Grimm, K. J.Ram, N.Hamagami, F.

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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).

Plots of height for 5 girls

Plots of height for 5 boys

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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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Nonlinear change patternsQuadratic growth

Latent basis growth

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Nonlinear latent curve modelsGompertz Logistic

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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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Results

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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