Presentation on theme: "Latent Growth Curve Models"— Presentation transcript:
1 Latent Growth Curve Models Patrick Sturgis,Department of Sociology, University of Surrey
2 Overview Random effects as latent variables Growth parameters Specifying time in LGC modelsLinear GrowthNon-linear growthExplaining GrowthFixed and time-varying predictorsBenefits of SEM framework
3 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.
4 A Single Latent Variable Model LVX11X12X13X14E11E2E3E4same item at 4 time points4 different itemsConstrain factor loadingsEstimate factor loadingsEstimate mean and variance of trajectory of change over timeEstimate mean and variance of underlying factor
5 Repeated Measures & Random Effects We have average (or ‘fixed’) effects for the population as a wholeAnd individual variability (or ‘random’) effects around these average coefficients
6 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.
7 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.
8 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.
9 A Linear Growth Curve Model Constraining values of the intercept to 1 makes this parameter indicate initial statusConstraining values of the slope to 0,1,2,3 makes this parameter indicate linear change1123
10 Quadratic Growth Add additional latent variables with factor X1X2X3X4Add additional latentvariables with factorloadings constrainedto powers of the linearslope1123941ICEPTSLOPEQUAD
11 File structure for LGCFor 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.
12 A (made up) ExampleWe 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.
13 Linear Growth Example mean=11.2 (1.4) p<0.001 23mean=11.2 (1.4) p<0.001variance =4.1 (0.8) p<0.001mean=1.3 (0.25) p<0.001variance =0.6 (0.1) p<0.001
14 InterpretationThe average level of knowledge at time point one was 11.2There 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.
15 Explaining GrowthUp 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.
16 Time-Invariant Predictors Does initial status influence rate of growth?1123Do men have a different initial status than women?Do men grow at a different rate than women?Gender(women = 0; men=1)
18 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 modelHLMMlWinStata (RE, FE)
19 Fixed Effects/Unit Heterogeneity A fixed effects specification removes ‘unit effects’This controls for all observed and unobserved invariant unit characteristicsHighly desirable when one’s interest is in the effect of time varying variables on the outcomeThis is done by allowing the random effect to be correlated with all observed covariatesDownside=no information about effect of time invariant variables, possible efficiency lossSEM allows various hybrid models which fall between the classic random and fixed effect specifications
22 Hybrid model Remove equality constraint on beta weights Allow correlatederrorsZIntroduce Time-InvariantCovariate that has indirectEffect on X
23 Multiple Indicator LGC Models Single indicators assume concepts measured without errorMultiple indicators allow correction for systematic and random errorReduced likelihood of Type II errors (failing to reject false null)Tests for longitudinal meaning invarianceAllows modeling of measurement error covariance structure
25 Other Benefits of SEM Global tests and assessments of model fit Full Information Maximum Likelihood for missing dataDecomposition of effects – total, direct and indirectProbability weightsComplex sample data
26 Multiple Process Models SLOPE1ICEPT1X1t1X1t2X1t3X1t4E11E2E3E423ICEPT2SLOPE2Y1t1Y1t2Y1t3X1t4E51E6E7E823Does rate of growth on one variable influence rate of growth on the other?Does initial status on one variable influence development on the other?