Principles of Latent Growth Curve Modeling

1
Principles of Latent Growth Curve Modeling
Todd D. Little, Director Quantitative Training
Program Department of Psychology University of
Kansas www.Quant.KU.edu
2
Longitudinal Difference Scores

• Use is to assess a type of change over time
• e.g., Gain scores YTime2 - YTime1
• Simple difference scores may be unreliable
• Based on reliability estimates of CTT, some
  concluded Dont bother using them (Cronbach
  Furby, 1970)
• Others argued that CTT assumptions vastly
  underestimate the reliability of difference
  scores (Williams Zimmerman, 1977)
• Simple difference scores may be invalid
• Validity of difference scores varies based on a
  variety of parameters (Williams Zimmerman,
  1982)
• E.g., t is stable and s varys, the difference is
  due to s.

3
Latent Difference Scores

• True Score Theory
• Ytn ytn etn
• observed raw score latent score error score

y1
Y1
Note The unique score, or error, is both item
specific variance, s, and random error.
e1
4
Latent SEM Difference Score
1.0
T2-T1
DV
e
e
e?
1.0
1.0
0
T1
T2
A
B
C
1.0
?x
?y
?w
?w
?x
?y
X1
Y2
W1
W2
X2
Y1
5
Latent SEM Residual Score
1.0
T2-bT1
DV
e
e
e?
T1 and residual necessarily uncorrelated
1.0
1.0
0
T1
T2
A
B
C
e
?x
?y
?w
?w
?x
?y
X1
Y2
W1
W2
X2
Y1
6
Computing Latent Difference Scores
(Long8.1.LatentDifference.PosAFF)
?33
Change
a
?31
1.0
?11
0
Positive T1
Positive T2
1.0
a
?21
?31
?11
?42
?52
?62
PA2
PA3
PA1
PA1
PA2
PA3
7
Raw Means SDs of manifest variables
(Long8.1.LatentDifference.PosAFF)
N 1684
8
Computing Latent Difference Scores
(Long8.1.LatentDifference.PosAFF)
.75
Change
.08
a
-.45
(r -.61)
1.0
0
.71
Positive T1
Positive T2
1.0
3.59
a
.95
1.05
.95
1.01
1.05
1.01
PA2
PA3
PA1
PA1
PA2
PA3
Model Fit ?2(9, n1684) 36.5 RMSEA
.043(.029.058) NNFI .992 CFI .995
9
Other uses for Latent Difference Scores

• Predicting / predicted by other variables in
  model
• What are antecedents of change?
• What are consequences of change?
• Multiple group comparisons
• Is there more change in one group than another?
• Might be especially useful in clinical trials
  (e.g., change in treatment vs. control groups)
• Represents the most basic growth curve model

10
Latent Difference Scores as Growth Curve Models
-.45 (r -.61)
(Long8.3.LatentDiffGrowthCurve.PosAFF)
.75
.71
Slope / Change
Intercept
.08
3.59
a
a
1.0
0
1.0
1.0
0
0
Positive T1
Positive T2
.95
1.05
.95
1.0
1.05
1.0
PA2
PA3
PA1
PA1
PA2
PA3
Model Fit ?2(9, n1684) 36.5 RMSEA
.043(.029.058) NNFI .992 CFI .995
11
Modeling Latent Change Growth Curve
12
Modeling Latent Change Growth Curve

• Traditional multi-level representation of growth
  curves
• Level 1 submodel
• Yit p0i p1i(Time) p2i(Time2) eit
• Level 2 submodel
• p0i ?00 ?0i ?00 average intercept
• p1i ?10 ?1i ?10 average slope
• Stochastic components
• s02 variance of intercepts
• s12 variance of slopes
• s01 covariance of intercept and slope
• se2 error variance

