Applied Longitudinal Data Analysis: An Introductory Course

Applied Longitudinal Data Analysis: An Introductory Course Emilio Ferrer UC Davis The Risk and Prevention in Education Sciences (RPES) Curry School of Education - UVA August 2005

Acknowledgments Materials for this workshop are the result of work interactions with: Jack McArdle John Nesselroade Aki Hamagami Kevin Grimm Nilam Ram Sy Miin Chow

Course Overview Day 1 Basis of latent growth curve and mixed-effects models Linear and nonlinear modeling Programming and fitting linear LGC models Programming and fitting nonlinear LGC models

Course Overview Day 2 Incomplete data, exogenous variables, and multiple groups Multivariate models Programming and fitting multiple groups Programming and fitting multivariate models

Course Overview Day 3 Introduction to dynamic systems and its application to developmental research Models for the analysis of individual processes Programming and fitting dynamic models 1: univariate models Programming and fitting dynamic models 2: multivariate models

Statistical Methods to Represent Growth and Change 1: Introduction to Growth Curve Modeling

Overview Introduction to growth curve modeling Basics of GCM Specification, estimation, and evaluation Examples Extensions

Objectives of Longitudinal Research (Nesselroade & Baltes, 1979) 1. Identification of intra-individual change 2. Direct identification of inter-individual differences in intraindividual change 3. Analysis of interrelationships in change 4. Analysis of causes (determinants) of intraindividual change 5. Analysis of causes (determinants) of interindividual differences in intra-change

Some Features of Longitudinal Studies Some of the same entities (at least some of them) are observed at repeated occasions The measurement and scaling of observations are known The ordering or time underlying the observations is known

Growth Curve Models Class of techniques used to study change They allow explicit testing of hypotheses regarding the structure of longitudinal data Step 1: A model of change is specified Step 2: Expectations about means and covariances are generated based on the specified model Step 3: Parameters are estimated Step 4: Model fit is evaluated (discrepancy between model expectations and observed data)

Development Origins - Rao (1958), Tucker (1958, 1966), Meredith & Tisak (1984) Expansions - Browne & DuToit, (1991), McArdle (1988), McArdle & Epstein (1987) Overviews - McArdle & Nesselroade (2003), Singer & Willet (2003), Bollen & Curran (forthcoming)

Longitudinal Individual Data 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Longitudinal Individual Data WISC-R data from N=204 children Repeated measurements at grades 1, 2, 4, and 6 WISC total means = 18.8, 26.6, 36.0, and 47.3 WISC total SDs = 6.4, 7.3, 7.7 and 10.4 WISC total correlations =.765 -.867

Describing the Growth 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Describing the Growth: Initial Level 80 70 60 WISC Score 50 40 30 µ 0 + σ 0 20 10 0 1 2 3 4 5 6 Grade at Testing

Describing the Growth: Slope 80 70 60 WISC Score 50 40 30 µ s + σ s 20 µ 0 + σ 0 10 0 1 2 3 4 5 6 Grade at Testing

Basic Growth Model Factor Model σ 0 2 σ 0,s σ s 2 y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[3] e y[4] e y[5] e y[6]

Basic Growth Model Factor Model F 0 F s u 1 u 2 u 3 u 4 u 5 u 6 Y 1 β 1 1 u σ 1 u 2 0 0 0 0 0 Y 2 1 β 2 u 0 σ 2 u 2 0 0 0 0 Λ = Y 3 1 β 3 Ψ = u 3 0 0 σ u 2 0 0 0 Y 4 1 β 4 u 0 0 0 σ 4 u 2 0 0 Y 5 1 β 5 u 0 0 0 0 σ 5 u 2 0 Y 6 1 β 6 u 0 0 0 0 0 6 σ 2 u F 1 F 2 Φ = F 1 σ 2 0 σ 0s y i = µ + Λf i + u F 2 σ 0s σ 2 s Ε = ΛΦΛ' + Ψ

Basic Growth Model With Means ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[3] e y[4] e y[5] e y[6]

First level model y 0 Basics of Growth Models Y [t]n = y 0n + B [t] y sn + e [t]n = latent score representing an individual s initial level B [t] = group basis parameters represent timing y s = latent slopes for the individual change over time e [t] = errors of measurements Second level model y 0n = µ 01 + e 0n y sn = µ s1 + e 1n the levels and slope scores have means (µ i,j ) and residuals (e 1 ), and the residuals have variance components (σ i 2 )

Basics of Growth Models Fixed or group terms: 1. µ 0 = the mean of the initial level scores y 0 2. µ s = the mean of the slope scores y s 3. B [t] = the basis coefficients of the slope scores y s Random or individual terms: 4. σ e2 = the variance of the residual score e [t] 5. σ 02 = the variance of the initial level scores y 0 6. σ s2 = the variance of the slope scores y s 7. σ 0s = the covariance of the level and slope scores

Basics of Growth Models These techniques go by a number of different names: Mixed-effects models (SAS PROC MIXED, NLMIXED, MIXNOR, MIXREG) Multi-level models (SPSS HLM, MLn) Random coefficient models (VARCL) Hierarchical linear models (SPSS HLM) Latent growth models (SEM LISREL, Mx, AMOS, etc.) These models are algebraically identical with varied statistical computations

LGC vs. RM (M)ANOVA Group effects vs. individual change or growth MANOVA needs balanced designs same number of observation per subject same interval across assessments (and across subjects) MANOVA can t handle missing data Time is treated as a categorical variable Limited handling of covariates

Growth Hypotheses Level Only Model Y [t]n = y 0n + e [t]n Linear Slope Model Y [t]n = y 0n + B [t] y sn + e [t]n with B [t] fixed = 0, 1, 2, t Latent Slope Model Y [t]n = y 0n + B [t] y sn + e [t]n with B [t] free More complex functional relations Y [t]n = y 0n + B 1[t] y s1n + B 2[t] y s2n + e [t]n

Level Only Model (Y [t]n = y 0n + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Level Only Growth Model µ 0 1 y 0 1 1 1 1 1 1 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[6]

No-Growth - Mean Expectations µ = 1 Y [1] µ 0 Y [2] µ 0 Y [4] µ 0 Y [6] µ 0

No-Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 2 e Y [2] 0 σ 2 e Y [4] 0 0 σ 2 e Y [6] 0 0 0 σ 2 e

No-Growth Model (with σ 02 ) y 0 * σ 0 µ 0 1 y 0 1 1 1 1 1 1 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[6]

No-Growth - Mean Expectations µ = 1 Y [1] µ 0 Y [2] µ 0 Y [4] µ 0 Y [6] µ 0

No-Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 2 0 + σ 2 e Y [2] σ 2 0 σ 2 0 + σ 2 e Y [4] σ 2 0 σ 2 0 σ 02 + σ 2 e Y [6] σ 2 0 σ 2 0 σ 2 0 σ 02 + σ 2 e

Linear Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Linear Growth ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[6]

Linear Growth Model The mean at any time is: µ [t] = µ 0 + µ 1 B [t] µ 0 = mean of the initial level. It is usually scaledependent µ 1 = mean of the slope. It is the average group change per unit of the basis B [t] B [t] = basis coefficients of the slope scores. The value of these coefficients define the shape of the average growth curve

Linear Growth - Mean Expectations µ = 1 Y [1] µ 0 + µ s β 1 Y [2] µ 0 + µ s β 2 Y [4] µ 0 + µ s β 4 Y [6] µ 0 + µ s β 6

Linear Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 0 2 + σ e 2 + λ 1 2 σ s 2 + 2σ 0s λ 1 Y [2] σ 2 0 σ 02 + σ 2 e + λ 1 σ 2 s λ 2 + λ 2 2 σ 2 s +2 λ 1 σ 0s λ 2 + 2σ 0s λ 2 Y [4] σ 2 0 σ 2 0 σ 2 0 + σ 2 e + λ 1 σ 2 s λ 4 + λ 2 σ 2 s λ 4 + λ 2 4 σ 2 s +2 λ 1 σ 0s λ 4 +2 λ 2 σ 0s λ 4 + 2σ 0s λ 4 Y [6] σ 2 0 σ 2 0 σ 2 0 σ 2 0 + σ 2 e + λ 1 σ 2 s λ 6 + λ 2 σ 2 s λ 6 + λ 4 σ 2 s λ 6 + λ 2 6 σ 2 s +2 λ 1 σ 0s λ 6 +2 λ 2 σ 0s λ 6 +2 λ 4 σ 0s λ 6 + 2σ 0s λ 6

Latent Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Quadratic Slope (Y [t]n = y 0n + B 1[t] y s1n +B 2[t] y s2n + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Extension Variables Initial Latent Growth Model Y [t]n = y 0n + B [t] y sn + e [t]n Prediction of individual level scores y 0n = G X n + H Z n + e 0n Prediction of individual slope scores y sn = J X n + K Z n + e sn Exactly the same logic as what are now termed hierarchical or multi-level models

Latent Growth in Groups Latent growth model with groups Y (1) [t]n = L (1) n + B (1) [t] S (1) n + U (1) [t]n Y (2) [t]n = L (2) n + B (2) [t] S (2) n + U (2) [t]n Y (g) [t]n = L (g) n + B (g) [t] S (g) n + U (g) [t]n

Statistical Methods to Represent Growth and Change 2: Nonlinear Models

Nonlinear Models Most psychological phenomena are nonlinear in nature Most psychological theories are described with nonlinear relationships Y = f(x), with the function f changing at different levels of X Some classic examples include learning curves, developmental stages, or the inverted function of arousal and performance Some more recent examples include nonlinear dynamics

