS Data
The JOBS data are from a Michigan University Prevention ResearchCenter study of interventions aimed at preventing poor mental health of unemployed workers and promoting high quality of reemployment. The intervention consisted of five half-day training seminars that focused on problem solving, decision making group processes, and learning and practicing job search skills. The control group received a booklet briefly describing job search methods and tips. Respondents wererecruited from the Michigan Employment Security Commission. After a series of screening procedures, 1801 were randomly assigned totreatment and control conditions. Of the 1249 in the treatment group, only 54% participated in the treatment.
The variables collected in the study include depression scores and outcome measures related to reemployment. Background variables include demographic and psychosocial variables.
51
101
JOBS Data (Continued)
Data for the analysis include the outcome variable of depression and the background variables of treatment status, age, education, marital status, SES, ethnicity, a risk score for depression, a pre-intervention depression score, a measure of motivation to participate, and a measure of assertiveness. A subset of 502 individuals classified as having high-risk of depression were analyzed.
The analysis replicates that of Little and Yau (1998).
102
c
depress
depbase txrisk
ageeducmotivateeconassertsinglenonwhite
52
103
Input For Complier Average Causal Effect(CACE) Model
TYPE = MIXTURE;ANALYSIS:
TECH8;OUTPUT:
%OVERALL%
depress ON Tx risk depbase;c#1 ON age educ motivate econ assert single nonwhite;
%C#2% !c#2 is the noncomplier class (noshows)
[depress];
depress ON Tx@0;
MODEL:
NAMES ARE depress risk Tx depbase age motivate educ assert single econ nonwhite x10 c1 c2;
USEV ARE depress risk Tx depbase age motivate educ assert single econ nonwhite c1-c2;
CLASSES = c(2);TRAINING = c1-c2;
VARIABLE:
FILE IS wjobs.dat;DATA:
Complier Average Causal Effect (CACE) estimation in a randomized trial.
TITLE:
104
Tests Of Model FitLoglikelihood
H0 Value -729.414
Information Criteria
Number of Free Parameters 14Akaike (AIC) 1486.828Bayesian (BIC) 1545.888Sample-Size Adjusted BIC 1501.451
(n* = (n + 2) / 24)Entropy 0.727
Output Excerpts Complier Average Causal Effect(CACE) Model
53
105
Output Excerpts Complier Average Causal Effect(CACE) Model (Continued)
Model ResultsFINAL CLASS COUNTS AND PROPORTIONS OF TOTAL SAMPLE SIZE
0.54170271.93488Class 10.45830230.06512Class 2
CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY CLASS MEMBERSHIP
Class Counts and Proportions
0.55378278Class 10.44622224Class 2
Average Latent Class Probabilities for Most Likely Latent Class Membership (Row) by Latent Class (Column)
0.1000.900Class 10.9030.097Class 2
21
106
Output Excerpts Complier Average Causal Effect(CACE) Model (Continued)
Model Results (Continued)
6.068.2991.812DEPRESSIntercepts
Est./S.E.S.E.Estimates
.037
.181
.247
.130 -2.378-.310TXDepress ON
-8.077-1.463DEPBASE3.685.912RISK
Residual Variances13.742.506DEPRESS
Class 1
54
107
Output Excerpts Complier Average Causal Effect(CACE) Model (Continued)
Model Results (Continued)
5.977.2731.633DEPRESSIntercepts
Est./S.E.S.E.Estimates
.037
.181
.247
.000 .000.000TXDepress ON
-8.077-1.463DEPBASE3.685.912RISK
Residual Variances13.742.506DEPRESS
Class 2
108
Output Excerpts Complier Average Causal Effect(CACE) Model (Continued)
Model Results (Continued)LATENT CLASS REGRESSION MODEL PART
-5.4981.590-8.740C#1
C#1 ON
.317
.283
.143
.152
.157
.068
.015
-1.045-.159ECON4.243.667MOTIVATE4.390.300EDUC
1.908.540SINGLE-2.631-.376ASSERT
-1.571-.499NONWHITE
Intercepts
5.184.079AGE
55
109
Angrist, J.D., Imbens, G.W., Rubin, D.B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91, 444-445.