13
Modeling Latent Change Growth Curve
14
Raw Means Standard Deviations
(L9.Panas.4Wave.dat)
N 1684
15
Modeling Latent Change Growth Curve
a1
a2
y21
Intercept
Slope
y11
y22
l12
l41
l42
l11
l22
l31
l32
l21
Time 1
Time 2
Time 3
Time 4
q11
q22
q33
q44
16
Linear Model Negative Affect
(L9.1.GC.Linear.4Wave.NegAFF)
a1
a2
?21
Intercept
Slope
?11
?22
0
1
1
1
3
1
1
2
Time 1
Time 2
Time 3
Time 4
?
?
?
?
17
Linear Model Negative Affect
(L9.1.GC.Linear.4Wave.NegAFF)
a1
a2
2.09
-.10
-.05
Intercept
Slope
.016
.30
0
1
1
1
3
1
1
2
Time 1
Time 2
Time 3
Time 4
.30
.30
.30
.30
Model Fit ?2(8, n1684) 32.7 RMSEA
.043(.028.059) NNFI .984 CFI .979
18
Alternate Linear Model Negative Affect
(L9.2.AltLinear.4Wave.NegAFF)
a1
a2
1.94
-.05
-.02
Intercept
Slope
.004
.17
-3
1
1
1
3
1
-1
1
Time 1
Time 2
Time 3
Time 4
.30
.30
.30
.30
Model Fit ?2(8, n1684) 32.7 RMSEA
.043(.028 .059) NNFI .984 CFI .979 The
mean and variance of the intercept, and the
intercept-slope covariance, have changed and have
new interpretation. The mean variance of the
slope have changed in value but not significance.
19
Unconditional (AKA Level and Shape) Model
L9.3.Unconstrained.4Wave.NegAFF
a1
a2
2.09
-.10
-.058
Intercept
Slope
.018
.30
0
1
1
1
1
1
2.12
2.85
Time 1
Time 2
Time 3
Time 4
.30
.30
.30
.30
Model Fit ?2(6, n1684) 29.6 RMSEA
.048(.032.067) NNFI .980 CFI .980
20
Interpreting Unconditional GC Model

• Use tracing rules to reproduce means at each
  time
• Time 1 1(2.09) 0(-.10) 2.09 (observed
  2.08)
• Time 2 1(2.09) 1(-.10) 1.99 (observed
  2.00)
• Time 3 1(2.09) 2.12(-.10) 1.88 (observed
  1.89)
• Time 4 1(2.09) 2.85(-.10) 1.80 (observed
  1.78)
• Values perfectly reproduce means, within rounding
  error
• Can plot values to get picture of shape

2.00
1.90
1.80
T1
T2
T3
T4
21
Testing adequacy of Linear GC Model

• Because Linear GC is restricted version of
  unconditional GC, can compare via nested model
  comparison
• Linear GC Model ?2(8, n1684)32.7
• Unconditional GC Model ?2(6, n1684)29.6
• Difference ??2(2, n1684)3.1, p gt .30
• The difference in fit between the unconditional
  model and the enforced linear model is
  nonsignificant. Because the difference in ?2 is
  non-significant, the constraints are supported.
  The mean growth trend can be adequately captured
  as linear (the more parsimonious model).

22
Raw Means Standard Deviations
(L10.Panas.4Wave.dat)
N 1684
23
Modeling growth in Positive Affect

• Because shape looks nonlinear, it will be
  important to test adequacy of our growth curve
  models
• Here (and maybe as a general practice), we will
  begin with the Unconditional GC Model.
• After fitting the Unconditional Model, we will
  attempt to fit the data as well with Constrained
  Models (beginning with Linear GC Model).

24
Unconditional Model Positive Affect
(L10.1.LevelShape.4Wave.PosAFF)
a1
a2
y21
Intercept
Slope
y11
y22
1
0
1
1
1
1
l22
l32
Time 1
Time 2
Time 3
Time 4
?
?
?
?
Note that here the slope is identified by fixing
the last loading. This changes the scale but not
the estimated shape or significance levels.
25
Unconditional Model Positive Affect
(L10.1.LevelShape.4Wave.PosAFF)
a1
a2
3.59
.075
-.12
Intercept
Slope
.13
.38
1
0
1
1
1
1
1.09
1.43
Time 1
Time 2
Time 3
Time 4
.42
.42
.42
.42
Model Fit ?2(6, n1684) 29.8 RMSEA
.049(.032.067) NNFI .983 CFI .983
26
Linear Model Positive Affect

• Now, we see if we can fit the data as well (not
  significantly worse) with a linear model.

