1. Growth curve approaches to longitudinal data in gerontology research Time-varying and Time-invariant Covariates in a Latent Growth Model of Negative Interactions and Depression in Widowhood Jason T. Newsom & D. Morgan Portland State University, Portland, OR Individual Differences in Memory Function Among Older Adults Richard N. Jones, K. Kleinman, J. Allaire, P. Malloy, A. Rosenberg, J.N. Morris, M. Marsiske. Hebrew Rehabilitation Center for Aged, Boston MA Change Point Models Allow for Estimation of the Time at Which Cognitive Decline Accelerates in Preclinical Dementia Charles B. Hall, R.B. Lipton, M. Sliwinski, J.Ying, M.Katz, L. Kuo, & H.Buschke Albert Einstein College of Medicine, New York, NY Dual Sensory Impairment and Change in Personal ADL Function Among Elderly Over Time: A SEM Latent Growth Approach Ya-ping Su, M. Brennan and A. Horowitz Lighthouse International, New York, NY Discussant Karen Bandeen-Roche School of Public Health John Hopkins University, Baltimore, MD

2. Growth Curve Analysis Purpose is to model change over time Linear or nonlinear models possible Variability in change over time by modeling individual growth curves Variability in initial or average levels Predictors can be used to account for variability Two general approaches Hierarchical linear models (HLM) Structural equation models (SEM)

3. t Y t Y Y t High Variability in Intercepts and Slopes Low Variability in Intercepts and Slopes Low Variability in Intercepts and High Variability in Slopes Example Growth Curves

4. HLM Approach to Growth Curves Conceptualization Two levels: within individual and between individual Regression equation for each level

5. HLM Approach to Growth Curves Level 1: Within Individual Examines change in the dependent variable as a function of time for each individual Intercepts and slopes obtained for each individual Intercept is initial or average value of the dependent variable for a given individual (depending on coding of time variable) Slope describes linear increase or decrease in the dependent variable over time of a given individual With predictors, intercepts and slopes represent adjusted means and slopes

6. Intercepts and slopes obtained from Level 1 serve as dependent variables With no predictors, Level 2 intercept represents average of intercepts or slopes from Level 1 With no predictors, Level 2 residual provides information about variance of intercepts or slopes across individuals Can incorporate predictors measured at the individual level (gender, income, etc.) Predictors explain variation in intercepts or slopes across individuals HLM Approach to Growth Curves Level 2: Between Individuals

7. SEM Approach to Growth Curves General conceptualization and interpretation the same as HLM approach Use latent variables and their loadings to represent Level 1 parameters Possible with any SEM software program Requires “time structured data” but can model complex error structures or latent variables over time

8. SEM Approach to Growth Curves Example of a latent growth curve model with four time points y t1  0 (Intercept)   (Slope) 1 0 1 y t2 y t3 y t4 231 1 1

9. SEM Approach to Growth Curves Output Structural means must be estimated Mean of intercept latent variable represents average initial value or average mean value across individuals Mean of slope latent variable represents average slope Variance of intercept latent variable represents variability of initial or average value across individuals Variance of slope latent variable represents variability in growth across individuals Correlation between intercept and slope variables represents association between initial value and growth