Theoretical Curves of Gf-Gc (Cattell, 1971, 1987)

Empirical Nonlinear Data 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability 520 500 480 460 440 0 10 20 30 40 50 60 70 80 90 100 Age

Fitted Curves of Fluid and Crystallized WJ-R Factors General Fluid Ability (Gf) score as a function of Age General Crystallized Ability (Gc) score as a function of Age 60 60 General Fluid Ability score 40 20 0-20 -40 General Crystallized Ability score 40 20 0-20 -40-60 -60 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing (McArdle, Ferrer, Hamagami, & Woodcock, 2002)

Nonlinear Models Exponential functions 70 60 Cross-sectional data Visual Matching Scores 50 40 30 20 10 0 6 8 10 12 Age (yr) 40 Cross-sectional data 30 Cross Out Scores 20 10 0 6 8 10 12 Age (yr) Y = a be c*age

Nonlinear Models There are some theoretical nonlinear curves such as Verhulst s logistic, Gompertz, von Bertalanffy (competition) Rao (1958) and Tucker (1966) principal components of repeated measures There are also mathematical (nonlinear) functions that can be fitted to the data with no theoretical basis An alternative approach is to estimate a set of latent coefficients based on the data

Nonlinear Models Fixed Coefficients One option is to use the basis coefficients to specify a particular function Λ[t] = [1, 1, 2, 2, 3, 3] for steps Λ[t] = [1, 2, 3, 3, 2, 1] for up and down Λ[t] = [1, -1, 1, -1, 1, -1] for cycles Another possibility is to specify the basis coefficients as unknown but functionally related constants Λ[t] = q[t] In all these cases, the parameter estimates may be altered but other features remain the same: the value of the model expectations, the goodness-of-fit, and the change in goodness-of-fit due to a latent slope

Nonlinear Growth Fixed Coefficients ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 0 1 1 1 1 1 1 1 1 2 2 y s 3 3 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[6]

Nonlinear Models Polynomials Quadratic model Y [t]n = y 0n + B 1[t] y s1n + B 2[t] y s2n + e [t]n y 0n = latent score representing an individual s initial level B 1[t] = fixed linear weights with slopes y s1n B 2[t] = fixed quadratic weights with slopes y s2n e [t] = errors of measurements Second level model y 0n = µ 01 + e 0n y s1n = µ s1 + e 1n y s2n = µ s2 + e 2n the levels and slope scores have means (µ ij ) and residuals (e 1 ), and the residuals have variance components (σ i 2 )

Quadratic Growth ρ 0s2 ρ 0s1 ρ s1,s2 y 0 * y s1 * y s2 * σ 0 1 µ s2 σ s1 σ s2 µ 0 µ s1 y 0 B [t] y s1 y s2 1/2 B [t] 2 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[6]

Quadratic Slope (Y [t]n = y 0n + B 1[t] y s1n +B 2[t] y s2n + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Nonlinear Models Splines By defining a knot point k, time can be divided in segments and a nonlinear curve expressed as Y [t]n = y 0n + B 1[t] y s1n + B 2[t] y s2n + e [t]n where B 1[t] = T k, iff t < k, and B 2[t] = T k, iff t > k y 0n = intercept the predicted score of Y [0] at k (B 1[t] = B 2[t] = 0) y s1 = slope term before k change in the predicted score of Y [t] for one unit change in B 1[t] before k y s2 = slope term after k change in the predicted score of Y [t] for one unit change in B 2[t] after k e [t] = errors of measurements the part of Y [t] that unpredicted and independent of the specification B [t]

Nonlinear Models Splines Linear spline model (piecewise model) Y tn = y 0n + B 1 (t<k) y s1n + B 2 (t >k) y s2n + e tn For example, given T = 6 and k = 4 B 1[t] = [-3, -2, -1, 0, 0, 0], and B 2[t] = [ 0, 0, 0, 0, 1, 2], and Y 0n = intercept at k = 4 (B 1[t] = B 2[t] = 0) This model can be compared with a single-slope model via χ 2 and df It is possible to find k from the data, with individual differences (Cudeck & Klebe, 2002)

Nonlinear Models Splines 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability 520 500 480 460 440 0 10 20 30 40 50 60 70 80 90 100 Age

Nonlinear Models Splines 140 Heart Rate During Gazing Task -- Non-Attached 140 Heart Rate During Gazing Task -- Attached 130 130 120 120 110 110 Heart Rate 100 90 80 Heart Rate 100 90 80 70 70 60 60 50 50 40 2 4 6 8 10 12 14 16 18 20 22 24 Time (seconds) 40 2 4 6 8 10 12 14 16 18 20 22 24 Time (seconds)

Nonlinear Models Splines HR -- Non-Attached 110 100 90 80 110 Heart Rate During Gazing Task r t1,t1 =.22 r t1,t1 =.17 ns 2 4 6 8 10 12 14 16 18 20 22 24 Time (s) HR -- Attached 100 90 r t1,t1 =.43 r t1,t1 =.58 80 2 4 6 8 10 12 14 16 18 20 22 24 Time (s)

Nonlinear Models Residuals It is possible to model the structure of the residuals This is often used to account for changes in the individual differences (covariances) that are not reflected in the group trends (means) over time This approach uses time-series concepts about changes over time and can easily improve the fit It is easy to apply with current programs but it is important to evaluate its use

Nonlinear Models Residuals AR(1) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] σ e σ e σ e σ e e y[1] e y[2] e y[4] e y[2] σ e ey[4] β β β β β σ e e y[6]

Nonlinear Models Residuals AR(2) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] σ e σ e σ e σ e Y [6] σ e e y[1] e y[2] e y[4] e y[2] σ e ey[4] β 1 β 1 β 1 β 1 β 1 e y[6] β 2 β 2 β 2 β 2

Nonlinear Models Residuals (other) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 β 1 β 2 y s β 3 β 4 β 5 β 6 Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] σ e1 σ e2 σ e3 σ e4 σ e5 σ e6 β 1 β 1 β 1 β 1 β 1 e y[1] e y[2] e y[4] e y[2] e y[4] e y[6] β 2 β 2 β 2 β 2

Nonlinear Models Latent Coefficients It is also possible to estimate the basis coefficients as latent values (based on the data) as in a common factor model (see Rao, 1958, Tucker, 1966, Meredith & Tisak, 1990, McArdle, 1986) This requires identification constraints, e.g., Λ[t] = [0 =, β 2, β 3, β 4, β 5, 1 = ] The fixed values are used for centering (β 1 =0) and scaling (β 1 =1), and the other coefficients are estimated from the data to define the best generalized curve This model is exploratory but comparable with other alternatives via goodness-of-fit

Nonlinear Models Latent Basis ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 0 1 1 1 1 1 1 0 β 2.4 β 4 y s.8 1 Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] σ e σ e σ e σ e e y[1] e y[2] e y[4] e y[2] σ e ey[4] β β β β β σ e e y[6]

Latent Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Latent Growth - Mean Expectations µ = 1 Y [1] µ 0 + µ s β 1 Y [2] µ 0 + µ s β 2 Y [4] µ 0 + µ s β 4 Y [6] µ 0 + µ s β 6

Latent Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 0 2 + σ e 2 + λ 1 2 σ s 2 + 2σ 0s λ 1 Y [2] σ 2 0 σ 02 + σ 2 e + λ 1 σ 2 s λ 2 + λ 2 2 σ 2 s +2 λ 1 σ 0s λ 2 + 2σ 0s λ 2 Y [4] σ 2 0 σ 2 0 σ 2 0 + σ 2 e + λ 1 σ 2 s λ 4 + λ 2 σ 2 s λ 4 + λ 2 4 σ 2 s +2 λ 1 σ 0s λ 4 +2 λ 2 σ 0s λ 4 + 2σ 0s λ 4 Y [6] σ 2 0 σ 2 0 σ 2 0 σ 2 0 + σ 2 e + λ 1 σ 2 s λ 6 + λ 2 σ 2 s λ 6 + λ 4 σ 2 s λ 6 + λ 2 6 σ 2 s +2 λ 1 σ 0s λ 6 +2 λ 2 σ 0s λ 6 +2 λ 4 σ 0s λ 6 + 2σ 0s λ 6

Parameters & Fit Indices Level Linear Latent Slope Loadings β [0] β [1] β [2] β [3] β [4] β [5] 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=).2 (=).4 (=).6 (=).8 (=) 1.0 (=) 0 (=).27 (30).4 (=).60 (68).8 (=) 1.0 (=) Means/Intercepts Level µ 0 / γ 01 Slope µ s / γ s1 Mother s ED µ x 32.2 (62) 0 (=) ---- 19.7 (46) 27.7 (62) ---- 18.8 (42) 28.6 (61) ---- Regressions from X Level γ 0x Slope γ sx ---- ---- ---- ---- ---- ---- Deviations/Variances Level σ 0 Slope σ s Mother s ED σ s 2 Unique Deviation σ e 3.68 (5) 0 (=) ---- 12.9 (35) 5.63 (17) 4.85 (11) ---- 3.19 (29) 5.61 (17) 5.27 (12) ---- 2.95 (29) Correlation ρ 0s 0 (=).65 (6).55 (6) Goodness-of-Fit Parameters Degrees of freedom Likelihood Ratio L 2 RMSEA e a (p-close fit) CFI TLI Fit Changes χ 2 / df (RMSEA ) 3 11 1694.70 (.00).45.45 ---- ---- 6 8 78.20 (.00).98.98 1616/3 (1.63) 8 6 17.10 (.06) 1.00 1.00 1677/5 (1.28) 61/2 (.381)

