Exogenous Variables and Multiple Groups

Transcription

1 Exogenous Variables and Multiple Groups

2 LGC -- Extension Variables McArdle & Epstein (1987)

3 Growth Model with Exogenous Variable ω 0s z y0 * z ys * ω 0 ω s γ 01 1 γ s1 µ x γ 0x γ sx X σ x 2 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]

4 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 = γ 01 + γ 0x X n + e 0n Prediction of individual slope scores y sn = γ s1 + γ sx X n + e sn Exactly the same logic as what are now termed hierarchical or multi-level models

5 Extension Variables (Mplus) Initial Latent Growth Model Y [t]n = y 0n + B [t] y sn + e [t]n Prediction of individual level scores y 0n = γ 01 + γ 0x X n + e 0n Prediction of individual slope scores y sn = γ s1 + γ sx X n + e sn Exactly the same logic as what are now termed hierarchical or multi-level models

6 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

7 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

8 Latent Slope with Exogenous Variable 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

9 Linear Model with Exogenous Variable!level loadings fixed at 1 level BY lwisc1-lwisc6@1 ;!slope loadings fixed at linear estimates (0-1) slope BY lwisc1@0 lwisc2@.2 lwisc3@.4 lwisc4@.6 lwisc5@.8 lwisc6@1;!effects of mother on level and slope level ON momed*1.2; slope ON momed*.45;!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);

10 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;

11 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

12 Multiple Groups A common research question: Are the model parameters equal across subpopulations? Is the process under investigation equal across samples? MG analysis was initiated by Joreskog (1971) and Sorbom (1974, 1978)

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17 Multiple Groups 1: Multi-level Grouping Modeling

18 LGC with Group Variable ω 0s z y0 * z ys * ω 0 ω s γ 01 1 γ s1 µ g γ 0g γ sg G σ x 2 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]

19 LGC with Group Variable - Multilevel Individual Growth Model Y [t]n = y 0n + B [t] y sn + e [t]n Prediction of individual level and slope scores y 0n = γ 01 + γ 0g G n + e 0n y sn = γ s1 + γ sg G n + e sn Interpretation depends on the centering and scaling Do measured group-variables G account for systematic variation in the latent levels or slopes? This is exactly the logic of multi-level models

20 LGC with Group Variable - Multilevel!level loadings fixed at 1 level BY lwisc1-lwisc6@1 ;!slope loadings fixed at linear estimates (0-1) slope BY lwisc1@0 lwisc2@.2 lwisc3@.4 lwisc4@.6 lwisc5@.8 lwisc6@1;!effects of mother on level and slope (WITH MOM_GRAD = 0,1) level ON mom_grad*1.2; slope ON mom_grad*.45;!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);

21 Multiple Groups 2: LGC with Multiple Groups

22 LGC with Multiple Groups ρ 0s y 0 * y s * ρ 0s y 0 * y s * σ 0 1 σ s σ 0 1 σ s µ 0 µ s µ 0 µ s y β 1 β 2 y s β 3 β 4 β 5 β 6 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] Y [1] Y [2] Y [3] Y [4] Y [5] Y [6] σ e σ e σ e σ e σ e σ e σ e σ e σ e σ e σ e σ e e y[1] e y[2] e y[3] e y[4] e y[5] e y[6] e y[1] e y[2] e y[3] e y[4] e y[5] e y[6] Group 1 Group 2

23 Multiple Groups Known Groups Similar approach as longitudinal factor analysis strong emphasis on factorial invariance Groups can differ in: - means of the group factors µ 0, µ s - variance components σ 02, σ s2, σ e2, σ 0s - shape of growth B [t]

24 Multiple Groups - Steps The data are scaled and split up into G independent groups of subjects A model is used to examine processes both within and between groups Covariance matrices (or raw data) must be used for multiple-group analysis (due to equality) Means should also be included Using strong sampling theory, the loadings are forced to be equal (invariant) over groups Misfit indices for each group can be added to form a misfit index for the overall model

