multiple-indicator latent growth curve models: an...
TRANSCRIPT
Multiple-Indicator Latent Growth Curve
Models: An Analysis of the Second-
Order Growth Model and Two
Less Restrictive Alternatives
Jacob Bishop and Christian Geiser
Presentation Overview
• Part I – Theoretical Background &
Purpose of Research
• Part II – Model Formulation
• Part III – Model Comparison
• Part IV – Summary & Discussion
Part I – Theoretical Background &
Purpose of Research
I. Background & Purpose:
Growth Process
Time
Observed Score Measurement Error Trait/Growth
• McArdle & Eppstein (1987) • Meredith & Tisak (1990)
I. Background & Purpose:
State-Trait Process
Time
Observed Score Measurement Error State Residual Trait
State
• Steyer (1992)
I. Background & Purpose
Hybrid LST-Growth Process
Time
Observed Score Measurement Error State Residual Trait/Growth
State
I. Background & Purpose:
(Multiple-Indicator) Latent Growth Curve Models
• Advantages
– Separate systemic variability from true trait change and measurement error.
– Test for measurement equivalence of indicators across time.
– Obtain less biased estimates of indicator reliabilities.
– Test for indicator-specific (method) effects.
– Obtain greater power to detect individual differences in change.
– Greater flexibility in modeling complex patterns of change.
I. Background & Purpose:
(Multiple-Indicator) Latent Growth Curve Models
• Disadvantages
– Not widely adopted (Leite, 2007).
– Not well understood (Ferrer, Balluerka, Widaman, 2008).
• Theoretical foundation unclear.
• Second-order growth model (SGM; McArdle, 1988) often
viewed as only option.
• Less restrictive alternatives exist (GSGM, ISGM; Eid,
Courvoisier, & Lischetzke, 2012; Eid & Hoffman,1998).
I. Background & Purpose:
Purpose of this Research
• Formulate Multiple-Indicator LGCMs based on
Latent State-Trait (LST) Theory
– Second-Order Growth Models (SGM)
– Generalized Second-Order Growth Model (GSGM)
– Indicator-Specific Growth Model (ISGM)
• Compare the Models
– Model assumptions, constraints, similarity, nesting,
etc.
Part II – Model Formulation
II. Model Formulation
Hybrid LST-Growth Process
Time
Observed Score Measurement Error State Residual Trait/Growth
State
II. Model Formulation:
Latent Variables in LST Theory Latent Mean
Observed Score
Measurement Error
Variable
Latent State
Residual Variable
Latent Trait Variable
Latent State Variable
• Y: Observed Score
• Test/subscale i (i = 1, … , j, … , m)
• Measurement time t (t = 1, … , s, … , n)
• τ (Latent State): Characterizes persons-in-
situations.
• ξ (Latent Trait): Characterizes the person
only.
• ζ (Latent State Residual): Characterizes
effects of the situation and/or person ×
situation interactions
• ε (Measurement Error): Characterizes
random measurement error.
II. Model Formulation:
Multiple-Indicator LST Base Model
II. Model Formulation:
Too Many Unknown Parameters!
Second-Order Growth Model (SGM)
Generalized Second-Order Growth Model (GSGM)
Indicator-Specific Growth Model (ISGM)
Restrictive
Assu
mp
tion
s
SGM
II. Model Formulation – SGM:
Assumptions
• Time-invariant congenerity of latent states:
• Linear trait growth:
• Where
Intercept factor
Linear slope factor
II. Model Formulation – SGM:
Starting Point – LST Base Model
II. Model Formulation – SGM:
Assumption 1 – Time-Invariant Congenerity of Latent States
II. Model Formulation – SGM:
Assumption 1 – Time-Invariant Congenerity of Latent States
II. Model Formulation – SGM:
Assumption 2 – Trait Growth
II. Model Formulation – SGM:
Assumption 2 – Trait Growth
GSGM
II. Model Formulation – GSGM:
Assumptions
• Time-invariant congenerity of latent traits:
• Time-invariant congenerity of latent state residuals.
