Grimm, K. J.
Ram, N.
Hamagami, F.
1
Road Map
 The role of growth models in developmental
studies
 Growth curve analysis
 Linear growth curve
 Nonlinear change patterns
 An example—The Berkeley Growth and Guidance
Studies
 Discussion
2
Growth models in
developmental research
 The focus of developmental research
 In psychological studies
 The use of growth model in developmental research
 Many applications only model linear change patterns
3
Mathematics achievement in grade 7-12 in the longitudinal study of
American Youth (Muthen, B., & Khoo, S-T., 1998).
4
Plots of height for 5 girls
Plots of height for 5 boys
5
 However, many developmental processes are more
complex
 Acknowledgement of the nonlinearity

But limited to polynomial models, such as quadratic and
cubic changes
 The gap between the models and change patterns
 Models should provide appropriate representations of
developmental theory
 There may not yet be strong theories
6
Growth curve analysis
 Linear model
 At least 3 time points for each individuals
 Differences between SEM and multilevel approaches


SEM--latent variables: (µi, σ2i), (µs, σ2s); factor loadings: 1, (tk1)/k2
Multilevel approach:

i n   n  u 0i
is   s  u1i
u oi ~ N (0,  i2 )
u1i ~ N (0,  s2 )
cov ariance:  i , s
7
 Nonlinear change patterns
 Quadratic growth
 Latent basis growth
 Nonlinear latent curve model


Additive nonlinear latent curve
Multiplicative nonlinear latent curve
8
 Nonlinear change patterns
 Quadratic growth
 Latent basis growth
9
10
Piecewise model
11
Nonlinear latent curve models
Gompertz
Logistic
12
 Richards curve
 Also termed as
generalized logistic
curve
A=o, K=1, B=1.5, Q=v=0,5, M=0.5
A: the lower asymptote
K: the upper asymptote
B: growth rate
13
 Nonlinear latent curve models
Further classified as
 Additive nonlinear latent curve
 Multiplicative nonlinear latent curve
14
Example
 Data
 The Berkeley Growth Study and Berkeley Guidance
Study
 Ages between 3 and 19 years old
 155 males and 167 females
15
 Models
 The Preece-Baines model
h1n: the individual adult height;
h2n: the height at the age when the individual grows fastest;
s0n: the growth rate during childhood;
s1n: the growth rate during puberty.
16
Results
17
Height_ Rate_
puberty child
Height_
Adult
Height_
Puberty
Rate_
Child
Rate_
Puberty
0.99
-0.51
Rate_ Age_
puberty puberty
-0.2
-0.4
0.55
0.55
0.58
-0.78
-0.44
18
 Gender
 Adult height: girls < boys (b=-13.53)
 Height during puberty: girls < boys (b=-11.47)
 Rate during childhood: girls > boys (b=.02)
 Rate during puberty: girls > boys (b=.13)
 Time experienced puberty: girls earlier (b=-2.11)
19
Discussion
 Advocate the use of multiplicative nonlinear curve
 Additional change models
 Multiphase models
 Latent difference score models
 Drawbacks:
 The approximation method to reduce the problem of
convergence
 Require more data/measurement occasions
20
Comments/thoughts for future
research
 The lack of the measurement part
 Comparisons between multilevel IRT model and 2-stage
growth curve model.
 It is impractical to develop a specific model for
individual characteristics.
 Is there a more general model fitting human
development?
 Comparisons between IRT approach and SEM approach in
longitudinal studies.
 Minimum numbers of sample sizes, observation occasions..
21