Nonlinear Models Exponential Models Another possibility is to specify the basis coefficients as unknown but functionally related constants Λ[t] = q[t] Setting β[t]=exp[(-t-1)π] gives a nonlinear exponential shape with rate of change π to be estimated (McArdle & Hamagami, 1996) Double-exponential model (McArdle et al., 2002) Y tn = y 0n + β(age t ) y s (τ 1,τ 2 ) n + e tn with β[t] = exp(-π b Age t ) - exp(-π a Age t ) β[t] = the accumulation of a latent age basis, π b = latent rate before the age peak, π a = latent rate after the age peak, and y s (τ 1,τ 2 ) n = the combined latent slope for person n Dual nonlinear exponential shape with two rates of change (π a, π a ) representing competing forces

Nonlinear Models Exponential Models ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 e (-0π) y s e (-1π) e (-2π) e (-3π) e (-4π) e (-5π) Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] σ e σ e σ e σ e e y[1] e y[2] e y[4] e y[2] σ e ey[4] β β β β β σ e e y[6]

Nonlinear Growth (SAS-NLMIXED) TITLE: Double Exponential Model ; PROC NLMIXED; PARMS m_level=-80 m_slope=120 m_rate_a=.100 m_rate_b =.001 v_error=20 v_level=80 v_slope=10 c_levslo=-.01 ; level = m_level + d_level ; slope = m_slope + d_slope ; rate_a = m_rate_a ; rate_b = m_rate_b; traject = level+slope*(exp(-rate_b*age)-exp(-rate_a*age)); MODEL y01 ~ NORMAL(traject, v_error); RANDOM d_level d_slope ~ NORMAL([0,0], [v_level, c_levslo, v_slope]) SUBJECT=id; RUN;

Raw Data Longitudinal 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability 520 500 480 460 440 0 10 20 30 40 50 60 70 80 90 100 Age

Raw-Data Multiple-Variable Comparison 60 60 Fluid Reasoning (Gf) score 40 20 0-20 -40 Crystallized Knowledge (Gc) score 40 20 0-20 -40-60 -60 0 10 20 30 40 50 60 70 80 90 100 Age at Testing 0 10 20 30 40 50 60 70 80 90 100 Age at Testing 60 60 Processing Speed (Gs) score 40 20 0-20 -40 Short-Term Memory (Gsm) score 40 20 0-20 -40-60 -60 0 10 20 30 40 50 60 70 80 90 100 Age at Testing 0 10 20 30 40 50 60 70 80 90 100 Age at Testing

LGC Nonlinear Models (McArdle et al.2002) (a) 50 (b) 50 Quartic 0 2-Segment 0-50 -50 (c) 50 0 50 100 Age-at-Testing (d) 50 0 50 100 Age-at-Testing 5-Segment 0 Dual-Exp 0-50 -50 0 50 100 Age-at-Testing 0 50 100 Age-at-Testing

Double-Exponential Model General Fluid Ability (Gf) score as a function of Age General Crystallized Ability (Gc) score as a function of Age 60 60 General Fluid Ability score 40 20 0-20 -40 General Crystallized Ability score 40 20 0-20 -40-60 -60 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing (McArdle, Ferrer, Hamagami, & Woodcock, 2002)

Growth Curve of Fluid Reasoning Gf General Fluid Ability (Gf) score as a function of Age 60 40 General Fluid Ability score 20 0-20 -40-60 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing

Individual Modeling Predicted change in Broad Cognitive Ability (BCA) score as a function of first Age of testing 60 Predicted Broad Cognitive Ability score 40 20 0-20 -40-60 0 10 20 30 40 50 60 70 80 90 100 Age-at-Testing (first age is real data, second age is predicted scores)

Nonlinear Models Fluctuations 5 4 positive affect 3 2 1 0 1 31 61 91 121 151 18 time in days

Nonlinear Models More Complex Functions

Parameters & Fit Indices Level Linear Latent Latent with Exogenous Slope Loadings β [0] β [1] β [2] β [3] β [4] β [5] 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=) 0 (=).2 (=).4 (=).6 (=).8 (=) 1.0 (=) 0 (=).27 (30).4 (=).60 (68).8 (=) 1.0 (=) 0 (=).27 (30).4 (=).60 (68).8 (=) 1.0 (=) Means/Intercepts Level µ 0 / γ 01 Slope µ s / γ s1 Mother s ED µ x 32.2 (62) 0 (=) ---- 19.7 (46) 27.7 (62) ---- 18.8 (42) 28.6 (61) ---- 5.78 (4) 23.5 (13) 10.8 (57) Regressions from X Level γ 0x Slope γ sx ---- ---- ---- ---- ---- ---- 1.21 (9) 0.47 (3) Deviations/Variances Level σ 0 Slope σ s Mother s ED σ s 2 Unique Deviation σ e 3.68 (5) 0 (=) ---- 12.9 (35) 5.63 (17) 4.85 (11) ---- 3.19 (29) 5.61 (17) 5.27 (12) ---- 2.95 (29) 4.57 (16) 5.12 (12) 7.28 (10) 2.95 (29) Correlation ρ 0s 0 (=).65 (6).55 (6).52 (5) Goodness-of-Fit Parameters Degrees of freedom Likelihood Ratio L 2 RMSEA e a (p-close fit) CFI TLI Fit Changes χ 2 / df (RMSEA ) 3 11 1694.70 (.00).45.45 ---- ---- 6 8 78.20 (.00).98.98 1616/3 (1.63) 8 6 17.10 (.06) 1.00 1.00 1677/5 (1.28) 61/2 (.381) 12 8 22.09 (.05) 1.00 1.00 ---- ----

Fitting Latent Growth Models

Fitting Latent Growth Models 1: Univariate Models

Different Input and Output Slightly different data inputs required for different computer programs Assuming N individuals on T repeated occasions Most programs input is based on flat (N x T) raw data matrix or T means and (T x T) covariances Many mixed models (e.g., SAS PROC MIXED) use relational input of T vectors (rows) per person (T x N) with same ID code Outputs also differ, but basic model parameters and indexes are available from all programs

Example (McArdle & Epstein, 1987) Data from longitudinal study of WISC-R by on N=204 children (Osborne & Suddick, 1972) Repeated measurements at grades 1, 2, 4, and 6 WISC total means = 18.8, 26.6, 36.0, and 47.3 WISC total SDs = 6.4, 7.3, 7.7 and 10.4 WISC total correlations =.765 -.867 Fit alternative models of change to these data

WISC Data (Individual Scores) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Level Only Growth Model y 0 * σ 0 µ 0 1 y 0 1 1 1 1 1 1 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Level Only Model (AMOS input) Sem.BeginGroup "wisc.sav" Sem.Structure "total1 = (0) + (1) LEVEL + (1) E1 " Sem.Structure "total2 = (0) + (1) LEVEL + (1) E2 " Sem.Structure "total3 = (0) + (1) LEVEL + (1) E3 " Sem.Structure "total4 = (0) + (1) LEVEL + (1) E4 " Sem.Structure "total5 = (0) + (1) LEVEL + (1) E5 " Sem.Structure "total6 = (0) + (1) LEVEL + (1) E6 " Sem.Mean "LEVEL", "mn_level" Sem.Structure "E1 (v_uniq) " Sem.Structure "E2 (v_uniq) " Sem.Structure "E3 (v_uniq) " Sem.Structure "E4 (v_uniq) " Sem.Structure "E5 (v_uniq) " Sem.Structure "E6 (v_uniq)

Level Only Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- total1 <----- LEVEL 1.000 total2 <----- LEVEL 1.000 total3 <----- LEVEL 1.000 total4 <----- LEVEL 1.000 total5 <----- LEVEL 1.000 total6 <----- LEVEL 1.000 Means: Estimate S.E. C.R. Label ------ -------- ------- ------- ------- LEVEL 32.164 0.519 61.953 mn_leve Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- ------- LEVEL 13.305 5.927 2.245 E1 165.646 9.493 17.450 v_uniq E2 165.646 9.493 17.450 v_uniq E3 165.646 9.493 17.450 v_uniq E4 165.646 9.493 17.450 v_uniq E5 165.646 9.493 17.450 v_uniq E6 165.646 9.493 17.450 v_uniq Chi-square = 1697.040 Degrees of freedom = 11 Probability level = 0.000

Estimates from a Level Only Model y 0 * 3.68 32.2 1 y 0 1 1 1 1 1 1 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 12.9 12.9 12.9 12.9 12.9 12.9 e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (11) = 1697

Level Only Model (Y [t]n = y 0n + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Linear Growth Model ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 0.2 y s.4.6.8 1 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Linear Growth Model (AMOS input) Sem.BeginGroup "wisc.sav" Sem.Structure "total1 = (0) + (1) LEVEL + (0) SLOPE + (1) E1 " Sem.Structure "total2 = (0) + (1) LEVEL + (.2) SLOPE + (1) E2 " Sem.Structure "total3 = (0) + (1) LEVEL + (.4) SLOPE + (1) E3 " Sem.Structure "total4 = (0) + (1) LEVEL + (.6) SLOPE + (1) E4 " Sem.Structure "total5 = (0) + (1) LEVEL + (.8) SLOPE + (1) E5 " Sem.Structure "total6 = (0) + (1) LEVEL + (1) SLOPE + (1) E6 " End Sub Sem.Mean "LEVEL", "mn_level" Sem.Mean "SLOPE", "mn_slope" Sem.Structure "LEVEL<>SLOPE (c_ls) " Sem.Structure "E1 (v_uniq) " Sem.Structure "E2 (v_uniq) " Sem.Structure "E3 (v_uniq) " Sem.Structure "E4 (v_uniq) " Sem.Structure "E5 (v_uniq) " Sem.Structure "E6 (v_uniq) "

