1. University Rennes 2, CRPCC, EA 1285
Latent variable modeling of psychological longitudinal data: taking into account the unobserved heterogeneity using Mplus
Jacques Juhel
University Rennes 2, CRPCC, EA 1285
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

2. Studying individual differences in learning, change and development
A double compromise :
random effect model,
classification techniques.
Introduction
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

3. (among other methods) the GMM approach of Muthén and colleagues
A technique for longitudinal data that :
combines categorical and continuous latent variables in the same model (“beyond SEM”),
accommodates unobserved heterogeneity in the sample,
allows for each class membership latent growth parameters to be influenced by time-varying covariates and time-invariant predictor variables,
incorporates consequent outcomes predicted by the latent class variable.
Introduction
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

4. Factor analysis measurement model (level 1) :
LGM specifications
The LGM for a continuous outcome : the multivariate latent variable approach
Factor analysis measurement model (level 1) :
Yi (mx1) repeated measures over fixed time points,
n (mx1) intercepts in the regression from Yi on hi ,
hi (px1) latent growth factors,
L (mxp) design matrix of factor loadings,
ei (mx1) residuals in the regression of Yi on hi (covariance matrix Q).
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

5. Structural regression model (level 2) :
LGM specifications
The LGM for a continuous outcome : the multivariate latent variable approach
Structural regression model (level 2) :
a (px1) means of hi or intercepts in the regression of hi on hi ,
B (pxp) regression coefficients in the regression of hi on hi ,
hi (px1) latent growth factors,
zi (px1) residuals in the regression of hi on hi (covariance matrix Y).
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

6. e, z and h are mutually uncorrelated, E[e] and E[z] equal 0.
The LGM for a continuous outcome : the multivariate latent variable approach
The covariance and mean structure are derived for the population with the hypothesis that :
e, z and h are mutually uncorrelated,
E[e] and E[z] equal 0.
LGM assumptions
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

7. The unconditional linear LGM Free parameters (Mplus output)
SEM representation
The unconditional linear LGM
Free parameters (Mplus output) y1 y2 y3 y4 a
Means of h0 and h1, Y
var(h0)
var(h1)
cov(h0,h1)
res. var(y)
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

8. The LGM with time-varying covariates
LGM specifications
The LGM with time-varying covariates
Factor analysis measurement model (level 1) :
Yi (mx1) repeated measures over fixed time points,
n (mx1) intercepts in the regression from Yi on hi ,
hi (px1) latent growth factors,
L (mxp) design matrix of factor loadings,
K (mxr) coefficients in the regression from Yi on time-varying covariates ai.
ei (mx1) residuals in the regression of Yi on hi (covariance matrix Q).
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

9. Regression coefficients from y on a
SEM representation
Linear LGM with time-varying covariates
Free parameters (Mplus output) y1 y2 y3 y4 a1 a2 a3 a4 Y
var(h0)
var(h1)
cov(h0,h1)
res.var(y)
cov(a, h0)
cov(a, h1) B
Regression coefficients from y on a a
Means of h0 and h1,
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

10. The LGM with time-invariant covariates
LGM specifications
The LGM with time-invariant covariates
Structural regression model (level 2), with vector of predictors x :
hi (px1) latent growth factors,
a (px1) means of hi or intercepts in the regression of hi on hi ,
B (pxp) regression coefficients in the regression of hi on hi ,
Xi (qx1) time-invariant covariate predictors of change,
G (pxq) regression coefficients in the regression from h on X,
zi (px1) residuals in the regression of hi on hi (covariance matrix Y).
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

11. The linear LGM with time-varying and time-invariant covariates
SEM representation
The linear LGM with time-varying and time-invariant covariates
Free parameters (Mplus output) y1 y2 y3 y4 x1 x2 x3 a1 a2 a3 a4 B
Regression coefficients from y on a
Regression coefficients from h0 and h1on X a
Intercepts of h0 and h1,
Means of a1-a4 Y
res.var(h0)
res. var(h1)
res. cov(h0,h1)
res. var(y)
cov(a, h0)
cov(a, h1)
cov(a, x)
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

12. Zi (dx1) vector of distal outcomes of change,
LGM specifications
The linear LGM with time-varying, time-invariant covariates and a distal outcome
Consequences of change as outcomes can be predicted by the latent growth factors :
Zi (dx1) vector of distal outcomes of change,
b (dxp) matrix of regression coefficients from Z on h,
w (dx1) vector of regression intercepts for Z,
xi (px1) residuals in the regression of Zi on hi (covariance matrix Y).
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