a1
a2
y21
Intercept
Slope
y11
y22
-3
1
1
1
3
1
-1
1
Time 1
Time 2
Time 3
Time 4
?
?
?
?
(L10.2.Linear.4Wave.PosAFF)
27
Linear Model Positive Affect
(L10.2.Linear.4Wave.PosAFF)
a1
a2
3.66
.016
-.003ns
Intercept
Slope
.002
.27
-3
1
1
1
3
1
-1
1
Time 1
Time 2
Time 3
Time 4
.45
.45
.45
.45
Model Fit ?2(8, n1684) 75.3 RMSEA
.071(.057.086 NNFI .963 CFI .951 Note
??2(2, n1684) 45.5, p lt .001
28
Adding a Quadratic Term Positive Affect
(L10.3.Quadratic.4Wave.PosAFF)
y31
a2
a1
a3
y32
y21
Intercept
Linear
Quad -ratic
29
Orthogonal Contrast Codes
30
Alt. Orthogonal Contrast Codes
Create.Orthogonal.Codes.SAS
31
Adding a Quadratic Term Positive Affect
(L10.3.Quadratic.4Wave.PosAFF)
-.003ns
a2
a1
a3
3.66
.016
-.020
-.003ns
-.006
Intercept
Quad -ratic
Linear
.29
.006
.033
1
1
1
1
1
1
-1
-1
-3
3
1
-1
Time 1
Time 2
Time 3
Time 4
.38
.38
.38
.38
Model Fit ?2(4, n1684) 12.5 RMSEA
.035(.014.059) NNFI .991 CFI .994
32
Adding a Quadratic Term Positive Affect
(L10.3.1.Quadratic.Atl.Ortho.4Wave.PosAFF)
.014
a2
a1
a3
3.606
.033
-.020
-.041
-.012
Intercept
Quad -ratic
Linear
.360
.023
.033
1
1
1
1
1
1
-1
-1
0
3
2
1
Time 1
Time 2
Time 3
Time 4
.38
.38
.38
.38
Model Fit ?2(4, n1684) 12.5 RMSEA
.035(.014.059) NNFI .991 CFI .994
33
Comparing Quadratic to Linear GC Model

• Compare via nested model comparison
• Linear GC Model ?2(8, n1684) 75.3
• Quadratic GC Model ?2(4, n1684) 12.5
• Difference ??2(4, n1684)62.8, p lt .001
• Based on model fit, we conclude that adding the
  Quadratic term significantly improves upon the
  Linear-only model.
• Also should consider significance of parameters
• Quadratic term had significant mean, variance,
  and covariance (with linear term).
• If any of the parameters are significant, then
  the quadratic term offers meaningful information.

34
Why Linear Models are Nested within Quadratic
Models
y310
a2
a1
a30
y21
y320
Intercept
Quad -ratic
Linear
y11
y330
y22
1
1
1
1
1
1
-1
-1
-3
3
1
-1
Time 1
Time 2
Time 3
Time 4
?
?
?
?
By constraining the mean, variance, and
covariances associated with the quadratic term,
this Quadratic model becomes a Linear GC Model
35
How complex of models can one fit?

• Statistical limits depend on number of time
  points
• Saturated growth model will perfectly reproduce
  observed means
• Unsaturated growth there is the possibility of
  misfit of the means
• Practical limits depend on what is conceptually
  meaningful.

36
Multivariate Models Multiple indicators

• Previously, we considered linear growth in
  Negative Affect in which each Time point had a
  single indicator.

a1
a2
2.09
-.10
-.05
Intercept
Slope
.016
.30
0
1
1
1
3
1
1
2
Time 1
Time 2
Time 3
Time 4
.30
.30
.30
.30
Model Fit ?2(8, n1684) 32.7 RMSEA
.043(.028.059) NNFI .984 CFI .979
(L9.1.GC.Linear.4Wave.NegAFF)
37
Multivariate Models Multiple indicators

• Modeling latent variables at each occasion
• Allows for evaluation of measurement invariance
• Useful for any longitudinal research
• Ensures that growth is not due simply to change
  in measurement properties over time (e.g., a
  measure that becomes increasingly influenced by a
  high frequency item will appear to increase over
  time).
• Allows more precise models of growth
• Models growth only in true construct, not
  specific components
• Model of growth is free of unreliable error.