Linear Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- total1 <----- LEVEL 1.000 total2 <----- LEVEL 1.000 total3 <----- LEVEL 1.000 total4 <----- LEVEL 1.000 total5 <----- LEVEL 1.000 total6 <----- LEVEL 1.000 total1 <----- SLOPE 0.000 total2 <----- SLOPE 0.200 total3 <----- SLOPE 0.400 total4 <----- SLOPE 0.600 total5 <----- SLOPE 0.800 total6 <----- SLOPE 1.000 Means: Estimate S.E. C.R. Label ------ -------- ------- ------- ------- LEVEL 19.698 0.430 45.784 mn_leve SLOPE 27.704 0.447 62.008 mn_slop Covariances: Estimate S.E. C.R. Label ------------ -------- ------- ------- ------- LEVEL <-----> SLOPE 17.637 2.877 6.130 c_ls Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- ------- LEVEL 31.580 3.753 8.414 SLOPE 23.395 4.198 5.573 E1 10.104 0.709 14.248 v_uniq

Estimates from a Linear Slope Model.65 y 0 * y s * 5.63 19.7 1 27.7 4.85 y 0 0.2 y s.4.6.8 1 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 3.19 3.19 3.19 3.19 3.19 3.19 e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (8) = 79 χ 2 (3) = 1616

Linear Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) 80 70 60 WISC Score 50 40 30 20 10 0 1 2 3 4 5 6 Grade at Testing

Latent Growth Model ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 0 1 1 1 1 1 1 0 β 2.4 β 4 y s.8 1 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Latent Growth Model (AMOS input) Sem.BeginGroup "wisc.sav" Sem.Structure "total1 = (0) + (1) LEVEL + (0) SLOPE + (1) E1 " Sem.Structure "total2 = (0) + (1) LEVEL + (b_1) SLOPE + (1) E2 " Sem.Structure "total3 = (0) + (1) LEVEL + (.4) SLOPE + (1) E3 " Sem.Structure "total4 = (0) + (1) LEVEL + (b_2) SLOPE + (1) E4 " Sem.Structure "total5 = (0) + (1) LEVEL + (.8) SLOPE + (1) E5 " Sem.Structure "total6 = (0) + (1) LEVEL + (1) SLOPE + (1) E6 " Sem.Mean "LEVEL", "mn_level" Sem.Mean "SLOPE", "mn_slope" Sem.Structure "LEVEL<>SLOPE (c_ls) " End Sub Sem.Structure "E1 (v_uniq) " Sem.Structure "E2 (v_uniq) " Sem.Structure "E3 (v_uniq) " Sem.Structure "E4 (v_uniq) " Sem.Structure "E5 (v_uniq) " Sem.Structure "E6 (v_uniq) "

Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- ------- LEVEL 31.289 3.681 8.499 SLOPE 27.676 4.433 6.243 E1 8.675 0.609 14.248 v uniq Latent Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- total1 <----- LEVEL 1.000 total2 <----- LEVEL 1.000 total3 <----- LEVEL 1.000 total4 <----- LEVEL 1.000 total5 <----- LEVEL 1.000 total6 <----- LEVEL 1.000 total1 <----- SLOPE 0.000 total2 <----- SLOPE 0.271 0.009 30.002 b_1 total3 <----- SLOPE 0.400 total4 <----- SLOPE 0.597 0.009 67.809 b_2 total5 <----- SLOPE 0.800 total6 <----- SLOPE 1.000 Means: Estimate S.E. C.R. Label ------ -------- ------- ------- ------- LEVEL 18.813 0.443 42.441 mn_leve SLOPE 28.590 0.471 60.734 mn_slop Covariances: Estimate S.E. C.R. Label ------------ -------- ------- ------- ------- LEVEL <-----> SLOPE 16.298 2.919 5.583 c_ls

Estimates from a Latent Slope Model.55 y 0 * y s * 5.61 18.8 1 28.6 5.27 y 0 y s 0.27.4.60 1.8 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 2.95 2.95 2.95 2.95 2.95 2.95 e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (6) = 17 χ 2 (2) = 61

Latent Slope Model with Exogenous Variable (Mother s Education) ω 0s z y0 * z ys * ω 0 ω s γ 01 1 γ s1 µ x X σ x 2 γ 0x γ sx y 0 1 1 1 1 1 1 β 0 β 1 y s β 2 β 3 β 5 β 4 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Latent Slope with Exogenous Variable Sem.BeginGroup "wisc.sav" Sem.Structure "total1 = (0) + (1) LEVEL + (0) SLOPE + (1) E1 " Sem.Structure "total2 = (0) + (1) LEVEL + (b_1) SLOPE + (1) E2 " Sem.Structure "total3 = (0) + (1) LEVEL + (.4) SLOPE + (1) E3 " Sem.Structure "total4 = (0) + (1) LEVEL + (b_2) SLOPE + (1) E4 " Sem.Structure "total5 = (0) + (1) LEVEL + (.8) SLOPE + (1) E5 " Sem.Structure "total6 = (0) + (1) LEVEL + (1) SLOPE + (1) E6 " Sem.Structure "LEVEL = (int_level) + (mom_l) momed + (1) var_level " Sem.Structure "SLOPE = (int_slope) + (mom_s) momed + (1) var_slope " Sem.Structure "momed = (int_momed) + (1) var_momed Sem.Structure "var_level<>var_slope (c_ls) " Sem.Structure "E1 (v_uniq) " Sem.Structure "E2 (v_uniq) " Sem.Structure "E3 (v_uniq) " Sem.Structure "E4 (v_uniq) " Sem.Structure "E5 (v_uniq) " Sem.Structure "E6 (v_uniq) " End Sub

Latent Slope with Exogenous Variable Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- LEVEL <------------ momed 1.206 0.134 9.012 mom_l SLOPE <------------ momed 0.475 0.168 2.821 mom_s total1 <----------- LEVEL 1.000 total2 <----------- LEVEL 1.000 total1 <----------- SLOPE 0.000 total2 <----------- SLOPE 0.271 0.009 29.995 b_1 total3 <----------- SLOPE 0.400 total4 <----------- SLOPE 0.597 0.009 67.843 b_2 total5 <----------- SLOPE 0.800 total6 <----------- SLOPE 1.000 Intercepts: Estimate S.E. C.R. Label ----------- -------- ------- ------- ------- momed 10.811 0.189 57.228 int_mom LEVEL 5.776 1.494 3.867 int_lev SLOPE 23.457 1.875 12.508 int_slo Covariances: Estimate S.E. C.R. Label ------------ -------- ------- ------- ------- var_level <---> var_slope 12.151 2.404 5.054 c_ls Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- ------- var_momed 7.245 0.719 10.075 var_level 20.756 2.644 7.851 var_slope 26.034 4.274 6.090 E1 8.676 0.609 14.248 v_uniq E2 8.676 0.609 14.248 v_uniq

Estimates from a Latent Slope Model with Mother s Education.52 4.57 z y0 * z ys * y 0 5.78 1 10.8 23.5 y s 5.12 1.21 0.47 0.27.4.60 1.8 X 7.25 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 2.95 2.95 2.95 2.95 2.95 2.95 e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (8) = 22

LDS Fit Statistics Goodness Model 1 Model 2 Model 3 Model 4 of Fit Level Linear Latent Mothed LRT (χ 2 ) 1694 78 17 22 df 11 8 6 8 RMSEA.70.20.10.09 p-close fit.00.00.06.05 χ 2 / df ---- 1616/3 61/2 5/2

Conclusions A latent model with unequal growth over time seems more reasonable for these data than models with flat or linear trajectories Mother s education have a positive influence on both the level and slope Other modeling alternatives are possible (e.g., age vs. grade)

Other Programs: Mplus

Linear Growth Model (Mplus input) TITLE: Linear Growth Models --WISC Data DATA: FILE IS wiscraw.dat; VARIABLE: NAMES ARE id wisc1 wisc2 wisc4 wisc6; USEVAR = wisc1 wisc2 wisc4 wisc6; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL:!creating latent variables to deal with incomplete data lwisc1 by wisc1@1; lwisc2 by wisc2@1; lwisc3 by wisc1@0; lwisc4 by wisc4@1; lwisc5 by wisc2@0; lwisc6 by wisc6@1;

Linear Growth Model (Mplus input cont.)!level loadings fixed at 1 level BY lwisc1-lwisc6@1 ;!slope loadings fixed at linear estimates (0-1); relax this for a latent model (*) slope BY lwisc1@0 lwisc2@.2 lwisc3@.4 lwisc4@.6 lwisc5@.8 lwisc6@1;!level and slope means with starting values; other means set to 0 [level*19 slope*27 wisc1-wisc6@0 lwisc1-lwisc6@0];!level and slope variances and covariance (r= cov/sd*sd) level*25 slope*25 ; level with slope*17 ;!equal unique variances wisc1-wisc6*10 (1);!latent variances to 0 lwisc1-lwisc6@0 ; OUTPUT: SAMPSTAT STANDARDIZED TECH1;

