lecture20_s10 growth

lecture20_s10 growth - Growth curve modeling Psychology...

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Growth curve modeling Psychology 588: Covariance structure and factor models Apr 23, 2010
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Alternative formations of longitudinal data 2 11 12 1 21 22 2 1 2 T T N N NT y y y y y y y y y # # # # Subject 1 Subject 2 Subject N 11 12 1 21 22 2 12 T T NN N T yy y y y ## # % Multilevel, univariate format Single-level, multivariate format N subjects × T times
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Basic Data Relation Matrix – Cattell (1946) Subjects R-technique S-technique
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• Types of covariances analyzed by different techniques: At one occasion: ¾ R: variables x variables computed over subjects ¾ Q: subjects x subjects computed over variables At one subject: ¾ P: variables x variables computed over occasions ¾ O: occasions x occasions computed over variables At one variable: ¾ T: subjects x subjects computed over occasions ¾ S: occasions x occasions computed over subjects • Any analysis of such one covariance matrix deals with one aspect of the covariation in the data box --- data for S- technique are what we’re interested in GCM
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2-level regression model by HLM 5 ( ) 2 LQ L it i i i t it it ii i i yt t v w w αα ββ α γ ε ζ =+ + + + + + + " t = 0, 1, 2, …, T –1 (measurement time common to all individuals, alternatively may be replaced by, e.g., t it = t + t 0 i , t 0 i = i -th age at t = 0 ), v and w are respectively time-variant and -invariant predictors • Individually varying intercepts α i and linear slopes β L i of growth trends are regressed on time-invariant, level-2 predictor w --- random effects (i.e., random variables) in that α i and β L i are unobservable, individually varying quantities, with some assumed parametric distribution Level 1: Level 2:
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• In contrast, all level-1 parameters without a subscript for individuals (e.g., γ t ) are fixed effects which are common to all individuals (the level-2 units) • Limitation in modeling growth curve polynomials --- statistically testable up to T –1 trend components (e.g., up to the cubic component given 4 time points per individual) • Linear scaling of time for a linear growth curve: () L0 LL 0 , it i i it ii i i t yt t c t d dc t α βε ββ ε =+ + = + + + A growth curve is defined as a function of time (with or without some time-specific adjustment by v it ), and so its interpretation should depend on the used time scale
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lecture20_s10 growth - Growth curve modeling Psychology...

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