Jo, B. (2002). Estimation of intervention effects with noncompliance: Alternative model specifications. Journal of Educational and Behavioral Statistics, 27, 385-409.
Jo, B. (2002). Statistical power in randomized intervention studies with noncompliance. Psychological Methods, 7, 178-193.
Jo, B., Asparouhov, T., Muthén, B., Ialongo, N. & Brown, H. (2007). Cluster randomized trials with treatment noncompliance. Acceptedfor publication in Psychological Methods.
Little, R.J. & Yau, L.H.Y. (1998). Statistical techniques for analyzing data from prevention trials: treatment of no-shows using Rubin's causal model. Psychological Methods, 3, 147-159.
Further Readings OnCACE
110
Causal Inference
56
111
Causal Inference Concepts
• Potential outcomes
• Principal Stratification
• Finite mixtures
112
Potential Outcomes Framework
• Treatment variable X (e.g., X dichotomous with X=1 or X=0)
• Observed outcome Y, potential outcome variables Y(1), Y(0)
• Observed outcome under selected trmt x equals potential outcome under trmt assignment X=x : yi = yi(x) if xi = x
ACEMeans
yN = yN(1)yN(1)- yN(0)yN(0)yN(1)xN = 1N………………
y2 = y2(0)y2(1)- y2(0)y2(0)y2(1)x2 = 02
y1 = y1(1)y1(1)- y1(0)y1(0)y1(1)x1 = 11
YCausal EffectY(0)Y(1)XSubject #
57
113
Causal Inference And Non-Compliance
114
Causal Effects: The AIR (1996) Vietnam Draft Example
Angrist, Imbens & Rubin (1996) in JASA • Conscription into the military randomly allocated via
draft lottery
TREATMENT TAKEN
D
TREATMENT
ASSIGNMENTZ Y
OUTCOME
58
115
Causal Effects: The AIR (1996) Vietnam Draft Example (Continued)
• Z: treatment assignment (draft status) • Z = 1: assigned to serve in the military (for low lottery numbers)• Z = 0: not assigned to serve (for high lottery numbers)
• D: treatment taken (veteran status)• D = 1: served in the military
• D = 0: did not serve in the military
• Y: health outcome (mortality after discharge) • Note that D is not always = Z
• avoid the draft (or deferred for medical reasons); non-compliance: Z = 1, D=0
• volunteer for military service: Z = 0, D = 1
116
Causal Effect Of Z On Y, Yi(1, Di(1)) – Yi(0, Di(0)), For The Population Of Units Classified By Di(0) And Di(1)
wherepc+a is the proportion in the treatment group who take the treatmentpa is the proportion in the control group who take the treatment
In JOBS (Little & Yau, 1998), there are no always-takers (could not get into the seminars if not assigned), so
pa = 0
which is the Bloom (1984) IV estimate (the less compliance, the more attenuated the treatment and the more you upweight the mean difference).
),/()( 01 aac ppyy −− +
,/)( 01 cpyy −
Causal Effect of D Y Continued
118
Analysis With Missing Data
60
119
Analysis With Missing Data
Used when individuals are not observed on all outcomes in theanalysis to make the best use of all available data and to avoidbiases in parameter estimates, standard errors, and tests of model fit.
