Applied Longitudinal Data Analysis: An Introductory Course

Transcription

1 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

2 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

3 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

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

5 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

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

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

8 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

9 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

10 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)

11 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)

12 Longitudinal Individual Data WISC Score Grade at Testing

13 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 =

14 Describing the Growth WISC Score Grade at Testing

15 Describing the Growth: Initial Level WISC Score µ 0 + σ Grade at Testing

16 Describing the Growth: Slope WISC Score µ s + σ s 20 µ 0 + σ Grade at Testing

17 Basic Growth Model Factor Model σ 0 2 σ 0,s σ s 2 y 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]

18 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 Y 2 1 β 2 u 0 σ 2 u Λ = Y 3 1 β 3 Ψ = u σ u Y 4 1 β 4 u σ 4 u Y 5 1 β 5 u σ 5 u 2 0 Y 6 1 β 6 u 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

19 Basic Growth Model With Means ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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]

20 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 )

21 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

22 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

23 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

24 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

25 Level Only Model (Y [t]n = y 0n + e [t]n ) WISC Score Grade at Testing

26 Level Only Growth Model µ 0 1 y 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]

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

28 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] σ 2 e

29 No-Growth Model (with σ 02 ) y 0 * σ 0 µ 0 1 y 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]

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

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

32 Linear Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) WISC Score Grade at Testing

33 Linear Growth ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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]

34 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

35 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

36 Linear Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 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 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 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

37 Latent Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) WISC Score Grade at Testing

38 Quadratic Slope (Y [t]n = y 0n + B 1[t] y s1n +B 2[t] y s2n + e [t]n ) WISC Score Grade at Testing

39 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

40 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

41 Statistical Methods to Represent Growth and Change 2: Nonlinear Models

42 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

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

44 Empirical Nonlinear Data 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability Age

45 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 General Fluid Ability score General Crystallized Ability score Age-at-Testing Age-at-Testing (McArdle, Ferrer, Hamagami, & Woodcock, 2002)

46 Nonlinear Models Exponential functions Cross-sectional data Visual Matching Scores Age (yr) 40 Cross-sectional data 30 Cross Out Scores Age (yr) Y = a be c*age

47 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

48 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

49 Nonlinear Growth Fixed Coefficients ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 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]

50 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 )

51 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]

52 Quadratic Slope (Y [t]n = y 0n + B 1[t] y s1n +B 2[t] y s2n + e [t]n ) WISC Score Grade at Testing

53 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]

54 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)

55 Nonlinear Models Splines 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability Age

56 Nonlinear Models Splines 140 Heart Rate During Gazing Task -- Non-Attached 140 Heart Rate During Gazing Task -- Attached Heart Rate Heart Rate Time (seconds) Time (seconds)

57 Nonlinear Models Splines HR -- Non-Attached Heart Rate During Gazing Task r t1,t1 =.22 r t1,t1 =.17 ns Time (s) HR -- Attached r t1,t1 =.43 r t1,t1 = Time (s)

58 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

59 Nonlinear Models Residuals AR(1) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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]

60 Nonlinear Models Residuals AR(2) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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

61 Nonlinear Models Residuals (other) ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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

62 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

63 Nonlinear Models Latent Basis ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 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]

64 Latent Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) WISC Score Grade at Testing

65 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

66 Latent Growth - Covariance Expectations Σ = Y [1] Y [2] Y [4] Y [6] Y [1] σ 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 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 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

67 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 (=) (46) 27.7 (62) (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 (=) (35) 5.63 (17) 4.85 (11) (29) 5.61 (17) 5.27 (12) (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 ) (.00) (.00) /3 (1.63) (.06) /5 (1.28) 61/2 (.381)

68 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

69 Nonlinear Models Exponential Models ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y 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]

70 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;

71 Raw Data Longitudinal 560 WJ Fluid Ability as a Function of Age 540 WJ Fluid Ability Age

72 Raw-Data Multiple-Variable Comparison Fluid Reasoning (Gf) score Crystallized Knowledge (Gc) score Age at Testing Age at Testing Processing Speed (Gs) score Short-Term Memory (Gsm) score Age at Testing Age at Testing