38
Multivariate Models Multiple indicators
a1
a2
Intercept
Slope
0
1
1
1
3
1
1
2
Time 1
Time 2
Time 3
Time 4
(L10.4.LatentGC.4Wave.NegAFF)
39
Multivariate Models Latent Negative Affect
a1
a2
2.08
-.10
-.048
Intercept
Slope
.015
.28
0
1
1
1
3
1
1
2
Time 1
Time 2
Time 3
Time 4
.23
.23
.23
.23
.95
1.04
1.02
Model Fit ?2(50, n1684) 391.8 RMSEA
.063(.057.069) NNFI .971 CFI .978
(L10.4.LatentGC.4Wave.NegAFF)
40
Multivariate growth curves Positive Negative
Affect
Time 1
Time 2
Time 3
Time 4
Neg Intercept
Neg Linear
(L10.5.GC.4Wave.PosNegAFF)
a
a
a
a
a
Pos Intercept
Pos Quad
Pos Linear
1
-1
41
Unbalanced designs in LGM

• It is commonly said that
• LGM are better for balanced designs (due to all
  the advantages of LATENT growth curve modeling
• MLM are better for unbalanced designs

42
Unbalanced designs in LGM

• This statement is NOT true!
• Unbalanced designs can be viewed as a balanced
  design with missing data
• This missingness can be handled in two ways
• Multiple group models
• Data imputation

43
Unbalanced designs in LGM

• Data imputation approach
• Simply treat missing time points as missing data,
  and impute (perhaps multiple times) this missing
  data
• Can then fit LGM to imputed (complete) data set!

44
Accelerated Longitudinal Designs

• Intentional pattern of missing data
• Collect short-term longitudinal measures of
  multiple overlapping cohorts in order to
  approximate long-term longitudinal study
• Allows for assessment of long-term growth curves
  using shorter timeframe

45
Accelerated Longitudinal Designs
Grade
46
Accelerated Growth Curve Model
(L5.1.GC.LevelCUBIC.Accelerated.LS8)
47
Plot of Estimated Trends
48
Accelerated Longitudinal Designs
Age
49
Accelerated Longitudinal Designs
Age
50
Modeling Discontinuous Growth with Spline Models
Knot point
Growth after
Growth before
51
Raw Means Standard Deviations
(L11.Panas.6Wave.dat)
N 1684
52
Modeling Discontinuous Growth with Spline Models

• The knot point can be chosen empirically
• Often can see point in data where growth changes.
  
• Could repeatedly fit models with different knot
  points, then compare model fit (non-nested, so
  would need to rely on AIC or similar criteria)
  and interpretability.
• However, it is better if point chosen on
  conceptual grounds
• Beginning of treatment
• Life transitions (e.g., puberty, marriage, birth
  of child, retirement)
• Contextual transitions (e.g., changing school,
  immigrating)
• The knot point may be used to as frame of
  reference for time (i.e., everyone aligned at
  Time0 at knot point, with other Time represented
  as period before or after event)

53
Modeling Discontinuous Growth with Spline Models
(L11.1.Spline.6Wave.PosAFF)
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
Possible Knot point?
54
Modeling Discontinuous Growth with Spline Models
(L11.1.Spline.6Wave.PosAFF)
a2
a1
a3
Slope 1
Knot
Slope 2
0
1
1
0
1
1
1
-1
2
1
-3
-2
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
55
Modeling Discontinuous Growth with Spline Models
(L11.1.Spline.6Wave.PosAFF)
a2
a1
a3
Slope 1
Knot
Slope 2
0
1
1
0
1
1
1
-1
2
1
-3
-2
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
Model Fit ?2(17, n1684) 136.8 RMSEA
.065(.056.076) NNFI .966 CFI .962
56
Modeling Discontinuous Growth with Spline Models
(L11.1.Spline.6Wave.PosAFF)
How means are reproduced
a2
a1
a3
.05
3.70
-.09
Slope 1
Knot
Slope 2
0
1
1
0
1
1
1
-1
2
1
-3
-2
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
(-3)(.05) (1)(3.70) (0)(-.09) 3.55 (3.54)
(-2)(.05) (1)(3.70) (0)(-.09) 3.60 (3.59)
(0)(.05) (1)(3.70) (2)(-.09) 3.52 (3.52)
(0)(.05) (1)(3.70) (0)(-.09) 3.70 (3.69)
(-1)(.05) (1)(3.70) (0)(-.09) 3.65 (3.66)
(0)(.05) (1)(3.70) (1)(-.09) 3.61 (3.61)
57
Modeling Discontinuous Growth with Spline Models
(L11.2.SplineLinear.6Wave.PosAFF)
Does the spline fit better than linear?
Fix to r 1 (?21 ?11 and ?22)
equate
equate
a2
a1
a3
Slope 1
Knot
Slope 2
0
1
1
0
1
1
1
-1
2
1
-3
-2
1
equate
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
58
Modeling Discontinuous Growth with Spline Models
(L11.2.SplineLinear.6Wave.PosAFF)
.004
a2
a1
a3
-.001
3.60
-.001
-.005
-.005
Slope 1
Knot
Slope 2
.26
.004
.004
0
1
1
0
1
1
1
-1
2
1
-3
-2
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
.48
.48
.48
.48
.48
.48
Model Fit ?2(21, n1684) 243.0 RMSEA
.080(.072.090) NNFI .950 CFI .929 Model
comparison ??2(4) 243.0 136.8 106.2, plt
.001 Conclusion The Spline model fits the data
better than a Linear model
59
Modeling Discontinuous Growth with Spline Models