Types of Missingness
• MCAR -- missing completely at random• Variables missing by chance• Missing by randomized design• Multiple cohorts assuming a single population
• MAR -- missing at random• Missingness related to observed variables• Missing by selective design
• Non-Ignorable• Missingness related to values that would have been observed• Missingness related to latent variables
120
Estimation With Missing Data
Types of Estimation (Little & Rubin, 2002)
• Estimation using listwise deleted sample• When MCAR is true, parameter estimates and s.e.’s are
consistent but estimates are not efficient• When MAR is true but not MCAR, parameter estimates and
s.e.’s are not consistent• Maximum likelihood using all available data
• When MCAR or MAR is true, parameter estimates and s.e.’sare consistent and estimates are efficient
• Imputation• Mean and regression imputation – underestimation of
variances and covariances• Multiple imputation using all available data – a Bayesian
approach – credibility intervals are Bayesian justifiable under MCAR and MAR
• Pattern-mixture – used for non-ignorable missingness
61
121
Weighted Least Squares Estimation With Missing Data
Weighted least squares for categorical and censored outcomes
• Assumes MCAR when there are no covariates
• Allows MAR when missingness is a function of covariates
122
MCAR: Missing By Design
η
y2
y3
y1
η
y2
y3
y1
y1 y2 y3 η
62
123
Two-Cohort Growth Model
η0
y7
η1
y8 y9 y10 y11 y12
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5
η0
y7
η1
y8 y9 y10 y11 y12
124
MAR
x
y
L H
yi = α + βxi + ζi
E(ζ) = 0, V(ζ) = E(x) = μx, V(x) =
2ζσ
2xσ
▫ Data Matrix:
Complete DataGroup
Missing Data
x y
nH
nL
63
125
Missing At Random (MAR): Missing On y In Bivariate Normal Case
xi / (nL + nH) =nL + nH
i = 1
nL xL + nH xHμx = Σ nL + nH
, (52)
nL + nH
(xi - μx)2 / (nL + nH)i = 1
σxx = Σ . (53)
126
estimated by the complete-data (listwise present) sample (sample size nH)
α = y – β x , (55)β = syx / sxx , (56)
σζζ = syy – / sxx . (57)2yxs
This gives the ML estimates of μy and σyy, adjusting the complete-data sample statistics:
μy = α + β μx = y + β (μx – x), (58)
σyy = σζζ + β2 σxx = syy + β2 (σxx – sxx). (59)
Missing At Random (MAR): Missing On y In Bivariate Normal Case (Continued)
Consider the regressionyi = α + β xi + ζi (54)
64
127
Correlates Of Missing Data• MAR is more plausible when the model includes covariates
influencing missing data
• Correlates of missing data may not have a “causal role” in the model, i.e. not influencing dependent variables, in which case including them as covariates can bias model estimates• Multiple imputation (Bayes; Schafer, 1997) with two
different sets of observed variables− Imputation model− Analysis model
• Modeling (ML)− Including missing data correlates not as x variables but as
“y variables,” freely correlated with all other observed variables
Recent overview in Schafer & Graham (2002).
128
Missing On X
• Regular modeling concerns the conditional distribution
[y | x] (1)
that is, as in regular regression the marginal distribution of [x] is not involved. This is fine if there is no missing on x in which case considering
[y | x]
gives the same estimates as (Joreskog & Goldberger, 1975) considering the joint distribution
[y, x] = [y | x] [x]
65
129
Missing On X (Continued)
• With missing on x, ML under MAR must make a distributional assumption about [x], typically normality. The modeling then concerns
[y, x] = [y | x] [x] (2)
which with missing on [x] is an expanded model that makes stronger assumptions as compared to (1).
