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Latent Growth Curve Models

Patrick Sturgis,

Department of Sociology, University of Surrey

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Overview

• Random effects as latent variables• Growth parameters• Specifying time in LGC models• Linear Growth• Non-linear growth• Explaining Growth• Fixed and time-varying predictors• Benefits of SEM framework

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SEM for Repeated Measures

• The SEM framework can be used on repeated measured data to model individual growth trajectories.

• For cross-sectional data latent variables are specified as a function of different items at the same time point.

• For repeated measures data, latent variables are specified as a function of the same item at different time points.

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LV

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A Single Latent Variable Model

4 different items

same item at 4 time points

Estimate mean and variance of underlying factor

Estimate mean and variance of trajectory of change over time

Estimate factor loadingsConstrain factor loadings

LV

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Repeated Measures & Random Effects

• We have average (or ‘fixed’) effects for the population as a whole

• And individual variability (or ‘random’) effects around these average coefficients

ittiiit xy

ii ii

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Random Effects as Latent Variables

• In LGC:• The mean of the latent variable is the fixed

part of the model.– It indicates the average for the parameter in the

population.

• The variance of the latent variable is the random part of the model.– It indicates individual heterogeneity around the

average.– Or inter-individual difference in intra-individual

change.

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Growth Parameters

• The earlier path diagram was an over-simplification.

• In practice we require at least two latent variables to describe growth.

• One to estimate the mean and variance of the intercept.

• And one to estimate the mean and variance of the slope.

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Specifying Time in LGC Models

• In random effects models, time is included as an independent variable:

• In LGC models, time is included via the factor loadings of the latent variables.

• We constrain the factor loadings to take on particular values.

• The number of latent variables and the values of the constrained loadings specify the shape of the trajectory.

ijijiiij xy 10

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ICEPT SLOPE

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A Linear Growth Curve Model

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Constraining values of the intercept to 1 makes this parameter indicate initial status

Constraining values of the slope to 0,1,2,3 makes this parameter indicate linear change

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Quadratic Growth

ICEPT SLOPE

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QUAD

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Add additional latentvariables with factor loadings constrained to powers of the linearslope

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File structure for LGC

• For random effect models, we use ‘long’ data file format.

• There are as many rows as there are observations.

• For LGC, we use ‘wide’ file formats.

• Each case (e.g. respondent) has only one row in the data file.

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A (made up) Example

• We are interested in the development of knowledge of longitudinal data analysis.

• We have measures of knowledge on individual students taken at 4 time points.

• Test scores have a minimum value of zero and a maximum value of 25.

• We specify linear growth.

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Linear Growth Example

mean=11.2 (1.4) p<0.001

variance =4.1 (0.8) p<0.001

mean=1.3 (0.25) p<0.001

variance =0.6 (0.1) p<0.001

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ICEPT SLOPE

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Interpretation

• The average level of knowledge at time point one was 11.2

• There was significant variation across respondents in this initial status.

• On average, students increased their knowledge score by 1.2 units at each time point.

• There was significant variation across respondents in this rate of growth.

• Having established this descriptive picture, we will want to explain this variation.

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Explaining Growth

• Up to this point the models have been concerned only with describing growth.

• These are unconditional LGC models.• We can add predictors of growth to explain why

some people grow more quickly than others.• These are conditional LGC models.• This is equivalent to fitting an interaction

between time and predictor variables in random effects models.

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Time-Invariant Predictors

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Do men have a different initial status than women?

Do men grow at a different rate than women?

Gender

(women = 0; men=1)

Does initial status influence rate of growth?

ICEPT SLOPE

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Time-varying predictors of growth

ICEPT

SLOPE

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Why SEM?

• Most of this kind of stuff could be done using random/fixed effects.

• SEM has some specific advantages which might lead us to prefer it over potential alternatives:– SPSS linear mixed model– HLM– MlWin– Stata (RE, FE)

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Fixed Effects/Unit Heterogeneity

• A fixed effects specification removes ‘unit effects’• This controls for all observed and unobserved invariant

unit characteristics• Highly desirable when one’s interest is in the effect of

time varying variables on the outcome• This is done by allowing the random effect to be

correlated with all observed covariates• Downside=no information about effect of time invariant

variables, possible efficiency loss• SEM allows various hybrid models which fall between

the classic random and fixed effect specifications

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Random effect model

ICEPT

SLOPE

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E1 E2 E3 E4

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w1 w2 w3 w4

b b b b

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Fixed effect model

ICEPT

SLOPE

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w1 w2 w3 w4

b b b b

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Hybrid model

ICEPT

SLOPE

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w1 w2 w3 w4

b b b b

Z

Introduce Time-Invariant Covariate that has indirect Effect on X

Remove equality constrainton beta weights

Allow correlatederrors

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Multiple Indicator LGC Models

• Single indicators assume concepts measured without error

• Multiple indicators allow correction for systematic and random error

• Reduced likelihood of Type II errors (failing to reject false null)

• Tests for longitudinal meaning invariance• Allows modeling of measurement error

covariance structure

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Multiple Indicator LGC Models

MI1

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MI2

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MI3

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INT SLOPE

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Other Benefits of SEM

• Global tests and assessments of model fit

• Full Information Maximum Likelihood for missing data

• Decomposition of effects – total, direct and indirect

• Probability weights

• Complex sample data

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SLO

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ICE

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Does initial status on one variable influence development on the other?

Does rate of growth on one variable influence rate of growth on the other?

Multiple Process Models