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Latent Growth Modeling Chongming Yang Research Support Center FHSS College

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Objectives Understand the basics of LGM Learn about some applications Obtain some hands-on experience

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Limitations of Traditional Repeated ANOVA / MANOVA / GLM Concern group-mean changes over time Variances of changes not explicit parameters List-wise deletion of cases with missing values Can’t incorporate time-variant covariate

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Recent Approaches Individual changes Multilevel/Mixed /HL modeling Generalized Estimating Equations (GEE) Structural equation modeling (latent growth (curve) modeling)

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Long Format Data Layout—Trajectory(T) (for Multilevel Modeling) IDDV Y Time IV X 16.504.5 17.016.4 18.024.8 18.536.7 28.805.7 29.016.8 29.227.2 29.437.5 …

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Run Linear Regression for each case y it = i + i T + it – i = individual – T = time variable

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Intercept & Slope

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Individual Level Summary Linear Regression id /classinterceptslope … 17.722.50 28.513.26 37.644.07 416.250.92 513.171.27 611.213.85 79.054.21 817.111.32 915.322.11

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Model Intercepts and Slopes = i + i = s + s IF variance of i = 0, Then = i, starting the same IF variance of s = 0, Then = s, changing the same Thus variances of i and s are important parameters

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Unconditional Growth Model-- Growth Model without Covariates y t = + T + t = i + i (i = intercept here) = s + s

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Estimating Different Trajectories IDDependent Variable LinearNon- equidistant Quadratic curve Logarithmic curve Exponential curve 16.50.0000 17.01.1.69.172 18.02.2.41.10.639 18.53.3.91.391.909 28.80.0000 29.01.1.69.172 29.22.5.25.41.10.639 29.43.35.91.391.909 …

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Conditional Growth Model-- Growth Model with Covariates y t = i + i T + t 3 + t i = i + i1 1 + i2 2 + i i = s + s1 1 + s2 2 + s Note: i=individual, t = time, 1 and 2 = time-invariant covariates, 3 = time- variant covariate. i and I are functions of 1,2…n, y it is also a function of 3i.

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Limitations of Multilevel/Mixed Modeling No latent variables Growth pattern has to be specified No indirect effect No time-variant covariates

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Latent Growth Curve Modeling within SEM Framework Data—wide format idx1x2t1y1t2y1t3y1… 125123 234345 343678

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Measurement Model of Y y = + +

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Specific Measurement Models y 1 = 1 + 1 + 1 y 2 = 2 + 2 + 2 y 3 = 3 + 3 + 3 y 4 = 4 + 4 + 4 = i + i = s + s

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Unconditional Latent Growth Model y = + + y = 0 + 1* i + s +

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Five Parameters to Interpret Mean & Variance of Intercept Factor (2) Mean & Variance of Slope Factor (2) Covariance /correlation between Intercept and Slope factors (1)

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Interchangeable Concepts Intercept = initial level = overall level Slope = trajectory = trend = change rate Time scores: factor loadings of the slope factor

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Growth Pattern Specification (slope-factor loadings) Linear: Time Scores = 0, 1, 2, 3 … (0, 1, 2.5, 3.5…) Quadratic: Time Scores = 0,.1,.4,.9, 1.6 Logarithmic: Time Scores = 0, 0.69, 1.10, 1.39… Exponential: Time Scores = 0,.172,.639, 1.909, To be freely estimated: Time Scores = 0, 1, blank, blank…

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Parallel Growths

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Cross-lagged Model

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Parallel Growth with Covariates

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Antecedent and Subsequent (Sequential) Processes

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Control Group Experimental Group

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Cohort 1 Cohort 2 Cohort 3

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Piecewise Growth Model Slope1 Slope2

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Two-part Growth Model (for data with floor effect or lots of 0) Dummy- Coding 0-1 Original Rating 0-4 Continuous Indicators Categorical Indicators

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Mixture Growth Modeling Heterogeneous subgroups in one sample Each subgroup has a unique growth pattern Differences in means of intercept and slopes are maximized across subgroups Within-class variances of intercept and slopes are minimized and typically held constant across all subgroups Covariance of intercept and slope equal or different across groups

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Growth Mixtures

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T-scores approach Use a variable that is different from the one that indicates measurement time to examine individual changes Example – Sample varies in age – Measurement was collected over time – Research question: How measurement changes with age?

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Advantage of SEM Approach Flexible curve shape via estimation Multiple processes Indirect effects Time-variant and invariant covariates Model indirect effects Model growth of latent constructs Multiple group analysis and test of parameter equivalence Identify heterogeneous subgroups with unique trajectories

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Model Specification growth of observed variable ANALYSIS: MODEL: I S | y1@0 y2@1 y3 y4 ;

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Specify Growth Model of Factors with Continuous Indicators MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2); (invariant measurement over time) [Y11-Y13@0 Y21-Y23@0 Y31-Y33@0 F1-F3@0]; (intercepts fixed at 0) I S | F1@0 F2@1 F3 F4 ;

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Why fix intercepts at 0 ? Y = 1 + F1 F1 = 2 + Intercept Y = ( 1 = 2 =0) + Intercept

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Specify Growth Model of Factors with Categorical Indicators MODEL: F1 BY y11 y12(1) y13(2); F2 BY y21 y22(1) y23(2); F3 BY y31 y32(1) y33(2); [Y11$1-Y13$1](3); [Y21$1-Y23$1](4); [Y31$1-Y33$1](5); (equal thresholds) [F1-F3@0]; (intercepts fixed at 0) [I@0]; (initial mean fixed 0, because no objective measurement for I) I S | F1@0 F2@1 F3 F4 ;

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Practical Tip Specify a growth trajectory pattern to ensure the model runs Examine sample and model estimated trajectories to determine the best pattern

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Practical Issues Two measurement—ANCOVA or LGCM with variances of intercept and slope factors fixed at 0 Three just identified growth (specify trajectory) Four measurements are recommended for flexibility in Test invariance of measurement over time when estimating growth of factors Mean of Intercept factor needs to be fixed at zero when estimating growth of factors with categorical indicators Thresholds of categorical indicators need to be constrained to be equal over time

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Unstandardized or Standardized Estimates? Report unstandardized If the growth in observed variable is modeled, If latent construct measured with indicators are, report standardized

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Resources Bollen K. A., & Curren, P. J. (2006). Latent curve models: A structural equation perspective. John Wiley & Sons: Hoboken, New Jersey Duncan, T. E., Duncan, S. C., Strycker, L. A., Li, F., & Alpert A. (1999). An introduction to latent variable growth curve modeling: Concepts, issues, and applications. Lawrence Erlbaum Associates, Publishers: Mahwah, New Jersey www.statmodel.com Search under paper and discussion for papers and answers to problems www.statmodel.com

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Practice 1.Estimate an unconditional growth model 2.Compare various trajectories, linear, curve, or unknown to determine which growth model fit the data best 3.Incorporate covariates 4.Use sex or race as grouping variable and test if the two groups have similar slopes. 5.Explore mixture growth modeling

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HSRP 734: Advanced Statistical Methods July 24, 2008.

HSRP 734: Advanced Statistical Methods July 24, 2008.

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