25 Multiple Groups Regression In the traditional analysis of covariance, the model can be written as Y n = B 0 1 n + B g G n + B x X n + B i (G n X n )+ e n where G is a binary variable (yes or no) and GX is a product variable G n X n If dummy (0,1) codes are used, the model can be written as Y n : [G n =0]= B 0 1 n + B g 0 + B x X n + B i (0 X n )+ e n = B 0 1 n + B x X n + e n Y n : [G n =1] = B 0 1 n + B g 1 + B x X n + B i (1 X n )+ e n = B 0 1 n + B g 1 + B x X n + B i (1 X n )+ e n where B 0 and B x are the intercept and slope for the group coded 0, and B g and B i are the change in the intercept and change in the slope for the group coded 1

26 Multiple Groups Regression Given the covariance model Y n = B 0 1 n + B g G n + B x X n + B i (G n X n )+ e n Such a model can be re-written for two groups (0,1) as Y (0) n = B (0) 0 1 (0) n + B (0) 1 X (0) n + e (0) n, and Y (1) n = B (1) 0 1 (1) n + B (1) 1 X (1) n + e (1) n Means and slopes can be constrained to be invariant as Y (0) n = B (=) 0 1 (0) n + B (=) 1 X (0) n + e (0) n, and Y (1) n = B (=) 0 1 (1) n + B (=) 1 X (1) n + e (1) n In this approach B g = 0 is tested by B (0) 0 = B (1) 0, and B i = 0 is tested by B (0) 1 = B (1) 1

27 Multiple Groups Factor Analysis Factorial invariance initially defined by Thurstone (1947) and Meredith (1964) Given a general factor model Y (g) = Λ (g) f (g) + u (g), for g =1 to G Configural invariance refers to the pattern of loadings; it requires the same number of factors and the same pattern of zeros across groups y m (1) n = Λ m (1) f (1) n + u m (1) n y m (2) n = Λ m (2) f (2) n + u m (2) n y m (g) n = Λ m (g) f (g) n + u m (g) n

28 Multiple Groups Factor Analysis And metric invariance implies equality of the loadings (zero and nonzero) across groups Λ (1) = Λ (2), and this is strong factor theory y m (1) n = Λ m (*) f (1) n + u m (1) n y m (2) n = Λ m (*) f (2) n + u m (2) n y m (g) n = Λ m (*) f (g) n + u m (g) n

29 Multiple Groups Implications Given a common factor model over time Y [t]n = Λ [t] f [t]n + u [t]n, for Λ [1] = Λ [2] If this model does not fit, it might be evidence for a qualitative change, with possible important theoretical implications One might consider some sampling theory of variables and a more complex factor pattern Or one might find that some specific variables may be the problem and only a model with partial invariance may be needed

30 Multiple Groups Likelihood Calculations We can write a log likelihood (L 2 ) based on the means (observed m and expected µ) and covariances (C and Σ) for N individuals as L 2 = N * f{[m - µ], [C - Σ]} so the differences L 2 [1] - L 2 [2] ~ χ 2 (P [2] - P [1] ) Now consider 2 independent groups (A and B) with data on the same variables. We write a model for group A as [µ (a), Σ (a) ] and a model for group B as [µ (b), Σ (b) ] We write the the joint likelihood as L 2 = L 2(a) + L 2(b) where L 2(a) = N (a) * f{[m (a) - µ (a)], [C (a) - Σ (a) ]}, and L 2(b) = N (b) * f{[m (b) - µ (b)], [C (b) - Σ (b) ]}

31 Multiple Groups Fit Evaluation If no parameters are invariant over groups then the maximum likelihood is achieved but using the most parameters L 2 = L 2(a) + L 2(b) but P = P[µ (a) +Σ (a) +µ (b) +Σ (b) ] If some parameters are invariant over groups then less than the maximum likelihood is achieved with fewer than the maximum number of parameters L 2 > L 2(a) + L 2(b) but P < P[µ (a) +Σ (a) +µ (b) +Σ (b) ] If all parameters are invariant over groups then the overall model with minimum likelihood is achieved with the minimum number of parameters L 2 = N * f {[m - µ], [C-Σ]} and P =P[µ +Σ]