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

74 Double-Exponential Model General Fluid Ability (Gf) score as a function of Age General Crystallized Ability (Gc) score as a function of Age General Fluid Ability score General Crystallized Ability score Age-at-Testing Age-at-Testing (McArdle, Ferrer, Hamagami, & Woodcock, 2002)

75 Growth Curve of Fluid Reasoning Gf General Fluid Ability (Gf) score as a function of Age General Fluid Ability score Age-at-Testing

76 Individual Modeling Predicted change in Broad Cognitive Ability (BCA) score as a function of first Age of testing 60 Predicted Broad Cognitive Ability score Age-at-Testing (first age is real data, second age is predicted scores)

77 Nonlinear Models Fluctuations 5 4 positive affect time in days

78 Nonlinear Models More Complex Functions

79

80 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 (=) (46) 27.7 (62) (42) 28.6 (61) (4) 23.5 (13) 10.8 (57) Regressions from X Level γ 0x Slope γ sx (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 (=) (35) 5.63 (17) 4.85 (11) (29) 5.61 (17) 5.27 (12) (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 ) (.00) (.00) /3 (1.63) (.06) /5 (1.28) 61/2 (.381) (.05)

81 Fitting Latent Growth Models

82 Fitting Latent Growth Models 1: Univariate Models

83 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

84 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 = Fit alternative models of change to these data

85 WISC Data (Individual Scores) WISC Score Grade at Testing

86 Level Only Growth Model y 0 * σ 0 µ 0 1 y 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]

87 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)

88 Level Only Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label total1 <----- LEVEL total2 <----- LEVEL total3 <----- LEVEL total4 <----- LEVEL total5 <----- LEVEL total6 <----- LEVEL Means: Estimate S.E. C.R. Label LEVEL mn_leve Variances: Estimate S.E. C.R. Label LEVEL E v_uniq E v_uniq E v_uniq E v_uniq E v_uniq E v_uniq Chi-square = Degrees of freedom = 11 Probability level = 0.000

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

90 Level Only Model (Y [t]n = y 0n + e [t]n ) WISC Score Grade at Testing

91 Linear Growth Model ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y y s 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]

92 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) "

93 Linear Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label total1 <----- LEVEL total2 <----- LEVEL total3 <----- LEVEL total4 <----- LEVEL total5 <----- LEVEL total6 <----- LEVEL total1 <----- SLOPE total2 <----- SLOPE total3 <----- SLOPE total4 <----- SLOPE total5 <----- SLOPE total6 <----- SLOPE Means: Estimate S.E. C.R. Label LEVEL mn_leve SLOPE mn_slop Covariances: Estimate S.E. C.R. Label LEVEL <-----> SLOPE c_ls Variances: Estimate S.E. C.R. Label LEVEL SLOPE E v_uniq

94 Estimates from a Linear Slope Model.65 y 0 * y s * y y s Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 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

95 Linear Slope (Y [t]n = y 0n + B [t] y sn + e [t]n ) WISC Score Grade at Testing

96 Latent Growth Model ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 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]

97 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) "

98 Variances: Estimate S.E. C.R. Label LEVEL SLOPE E v uniq Latent Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label total1 <----- LEVEL total2 <----- LEVEL total3 <----- LEVEL total4 <----- LEVEL total5 <----- LEVEL total6 <----- LEVEL total1 <----- SLOPE total2 <----- SLOPE b_1 total3 <----- SLOPE total4 <----- SLOPE b_2 total5 <----- SLOPE total6 <----- SLOPE Means: Estimate S.E. C.R. Label LEVEL mn_leve SLOPE mn_slop Covariances: Estimate S.E. C.R. Label LEVEL <-----> SLOPE c_ls

99 Estimates from a Latent Slope Model.55 y 0 * y s * y 0 y s Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] 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

100 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 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]

