1. Multivariate Statistics Latent Growth Curve Modelling. Random effects as latent variables: SEM for repeated measures data Dr Patrick Sturgis University of Surrey W. M. van der Veld University of Amsterdam

2. Overview Random effects (multi-level) models for repeated measures data. Random effects as latent variables. Specifying time in LGC models. Growth parameters. Linear Growth. Predicting Growth. An example: Issue Voting

3. Repeated Measures & Random Effects A problem when analysing panel data is how to account for the correlation between observations on the same subject. Why is this a problem? Say we have measured variable A at 3 points in time, using the same question and the same respondents. The correlation is now due to: –The variable that is measured, which is good, but also –The respondent who answered the question repeatedly, which might lead to ‘extra correlation’.

4. Repeated Measures & Random Effects Different approaches handle this problem in different ways. Here we discuss the multilevel approach, and more specific, the random effects model. In that model it is assumed that: –correlations between repeated observations arise because the regression coefficients vary across subjects. –So, we have average (or ‘fixed’) effects for the population as a whole. –And individual variability (or ‘random’) effects around these average coefficients. This is also known as (latent) growth curve modelling. Where the growth curves or individual changes across time are modelled. –Each time point represents the fixed effect. –Individual change represents the random effect.

5. Regression Assumptions We now make the basic assumptions that in the model that are needed for correct OLS estimation: y i = α + βx + ε i where i=1,2,….,n 1.ε i is a random variable with E(ε i ) = 0 and var(ε i ) = σ 2 2.cov(ε i,ε j ) = 0where i ≠ j 3.cov(ε i,x i ) = 0, no spurious relations 4.ε i is normally distributed with mean zero and variance σ 2.

6. SEM for Repeated Measures We have already discussed the confirmatory factor model and how it is used in a cross-sectional analysis. The same framework can be used on repeated measured data to overcome the correlated residuals problem. The mean of a latent variable is used to estimate to the average or ‘fixed’ effect. The variance of a latent variable represents individual heterogeneity around the fixed coefficient – the ‘random’ effect. 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.

7. LV X11X12X13X14 E1 1 E2 1 E3 1 E4 1 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

8. Random Effects as Latent Variables So, it turns out that another way of estimating growth trajectories on repeated measures data is as latent variables in SEM models. 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.

9. 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 (usually denoting initial status). And one to estimate the mean and variance of the slope (denoting change over time).

10. Specifying Time in LGC Models In random effect 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.

11. A Linear Growth Curve Model 1 1 1 1 1 0 2 3 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

12. 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.

13. An Example We are interested in the development of knowledge of SEM during a course. 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.

14. 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 1 1 1 1 1 0 2 3

15. 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.

16. Graphical Displays It is useful to graph observed and fitted growth trajectories. This gives us a clear picture of heterogeneity in individual development. This is useful for determining which time function(s) to specify. And can highlight model mis-specifications in a way that is difficult to spot with just the numerical estimates.

17. Observed Individual Trajectories

18. Fitted Trajectories

19. 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.

20. Predicting Growth Some predictors of growth do not change during the period of observation. E.g. sex, parental social class, date of birth. These are referred to as ‘fixed’ or time-constant. Other predictors change over time and may influence the outcome variable. E.g. parental status, health status. These are referred to as time-varying covariates.

21. Fixed Predictors of Growth 1 1 1 1 1 0 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) Does initial status influence rate of growth?

22. Example: Issue Voting Proximity to parties on issue dimensions strongly related to political preferences. All previous investigations use between-person analysis of cross-sectional data. Can individual change in issue proximity predict individual change in party evaluation over the 5 years of the panel? Is this relationship moderated by level of political knowledge? British Election Panel Study 1997-2001.

23. Expectation In cross-sectional analysis it is found that issue proximity does predict party evaluation, and when the political knowledge is taken into account, then for those with –High knowledge the prediction is better, –Low knowledge the prediction is still good. What would happen if we take into account individual change across time, using a LGC model? –If the change is systematic we will find the same result as in the cross-sectional analysis. –If the change is random across time, we will find different results.

24. Direction and Proximity P1P2VP3 Proximity – voter prefers party closest to them Direction – voter prefers party strongest on same side of issue as them LeftRight Penalty applied to parties outside ‘region of acceptability’

25. Direction and Proximity P1P2VP3 Proximity – voter prefers party closest to them Direction – voter prefers party strongest on same side of issue as them LeftRight Penalty applied to parties outside ‘region of acceptability’

26. Issue Dimensions European Integration –Some people feel that Britain should do all it can to unite fully with the European Union. Other people feel that Britain should do all it can to protect its independence from the European Union. Taxation and spending –Some people feel that the government should put up taxes a lot and spend much more on health and social services. Other people feel that the government should cut taxes a lot and spend much less on health and social services.

27. Issue Dimensions Income redistribution –Some people feel that government should make much greater efforts to make people’s incomes more equal. Other people feel that government should be much less concerned about how equal people’s incomes are. Unemployment and Inflation –Some people feel that getting people back to work should be the government's top priority. Other people feel that keeping prices down should be the government's top priority.

28. Derive proximity measure Issue preferences should be measured or recoded so that there is a zero midpoint. In order to allow negative and positive preferences. Spatial (general proximity measure): –Take the differences between personal and party stance Directional –Take the product of personal and party stance Proximity measure, method 1: –Use respondents issue preference –Use respondent placement of party-stance on issue Proximity measure, method 2: –Use respondents issue preference –Use the sample mean placement of party-stance on the issue. Let’s focus on the spatial measure

29. Spatial & Directional Scores 97-01

30. Party Evaluations 1997-2001 Choose a phrase from this scale to say how you feel about the Labour/Conservative/Liberal Democrat party 5. Strongly Against 4. Against 3. Neither/Nor 2. In Favour 1. Strongly in Favour

31. Path Diagram for LGC Models

32. Cross-Sectional Betas of party evaluation on proximity by Knowledge of Party Positions

33. Betas of party evaluation slope on proximity slope from LGC models by Knowledge of Party Positions

34. Conclusions For more sophisticated voters, change in policy proximity correlated with change in evaluation. No relationship between change in policy proximity and evaluation for least sophisticated. Cross-sectional parameters tell us nothing about temporal dimension of relationships.