32 Multiple Groups More In theory, it seems reasonable to specify experimental concepts using a multiple-group approach Among other benefits, it emphasizes invariance and replication In practice, however, there is a variety of problems using programs for multiple-group analyses The multiple-group approach is also applicable to incomplete data; model parameters can be recovered by combining different data sets using a single model But multiple-group analyses can entail complexities and computer programs do not always work As in all SEM models, there is no causal inference without good design

33 LGC with Multiple Groups McArdle (1994); McArdle & Hamagami (1991, 1992) Find different patterns of incompleteness and treat each pattern as a group Parameters are estimated simultaneously for all groups using all available data Involves assumptions of metric factorial invariance Other procedures are also available (e.g., Multiple Imputation)

34 LGC with Multiple 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

35 McArdle & Hamagami (1991)

36 LGC with Multiple Groups (McArdle & Anderson, 1990) Purpose: Modeling applications in research on aging WAIS-Total score data on four groups of adults (N=87) over four occasions Relationship among cognitive growth, age, and occasion of measurement

37 WAIS scores from 4 groups

38 ML Estimates for Multiple Groups

39 Mplus SEM for Observed Groups TITLE: Latent Growth Model for WISC data with multiple groups DATA: FILE IS d:\uva_2005\data\wisc4vpe.dat; VARIABLE: NAMES ARE V6 V7 V9 V11 P6 P7 P9 P11 moeducat; USEVAR=V6 V7 V9 V11; GROUPING is moeducat (1=nohs, 2=somehs, 3=hsgrad) ; ANALYSIS: TYPE=MEANSTRUCTURE; MODEL: y0 BY ys BY 6 V7 * 7 V9 * 9 11; V6-V11 * 10 (1); 0]; [y0*50 ys*5]; MODEL nohs: ys BY V7 * 7 V9 * 9; V6-V11 * 10 (1); [y0*50 ys*5]; MODEL somehs: ys BY V7 * 7 V9 * 9; V6-V11 * 10 (1); [y0*50 ys*5]; MODEL hsgrad: ys BY V7 * 7 V9 * 9; V6-V11 * 10 (1); [y0*50 ys*5]; OUTPUT: SAMPSTAT;

40 Multiple Groups 3: LGC with Latent Multiple Groups

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42 Latent-Class Models for Group Differences We write the same growth curve model but we add information about groups by writing a growth-mixture model (Muthen & Muthen) for c = 1 to C classes: (a) LGM: Y [t]n (c) = y 0n (c) + B [t] y sn (c) + e [t]n (c) (b) Mixture: Y [t]n = Σπ(c) n * {y [t]n (c) + e [t]n (c) } where Σπ(c) n = 1 for each person Questions: (a) Is there any empirical evidence for multiple latent classes? (2) Does the shape of the basis trajectory change over the latent classes, (3) Which individuals are likely to be in what class, (4) What are the features associated with class membership?

43 Mplus SEM for Latent Groups TITLE: Growth-Mixture Model for WISC data DATA: FILE IS d:\uva_2005\data\wisc4vpe.dat; VARIABLE: NAMES ARE V6 V7 V9 V11 P6 P7 P9 P11 moeducat; USEVAR=V6 V7 V9 V11; CLASSES = c(2); ANALYSIS: TYPE=MIXTURE; MODEL: %OVERALL% y0 BY ys BY 6 V7 * 7 V9 * 9 11; V6-V11 * 10 (1); 0]; [y0*50 ys*5]; %c#2% [y0*50 ys*5]; OUTPUT: SAMPSTAT;

44 Multivariate Models

45 Longitudinal Factorial Invariance Multivariate longitudinal factorial invariance Y (t)n = λ y f (t)n + e y(t)n, and X (t)n = λ x f (t)n + e x(t)n, and W (t)n =λ w f (t)n +e w(t)n, and Z (t)n = λ z f (t)n + e z(t)n