Linear Growth Model (Mplus output) TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 79.101 Degrees of Freedom 8 P-Value 0.0000 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.209 90 Percent C.I. 0.168 0.252 Probability RMSEA <=.05 0.000 MODEL RESULTS Estimates S.E. Est./S.E. Means LEVEL 19.698 0.429 45.895 SLOPE 27.704 0.446 62.164 Variances LEVEL 31.584 3.744 8.435 SLOPE 23.391 4.187 5.587 SLOPE BY LWISC1 0.000 0.000 0.000 LWISC2 0.200 0.000 0.000 LWISC3 0.400 0.000 0.000 LWISC4 0.600 0.000 0.000 LWISC5 0.800 0.000 0.000 LWISC6 1.000 0.000 0.000 LEVEL WITH SLOPE 17.629 2.870 6.143 Residual Variances WISC1 10.104 0.707 14.283 WISC2 10.104 0.707 14.283

Latent Growth Model (Mplus output) Chi-Square Test of Model Fit Value 17.485 Degrees of Freedom 6 P-Value 0.0076 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.097 90 Percent C.I. 0.046 0.151 Probability RMSEA <=.05 0.063 Means LEVEL 18.813 0.442 42.544 SLOPE 28.590 0.470 60.884 Variances LEVEL 31.292 3.673 8.520 SLOPE 27.677 4.422 6.259 SLOPE BY LWISC1 0.000 0.000 0.000 LWISC2 0.271 0.009 30.079 LWISC3 0.400 0.000 0.000 LWISC4 0.597 0.009 67.978 LWISC5 0.800 0.000 0.000 LWISC6 1.000 0.000 0.000 SLOPE WITH LEVEL 16.296 2.912 5.596 Residual Variances WISC1 8.675 0.607 14.283 WISC2 8.675 0.607 14.283 WISC4 8.675 0.607 14.283 WISC6 8.675 0.607 14.283

Other Programs: SAS

Making a Multiple-Record Data File (SAS) TITLE 'Making a Multiple-Record Data File'; DATA temp1; SET wiscraw; age1=6; age2=6.95; age4=8.8; age6=10.8; grade1=0; grade2=1; grade4=3; grade6=5; FILE outfile LRECL=200 LINESIZE=200; PUT #1 id age1 grade1 verbal1 nv1 wisc1 mothed #2 id age2 grade2 verbal2 nv2 wisc2 mothed #3 id age4 grade4 verbal4 nv4 wisc4 mothed #4 id age6 grade6 verbal6 nv6 wisc6 mothed ; RUN; DATA temp2; INFILE outfile LRECL=200 LINESIZE=200; INPUT id age grade verbal nv wisc mothed ; agec=age-6; agec2=agec*agec; age2=age*age; RUN;

Age-Based Linear Models (SAS Mixed) TITLE : Initial Baseline Variance'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = / SOLUTION; RUN; TITLE : No Growth'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = / SOLUTION; RANDOM INTERCEPT / SUBJECT=id TYPE=UN GCORR; RUN; TITLE: 'Linear Age'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = age / SOLUTION; RANDOM INTERCEPT age / SUBJECT=id TYPE=UN GCORR; RUN;

Model 0: Initial Baseline Covariance Parameters 1 Observations Used 816 Observations Not Used 0 Total Observations 816 Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z Residual 179.17 8.8757 20.19 <.000 Fit Statistics -2 Res Log Likelihood 6548.1 AIC (smaller is better) 6550.1 AICC (smaller is better) 6550.1 BIC (smaller is better) 6554.8 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept 32.1644 0.4686 815 68.64 <.0001

Model 1: No Growth Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 13.5762 5.9492 2.28 0.0112 Residual 165.64 9.4693 17.49 <.0001 Fit Statistics -2 Res Log Likelihood 6541.7 AIC (smaller is better) 6545.7 AICC (smaller is better) 6545.7 BIC (smaller is better) 6552.3 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept 32.1644 0.5192 203 61.95 <.0001

Model 2: Linear Growth Model: Linear Age Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept 1 1.0000 0.6408 2 agec 1 0.6408 1.0000 Covariance Parameter Estimates Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 31.8714 3.7715 8.45 <.0001 UN(2,1) id 3.7019 0.6060 6.11 <.0001 UN(2,2) id 1.0471 0.1851 5.66 <.0001 Residual 10.0249 0.7019 14.28 <.0001 Fit Statistics -2 Res Log Likelihood 4916.4 AIC (smaller is better) 4924.4 Solution for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept 19.7878 0.4303 203 45.99 <.0001 agec 5.7902 0.09366 611 61.82 <.0001

Linear Growth Models in SAS (NLMIXED) TITLE 'Linear Growth Curve Model With Basis'; PROC NLMIXED; level= m_level + d_level; slope= m_slope + d_slope; IF (grade=0) THEN basis= 0; IF (grade=1) THEN basis=.2; IF (grade=3) THEN basis=.6; IF (grade=5) THEN basis= 1; traject = level + slope * basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM d_level d_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN; PARMS m_level=20 m_slope=27 v_level=10 v_slope=2 c_ls=0 v_error=10;

Age-Based Linear Model (NLMixed) TITLE: 'Linear Growth Curve Model'; PROC NLMIXED; PARMS m_level=20 m_slope=5 v_level=10 v_slope=2 c_ls=0 v_error=10; level= m_level + d_level; slope= m_slope + d_slope; age1=6; age2=6.95; age4=8.8; age6=10.8; basis2=(age2-age1)/(age6-age1); basis3=(age4-age1)/(age6-age1); IF (age=6) THEN basis=0; IF (age=6.95) THEN basis=basis2; IF (age=8.8) THEN basis=basis3; IF (age=10.8) THEN basis=1; traject = level + slope*basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM d_level d_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN;

Age-Based Linear Model (NLMixed) Linear Growth Curve Model Fit Statistics -2 Log Likelihood 4913.6 AIC (smaller is better) 4925.6 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level 19.7878 0.4292 202 46.10 <.0001 0.05 18.9415 m_slope 27.7929 0.4484 202 61.98 <.0001 0.05 26.9086 v_level 31.6863 3.7442 202 8.46 <.0001 0.05 24.3036 v_slope 23.9242 4.2348 202 5.65 <.0001 0.05 15.5741 c_ls 17.7192 2.8884 202 6.13 <.0001 0.05 12.0239 v_error 10.0249 0.7019 202 14.28 <.0001 0.05 8.6409 Parameter Estimates Parameter Upper Gradient m_level 20.6342 0.000022 m_slope 28.6771 6.105E-6 v_level 39.0689 1.226E-6 v_slope 32.2744 1.064E-6 c_ls 23.4144 2.927E-6 v_error 11.4089-6.42E-6

Latent Growth Model (NLmixed) TITLE 'Latent Growth Curve Model With Basis'; PROC NLMIXED; level= m_level + d_level; slope= m_slope + d_slope; IF (grade=0) THEN basis= 0; IF (grade=1) THEN basis=basis2; IF (grade=3) THEN basis=basis4; IF (grade=5) THEN basis= 1; traject = level + slope * basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM d_level d_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN; PARMS m_level=20 m_slope=27 basis2=.2 basis4=.6 v_level=10 v_slope=2 c_ls=0 v_error=10;

Latent Growth Model (NLmixed) Fit Statistics -2 Log Likelihood 4854.2 AIC (smaller is better) 4870.2 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level 18.8127 0.4423 202 42.54 <.0001 0.05 17.9406 m_slope 28.5896 0.4695 202 60.90 <.0001 0.05 27.6639 v_level 31.2904 3.6723 202 8.52 <.0001 0.05 24.0495 v_slope 27.6928 4.4300 202 6.25 <.0001 0.05 18.9578 c_ls 16.2969 2.9125 202 5.60 <.0001 0.05 10.5540 v_error 8.6742 0.6074 202 14.28 <.0001 0.05 7.4766 basis2 0.2714 0.009020 202 30.09 <.0001 0.05 0.2536 basis3 0.5967 0.008785 202 67.91 <.0001 0.05 0.5793 Parameter Estimates Parameter Upper Gradient m_level 19.6847-0.00026 m_slope 29.5154 0.000032 v_level 38.5312-0.00014 v_slope 36.4278 0.000857 c_ls 22.0397-0.0002 v_error 9.8718-0.00002 basis2 0.2892-0.00837 basis3 0.6140 0.003214

Age-Based Latent Model (SAS NLMixed) TITLE 'Latent Growth Curve Model'; PROC NLMIXED; PARMS m_level=20 m_slope=27 v_level=31 v_slope=24 c_ls=0 v_error=10 basis2=.2 basis3=.6; level= m_level + d_level; slope= m_slope + d_slope; IF (age=6) THEN basis=0; IF (age=6.95) THEN basis=basis2; IF (age=8.8) THEN basis=basis3; IF (age=10.8) THEN basis=1; traject = level + slope*basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM d_level d_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN;

Age-Based Latent Model (SAS NLMixed) Latent Growth Curve Model Fit Statistics -2 Log Likelihood 4854.2 AIC (smaller is better) 4870.2 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level 18.8127 0.4423 202 42.54 <.0001 0.05 17.9406 m_slope 28.5896 0.4695 202 60.90 <.0001 0.05 27.6639 v_level 31.2904 3.6723 202 8.52 <.0001 0.05 24.0495 v_slope 27.6928 4.4300 202 6.25 <.0001 0.05 18.9578 c_ls 16.2969 2.9125 202 5.60 <.0001 0.05 10.5540 v_error 8.6742 0.6074 202 14.28 <.0001 0.05 7.4766 basis2 0.2714 0.009020 202 30.09 <.0001 0.05 0.2536 basis3 0.5967 0.008785 202 67.91 <.0001 0.05 0.5793 Parameter Estimates Parameter Upper Gradient m_level 19.6847-0.00026 m_slope 29.5154 0.000032 v_level 38.5312-0.00014 v_slope 36.4278 0.000857 c_ls 22.0397-0.0002 v_error 9.8718-0.00002 basis2 0.2892-0.00837 basis3 0.6140 0.003214