• The LHS of (2) shows that y and x are treated the same -they are both “y variables” in Mplus terminology. This is the default in Mplus when all y’s are continuous. In other cases, x’s can be turned into “y’s” e.g. by the model statement
x1-xq;
130
Technical Aspects Of Ignorable Missing Data:ML Under MAR
Likelihood: log [yi | xi]. (87)i = 1Ση
With missing data on y, the ith term of (87) expands into[yi , yi , mi | xi], (88)
where mi is a 0/1 indicator vector of the same length as yi .The likelihood focuses on the observed variables,
[yi , mi | xi] = [yi , yi | xi] [mi | yi yi , xi] dyi , (89)which, when assuming that missingness is not a function ofyi (that is, assuming MAR),
With distinct parameter sets in (91), the last term can be ignored and maximization can focus on the [yi | xi] term. This leads to the standard MAR ignorable missing data procedure.
obs mis mis obs
obs obs
obs
Technical Aspects Of Ignorable Missing Data:ML Under MAR (Continued)
132
AMPS DataThe data are taken from the Alcohol Misuse Prevention Study(AMPS). Forty-nine schools with a total of 2,666 studentsparticipated in the study. Students were measured seven timesstarting in the Fall of Grade 6 and ending in the Spring of Grade 12.
Data for the analysis include the average of three items related toalcohol misuse:
During the past 12 months, how many times did you
drink more than you planned to?feel sick to your stomach after drinking?get very drunk?
Responses: (0) never, (1) once, (2) two times,(3) three or more times
Four of the seven timepoints are studied: Fall Grade 6, SpringGrade 6, Spring Grade 7, and Spring Grade 8.
Output Excerpts AMPS Growth ModelWith Missing Data (Continued)
0.860AMOVER30.567AMOVER20.594AMOVER10.848AMOVER0
R-SquareVariableObserved
R-SQUARE
142
0.6
MARListwiseam
over
timepoint0 1 3 5
0
0.1
0.2
0.3
0.4
0.5
AMPS: Estimated Growth Curves
72
143
Outcome
Escalating
Early Onset
NormativeAge
i
y2 y3 y4
s
x c
y1
u1 u2 u3 u4
Growth Mixture ModelingWith Ignorable Missingness
144
Selection modeling: [y | x] [m | y, x]. Different approaches to [m | y, x]:
Little & Rubin (2002) book: overviewDiggle & Kenward (1994) in Applied Statistics:
using y, y* (non-ignorable dropout)Wu & Carroll (1988), Wu & Bailey (1989) in Biometrics:
using the slope s Frangakis & Rubin (1999) in Biometrika:
using a latent class variable c (compliance)Muthen, Jo, Brown (2003) in JASA:
using c and s (GMM)
Pattern-mixture modeling: [m | x] [y | m, x]
Little & Rubin (2002): overviewRoy (2003) in Biometrics:
using a latent class variable c (missing data patterns)
Non-Ignorable Missing DataModeling Approaches And References
73
145
Outcome
Escalating
Early Onset
NormativeAge
i
y2 y3 y4
s
x c
y1
u1 u2 u3 u4
Growth Mixture Modeling WithNon-Ignorable Missingness As A Function Of y
146
Outcome
Escalating
Early Onset
NormativeAge
i
y2 y3 y4
s
x c
y1
u1 u2 u3 u4
Growth Mixture Modeling WithNon-Ignorable Missingness As A Function Of s
74
147
Outcome
Escalating
Early Onset
NormativeAge
i
y2 y3 y4
s
x c
y1
u1 u2 u3 u4
Growth Mixture Modeling WithNon-Ignorable Missingness As A Function Of c
148
Outcome
Escalating
Early Onset
NormativeAge
i
y2 y3 y4
s
x cy
y1
u1 u2 u3 u4
cu
Growth Mixture Modeling WithNon-Ignorable Missingness As A Function Of cu
75
149
Hedeker, D. & Rose, J.S. (2000). The natural history of smoking: A pattern-mixture random-effects regression model. Multivariate applications in substance use research, J. Rose, L. Chassin, C. Presson & J. Sherman (eds.), Hillsdale, N.J.: Erlbaum, pp. 79-112.
Little, R.J., & Rubin, D.B. (2002). Statistical analysis with missing data. 2nd edition. New York: John Wiley & Sons.