46 Longitudinal Factorial Invariance σ f0 2 σ ft 2 f [0] f [t] λ x λ y λ w λ z λ x λ y λ w λ z X [0] Y [0] W [0] Z [0] X [t] Y [t] W [t] Z [t] e x[0] e y[0] e w[0] e z[0] e y[0] e y[1] e y[2] e y[3] σ ex0 2 σ ey0 2 σ ew0 2 σ ez0 2 σ ext 2 σ eyt 2 σ ewt 2 σ ezt 2

47 Dynamic Factor Analysis (Molenaar, 1985) The DFM incorporates q factors and s lags of manifest variables on common factors (DFM [q,s] ) is specified as z (t) = Λ (0) η (t) + Λ (1) η (t-1) + + Λ (s) η (t-s) + ε (t) where z (t) is the observed p-variate time-series measured at time t, η (t) is the latent q-variate factor time-series, Λ (u), u = 0, 1,, s-1 are p x q matrices of lagged factor loadings, and ε (t) is a p-variate noise time-series

48 U 1t-1 U 2t-1 U 3t-1 U 1t U 2t U 3t U 1t+1 U 2t+1 U 3t+1 M 1t-1 M 2t-1 M 3t-1 M 1t M 2t M 3t M 1t+1 M 2t+1 M 3t+1 F1 t-3 F1 t-2 F1 t-1 F1 t F1 t+1 F2 t-3 F2 t-2 F2 t-1 F2 t F2 t+1 M 4t-1 M 5t-1 M 6t-1 M 4t M 5t M 6t M 4t+1 M 5t+1 M 6t+1 U 4t-1 U 5t-1 U 6t-1 U 4t U 5t U 6t U 4t+1 U 5t+1 U 6t+1 t-3 t-2 t-1 t t+1 Time

49 Curve of Factors (CUFFS) Multivariate curve of factors model (McArdle, 1988) Y (t)n = λ y f (t)n + e y(t)n, and X (t)n = λ x f (t)n + e x(t)n, and W (t)n =λ w f (t)n + e w(t)n, and Z (t)n = λ z f (t)n + e z(t)n, with f (t)n = f 0n + β (t) f sn + z (t)n

50 Curve of Factors (CUFFS) σ f0 2 1 µ f0 µ fs σ fs 2 σ f0s f 0 f s σ z β 0 β t σ zt 2 f [0] f [t] λ x λ y λ w λ z λ x λ y λ w λ z X [0] Y [0] W [0] Z [0] X [t] Y [t] W [t] Z [t] e x[0] e y[0] e w[0] e z[0] e y[0] e y[1] e y[2] e y[3] σ ex0 2 σ ey0 2 σ ew0 2 σ ez0 2 σ ext 2 σ eyt 2 σ ewt 2 σ ezt 2

51 Curve of Factors

52 Multivariate LDE Structural Model (Boker)

53 LGC Extensions--Multivariate Models Multivariate LGC model (parallel growth curves) Y(Age t ) n = µ y + λ y g(age t ) n + µ y (Age t ) n X(Age t ) n = µ x + λ x g(age t ) n + µ y (Age t ) n with g(age t ) n = g 0n + β(age t ) Sg n + e(age t ) n

54 Multivariate LGC model

55 Structural Model of Latent Growth Curves Multivariate LGC model (correlated or associative growth curves) Y (t)n = y 0n + λ y(t) y sn + e y(t)n, and X (t)n = x 0n + λ x(t) x sn + e x(t)n

56 Structural Model of Latent Growth Curves ρ y0x0 x s * ρ ysxs y s * x 0 * y 0 * σ x0 σ xs µ x0 1 µ y0 σ ys σ y0 µ xs µ ys x s x 0 y s y 0 λ x0 λx1 λ x2 λ x3 λ y0 λ y1 λ y2 λ y3 X [0] X [1] X [2] X [3] Y [0] Y [1] Y [2] Y [3] e x[0] e x[1] e x[2] e x[3] e y[0] e y[1] e y[2] e y[3] σ 2 ex σ 2 ex σ 2 ex σ 2 ex σ 2 ey σ 2 ey σ 2 ey σ 2 ey

57 Dynamics of Co-regulation (Hazan, Ferrer, & Sbarra, 2005) 140 Heart Rate During Gazing Task -- Non-Attached 140 Heart Rate During Gazing Task -- Attached Heart Rate Heart Rate Time (seconds) Time (seconds)