Polynomial Models (SAS Mixed) TITLE Quadratic Model'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = agec agec2 / SOLUTION DDFM=BW; RANDOM INTERCEPT agec agec2 / SUBJECT=id TYPE=UN; RUN; TITLE Cubic Model With Restrictions'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = agec agec2 /SOLUTION DDFM=BW; RANDOM INTERCEPT agec agec2/ SUBJECT=id TYPE=UN GCORR; PARMS (31.5) (.01) (27.5) (0) (0) (1.1) (15) / EQCONS=4,5; RUN;

Linear Model with Exogenous Variable TITLE 'Linear Grade with Covariate on Level'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = grade mothed / SOLUTION DDFM=BW; RANDOM INTERCEPT grade/ SUBJECT=id TYPE=UN; RUN; TITLE 'Linear Grade with Covariate on Level and Slope'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = grade mothed grade*mothed/solution DDFM=BW; RANDOM INTERCEPT grade / SUBJECT=id TYPE=UN GCORR; RUN;

Linear Model with Covariate Model: Linear Age plus Covariate on Level and Slope Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept 1 1.0000 0.6263 2 agec 1 0.6263 1.0000 Covariance Parameter Estimates Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 21.0872 2.7166 7.76 <.0001 UN(2,1) id 2.8619 0.5018 5.70 <.0001 UN(2,2) id 0.9901 0.1800 5.50 <.0001 Residual 10.0249 0.7019 14.28 <.0001 Fit Statistics -2 Res Log Likelihood 4853.4 AIC (smaller is better) 4861.4 Solution for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept 6.5492 1.5054 202 4.35 <.0001 agec 4.7640 0.3814 610 12.49 <.0001 mothed 1.2245 0.1351 202 9.06 <.0001 agec*mothed 0.09492 0.03424 610 2.77 0.0057

Latent Model with Exogenous Variable TITLE4 'Latent Growth Curve Model With Exogenous Variable'; PROC NLMIXED; level= i_level + beta_lm * mothed + e_level; slope= i_slope + beta_sm * mothed + e_slope; IF (grade=0) THEN basis= 0; IF (grade=1) THEN basis=basis2; IF (grade=3) THEN basis=basis4; IF (grade=5) THEN basis= 1; traject = level + slope * basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM e_level e_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN; PARMS i_level=5 i_slope=25 basis2=.2 basis4=.6 beta_lm=.1 beta_sm=.1 v_level=20 v_slope=25 c_ls=.1 v_error=20;

Latent Model with Exogenous Variable Outcome Y = Wisc Total Latent Growth Curve Model with Mothered The NLMIXED Procedure Fit Statistics -2 Log Likelihood 4784.0 AIC (smaller is better) 4804.0 AICC (smaller is better) 4804.2 BIC (smaller is better) 4837.1 Parameter Estimates Parameter Estimate SE DF t Value Pr > t Alpha Lower Upper i_level 5.7959 1.4916 202 3.89 0.0001 0.05 2.8548 8.7371 i_slope 23.5294 1.8723 202 12.57 <.0001 0.05 19.8377 27.2211 v_level 20.7998 2.6453 202 7.86 <.0001 0.05 15.5839 26.0157 v_slope 26.1399 4.2872 202 6.10 <.0001 0.05 17.6865 34.5934 c_ls 12.2034 2.4075 202 5.07 <.0001 0.05 7.4563 16.9505 v_error 8.6722 0.6071 202 14.29 <.0001 0.05 7.4752 9.8692 beta_lm 1.2039 0.1336 202 9.01 <.0001 0.05 0.9405 1.4672 beta_sm 0.4686 0.1680 202 2.79 0.0058 0.05 0.1372 0.7999 basis2 0.2714 0.009017 202 30.09 <.0001 0.05 0.2536 0.2891 basis3 0.5970 0.008785 202 67.96 <.0001 0.05 0.5797 0.6143

Estimates from a Latent Slope Model with Mother s Education 20.8.52 z y0 * z ys * 26.1 5.80 1 10.8 23.5 1.20 0.47 X 7.28 y 0 y s 0.27.4.60 1.8 Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] -2LL = 4784 e y[1] e y[2] e y[3] e y[4] e y[5] e y[6] 8.67 8.67 8.67 8.67 8.67 8.67

Other Programs: LISREL

Level Only Model (LISREL input) Growth Curves WISC Data (McArdle & Epstein, 1987)!Model0: Level Only Model DA NO=204 NI=15 MA=MM RAw_data FI=wiscraw.dat!Select the variables wisc1 wisc2 wisc4 wisc6 constant SE 11 12 13 14 15 / MO NY=5 NE=17 LY=FI,FU BE=FI,FU PS=FI,SY TE=ZE!Labels of All the Variables LE wisc1 wisc2 wisc3 wisc4 wisc5 wisc6 cte e1 e2 e3 e4 e5 e6 level slope level* slope*

Level Only Model (LISREL cont.)!filter MATRIX (Observed vs Latent Variables) ST 1 LY(1,1) LY(2,2) LY(3,4) LY(4,6) LY(5,7)!BETA MATRIX (One-Headed Arrows)!Level Loadings Fixed at 1 ST 1 BE(1,14) BE(2,14) BE(3,14) BE(4,14) BE(5,14) BE(6,14)!Slope Loadings Fixed at 0 ST 0 BE(1,15) BE(2,15) BE(3,15) BE(4,15) BE(5,15) BE(6,15)!Level Mean (Slope Mean = 0) FR BE(14,7) ST 17 BE(14,7)!Level Deviation (Slope Deviation = 0) FR BE(14,16) BE(15,17) ST 1 BE(14,16) BE(15,17)

Level Only Model (LISREL cont.)!psi MATRIX (Two-Headed Arrows) FR PS(7,7) ST 1 PS(7,7)!Error Deviations Free and Equal FR BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13) ST 3 BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13) EQ BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13)!Error Variances Fixed at 1 ST 1 PS(8,8) PS(9,9) PS(10,10) PS(11,11) PS(12,12) PS(13,13)!Level Variance Fixed at 1 ST 1 PS(16,16) OU NS ML PT PC RS IT=100 AD=OFf XM ND=2

Level Only Model (LISREL output) LISREL Estimates (Maximum Likelihood) BETA cte e1 e2 e3 e4 e5 -------- -------- -------- -------- -------- -------- wisc1 - - 12.87 - - - - - - - - (0.37) 34.90 level 32.16 - - - - - - - - - - (0.52) 61.80 slope - - - - - - - - - - - - e6 level slope level* slope* -------- -------- -------- -------- -------- wisc1 - - 1.00 - - - - - - wisc2 - - 1.00 - - - - - - wisc3 - - 1.00 - - - - - - wisc4 - - 1.00 - - - - - - wisc5 - - 1.00 - - - - - - wisc6 12.87 1.00 - - - - - - (0.37) 34.90 level - - - - - - 3.68 - - (0.81) 4.56 slope - - - - - - - - - -

Level Only Model (LISREL output cont.) Goodness of Fit Statistics Degrees of Freedom = 11 Minimum Fit Function Chi-Square = 1694.32 (P = 0.00) Root Mean Square Error of Approximation (RMSEA) = 0.70 90 Percent Confidence Interval for RMSEA = (0.67 ; 0.74) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00

Estimates from a Level Only Model y 0 * 3.68 32.2 1 y 0 1 1 1 1 1 1 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 12.9 12.9 12.9 12.9 12.9 12.9 e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (11) = 1694

Linear Growth Model ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 0 1 1 1 1 1 1 0.2 y s.4.6.8 1 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Linear Growth Model (LISREL input) Growth Curves WISC Data (McArdle & Epstein, 1987)!Model1: Linear Growth with Incomplete Data DA NO=204 NI=15 MA=MM RAw_data FI=wiscraw.dat!Select the variables wisc1 wisc2 wisc4 wisc6 constant SE 11 12 13 14 15 / MO NY=5 NE=17 LY=FI,FU BE=FI,FU PS=FI,SY TE=ZE!Labels of All the Variables LE wisc1 wisc2 wisc3 wisc4 wisc5 wisc6 cte e1 e2 e3 e4 e5 e6 level slope level* slope*

Linear Growth Model (LISREL cont.)!filter MATRIX (Observed vs Latent Variables) ST 1 LY(1,1) LY(2,2) LY(3,4) LY(4,6) LY(5,7)!BETA MATRIX (One-Headed Arrows)!Level Loadings Fixed at 1 ST 1 BE(1,14) BE(2,14) BE(3,14) BE(4,14) BE(5,14) BE(6,14)!Slope Loadings Fixed at Linear Estimates (relax this for latent models) ST 0 BE(1,15) ST.2 BE(2,15) ST.4 BE(3,15) ST.6 BE(4,15) ST.8 BE(5,15) ST 1 BE(6,15)!Level and Slope Means FR BE(14,7) BE(15,7) ST 17 BE(14,7) ST 28 BE(15,7)!Level and Slope Deviation FR BE(14,16) BE(15,17) ST 1 BE(14,16) BE(15,17)