Muthén, B., Kaplan, D., & Hollis, M. (1987). On structural equationmodeling with data that are not missing completely at random. Psychometrika, 42, 431-462. (#17)
Muthén, B., Jo, B. & Brown, H. (2003). Comment on the Barnard, Frangakis, Hill & Rubin article, Principal stratification approach to broken randomized experiments: A case study of school choice vouchers in New York City. Journal of the American Statistical Association, 98, 311-314.
Output Excerpts LSAY Two-Level Growth ModelWith Free Time Scores And Covariates (Continued)
R-SquareWithin Level
0.036SW
R-SquareLatentVariable
0.774MATH100.834MATH90.826MATH80.803MATH7
R-SquareObservedVariable
0.097IW
83
165
Output Excerpts LSAY Two-Level Growth ModelWith Free Time Scores And Covariates (Continued)
R-SquareBetween Level
0.23207E+01UndefinedSW
R-SquareLatentVariable
0.933MATH100.994MATH90.961MATH80.847MATH7
R-SquareObservedVariable
0.875IW
166
Muthén, B. (1997). Latent variable modeling with longitudinal and multilevel data. In A. Raftery (ed), Sociological Methodology (pp. 453-480). Boston: Blackwell Publishers. (#73)
Raudenbush, S.W. & Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods. Second edition. Newbury Park, CA: Sage Publications.
Snijders, T. & Bosker, R. (1999). Multilevel analysis. An introduction to basic and advanced multilevel modeling. Thousand Oakes, CA: Sage Publications.
Further Readings On Three-Level Growth Modeling
84
167
y1 y2 y3 y4
iw sws
Student (Within)
w
s ib sb
y1 y2 y3 y4
School (Between)
Multilevel Modeling With A RandomSlope For Latent Variables
Poor Development: 20% Moderate Development: 28% Good Development: 52%
69% 8% 1%Dropout:
7 8 9 10
4060
8010
0
Grades 7-107 8 9 10
4060
8010
0
Grades 7-107 8 9 10
4060
8010
0
Grades 7-10
Growth Mixture Modeling:LSAY Math Achievement Trajectory ClassesAnd The Prediction Of High School Dropout
88
175mstrat
hsdrop
c
iw sw
math7 math8 math9 math10
math7 math9 math10
ib lunch
math8
hsdrop
C#1
C#2
female
hispanic
black
mother’s ed.
home res.
expectations
drop thoughts
arrested
expelled Within
Between
176
FILE = lsayfull_Dropout.dat;DATA:
NAMES = female mothed homeres math7 math8 math9 math10 expel arrest hisp black hsdrop expect lunch mstratdroptht7 schcode;!lunch = % of students eligible for full lunch!assistance (9th)!mstrat = ratio of students to full time math!teachers (9th)MISSING = ALL (9999);CATEGORICAL = hsdrop;CLASSES = c (3);CLUSTER = schcode;WITHIN = female mothed homeres expect droptht7 expel arrest hisp black;BETWEEN = lunch mstrat;
VARIABLE:
multilevel growth mixture model for LSAY math achievement
TITLE:
Input For A Multilevel Growth Mixture ModelFor LSAY Math Achievement
89
177
TYPE = PLOT3;SERIES = math7-math10 (s);
PLOT:
SAMPSTAT STANDARDIZED TECH1 TECH8;OUTPUT:
TYPE = MIXTURE TWOLEVEL MISSING;ALGORITHM = INTEGRATION;
ANALYSIS:
lunch = lunch/100;mstrat = mstrat/1000;
DEFINE:
Input For A Multilevel Growth Mixture ModelFor LSAY Math Achievement (Continued)
178
%WITHIN%
%OVERALL%
