Title: Growth Models: A Practical Guide
1Growth ModelsA Practical Guide
- Sarah O. Meadows
- Center for Research on Child Wellbeing
- Princeton University
- October 15, 2007
2Outline of Presentation
- What are growth models?
- Nuts and Bolts
- Hands-On Example
- Additional Issues
3Part I What is a Growth Model?
4What is a Growth Model?
- A way to assess individual stability and change,
both growth and decay, over time. - A two-level, hierarchical model that that models
(1) within individual change over time and (2)
between individual differences in patterns of
growth.
5A Rose by Any Other Name . . .
- Growth Models
- Trajectory Models
- Growth Curve Models
- Latent GM
- Latent TM
- Latent GCM
- Hierarchical Models
- Random Intercept Models
- Random Coefficient Models
- Random Intercept/Random Slope Models
- Variance Component Models
6Why Latent?
- Because we assume that whatever process that is
underlying the thing we are modeling (or the
behavior we observe) is actually unobserved, or
latent. - The characteristics we observe are a
manifestation of this latent trajectory. - This language grew out of structural equation
modeling (SEM).
7Why use GMs?
- Everyone else is doing it!
- Education
- Criminology
- Psychology
- Sociology
- Public Health
- You have longitudinal data and are interested in
change over time. - You may want to explain those changes.
- You may also believe that not everyone follows
the same path.
8How Have Others Used GMs?
- Growth Trajectories of Sexual Risk Behavior in
Adolescence and Young Adulthood. Fergus,
Zimmerman, Caldwell. American Journal of
Public Health. 2007. - Individual Differences in the Onset of Tense
Marking A Growth Model Example. Hadley
Colt. Journal of Speech, Language, and Hearing
Research. 2006. - Ten-Year Stability of Depressive Personality
Disorder in Depressed Outpatients. Laptook,
Klein, Dougherty. The American Journal of
Psychiatry. 2006. - Verbal Learning and Everyday Functioning in
Dementia An Application of Latent Variable
Growth Curve Modeling. Mast Allaire. The
Journals of Gerontology. 2006. - You Make Me Sick Marital Quality and Health
Over the Life Course. Umberson, Williams,
Powers. Journal of Health and Social Behavior.
2006. - Parental Divorce and Child Mental Health
Trajectories. Strohschein. 2005. Journal of
Marriage and Family.
9A Detailed Example
- Stability and Change in Family Structure and
Maternal Health Trajectories. Meadows,
McLanahan, Brooks-Gunn. American Sociological
Review. Forthcoming. - We wanted to know whether changes in family
structure, including transitions into and out of
coresidential relationships, had short-term
impacts on health (i.e., crisis model) or
long-term impacts on health (i.e., resource
model).
10Example (cont.)
- Trajectories of maternal self-rated health and
mental health problems from one year after birth
to five years after birth. - Two measures of family structure change
- Level 1 Time-Varying
- Level 2 Time-Invariant
11Example (cont.)
- Results
- Transitions, especially exits from marriages,
resulted in short-term declines in physical
health and short-term increases in mental health
problems. - Little support for the resource model no growing
gap in well-being between mothers who remained
stably married and those remained stably single,
as well as mothers who made transitions.
12Figure 1. Mothers Mental Health Trajectories
13Figure 2. Mothers Household Income Trajectories
14Figure 3. Fathers Mental Health Trajectories
15Figure 4. Fathers Earnings Trajectories
16Part II Nuts and Bolts
17Where Did GMs Come From?
- Time Series Models (Autoregressive)
- Repeated Measures ANOVA
- (Duncan Duncan, 2004)
- SEM
- Multilevel Models (HLM)
18Hierarchical Models
- Traditional
- Level 1 Students
- Level 2 Schools
- Growth Models (a type of HM)
- Level 1 Repeated Observations
- Level 2 Individuals
19Unconditional Model
- Level 1 Within Individual
- Level 2 Between Individual
20A Latent Trajectory
Latent Depression Trajectory
ß
Depressive Symptoms
a
Time
21Time-Invariant Covariates
- Level 1 Within Individual
- Level 2 Between Individual
22Time-Varying Variables
- Level 1 Within Individual
- Level 2 Between Individual
-
Time-varying effect.
23Fixed vs. Random
- Fixed Means of the latent trajectory parameters
(i.e., intercept and slope) - Random Variance of the latent trajectory
parameters (i.e., indicates individual
heterogeneity around population means)
24Part III An Example
25Software
- MPlus SEM based
- HLM Hierarchical Modeling
- SAS Proc Traj
- STATA
26Data Requirements
- Three observations
- For polynomial curves you need d 2 repeated
measures, where d is the degree of the
polynomial. - Horizontal data file (i.e., one person, one row).
- Convert data to .dat file.
- Remember the order of the variables!!
27Self-Rated Health
- Mothers in FFCWS
- In general, how is your health?
- Excellent (5)
- Very Good (4)
- Good (3)
- Fair (2)
- Poor (1)
- Repeated measures one, three, and five years
after birth.
28Setting the Trajectory
Intercept
Slope
1
0
4
1
1
2
SRH 1
SRH 3
SRH 5
29Models
- Unconditional
- Model Fit
- Conditional
- Time-Invariant Covariates
- MPlus Graphs
- Selection and Causation
- Time-Varying Covariates
30Model Fit
- Chi-Square
- Not Significant, but almost always is.
- CFI (Comparative Fit Index)
- Range 0 1 1 is best.
- TLI (Tucker Lewis Index or NNFI)
- Range 0 1 1 is best.
- RMSEA (Root Mean Square Error of Approximation)
- Under .05 is good above .10 is bad.
31Time-Invariant Covariates
- Age at Baseline
- Education
- Race
- Biological Parents Mental Health Problem
- Lived with both Bio Parents at Age 15
- Number of Previous Relationships
- Baseline SRH
- Considered an Abortion
- Positive Marriage Attitude
- Prenatal Variables (medical care, drug and
alcohol use, smoking) - Baseline Marital Status
32Time-Invariant Covariates
a
ß
33Figure 5. Mothers Self-Rated Health
Trajectories.
34Selection Issues
- Intercept
- Third factor is responsible for where people
start. - Slope
- Third factor is responsible for where people go.
35Time-Varying Covariate
- Mental Health Problems
- Range 0-3
- Includes CIDI Major depressive episode, binge
drinking, and drug use. - All occurred in the past 12-months.
36MH 1
MH 3
MH 5
37Part IV Additional Issues
38Multi-Models
- Multi-Group
- Growth process may vary for each group.
- Multi-Process
- Models more than one trajectory.
39Measurement
- Latent Measures (Multiple Indicators)
- Dichotomous/Categorical Variables
- Count Variables
- ZIP Models
- Skewness
- Transform Variable
- Semi-Continuous Growth Model
40Age-Based Growth Model
- Synthetic cohort
- Sample members may contribute different amounts
of information at different times. - Missing Data
- Drop Cases (default)
- Multiple Imputation
- Full Information Maximum Likelihood (FIML)
- Analysis MISSING
41Mixture Models
- Latent Class Models (LCM/LCA)
- Group membership not known.
- Latent Class Growth Models (LCGM/LCGA)
- Group membership not known and is based on
trajectory patterns. - No variation is allowed within latent classes.
- Growth Mixture Models (GMM)
- Group membership is not known and is based on
trajectory patterns. - Allows for variation within latent classes.
42Contact Info
- smeadows_at_princeton.edu