Nonlinear growth curves in developmental research

1. Nonlinear growth curves in developmental research Grimm, K. J.Ram, N. Hamagami, F.

2. 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

3. 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

4. Mathematics achievement in grade 7-12 in the longitudinal study of American Youth (Muthen, B., & Khoo, S-T., 1998).

5. Plots of height for 5 girls Plots of height for 5 boys

6. 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

7. 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, (t-k1)/k2 • Multilevel approach:

8. Nonlinear change patterns • Quadratic growth • Latent basis growth • Nonlinear latent curve model • Additive nonlinear latent curve • Multiplicative nonlinear latent curve

9. Nonlinear change patterns • Quadratic growth • Latent basis growth

11. Piecewise model

12. Nonlinear latent curve models Gompertz Logistic

13. A=o, K=1, B=1.5, Q=v=0,5, M=0.5 • Richards curve • Also termed as generalized logistic curve A: the lower asymptote K: the upper asymptote B: growth rate

14. Nonlinear latent curve models Further classified as • Additive nonlinear latent curve • Multiplicative nonlinear latent curve

15. Example • Data • The Berkeley Growth Study and Berkeley Guidance Study • Ages between 3 and 19 years old • 155 males and 167 females

16. 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.

17. Results

19. 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)

20. 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

21. 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..