58 Dynamics of Co-regulation (Hazan, Ferrer, & Sbarra, 2005) 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)

59 Factor of Curves (FOCUS) Multivariate curve of factors model (McArdle, 1988) Y (t)n = y 0n + λ y(t) y sn + e y(t)n, and X (t)n = x 0n + λ x(t) x sn + e x(t)n, with m 0n = α m f 0n + v 0n, and m sn = β m f sn + w sn,

60 Factor of Curves (FOCUS) σ f0 2 1 µ f0 µ fs σ fs 2 f 0 σ f0s f s α x0 α y0 β xs α ys σ x0 2 σ y0 2 σ y0 2 σ ys 2 x s x 0 y 0 y s λ x0 λx1 λ x2 λ x3 λ y0 λ λ λ y3 y1 y2 X [0] X [1] X [2] X [3] Y [0] Y [1] Y [2] Y [3] e x[0] e x[1] e x[2] e x[3] e y[0] e y[1] e y[2] e y[3] σ 2 ex σ 2 ex σ 2 ex σ 2 ex σ 2 ey σ 2 ey σ 2 ey σ 2 ey

61 Cross-Lagged Regression Model Multivariate cross-lagged regression model Y (t)n = β y y (t-1)n + γ y x (t-1)n + d y(t)n, and X (t)n = β x x (t-1)n + γ x y (t-1)n + d x(t)n, and

62 Cross-Lagged Regression Model σ 2 ey σ 2 ey σ 2 ey e y[0] e y[1] e y[2] e y[3] Y [0] Y [1] Y [2] Y [3] σ 2 y0 σ y0,x0 β y β y β y y [0] y [1] y [2] y [3] γ y σ 2 σ 2 dy dy σ 2 dy d y[1] d y[2] d y[3] γ y ρ dyx ρ dyx ρ dyx γ y γ x γ γ x x d x[1] d x[2] d x[3] σ 2 x0 x [0] x [1] x [2] x [3] β x β x β x σ 2 dx σ 2 dx σ 2 dx X [0] X [1] X [2] X [3] e x[0] e x[1] e x[2] e x[3] σ 2 ex σ 2 ex σ 2 ex

63 Latent Difference Score (LDS) Model Assume the observed trajectories are written as Y [t]n = y 0n + (Σ i=1,t y [i]n ) + e y[t]n and X [t]n = x 0n + (Σ i=1,t x [i]n ) + e x[t]n Any model for the latent changes can be written as y [t]n = α y y sn + β y y [t-1]n + γ y x [t-1]n, and x [t]n = α x x sn + β x x [t-1]n + γ x y [t-1]n Key hypotheses about the changes include coupling (a) γ y = 0, (b) γ x = 0, and (c) γ yx = γ xy = 0

64 σ 2 ey σ 2 ey σ 2 ey σ 2 ey e y[0] e y[1] e y[2] e y[3] Y [0] Y [1] Y [2] Y [3] y 0 * y [0] y [1] y [2] y [3] y s * y 0 γ x β y γ x β y y[1] y[2] y[3] γ x β y ρ y0x0 ρ ysxs * x s 1 µ y0 µ ys x 0 y s x s α y α x µ xs ρ eyex µ y0 γ y β x γ y x[1] x[2] x[3] β x γ y β x x 0 * x [0] x [1] x [2] x [3] X [0] X [1] X [2] X [3] e x[0] e x[1] e x[2] e x[3] σ 2 ex σ 2 ex σ 2 ex σ 2 ex

65 Y [0] Y [1] Y [t-1] Y [t] y 0 * y [0] y [1] y [t-1] y [t] y s * y 0 γ x y[1] γ x y[t-1] γ x y[t] 1 y s x s x s * x 0 * x 0 γ y x[1] γ x[t-1] y γ y x[t] x [0] x [1] x [t-1] x [t] X [0] X [1] X [t-1] X [t]

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67 Multivariate Dynamics Gf Gf Gc Gc g g Gq Gq Gak Gak General Factor Hypothesis

