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Road Map The role of growth models in
developmental studiesGrowth curve analysis
Linear growth curveNonlinear change patterns
An example—The Berkeley Growth and Guidance Studies
Discussion
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Growth models in developmental researchThe focus of developmental research
In psychological studiesThe use of growth model in developmental
researchMany applications only model linear change
patterns
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Mathematics achievement in grade 7-12 in the longitudinal study of American Youth (Muthen, B., & Khoo, S-T., 1998).
However, many developmental processes are more complexAcknowledgement 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 theoryThere may not yet be strong theories
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Growth curve analysis
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Linear model
At least 3 time points for each individualsDifferences between SEM and multilevel
approaches SEM--latent variables: (µi, σ2
i), (µs, σ2s); factor
loadings: 1, (t-k1)/k2 Multilevel approach:
inn ui 0 iss ui 1
si
si
ioi
ariance
Nu
Nu
,
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2
:cov
),0(~
),0(~
Nonlinear change patternsQuadratic growthLatent basis growthNonlinear latent curve model
Additive nonlinear latent curve Multiplicative nonlinear latent curve
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A=o, K=1, B=1.5, Q=v=0,5, M=0.5
Richards curveAlso termed as
generalized logistic curve
A: the lower asymptoteK: the upper asymptoteB: growth rate
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Nonlinear latent curve models Further classified as
Additive nonlinear latent curve
Multiplicative nonlinear latent curve
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ExampleData
The Berkeley Growth Study and Berkeley Guidance Study
Ages between 3 and 19 years old155 males and 167 females
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ModelsThe 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.
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Height_puberty
Rate_child
Rate_puberty
Age_puberty
Height_Adult
0.99 -0.51 -0.2 0.55
Height_Puberty
-0.4 0.55
Rate_Child
0.58 -0.78
Rate_Puberty
-0.44
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GenderAdult 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)
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DiscussionAdvocate the use of multiplicative nonlinear
curveAdditional change models
Multiphase modelsLatent difference score models
Drawbacks:The approximation method to reduce the
problem of convergenceRequire more data/measurement occasions
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Comments/thoughts for future researchThe 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