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

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)

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

HLM Approach to Growth Curves

Conceptualization

• Two levels: within individual and between individual

• Regression equation for each level

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

0 1ti ti tii iy x r

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

0 00 01 1 0i i iz u

HLM Approach to Growth Curves Level 2: Between Individuals

1 10 11 1 1i i iz u

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

SEM Approach to Growth CurvesExample of a latent growth curve model with four time points

yt1

0(Intercept)

(Slope)

1

01

yt2 yt3 yt4

2 31 11

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