i s | math7@0 math8@1 math9@2 math10@3;
i s ON female hisp black mothed homeres expect droptht7 expel arrest;
c#1 c#2 ON female hisp black mothed homeres expect droptht7 expel arrest;
hsdrop ON female hisp black mothed homeres expect droptht7 expel arrest;
MODEL:
Input For A Multilevel Growth Mixture ModelFor LSAY Math Achievement (Continued)
90
179
%c#1%[i*40 s*1];math7-math10*20;i*13 s*3;
%c#2%[i*40 s*5];math7-math10*30;i*8 s*3;i s ON female hisp black mothed homeres expectdroptht7 expel arrest;
%c#3%[i*45 s*3];math7-math10*10;i*34 s*2;i s ON female hisp black mothed homeres expectdroptht7 expel arrest;
Input For A Multilevel Growth Mixture ModelFor LSAY Math Achievement (Continued)
180
%BETWEEN%
%OVERALL%
ib | math7-math10@1; [ib@0];
ib*1; hsdrop*1; ib WITH hsdrop;math7-math10@0;
ib ON lunch mstrat;
c#1 c#2 ON lunch mstrat;
hsdrop ON lunch mstrat;
%c#1%[hsdrop$1*-.3];
%c#2%[hsdrop$1*.9];
%c#3%[hsdrop$1*1.2];
Input For A Multilevel Growth Mixture ModelFor LSAY Math Achievement (Continued)
91
181
Power For Growth Models
182
Designing Future Studies: Power
• Computing power for growth models using Satorra-Saris (Muthén & Curran, 1997; examples)
• Computing power using Monte Carlo studies (Muthén & Muthén, 2002)
• Power calculation web site – PSMG• Multilevel power (Miyazaki & Raudenbush, 2000;
This is an example of a Monte Carlo simulation study for a linear growth model for a continuous outcome with missing data where attrition is predicted by time-invariant covariates (MAR)
TITLE:
Input Power Estimation For Growth ModelsUsing Monte Carlo Studies
186
x1-x2@1;[x1-x2@0];i s | y1@0 y2@1 y3@2 y4@3;[i*1 s*2];i*1; s*.2; i WITH s*.1;y1-y4*.5;i ON x1*1 x2*.5;s ON x1*.4 x2*.25;
MODEL POPULATION:
[y1-y4@-1];y1 ON x1*.4 x2*.2;y2 ON x1*.8 x2*.4;y3 ON x1*1.6 x2*.8;y4 ON x1*3.2 x2*1.6;
MODEL MISSING:
Input Power Estimation For Growth ModelsUsing Monte Carlo Studies (Continued)
94
187
TYPE = MISSING H1;ANALYSIS:
i s | y1@0 y2@1 y3@2 y4@3;MODEL:
[i*1 s*2];i*1; s*.2; i WITH s*.1;y1-y4*.5;i ON x1*1 x2*.5;s ON x1*.4 x2*.25;
TECH9;OUTPUT:
Input Power Estimation For Growth ModelsUsing Monte Carlo Studies (Continued)
188
Output Excerpts Power Estimation ForGrowth Models Using Monte Carlo Studies
Model Results
0.08770.0349
0.15700.0579
0.00750.0013
0.02410.0036
0.9380.936
0.9520.936
0.08650.0366
0.15540.0598
0.8300.24690.250X21.0000.39800.400X1
S ON
0.9080.50760.500X21.0001.00321.000X1
I ON
M. S. E. 95% %SigS.E.ESTIMATES
Average Cover CoeffStd. Dev.AveragePopulation
95
189
ReferencesAsparouhov, T. & Muthen, B. (2006). Multilevel mixture models. Forthcoming in
Hancock, G. R., & Samuelsen, K. M. (Eds.). (2007). Advances in latent variable mixture models. Charlotte, NC: Information Age Publishing, Inc.
Brown, E.C., Catalano, C.B., Fleming, C.B., Haggerty, K.P. & Abbot, R.D. (2005). Adolescent substance use outcomes in the Raising Healthy Children Project: A two-part latent growth curve analysis. Journal of Consulting and Clinical Psychology, 73, 699-710.