13. Free parameters (Mplus output)
SEM representation
The linear LGM with time-varying, time-invariant covariates and a distal outcome
Free parameters (Mplus output) y1 y2 y3 y4 x1 x2 x3 a1 a2 a3 a4 B
Regression coefficients from y on a
Regression coefficients from h0 and h1on x
Regression coefficients from z on h0 and h1 a
Intercepts of h0 and h1,
Means of a1-a4
Intercept of z Y
res. var(h0)
res. var(h1)
res. cov(h0,h1)
res. var(y)
cov(a, h0)
cov(a, h1)
cov(a, x) z
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

14. Illustration : data set 1
Clinical symptomatology, performance on the TMT and consciousness disorders in schizophrenia
130 stabilized patients with schizophrenia (M=31.0 yr., QI>90, all with neuroleptic medication).
Time to complete TMT parts A and B separately at 4 equally spaced time points (t=0, t=2, t=4 and t=6 months).
t=-1 : scores to the Positive and Negative Syndrome Scale.
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

15. Illustration: data set 1
Trail Making Test : Responding time (t0  t3, N = 102 complete, only!)
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

16. Illustration: data set 1
Fitting a linear LGM with time-varying and time-invariant covariates to TMT data (N=102) B1 B2 B3 B4 A1 A2 A3 A4
Dis
Pos
Neg
Host
Anx i s
TMT form B
TMT form A
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

17. Illustration: data set 1
Is the linear growth model tenable?
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

18. Illustration: data set 1
Conditional LGM : results
ML estimation
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I ON
DISORG
POS
NEG
HOST
ANX
S ON
DISORG
POS
NEG
HOST
ANX
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

19. Illustration: data set 1
Conditional LGM : results
ML estimation
Two-Tailed
Estimate S.E. Est./S.E. P-Value
B ON A
B ON A
B ON A
B ON A
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

20. Illustration: data set 1
Conditional LGM : results
ML estimation
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Intercepts B B B B I S
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

21. Illustration: data set 1
Conditional LGM : results
ML estimation
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Residual Variances B B B B I S
R-SQUARE B B B B I S
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

22. Representing heterogeneity with respect to the growth factors and covariates.
GMM specifies a separate LGM for each of the K latent class simultaneously :
and
GMM specification
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

23. with the reference class K,
GMM specification
Modeling predictive effects of time-invariant covariates on latent class membership
Mixture components (c) are related to covariates through a multinomial logistic regression model :
with the reference class K,
(1xq) vector of logistic regression coefficients from C on X,
p0k logistic regression intercept for class k relative to class K.
Xi (qx1) vector of time-invariant covariate predictors of change.
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

24. Indices for determining the “best” GMM Information-based criteria :
GMM selection
Indices for determining the “best” GMM
Information-based criteria :
BIC, SABIC
- Nested model Likelihood Ratio Test :
LMR (Low-Mendell-Rubin) LRT, bootstrapped LRT
Latent classification accuracy :
Entropy, average latent class probabilities for most likely latent class membership
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

25. Illustration: data set 1
Mplus representation of a linear GMM fitted to TMT data (N=102). B1 B2 B3 B4 A1 A2 A3 A4 i s c
Disorg
Pos
Neg
Host
Anx x
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

26. Illustration: data set 1
Determining the “best” growth two-class model x c i s
differences
between
classes
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

27. Illustration: data set 1
GMM results : TMT data (N=102)
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES
Latent
Classes
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

28. Illustration: data set 1
GMM results : TMT data (N=102)
CLASSIFICATION QUALITY
Entropy
CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP
Class Counts and Proportions
Latent classes
Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

29. Illustration: data set 1
Growth Mixture model results : TMT data (N=102)
VUONG-LO-MENDELL-RUBIN LIKELIHOOD RATIO TEST FOR 1 (H0) VERSUS 2 CLASSES
H0 Loglikelihood Value
2 Times the Loglikelihood Difference
Difference in the Number of Parameters
Mean
Standard Deviation
P-Value
LO-MENDELL-RUBIN ADJUSTED LRT TEST
Value
P-Value
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

30. Illustration: data set 1
Growth Mixture model results : TMT data (N=102)
Categorical Latent Variables
Two-Tailed
Estimate S.E. Est./S.E. P-Value
C# ON
DISORG
POS
NEG
HOST
ANX
Intercepts C#
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

31. Illustration: data set 1
GMM: probability of class membership as function of value on each of covariates : TMT data (N=102)
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