101 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

102 Latent Slope with Exogenous Variable Regression Weights: Estimate S.E. C.R. Label LEVEL < momed mom_l SLOPE < momed mom_s total1 < LEVEL total2 < LEVEL total1 < SLOPE total2 < SLOPE b_1 total3 < SLOPE total4 < SLOPE b_2 total5 < SLOPE total6 < SLOPE Intercepts: Estimate S.E. C.R. Label momed int_mom LEVEL int_lev SLOPE int_slo Covariances: Estimate S.E. C.R. Label var_level <---> var_slope c_ls Variances: Estimate S.E. C.R. Label var_momed var_level var_slope E v_uniq E v_uniq

103 Estimates from a Latent Slope Model with Mother s Education z y0 * z ys * y y s X 7.25 Y [0] Y [1] Y [2] Y [3] Y [4] Y [5] e y[0] e y[1] e y[2] e y[3] e y[4] e y[5] χ 2 (8) = 22

104 LDS Fit Statistics Goodness Model 1 Model 2 Model 3 Model 4 of Fit Level Linear Latent Mothed LRT (χ 2 ) df RMSEA p-close fit χ 2 / df /3 61/2 5/2

105 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)

106 Other Programs: Mplus

107 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;

108 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 [email protected] [email protected] [email protected] [email protected] 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;

109 Linear Growth Model (Mplus output) TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom 8 P-Value RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= MODEL RESULTS Estimates S.E. Est./S.E. Means LEVEL SLOPE Variances LEVEL SLOPE SLOPE BY LWISC LWISC LWISC LWISC LWISC LWISC LEVEL WITH SLOPE Residual Variances WISC WISC

110 Latent Growth Model (Mplus output) Chi-Square Test of Model Fit Value Degrees of Freedom 6 P-Value RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= Means LEVEL SLOPE Variances LEVEL SLOPE SLOPE BY LWISC LWISC LWISC LWISC LWISC LWISC SLOPE WITH LEVEL Residual Variances WISC WISC WISC WISC

111 Other Programs: SAS

112 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;

113 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;

114 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 <.000 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001

115 Model 1: No Growth Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id Residual <.0001 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001

116 Model 2: Linear Growth Model: Linear Age Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept agec Covariance Parameter Estimates Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 UN(2,1) id <.0001 UN(2,2) id <.0001 Residual <.0001 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) Solution for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept <.0001 agec <.0001

117 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;

118 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;

119 Age-Based Linear Model (NLMixed) Linear Growth Curve Model Fit Statistics -2 Log Likelihood AIC (smaller is better) Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level < m_slope < v_level < v_slope < c_ls < v_error < Parameter Estimates Parameter Upper Gradient m_level m_slope E-6 v_level E-6 v_slope E-6 c_ls E-6 v_error E-6

120 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;

121 Latent Growth Model (NLmixed) Fit Statistics -2 Log Likelihood AIC (smaller is better) Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level < m_slope < v_level < v_slope < c_ls < v_error < basis < basis < Parameter Estimates Parameter Upper Gradient m_level m_slope v_level v_slope c_ls v_error basis basis

122 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;

123 Age-Based Latent Model (SAS NLMixed) Latent Growth Curve Model Fit Statistics -2 Log Likelihood AIC (smaller is better) Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level < m_slope < v_level < v_slope < c_ls < v_error < basis < basis < Parameter Estimates Parameter Upper Gradient m_level m_slope v_level v_slope c_ls v_error basis basis

124 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;

125 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;

126 Linear Model with Covariate Model: Linear Age plus Covariate on Level and Slope Estimated G Correlation Matrix Row Effect id Col1 Col2 1 Intercept agec Covariance Parameter Estimates Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id <.0001 UN(2,1) id <.0001 UN(2,2) id <.0001 Residual <.0001 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) Solution for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept <.0001 agec <.0001 mothed <.0001 agec*mothed

127 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;