OU NS ML PT PC RS IT=100 AD=OFf XM ND=2 Linear Growth Model (LISREL cont.)!psi MATRIX (Two-Headed Arrows) FR PS(7,7) ST 1 PS(7,7)!Error Deviations Free and Equal FR BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13) ST 3 BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13) EQ BE(1,8) BE(2,9) BE(3,10) BE(4,11) BE(5,12) BE(6,13)!Error Variances Fixed at 1 ST 1 PS(8,8) PS(9,9) PS(10,10) PS(11,11) PS(12,12) PS(13,13)!Mean and Slope Variances Fixed at 1 ST 1 PS(16,16) PS(17,17)!Level-Slope Correlation FR PS(16,17) ST.1 PS(16,17)

Linear Growth Model (LISREL output) LISREL Estimates (Maximum Likelihood) BETA cte e1 e2 e3 e4 e5 -------- -------- -------- -------- -------- -------- wisc1 - - 3.19 - - - - - - - - (0.11) 28.50 level 19.70 - - - - - - - - - - (0.43) 45.67 slope 27.70 - - - - - - - - - - (0.45) 61.86 e6 level slope level* slope* -------- -------- -------- -------- -------- wisc1 - - 1.00 - - - - - - wisc2 - - 1.00 0.20 - - - - wisc3 - - 1.00 0.40 - - - - wisc4 - - 1.00 0.60 - - - - wisc5 - - 1.00 0.80 - - - - wisc6 3.19 1.00 1.00 - - - - (0.11) 28.50 level - - - - - - 5.63 - - (0.33) 16.83 slope - - - - - - - - 4.85 (0.43) 11.15

Linear Growth Model (LISREL output cont.) PSI e6 level slope level* slope* -------- -------- -------- -------- -------- e6 1.00 level* - - - - - - 1.00 slope* - - - - - - 0.65 1.00 (0.10) 6.33 Goodness of Fit Statistics Degrees of Freedom = 8 Minimum Fit Function Chi-Square = 78.45 (P = 0.00) Root Mean Square Error of Approximation (RMSEA) = 0.20 90 Percent Confidence Interval for RMSEA = (0.16 ; 0.24) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00

wisc5 - - 1.00 0.80 - - - - wisc6 2.95 1.00 1.00 - - - - (0.10) 28.50 level - - - - - - 5.61 - - (0.33) 17.00 slope - - - - - - - - 5.27 (0.42) 12.49 Latent Growth Models (LISREL output) BETA cte e1 e2 e3 e4 e5 -------- -------- -------- -------- -------- -------- wisc1 - - 2.95 - - - - - - - - (0.10) 28.50 level 18.81 - - - - - - - - - - (0.44) 42.34 slope 28.59 - - - - - - - - - - (0.47) 60.59 e6 level slope level* slope* -------- -------- -------- -------- -------- wisc2 - - 1.00 0.27 - - - - (0.01) 29.93 wisc3 - - 1.00 0.40 - - - - wisc4 - - 1.00 0.60 - - - - (0.01) 67.65

Latent Growth Models (LISREL output cont.) PSI e6 level slope level* slope* -------- -------- -------- -------- -------- e6 1.00 level - - - - slope - - - - - - level* - - - - - - 1.00 slope* - - - - - - 0.55 1.00 (0.09) 5.84 Goodness of Fit Statistics Degrees of Freedom = 6 Minimum Fit Function Chi-Square = 17.40 (P = 0.0079) Normal Theory Weighted Least Squares Chi-Square = 17.68 (P = 0.0071) Estimated Non-centrality Parameter (NCP) = 11.68 90 Percent Confidence Interval for NCP = (2.68 ; 28.29) Minimum Fit Function Value = 0.086 Population Discrepancy Function Value (F0) = 0.058 90 Percent Confidence Interval for F0 = (0.013 ; 0.14) Root Mean Square Error of Approximation (RMSEA) = 0.098 90 Percent Confidence Interval for RMSEA = (0.047 ; 0.15) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.059

Fitting Growth Models Using Other Programs

Growth Models (RAMONA) USE WISCTOT ramona MANIFEST WISC1 WISC2 WISC3 WISC4 Momed K LATENT LEVEL SLOPE Zlevel Zslope E1 E2 E3 E4 MODEL WISC1 <- LEVEL(0,1) SLOPE(0,0) E1(1,5), WISC2 <- LEVEL(0,1) SLOPE(*,.3) E2(1,5), WISC3 <- LEVEL(0,1) SLOPE(*,.6) E3(1,5), WISC4 <- LEVEL(0,1) SLOPE(0,1) E4(1,5), SLOPE <- Zslope(*,5) K(*,23) Momed(*,.2), LEVEL <- Zlevel(*,10) K(*,10) Momed(*,.5), Momed <- K(*,10), E1 <-> E1(0,1), E2 <-> E2(0,1), E3 <-> E3(0,1), E4 <-> E4(0,1), Zlevel <-> Zlevel(0,1), Zslope(*,.5), Zslope <-> Zslope(0,1), Momed <-> Momed(*,7), K <-> K(*,1) print medium ESTIMATE / DISP=COVA, METHOD= MWL ncases=204 iter=300

Growth Models (Spss) TITLE "No Growth" Mixed wisc /print=solution /method=reml /fixed=intercept /random intercept subject(id). TITLE "Linear Growth" Mixed wisc with gradec /print=solution /method=reml /fixed=gradec /random intercept gradec subject(id) covtype(un). TITLE "Linear Growth with Mothed" Mixed wisc with mothed gradec /print=solution /method=ml /fixed=gradec mothed gradec*mothed /random intercept gradec subject(id) covtype(un).

Linear Growth Model (HLM)

Linear Growth with Extension Variable (HLM)

Linear Growth Model (MLwiN)

Linear Growth Model (MLwiN)

Linear Growth with Extension Variable (MLwiN)

Linear Growth with Extension Variable (MLwiN)

Final Conclusions Selection of program is the researcher s choice All models give similar results SAS Nlmixed is flexible and easy to program, allowing many linear and nonlinear possibilities

Linear Growth (SAS-NLMixed) TITLE: 'Baseline Model Intercepts'; PROC NLMIXED; ytraject = y0; MODEL yt ~ NORMAL(ytraject, ve); RANDOM y0 ~ NORMAL([m0],[v0]) SUBJECT=id; PARMS m0=15 ve=20 v0=80; RUN; TITLE: 'Linear Model Intercepts and Slopes'; PROC NLMIXED; ytraject = y0 + (ys * age) ; MODEL yt ~ NORMAL(ytraject, ve); RANDOM y0 ys ~ NORMAL([m0,ms], [v0, c0s, vs]) SUBJECT=id; PARMS m0=15 ms=10 ve=20 v0=80 vs=10 c0s=-.01 ; RUN;

Linear Growth (SAS-NLMixed) TITLE: 'Linear Covariate with Intercepts and Slopes'; PROC NLMIXED; ytraject = y0 + (ys * age) ; y0 = n0 + g0 * xvar + e0; ys = ns + gs * xvar + es; MODEL yt ~ NORMAL(ytraject, ve); RANDOM e0 es ~ NORMAL([0,0], [ev0, ec0s, evs]) SUBJECT=id; PARMS n0=15 ns=10 g0=.01 gs=.001 ve=20 ev0=10 evs=10 ec0s=-.01 ; RUN;

Fitting Latent Growth Models 2: Nonlinear Models

Latent Growth Model ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 0 1 1 1 1 1 1 0 β 2.4 β 4 y s.8 1 Y [0] Y [1] Y [2] Y [3] Y [4] σ e σ e σ e σ e σ e Y [5] σ e e y[0] e y[1] e y[2] e y[3] e y[4] e y[5]

Latent Growth Model (AMOS input) Sem.BeginGroup "wisc.sav" Sem.Structure "total1 = (0) + (1) LEVEL + (0) SLOPE + (1) E1 " Sem.Structure "total2 = (0) + (1) LEVEL + (b_1) SLOPE + (1) E2 " Sem.Structure "total3 = (0) + (1) LEVEL + (.4) SLOPE + (1) E3 " Sem.Structure "total4 = (0) + (1) LEVEL + (b_2) SLOPE + (1) E4 " Sem.Structure "total5 = (0) + (1) LEVEL + (.8) SLOPE + (1) E5 " Sem.Structure "total6 = (0) + (1) LEVEL + (1) SLOPE + (1) E6 " Sem.Mean "LEVEL", "mn_level" Sem.Mean "SLOPE", "mn_slope" Sem.Structure "LEVEL<>SLOPE (c_ls) " End Sub Sem.Structure "E1 (v_uniq) " Sem.Structure "E2 (v_uniq) " Sem.Structure "E3 (v_uniq) " Sem.Structure "E4 (v_uniq) " Sem.Structure "E5 (v_uniq) " Sem.Structure "E6 (v_uniq) "

Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- ------- LEVEL 31.289 3.681 8.499 SLOPE 27.676 4.433 6.243 E1 8.675 0.609 14.248 v uniq Latent Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- total1 <----- LEVEL 1.000 total2 <----- LEVEL 1.000 total3 <----- LEVEL 1.000 total4 <----- LEVEL 1.000 total5 <----- LEVEL 1.000 total6 <----- LEVEL 1.000 total1 <----- SLOPE 0.000 total2 <----- SLOPE 0.271 0.009 30.002 b_1 total3 <----- SLOPE 0.400 total4 <----- SLOPE 0.597 0.009 67.809 b_2 total5 <----- SLOPE 0.800 total6 <----- SLOPE 1.000 Means: Estimate S.E. C.R. Label ------ -------- ------- ------- ------- LEVEL 18.813 0.443 42.441 mn_leve SLOPE 28.590 0.471 60.734 mn_slop Covariances: Estimate S.E. C.R. Label ------------ -------- ------- ------- ------- LEVEL <-----> SLOPE 16.298 2.919 5.583 c_ls

