Autoregressive and Growth Curve Models Rachael Bedford Mplus

Autoregressive and Growth Curve Models Rachael Bedford Mplus: Longitudinal Analysis Workshop 26/09/2017

Overview � Examples � Brief using delinquency recap of SEM & autoregressive models � Growth Curve Models: ◦ Linear ◦ Quadratic � Summary

Delinquency Development � Delinquency � What 12 – 15 years might relate to changes in delinquency? ◦ Being bullied: victimization ◦ Substance use ◦ Deprivation index (e. g. free school meals)

Age-Crime curve This is the MEAN curve. 1 60 What about individual variability? 2 50 3 40 4 5 30 6 20 7 8 10 0 9 12 13 14 15 10 mean

SEM notation Observed variable Latent factor Effects of one variable on another (e. g. , regression) Covariance or correlation

Autoregressive Models � Analyse how the variance-covariance matrix of variables changes over time � First order autoregressive models for delinquency and victimisation at 12, 13, 14, 15 � Tell us if there is continuity over time. DEL 12 DEL 13 DEL 14 DEL 15 VIC 12 VIC 13 VIC 14 VIC 15

Autoregressive Models � It is likely that delinquency and victimisation at each age are correlated. DEL 12 DEL 13 DEL 14 DEL 15 VIC 12 VIC 13 VIC 14 VIC 15 � And they may be related over time (cross-lag) DEL 12 DEL 13 DEL 14 DEL 15 VIC 12 VIC 13 VIC 14 VIC 15

Mini Practical � What DEL 13 VIC 13 is the code for this model in Mplus? DEL 14 VIC 14 DEL 15 VIC 15 Model: DEL 14 on DEL 13; DEL 15 on DEL 14; VIC 14 on VIC 13; VIC 15 on VIC 14; DEL 13 with VIC 13; DEL 14 with VIC 14; DEL 15 with VIC 15;

Key Model Commands Mplus Code: [ ] – mean [X 1] EL 2 on EL 1; - variance X 1 EL 3 on EL 2; EL 4 on EL 3; with – correlation or covariance X 1 with X 2 by – factor is indicated by F 1 by X 1 X 2 X 3 on – regression Y 1 on X 1 EL 1 -EL 4 on group; e EL 1 ß e e e EL 2 EL 3 EL 4 ; - end of a command Y 1 on X 1; Group

Growth Curve Models (GCMs) � Like autoregressive models, GCMs explain change in the variance-covariance matrix over time. � But, GCMs enable inter- and intra-individual change to be quantified. � 3 time points-linear growth � 3+ time points-can fit curvilinear functions

Intercept & slope latent factors Expressive language 16 14 12 10 8 6 4 2 0 6 12 24 Age in months 36

Basic GCM explained Regression of intercept and slope on group. Mean of slope freely estimated. Regression coefficients of language on intercept fixed at 1 Regression coefficients of language on slope fixed at age/difference between visits.

Adding a quadratic term Del 12 intercept Del 13 Covariate slope Del 14 0 1 4 quadratic 9 Del 15

Equations! (Thank you Muthéns) i 13 14 s 15

Fixing slope coefficients � The slope pathways or coefficients can be fixed to reflect the time between visits ◦ i. e. 12, 13, 14, 15 months 0, 1, 2, 3 1 60 2 50 3 4 40 5 30 6 7 20 8 10 0 9 10 12 13 14 15 mean 1 60 2 50 3 4 40 5 30 6 7 20 8 10 0 9 10 0 1 2 3 mean

Example 1: Delinquency Growth Curve Model: Delinquency 12, 13, 14, 15 & 16 years. Linear growth Mplus code Plot : type = plot 3 ; Gives the most graphical options I always use this command. Model: i s | DEL 12@0 DEL 13@1 DEL 14@2 DEL 15@3; DEL 16@4; i with s; i with q; s with q; Specifies GCM Allows intercept and slope and quadratic factors to correlate

Example 1: Output Do the means look reasonable? Should we have included the quadratic term? Does the model fit the data well?

Example 1: Results

Practical 1 Growth Curve Model: Delinquency 12, 13, 14, 15 & 16 years. Linear growth Mplus code Plot : type = plot 3 ; Gives the most graphical options I always use this command. Model: i s | DEL 12@0 DEL 13@1 DEL 14@2 DEL 15@3; DEL 16@4; i with s; i with q; s with q; Specifies GCM Allows intercept and slope and quadratic factors to correlate

Example 2: Delinquency & covariate Growth Curve Model: Delinquency 12, 13, 14, 15, 16 years Latent factors (i s q) predicted by gender Mplus code Gender predicting latent factors i s q. Model: i s | DEL 12@0 DEL 13@1 DEL 14@2 DEL 15@3; DEL 16@4; i with s; i with q; s with q; i s q on gender;

Example 2: Output Does gender predict the intercept or slope of delinquency?

Practical 2 Growth Curve Model: Delinquency 12, 13, 14, 15, 16 years Latent factors (i s q) predicted by gender Mplus code Gender predicting latent factors i s q. Model: i s | DEL 12@0 DEL 13@1 DEL 14@2 DEL 15@3; DEL 16@4; i with s; i with q; s with q; i s q on gender;