128 Latent Model with Exogenous Variable Outcome Y = Wisc Total Latent Growth Curve Model with Mothered The NLMIXED Procedure Fit Statistics -2 Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Parameter Estimates Parameter Estimate SE DF t Value Pr > t Alpha Lower Upper i_level i_slope < v_level < v_slope < c_ls < v_error < beta_lm < beta_sm basis < basis <

129 Estimates from a Latent Slope Model with Mother s Education z y0 * z ys * X 7.28 y 0 y s 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]

130 Other Programs: LISREL

131 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 / 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*

132 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)

133 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

134 Level Only Model (LISREL output) LISREL Estimates (Maximum Likelihood) BETA cte e1 e2 e3 e4 e wisc (0.37) level (0.52) slope e6 level slope level* slope* wisc wisc wisc wisc wisc wisc (0.37) level (0.81) 4.56 slope

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

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

137 Linear Growth Model ρ 0s y 0 * y s * σ 0 µ 0 1 µ s σ s y y s 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]

138 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 / 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*

139 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)

140 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)

141 Linear Growth Model (LISREL output) LISREL Estimates (Maximum Likelihood) BETA cte e1 e2 e3 e4 e wisc (0.11) level (0.43) slope (0.45) e6 level slope level* slope* wisc wisc wisc wisc wisc wisc (0.11) level (0.33) slope (0.43) 11.15

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

143 wisc wisc (0.10) level (0.33) slope (0.42) Latent Growth Models (LISREL output) BETA cte e1 e2 e3 e4 e wisc (0.10) level (0.44) slope (0.47) e6 level slope level* slope* wisc (0.01) wisc wisc (0.01) 67.65

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

145 Fitting Growth Models Using Other Programs

146 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

147 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).

148 Linear Growth Model (HLM)

149 Linear Growth with Extension Variable (HLM)

150 Linear Growth Model (MLwiN)

151 Linear Growth Model (MLwiN)

152 Linear Growth with Extension Variable (MLwiN)

153 Linear Growth with Extension Variable (MLwiN)

154 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

155

156 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;

157 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;

158 Fitting Latent Growth Models 2: Nonlinear Models

159 Latent Growth Model ρ 0s y 0 * y s * σ 0 1 σ s µ 0 µ s y 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]

160 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) "

161 Variances: Estimate S.E. C.R. Label LEVEL SLOPE E v uniq Latent Growth Model (AMOS output) Regression Weights: Estimate S.E. C.R. Label total1 <----- LEVEL total2 <----- LEVEL total3 <----- LEVEL total4 <----- LEVEL total5 <----- LEVEL total6 <----- LEVEL total1 <----- SLOPE total2 <----- SLOPE b_1 total3 <----- SLOPE total4 <----- SLOPE b_2 total5 <----- SLOPE total6 <----- SLOPE Means: Estimate S.E. C.R. Label LEVEL mn_leve SLOPE mn_slop Covariances: Estimate S.E. C.R. Label LEVEL <-----> SLOPE c_ls

162 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;

163 Latent Growth Model (Mplus input cont.)!level loadings fixed at 1 level BY lwisc1-lwisc6@1 ;!slope loadings slope BY lwisc1@0 lwisc2*.2 [email protected] lwisc4*.6 [email protected] 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;

164 Latent Growth Model (Mplus output) Chi-Square Test of Model Fit Value Degrees of Freedom 6 P-Value RMSEA (Root Mean Square Error Of Approximation) Estimate Percent C.I Probability RMSEA <= Means LEVEL SLOPE Variances LEVEL SLOPE SLOPE BY LWISC LWISC LWISC LWISC LWISC LWISC SLOPE WITH LEVEL Residual Variances WISC WISC WISC WISC

165 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;

166 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;

167 Age-Based Latent Model (SAS NLMixed) Latent Growth Curve Model Fit Statistics -2 Log Likelihood AIC (smaller is better) Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > t Alpha Lower m_level < m_slope < v_level < v_slope < c_ls < v_error < basis < basis < Parameter Estimates Parameter Upper Gradient m_level m_slope v_level v_slope c_ls v_error basis basis

168 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;

169 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;

170 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;

171 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;

172 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