32. Illustration: data set 1
Growth Mixture model results : TMT data (N=102)
Latent class 1 = Latent class 2
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I ON
DiSORG
POS
NEG
HOST
ANX
S ON
DiSORG
POS
NEG
HOST
ANX
Illustration: data set 1
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

33. Illustration: data set 1
Growth Mixture model results : TMT data (N=102)
Nc#1= 7
Nc#2= 95
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

34. Illustration: data set 2
Data set 2 : Learning to read and development of phonological and morphological processing
344 children (6-7 years) tested 6 times (6 weeks between each measurement occasion)
t1-1: Raven Matrix (int)
t1 – t6 : 4 observed variables: Syllables Implicit Processing, Phonemes Implicit Processing , Syllables Explicit Processing, Phonemes Explicit Processing.
t6 + 1 week : Word reading (frequent words, rare words, pseudo-words)
Illustration: data set 2
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

35. Illustration: data set 2
t t t t t t5
Data set 2 : descriptive statistics
Illustration: data set 2
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

36. Illustration: data set 2
SEM representation of a quadratic GGMM with time invariant antecedents of change and a distal outcome (N=344)
Int
sip1
pip1
sep1
pep1 f1
sip2
pip2
sep2
pep2 f2
sip3
pip3
sep3
pep3 f3
sip4
pip4
sep4
pep4 f4
sip5
pip5
sep5
pep5 f5
sip6
sep6
pep6 f6 c i s q
Lect.
freq.
rare
pseudo
words
Illustration: data set 2
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

37. Multiple indicators GMM
Multiple indicator LGM
First-order factor scores : measurement model with (strong) invariance constraints
Second-order growth factors :
Factor scores as deviations from the group mean :
Second-order growth model:
Multiple indicators GMM
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

38. Multiple indicators GMM
Multiple indicator GMM
First-order constraints :
Differences between latent classes :
- means ,
- covariances ,
- parameters for representing growth
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

39. Illustration: data set 2
Unconditional GMM : 2 classes vs 3 classes
Illustration: data set 2
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

40. Illustration: data set 2
Three-class GMM with int as covariate, without (overall) and with (between) class differences
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

41. Illustration: data set 2
Conditional GMM: estimated means
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

42. Illustration: data set 2
GMM results : information criteria an quality of classification
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS
BASED ON ESTIMATED POSTERIOR PROBABILITIES
Latent
Classes
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

43. Illustration: data set 2
GMM results : information criteria an quality of classification
Entropy
CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP
Class Counts and Proportions
Latent
Classes
Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

44. Illustration: data set 2
GMM results : intercepts of i, s and q
Class 1
Intercepts I S Q
Residual Variances I S Q
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

45. Illustration: data set 2
GMM results : intercepts of i, s and q
Class 2
Intercepts I S Q
Residual Variances I S Q
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

46. Illustration: data set 2
GMM results : intercepts of i, s and q
Class 3
Intercepts I S Q
(linear trend in class 3 in fixing
Residual Variances I S Q
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

47. Illustration: data set 2
GMM results : coefficients regression from categorical variables c on covariate
Categorical Latent Variables
C# ON
INTNV
C# ON
INTNV
Intercepts C# C#
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

48. Illustration: data set 2
GMM results : probability of class membership
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

49. Illustration: data set 2
Estimated probabilities for c as a function of int level
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

50. Illustration: data set 2
GMM results : regression from i, s and q on covariate
Class 1
I ON
INTNV
S ON
INTNV
Q ON
INTNV
S WITH I
Q WITH I S
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

51. Illustration: data set 2
GMM results : regression from i, s and q on covariate
Class 2
I ON
INTNV
S ON
INTNV
Q ON
INTNV
S WITH I
Q WITH I S
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

52. Illustration: data set 2
GMM results : regression from i, s and q on covariate
Class 3
I ON
INTNV
S ON
INTNV
Q ON
INTNV
S WITH I
Q WITH I S
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

53. Illustration: data set 2
GMM results : reading proficiency level for each class
Class 1
Means
LECT
Class 2
LECT
Class 3
LECT
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data

54. Interest, limitations, cautions
Concluding remarks
Interest, limitations, cautions
GMM is a promising approach for modeling heterogeneous latent change across unobserved population subgroups.
But :
GMM is usually based on large samples.
The search for heterogeneity should be conducted in a principled and disciplined way; the best way to guide GMM selection is to test different models following theory-based models.
GMM always identify groups
The role that covariates play in the enumeration process has to be clarified.
An important question : how to model missing data on x variables?
June 2-4, Saint-Raphaël
INSERM workshop : Mixture modelling for longitudinal data