68 Dynamic Modeling: Introduction to Latent Difference Score Models

69 Structural Equation Modeling of Longitudinal Data Multivariate change models are widely used Include seminal SEM work by Joreskog (1971, 1974, 1977), and Sorbom (1975) Related to factor analytic change models of Nesselroade (1971, 1977) and Meredith (1990, 1991) More recent dynamic models of Molenaar (1985), Arminger (1986), and Browne (1990)

70 Dynamic Systems: Introduction Dynamic systems -- systems in which the focus is on time-dependent processes Dynamics center on motion and its underlying forces Process as key issue underlying development Individual differences in change LDS encourage formal considerations of change models and development --- What is your theory of change y [t] / [t] =?

71 Dynamic SEM Models Subtle differences in the choice among models define the nature of developmental process and change -- i.e., the dynamic systems The models presented here are new SEM extensions but all dynamic parameters can be estimated using standard SEM programs The traditional SEM methods provide a framework for the study of alternatives dynamic models

72 Latent Difference Score Models (McArdle, 2001; McArdle & Hamagami, 2001) Based on latent true scores y [t] and errors e [t] Y [t]n = y [t]n + e [t]n Latent difference scores can be defined as y [t]n = y [t-1]n + y n, or y n = y [t]n -y [t-1]n But different forms based on theory of change y/ t

73 A Latent Difference Score Model σ e1 2 σ e2 2 Y[1] Y[2] 1 1 y[1] y[2]

74 A Latent Difference Score Model σ e1 2 σ e2 2 Y[1] Y[2] 1 1 y[1] 1 y[2] 1 y

75 A Latent Difference Score Model σ e1 2 σ e2 2 Y[1] Y[2] 1 1 σ 1 2 y[1] 1 y[2] σ 1d µ µ d y σ d 2

76 Growth and Change Using Latent Difference Scores

77 Steps in LDS Models Based on latent true scores y [t] and errors e [t] Y [t]n = y [t]n + e [t]n Latent difference scores can be defined as y [t]n = y [t-1]n + y [t]n or y [t]n = y [t]n - y [t-1]n This leads to a component for the accumulation of the latent changes ( y [t] ) up to time t Y [t]n = y 0n + (Σ i=1,t y [i]n ) + e [t]n Any model for the latent changes can be written as y t / t = α + β. y t - t

78 Alternative LDS Models No change over time y t / t = 0 Additive change over time y t / t = α. y s Proportional change over time y t / t = β. y t- t Dual change over time y t / t = α. y s + β. y t- t Adding residuals to the change y t / t = α. y s + β. y t- t + z t

79 σ y0 2 y 0 µ y e 1 e 2 y[3] y[4] e 5 e 6 y[1] y[2] y[5] y[6] Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

80 σ y0 2 y 0 µ y0 1 y[2] y[3] y[4] y[5] y[6] y[1] y[2] y[5] y[6] e 1 e 2 y[3] y[4] e 5 e Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

81 Constant Change Score Model Consider a classical first-difference equation y [t] / [t] = α. y s By data design we define t =1, add individual differences in scores, and write the CCS with group parameter α and person scores y sn as y [t]n = α. y sn This model leads to the time series sequence y [t]n = y [t-1 ]n + y [t]n = y [t-1]n + α. y sn This LDS has the same particular solution or factorization as a linear growth model y [t]n = y [0]n + t. (α. y sn )

82 σ y0,ys σ y0 2 µ y0 µ 1 ys y 0 y s σ ys 2 α α α α α y[2] y[3] y[4] y[5] y[6] e 1 e 2 y[3] y[4] e 5 e 6 y[1] y[2] y[5] y[6] Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

83 σ y0,ys σ y0 2 µ y0 µ 1 ys y 0 y s σ ys 2 α 1 α 2 α 4 α 5 α 6 y[2] y[3] y[4] y[5] y[6] e 1 e 2 y[3] y[4] e 5 e 6 y[1] y[2] y[5] y[6] Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

