Latent Growth Curve Models Patrick Sturgis, Department of Sociology, University of Surrey

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

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.

A Single Latent Variable Model LV X11 X12 X13 X14 E1 1 E2 E3 E4 same item at 4 time points 4 different items Constrain factor loadings Estimate factor loadings Estimate mean and variance of trajectory of change over time Estimate mean and variance of underlying factor

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

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.

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.

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.

A Linear Growth Curve Model 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 1 1 2 3

Quadratic Growth Add additional latent variables with factor X1 X2 X3 X4 Add additional latent variables with factor loadings constrained to powers of the linear slope 1 1 2 3 9 4 1 ICEPT SLOPE QUAD

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.

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.

Linear Growth Example mean=11.2 (1.4) p<0.001 2 3 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

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.

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.

Time-Invariant Predictors Does initial status influence rate of growth? 1 1 2 3 Do men have a different initial status than women? Do men grow at a different rate than women? Gender (women = 0; men=1)

Time-varying predictors of growth

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)

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

Random effect model b b b b

Fixed effect model b b b b

Hybrid model Remove equality constraint on beta weights Allow correlated errors Z Introduce Time-Invariant Covariate that has indirect Effect on X

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

Multiple Indicator LGC Models

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

Multiple Process Models SLOPE1 ICEPT1 X1t1 X1t2 X1t3 X1t4 E1 1 E2 E3 E4 2 3 ICEPT2 SLOPE2 Y1t1 Y1t2 Y1t3 X1t4 E5 1 E6 E7 E8 2 3 Does rate of growth on one variable influence rate of growth on the other? Does initial status on one variable influence development on the other?