• Linear trait growth:
II. Model Formulation – GSGM:
Starting Point – LST Base Model
II. Model Formulation – GSGM:
Assumption 1 – Time-Invariant ξ-Congenerity
II. Model Formulation – GSGM:
Assumption 2 – Time-Invariant ζ-Congenerity
II. Model Formulation – GSGM:
Post-Assumption 2 Simplified Model
II. Model Formulation – GSGM:
Assumption 3 –Trait Growth
II. Model Formulation – GSGM:
Simplified Model
ISGM
II. Model Formulation – ISGM:
Assumptions
• Indicator-specific linear growth of latent trait variables:
• Time-invariant congenerity of latent state residuals:
II. Model Formulation – ISGM:
Starting Point – LST Base Model
II. Model Formulation – ISGM:
Assumption 1 – Indicator-Specific Growth
II. Model Formulation – ISGM:
Assumption 2 – Time-Invariant ζ-Congenerity
II. Model Formulation – ISGM:
Post-Assumption 2 Simplified Model
Part III – Model Comparison
III. Model Comparison
Second-Order Growth Model (SGM)
Generalized Second-Order Growth Model (GSGM)
Indicator-Specific Growth Model (ISGM)
III. Model Comparison:
Model Similarity and Nesting
Pre-Transformation Second-Order Growth Model (SGM)
Post-Transformation Second-Order Growth Model (SGM)
Transformation (Schmid & Leiman, 1957)
III. Model Comparison:
Model Similarity and Nesting
Second-Order Growth Model (SGM)
Generalized Second-Order Growth Model (GSGM)
State-Variability Components (Latent State Residual Factors)
III. Model Comparison:
Proportionality Constraint
Second-Order Growth Model (SGM) (Schmiedek & Li, 2004)
III. Model Comparison:
Proportionality Constraint
Generalized Second-Order Growth Model (GSGM)
III. Model Comparison:
Proportionality Constraint
Indicator-Specific Growth Model (ISGM)
III. Model Comparison:
Measurement Invariance (MI)
Generalized Second-Order Growth Model (GSGM)
Necessary for Meaningful Interpretation of Growth
Not Strictly Necessary
III. Model Comparison:
Heterogeneity of Indicators
Generalized Second-Order Growth Model (GSGM)
Indicator-Specific Growth Model (ISGM)
Trait: One slope and one intercept.
Trait: One slope and one intercept for each indicator.
Part IV – Summary & Discussion
IV. Summary & Discussion
• Why LST Theory? – Both state and trait/growth components are clearly separated.
– Variables have a clear meaning, based on concise mathematical definitions.
• What if there are actual state (not just trait) changes across measurement occasions? – Use Multiple-Indicator Latent Growth Curve Models.
• What if Model Includes External Factors? – Will likely have identification problems if using SGM (due to proportionality
constraint).
• What if SGM/GSGM doesn’t fit well, but ISGM does? – This means that indicators are NOT homogeneous (even if you wish they
were).
• Which model should I use?
– The most parsimonious model that still fits the data.
• What if I want to know more? – See Bishop, Geiser, & Cole (In Press)
References
Bishop, J., Geiser, C., & Cole, D. (In Press). Modeling latent growth with multiple indicators: A comparison of
three approaches. Psychological Methods.
Eid, M., Courvoisier, D. S., & Lischetzke, T. (2012). Structural equation modeling of ambulatory assessment data.
In M. R. Mehl & T. S. Connor (Eds.), Handbook of research methods for studying daily life (pp. 384–406).
New York, NY: Guilford.
Eid, M., & Hoffmann, L. (1998). Measuring variability and change with an item response model for polytomous
variables. Journal of Educational and Behavioral Statistics, 23, 193–215.
Ferrer, E., Balluerka, N., & Widaman, K. F. (2008). Factorial invariance and the specification of second-order
latent growth models. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 4, 22–36.
Leite, W. L. (2007). A comparison of latent growth models for constructs measured by multiple items. Structural Equation Modeling, 14, 581–610.
McArdle, J. J. (1988). Dynamic but structural equation modeling of repeated measures data. In J. R. Nesselroade
& R. B. Cattell (Eds.), Handbook of multivariate experimental psychology, Perspectives on individual
differences (2nd ed., pp. 561–614). New York, NY: Plenum Press.
McArdle, J. J., & Epstein, D. (1987). Latent Growth Curves within Developmental Structural Equation Models.
Child Development, 58, 110–133. doi:10.2307/1130295
Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107–122.
Schmid, J., & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53–61.
Schmiedek, F., & Li, S. C. (2004). Toward an Alternative Representation for Disentangling Age-Associated
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Steyer, R., Ferring, D., & Schmitt, M. J. (1992). States and traits in psychological assessment. European Journal of Psychological Assessment, 8, 79–98.