84 Proportional Change Score Model Start with another first-difference equation y [t] / [t] = β. y t- t Define t = 1 and add individual differences y [t]n = β. y [t-1 ]n Leading to the time series sequence y [t]n = y [t-1 ]n + y [t]n = (1 + β ). y [t-1 ]n With the particular solution (factorization) y [t]n = (1+β ) t. y [0]n Note that this is not the auto-regressive model

85 σ y0,ys σ y0 2 y 0 µ y0 1 y[2] y[3] y[4] y[5] y[6] β β β β β e 1 e 2 y[3] y[4] e 5 e 6 y[1] y[2] y[5] y[6] Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

86 Dual Change Score Model Start with another first-difference equation y [t] / [t] = β. y [t- t] + α. y s Define t =1 and add individual differences y [t]n = β. y [t-1]n + α. Y sn This leads to the time series sequence y [t]n = y [t-1]n + β. y [t-1]n + α. Y sn with the particular solution (factorization) y [t]n = (1+β ) t. y [0]n + Σ (1+β ) j. α. y sn If α =1 identical to Latent Partial Adjustment y [t] / [t] = π. (y [t- t] - y [a] )

87 σ y0,ys σ y0 2 µ y0 µ 1 ys y 0 y s σ ys 2 α α α α α y[2] y[3] y[4] y[5] y[6] β β β β β e 1 e 2 y[3] y[4] e 5 e 6 y[1] y[2] y[5] y[6] Y[1] Y 1 Y[2] Y 2 Y[3] V 3 Y[4] Y 4 VY[5] 5 Y[6] Y 6 Y 5 σ ue σ e σ e σ e σ e σ e e 1 e[1] e[2] e 2 e[3] e[4] e[5] e[6]

88 Multivariate Latent Difference Score Models

89 Bivariate LDS Models Assume the observed trajectories are written as Y [t]n = y 0n + (Σ i=1,t y [i]n ) + e y[t]n and X [t]n = x 0n + (Σ i=1,t x [i]n ) + e x[t]n Any model for the latent changes can be written as y [t]n = α y y sn + β y y [t-1]n + γ y x [t-1]n, and x [t]n = α x x sn + β x x [t-1]n + γ x y [t-1]n Key hypotheses about the changes include coupling (a) γ y = 0, (b) γ x = 0, and (c) γ yx = γ xy = 0 Do not usually test γ y = γ x due to lack of equal scaling, and not σ ys,xs = 0 due to lack of dynamic interpretation This LDS model is different from a latent cross-lagged model if there is growth/decline

90 Bivariate Dynamic System y [t]n = α y y sn + β y y [t-1]n + γ y x [t-1]n x [t]n = α x x sn + β x x [t-1]n + γ x y [t-1]n Change Linear Slope Feed- Forward = + + Coupling

91 Latent Difference Score (LDS) Model Y [0] Y [1] Y [2] Y [t] y 0 * y [0] y [1] y [2] y [t] y s * y 0 1 y s

92 Latent Difference Score (LDS) Model Y [0] Y [1] Y [2] Y [t] y 0 * y [0] y [1] y [2] y [t] y s * y 0 y[1] y[2] y[t] 1 y s

93 Latent Difference Score (LDS) Model Y [0] Y [1] Y [2] Y [t] y 0 * y [0] y [1] y [2] y [t] y s * y 0 y[1] y[2] y[t] 1 y s x s x s * x[1] x[2] x[t] x 0 x 0 * x [0] x [1] x [2] x [t] X [0] X [1] X [2] X [t]

94 LDS Models Path Diagram Y [0] Y [1] Y [t-1] Y [t] y 0 * y [0] y [1] y [t-1] y [t] y s * y 0 γ x β y β y β y y[1] γ x y[t-1] γ x y[t] 1 y s x s x s * γ y γ γ y y x[1] x[t-1] x[t] β x β x β x x 0 * x 0 x [0] x [1] x [t-1] x [t] X [0] X [1] X [t-1] X [t]