Dunn, G., Maracy, M., Dowrick, C., Ayuso-Mateos, J.L., Dalgard, O.S., Page, H., Lehtinen, V., Casey, P., Wilkinson, C., Vasquez-Barquero, J.L., & Wilkinson, G. (2003). Estimating psychological treatment effects from a randomized controlled trial with both non-compliance and loss to follow-up. British Journal of Psychiatry, 183, 323-331.
Jo, B. (2002). Statistical power in randomized intervention studies with noncompliance. Psychological Methods, 7, 178-193.
Jo, B., Asparouhov, T., Muthén, B., Ialongo, N. & Brown, H. (2007). Cluster randomized trials with treatment noncompliance. Accepted for publication in Psychological Methods.
190
References (continued)Jo, B., Asparouhov, T. & Muthén, B. (2007). Intention-to-treat analysis in cluster
randomized trials with noncompliance. Submitted for publication.Jo, B., Asparouhov, T., Muthén, B., Ialongo, N. & Brown, H. (2007). Cluster randomized trials with treatment noncompliance. Under review.
Kim, Y.K. & Muthén, B. (2007). Two-part factor mixture modeling: Application to an aggressive behavior measurement instrument. Submitted for publication.
Kreuter, F. & Muthen, B. (2007). Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling.
Muthén, B. (2004). Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data. In D. Kaplan (ed.), Handbook of quantitative methodology for the social sciences (pp. 345-368). Newbury Park, CA: Sage Publications.
Muthén, B. (2006). Latent variable hybrids: Overview of old and new models. Forthcoming in Hancock, G. R., & Samuelsen, K. M. (Eds.). (2007). Advances in latent variable mixture models. Charlotte, NC: Information Age Publishing, Inc.
96
191
Muthén, B. & Asparouhov, T. (2006). Growth mixture analysis: Models with non-Gaussian random effects. Forthcoming in Fitzmaurice, G., Davidian, M., Verbeke, G. & Molenberghs, G. (eds.), Advances in Longitudinal Data Analysis. Chapman & Hall/CRC Press.
Muthén, B. & Muthén, L. (2000). Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882-891.
Muthén, L. & Muthén, B. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 4, 599-620.
Muthén, B., Brown, C.H., Masyn, K., Jo, B., Khoo, S.T., Yang, C.C., Wang, C.P., Kellam, S., Carlin, J., & Liao, J. (2002). General growth mixture modeling for randomized preventive interventions. Biostatistics, 3, 459-475.
Nylund, K. (2007). Latent transition analysis: Modeling extensions and an application to peer victimization. Doctoral dissertation, University of California, Los Angeles.
References (continued)
192
Schaeffer, C.M., Petras, H., Ialongo, N., Poduska, J. & Kellam, S. (2003). Modeling growth in boys aggressive behavior across elementary school: Links to later criminal involvement, conduct disorder, and antisocial personality disorder. Developmental Psychology, 39, 1020-1035.
van Lier, P.A.C., Muthén, B., van der Sar, R.M. & Crijnen, A.A.M. (2004). Preventing disruptive behavior in elementary schoolchildren: Impact of a universal classroom-based intervention. Journal of Consulting and Clinical Psychology, 72, 467-478.
On Categorical Data Analysis:
Agresti, A. (2002). Categorical data analysis. Second edition. New York: John Wiley & Sons.
Agresti, A. (1996). An introduction to categorical data analysis. New York: Wiley.
Hosmer, D. W. & Lemeshow, S. (2000). Applied logistic regression. Second edition. New York: John Wiley & Sons.
Long, S. (1997). Regression models for categorical and limited dependent variables. Thousand Oaks: Sage.
References (continued)
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References (continued)On Growth Modeling:
Bollen, K.A. & Curran, P.J. (2007). Latent curve models. A structural equation perspective. New York: Wiley & Sons.
Raudenbush, S.W. & Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods. Second edition. Newbury Park, CA: Sage Publications.