Growth Mixture Modeling of Longitudinal Data

David Huang, Dr.P.H., M.P.H.

UCLA, Integrated Substance Abuse Program

Longitudinal Data

• Subjects have repeated measures on some characteristics over time, which could be

• Medical history (ex blood pressure)

• Children’s learning curve (ex. math score)

• Baby’s growth curve (ex. weight)

• Drug use history (ex. heroin use)

i d 50 216 257 1008

days use

0

5

10

15

20

25

30

Age

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Growth Curve Modeling

• Level 1 represents intra-individual difference in repeated measures over time. (individual growth curve).

• Level 2 represents variation in individual growth curves.

Growth Curve Model with One Class (N = 436)

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Class 1

Years Since The First Use

Days use per month

Limitation of Growth Curve Model

• Assume that growth curves are a sample from a single finite population. The growth model only represents a single average growth rate.

Growth Mixture Modeling

• Including latent classes into growth curve modeling.

• Modeling individual variation in growth rates.

• Classifying trajectories by latent class analysis.

Growth Mixture Model in Mplus

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

Source: Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

• This study is based on 436 male heroin addicts who were admitted to the California Civil Addict Program at 1964-1965 and were followed in the three follow-up studies conducted every ten years over 33 years.

Growth Curve Model with Two Classes (N = 436)

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Class 1 (N=63)Class 2 (N=373)

Years Since The First Use

Days use per month

Growth Curve Model with Three Classes (N = 436)

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Class 1 (N=56)Class 2 (N=78)Class 3 (N=302)

Years Since The First Use

Days of use per month

Growth Curve Model with Four Classes (N = 436)

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Class 1 (N=52)Class 2 (N=74)Class 3 (N=277)Class 4 (N=33)

Years Since The First Use

Days of use per month

Growth Curve Model with Five Classes (N = 436)

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Class 1 (N=70)Class 2 (N=66)Class 3 (N=249)Class 4 (N=34)Class 5 (N=17)

Years Since The First Use

Days of use per month

Goodness of fit

• Loglikelihood

• Akaike Information Criterion (AIC)

• Bayesian Information Criterion (BIC)

• Sample-size Adjusted BIC

• Entropy

Adjusted BIC Index by Latent Classes

45000

47000

49000

51000

53000

55000

0 1 2 3 4 5

Adjusted BIC

Latent Classes

Adjusted BIC

Difficulties in Model fitting

• EM algorithm reaches a local maxima, rather than a global maxima.

• Repeat EM algorithm with different sets of initial values.

• Use BIC to compare the goodness-of-fit of models

Example of Wrong Starting ValuesThree Classes (WRONG STRATING

VALUES)Three Classes

Member 1 Member 2 Member 3 Member 1 Member 2 Member 3

Intercept 21.44 3.43 14.11* 6.40 10.08** 26.06**

Slope -1.16 -1.25 0.67 -0.02 1.26** -0.58

Treatment on I -0.13 -0.01 0.16 0.16 -0.08 -0.04

Treatment on S 0.005 0.06 -0.01 -0.02 0.002 0.01

Class mean -2.43** -1.15 -- -0.71 -- 0.41

Treatment on class 0.05** -0.05 -- 0.03* -- 0.004

% of individual in each class (estimated)

161.8(0.32)

73.7 (0.14) 275.5 (0.54) 178.9(0.35)

121.8 (0.24)

210.3 (0.41)

% of individual in each class (observed)

162(0.32)

70 (0.14) 279 (0.54) 176(0.34)

123 (0.24) 212 (0.41)

Log Ho -31234.5 -31132.3

Akaike (AIC) 62533.0 62328.7

Bayesian (BIC) 62668.6 62464.3

Adjusted BIC 62567.0 62362.7

Entropy 0.897 0.890

use

02468

1012141618202224262830

Year s si nce t he fi r st her oi n use

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

11

1

1 11

11

11

11 1 1 1

22 2

2 2 22 2 2 2

22

22

2 2

3

3

33

3 3 3 3 3 3 33

33

33

Difficulties in Model fitting

• EM algorithm would NOT converge.

• Start with a simple model. Set variance of intercept and slope at zero. Assume residuals are constant across the classes.

Difficulties in Model fitting

• Individual classification is model dependent and initial value dependent. Individual classification could vary in different models.

use

02468

1012141618202224262830

Year s si nce t he fi r st her oi n use

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

1

11 1 1 1 1 1 1 1 1

11

11

2

2 2 2

22

2 2

22 2

2 2 22 2

3

3 33

3 33 3

3 33 3 3 3 3 3

use

02468

1012141618202224262830

Year s si nce t he fi r st her oi n use

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

11 1 1 1

1 11 1

11

1 1 1 1

2

22

2

2 22 2 2 2 2 2 2 2 2 2

3

3

3 33

3 3 3 3 3 33 3

33

3

References

• Terry Duncan (2002). Growth Mixture Modeling of Adolescent Alcohol Use Data. www.ori.org/methodology

• 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.