95 LDS Models Path Diagram Y [0] Y [1] Y [t-1] Y [t] y 0 * y [0] y [1] y [t-1] y [t] y s * 1 y 0 y s α y y[1] α y y[t-1] αy y[t] x s * x s α x α α x x x[1] x[t-1] x[t] x 0 * x 0 x [0] x [1] x [t-1] x [t] X [0] X [1] X [t-1] X [t]

96 LDS Models Path Diagram Y [0] Y [1] Y [t-1] Y [t] y 0 * y [0] y [1] y [t-1] y [t] β y β y β y y s * y 0 y[1] y[t-1] y[t] 1 y s x s x s * x 0 x[1] x[t-1] x[t] β x β x β x x 0 * x [0] x [1] x [t-1] x [t] X [0] X [1] X [t-1] X [t]

97 LDS Models Path Diagram Y [0] Y [1] Y [t-1] Y [t] y 0 * y [0] y [1] y [t-1] y [t] y s * y 0 γ x y[1] γ x y[t-1] γ x y[t] 1 y s x s x s * x 0 * x 0 γ y x[1] γ x[t-1] y γ y x[t] x [0] x [1] x [t-1] x [t] X [0] X [1] X [t-1] X [t]

98 LDS Models: Benefits LDS models generate structural expectations so can be fitted using standard SEM programs LDS are flexible and can be used with (a) multiple groups, (b) multiple measures, (c) multiple curves, and (d) can be used to make forecasts over time LDS overcome some critical problems of observed difference scores LDS are a good approach to study sequences and determinants (Nesselroade & Baltes, 1979)

99 LDS Models: Limitations LDS models apply directly to latent variables but only indirectly to observed variables LDS are written for differences in a latent score over discrete periods of time and are deterministic over time within an individual LDS are no more or less causal than any other dynamic observation, but they help describe determinants over time

100 Alternative Dynamic Extensions LDS over groups defined by another X y [t]n (g) = α (g).y sn (g) + β (g). y [t-1]n (g) LDS model with predictor of slopes y [t]n = α. y sn + β.y [t-1]n and y sn = ω 0 +ω 1. X n + u n LDS model with another predictor y [t]n = α.y sn + β.y [t-1]n + γ. X n But this is substantially more difficult when we have another time-varying predictor y [t]n = α.y sn + β.y [t-1]n + γ. x [t-1]n

101 LDS Model (DCS) σ s 2 1 µ ys y s µ 0 σ 0s α α α σ 0 2 y 0 y[1] y[t-1] y[t] β β β y [0] y [1] y [t-1] y [t] Y [0] Y [1] Y [t-1] Y [t] σ e 2 σ e 2 σ e 2 σ e 2

102 LDS Model with Extension Variable σ fs 2 σ 0,s σ x 2 X f s ω 0 ω s 1 µ ys y s γ x γ x µ 0 σ 0s γ x α α α σ f0 2 f 0 y 0 y[1] y[t-1] y[t] β β β y [0] y [1] y [t-1] y [t] Y [0] Y [1] Y [t-1] Y [t] σ e 2 σ e 2 σ e 2 σ e 2

103

104

105 Dynamic Modeling: Applications of Dynamic LDS Models to Longitudinal Data

106 The Dynamics of Cognitive Abilities and Achievement from Childhood to Early Adulthood

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

108 Growth Curves of Fluid and Crystallized WJ-R Factors (McArdle, Ferrer, Hamagami, & Woodcock, 2002) 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

109 The Investment Hypothesis In the school years fluid intelligence (Gf) is invested in the learning of perceptual, discriminatory, and executive skills As these skills are acquired they attach to perceptual and motor areas and become crystallized (Gc) The development of these skills enables the child to learn and improve in school tasks such as reading, writing, and arithmetic problems

110 Hypothesized Causal Action in The Investment Hypothesis (Cattell, 1967, 1987) Gf Gc Achievement

111 Methods Data from the National Growth and Change Study (NGCS: McArdle, ) Data on the Woodcock-Johnson tests (WJ-R) Time 1: N = 6,471 (norming sample, 1987) Time 2: N = 1,193 (retest sample, 1990) Study sample: N = 672 (from retest) Ages = 5-20 at Time 1; 5-24 at Time 2 Time lag between assessments (1 to 10 years)