Latent Growth Model (Mplus input) TITLE: Linear Growth Models --WISC Data DATA: FILE IS wiscraw.dat; VARIABLE: NAMES ARE id wisc1 wisc2 wisc4 wisc6; USEVAR = wisc1 wisc2 wisc4 wisc6; ANALYSIS: TYPE = MEANSTRUCTURE; MODEL:!creating latent variables to deal with incomplete data lwisc1 by wisc1@1; lwisc2 by wisc2@1; lwisc3 by wisc1@0; lwisc4 by wisc4@1; lwisc5 by wisc2@0; lwisc6 by wisc6@1;

Latent Growth Model (Mplus input cont.)!level loadings fixed at 1 level BY lwisc1-lwisc6@1 ;!slope loadings slope BY lwisc1@0 lwisc2*.2 lwisc3@.4 lwisc4*.6 lwisc5@.8 lwisc6@1;!level and slope means with starting values; other means set to 0 [level*19 slope*27 wisc1-wisc6@0 lwisc1-lwisc6@0];!level and slope variances and covariance (r= cov/sd*sd) level*25 slope*25 ; level with slope*17 ;!equal unique variances wisc1-wisc6*10 (1);!latent variances to 0 lwisc1-lwisc6@0 ; OUTPUT: SAMPSTAT STANDARDIZED TECH1;

Latent Growth Model (Mplus output) Chi-Square Test of Model Fit Value 17.485 Degrees of Freedom 6 P-Value 0.0076 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.097 90 Percent C.I. 0.046 0.151 Probability RMSEA <=.05 0.063 Means LEVEL 18.813 0.442 42.544 SLOPE 28.590 0.470 60.884 Variances LEVEL 31.292 3.673 8.520 SLOPE 27.677 4.422 6.259 SLOPE BY LWISC1 0.000 0.000 0.000 LWISC2 0.271 0.009 30.079 LWISC3 0.400 0.000 0.000 LWISC4 0.597 0.009 67.978 LWISC5 0.800 0.000 0.000 LWISC6 1.000 0.000 0.000 SLOPE WITH LEVEL 16.296 2.912 5.596 Residual Variances WISC1 8.675 0.607 14.283 WISC2 8.675 0.607 14.283 WISC4 8.675 0.607 14.283 WISC6 8.675 0.607 14.283

Polynomial Models (SAS Mixed) TITLE Quadratic Model'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = agec agec2 / SOLUTION DDFM=BW; RANDOM INTERCEPT agec agec2 / SUBJECT=id TYPE=UN; RUN; TITLE Quadratic Model With Restrictions'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = agec agec2 /SOLUTION DDFM=BW; RANDOM INTERCEPT agec agec2/ SUBJECT=id TYPE=UN GCORR; PARMS (31.5) (.01) (27.5) (0) (0) (1.1) (15) / EQCONS=4,5; RUN; TITLE Quadratic Model (Alt.)'; PROC MIXED NOCLPRINT COVTEST; CLASS id; MODEL wisc = agec agec*agec /SOLUTION DDFM=BW; RANDOM INTERCEPT agec agec*agec/ SUBJECT=id TYPE=UN; GCORR; RUN;

Nonlinear Models (SAS Mixed) TITLE4 'Latent Growth Curve Model With Basis'; PROC NLMIXED; level= m_level + d_level; slope= m_slope + d_slope; IF (grade=0) THEN basis= 0; IF (grade=1) THEN basis=basis2; IF (grade=3) THEN basis=basis4; IF (grade=5) THEN basis= 1; traject = level + slope * basis; MODEL wisc ~ NORMAL (traject, v_error); RANDOM d_level d_slope ~ NORMAL ([0,0], [v_level, c_ls, v_slope]) SUBJECT = id; RUN; PARMS m_level=20 m_slope=27 basis2=.2 basis4=.6 v_level=10 v_slope=2 c_ls=0 v_error=10;

Age-Based Latent Model (SAS NLMixed) Latent Growth Curve Model Fit Statistics -2 Log Likelihood 4854.2 AIC (smaller is better) 4870.2 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level 18.8127 0.4423 202 42.54 <.0001 0.05 17.9406 m_slope 28.5896 0.4695 202 60.90 <.0001 0.05 27.6639 v_level 31.2904 3.6723 202 8.52 <.0001 0.05 24.0495 v_slope 27.6928 4.4300 202 6.25 <.0001 0.05 18.9578 c_ls 16.2969 2.9125 202 5.60 <.0001 0.05 10.5540 v_error 8.6742 0.6074 202 14.28 <.0001 0.05 7.4766 basis2 0.2714 0.009020 202 30.09 <.0001 0.05 0.2536 basis3 0.5967 0.008785 202 67.91 <.0001 0.05 0.5793 Parameter Estimates Parameter Upper Gradient m_level 19.6847-0.00026 m_slope 29.5154 0.000032 v_level 38.5312-0.00014 v_slope 36.4278 0.000857 c_ls 22.0397-0.0002 v_error 9.8718-0.00002 basis2 0.2892-0.00837 basis3 0.6140 0.003214

Spline Models (SAS Mixed) TITLE Two segments out of age with knot at 19 ; knot =19; segb1 = (tage1 - amu); IF (tage1 GT amu) THEN segb1 = 0; sega1 = (tage1 - amu); IF (tage1 LT amu) THEN sega1 = 0; TITLE 'Model: Segmented Spline with Restricted Covariances'; PROC MIXED NOCLPRINT METHOD=ML COVTEST IC; CLASS id; MODEL wisc = segb01 sega01 retest / SOLUTION DDFM=BW CHISQ; RANDOM INTERCEPT segb01 sega01 retest / SUBJECT=id TYPE=UNR; PARMS (68) (1) (2) (1) (0) (2) (0) (0) (0) (5) (75) / EQCONS=5,7,8,9; RUN;

Double Exponential Model (SAS Nlmixed) TITLE 'Model: Dual Exponential Growth Model'; PROC NLMIXED ; PARMS m_level=-75 m_slope=110 m_rate_b=.0001 m_rate_a=.1165 v_level=90 v_slope=.620 c_levslo=-7 v_error=10 ; level = m_level + d_level ; slope = m_slope + d_slope ; rate_a = m_rate_a ; rate_b = m_rate_b ; traject = level + slope * ( EXP(-rate_b * tage01) - EXP(-rate_a * tage01) ); MODEL yt ~ NORMAL(traject, v_error); RANDOM d_level d_slope ~ NORMAL([0,0], [v_level, c_levslo, v_slope]) SUBJECT=id; RUN;

Double Exponential Model (SAS Nlmixed) TITLE 'Model: Dual Exponential Growth Model Plus Practice'; PROC NLMIXED ; PARMS m_level=-75 m_slope=110 m_prac=2 m_rate_b=.0001 m_rate_a=.1165 v_level=90 v_slope=.620 c_levslo=-7 v_error=10 ; level = m_level + d_level ; slope = m_slope + d_slope; prac = m_prac; rate_a = m_rate_a ; rate_b = m_rate_b ; traject = level + slope * ( EXP(-rate_b * tage01) - EXP(-rate_a * tage01) ) + prac*practice; MODEL y01 ~ NORMAL(traject, v_error); RANDOM d_level d_slope ~ NORMAL([0,0], [v_level, c_levslo, v_slope]) SUBJECT=id; RUN;

Double Exponential Model (SAS Nlmixed) TITLE Dual Competition Model with Two Slopes'; PROC NLMIXED; ytraject=y0 + ys1*a1t + ys2*at2; A1t=(EXP(-pi1*aget)); A2t=-(EXP(-pi2*aget)) ; MODEL yt ~ NORMAL(ytraject, ve); RANDOM y0 ys1 ys2 ~ NORMAL([m0,ms1,ms2], [v0, c0s1, vs1, c0s2, c0s12, vs2]) SUBJECT=id; PARMS m0 =-80 ms=120 pi1 =.10 pi2 =.001 ve=20 v0=10 c0s1=.01 vs1=1 c0s2=.01 c0s12=.001 vs2=.1; RUN;

Latent Transition Model (SAS Nlmixed) TITLE 'Transition Point Model (Cudeck & Klebe, 2002)'; PROC NLMIXED METHOD = FIRO ; PARMS ac0 = 2.9 ac1 = 1.3 dc1 = 4.6 tauc = 4.1 ae0 = 4.9 ae1 = 2.2 de1 = 0.6 taue = 5.2 v11 = 25.1 c21 = 02.9 v22 = 01.1 c31 = 00.1 c32 = 00.1 v33 = 04.6 v44 = 00.1 var_e = 7.5; RUN; IF (sex = 1) THEN bet_i1 = ac0 + u1 ; IF (sex = 1) THEN bet_i2 = ac1 + u2 ; IF (sex = 1) THEN bet_i3 = dc1 + u3 ; IF (sex = 1) THEN bet_i4 = tauc + u4 ; IF (sex = 2) THEN bet_i1 = ae0 + u1 ; IF (sex = 2) THEN bet_i2 = ae1 + u2 ; IF (sex = 2) THEN bet_i3 = de1 + u3 ; IF (sex = 2) THEN bet_i4 = taue + u4 ; fn = bet_i1 + bet_i2 * agec + bet_i3 * max(0, agec - bet_i4); MODEL vm ~ NORMAL(fn, var_e); RANDOM u1 u2 u3 u4 ~ NORMAL([0,0,0,0], [v11, c21, v22, c31, c32, v33, 0, 0, 0, v44]) SUBJECT=id