Multiple-Indicator Latent Growth Curve Models: An...

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

Differences in General and Specific Cognitive Abilities. Psychology and Aging, 19, 40–56.

Steyer, R., Ferring, D., & Schmitt, M. J. (1992). States and traits in psychological assessment. European Journal of Psychological Assessment, 8, 79–98.

Contact Information

• Email: jacob.bishop@usu.edu