• Extension Include multiple knots
• If data suggest or conceptually relevant (e.g.,
  multiple transitions)
• Extensions with what we have discussed so far
• Include higher order slope terms on each side of
  the spline.
• Fit spline models with latent representations at
  each occasion (would especially want to evaluate
  measurement equivalence before and after knot)
• Fit spline models to multiple constructs (examine
  across-construct relations between knot and both
  slope terms)
• Extensions with what we will learn later
• Fit and compare spline models in multiple (known)
  groups
• Fit spline models with unbalanced data

60
Regression discontinuity models

• Logic

If there was no event, growth might look like
But with event, growth looks like
Effect of event
61
Regression discontinuity models
(L11.Panas.6WaveDiscontinuity.dat)
N 1684
62
Regression discontinuity models
(L11.3.RegDiscont.6Wave.NegAFF)
a2
a1
a3
Intercept
Linear
Event
0
5
1
1
1
4
1
1
3
1
1
2
1
1
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
63
Regression discontinuity models
(L11.3.RegDiscont.6Wave.NegAFF)
-.014ns
a2
a1
a3
-.03
2.08
.024
1.33
-.02
Intercept
Linear
Event
.003
.074
.26
0
5
1
1
1
4
1
1
3
1
1
2
1
1
1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
.46
.46
.46
.46
.46
.46
Model Fit ?2(17, n1684) 145.2 RMSEA
.067(.057.077) NNFI .973 CFI .969
64
Modeling Acceleration with Difference Scores
65
Modeling Acceleration with Difference Scores
Difference
Difference
Difference
ß
ß
ß
1.0
1.0
1.0
Time 1
Time 2
Time 3
Time 4
1.0
1.0
1.0
?
?
?
?
66
Modeling Acceleration with Difference Scores
a
Avg Change
Intercept
1
1
1
1.0
Difference
Difference
Difference
0
0
0
ß
ß
ß
1.0
1.0
1.0
Time 1
Time 2
Time 3
Time 4
1.0
1.0
1.0
?
?
?
?
67
Modeling Acceleration with Difference Scores
a
Avg Change
Accel- eration
Intercept
0
1
1
1
1
2
1.0
Difference
Difference
Difference
0
0
0
ß
ß
ß
1.0
1.0
1.0
Time 1
Time 2
Time 3
Time 4
1.0
1.0
1.0
?
?
?
?
68
Modeling Acceleration with Difference Scores

• Potential
• Allows one to model complex nonlinear change
  (i.e., change in change)
• Seems to be popular among some systems theorists
• Problems
• Difficult to interpret.
• Not clear if better representation than other
  nonlinear models (e.g., quadratic) not nested
  so not easily comparable.
• Our advice
• Consider these if acceleration is critical in
  your theory of change.
• Look at work by McArdle, Boker, Deboeck

69
Comparing groups on latent growth parameters

• Can compare two or more (known) groups on
• 1) Growth shape
• 2) Means of growth parameters
• 3) Variances of growth parameters
• 4) Associations among growth parameters

70
Comparing groups on shapes of growth curve
71
Comparing groups on shapes of growth curves
(L12.Panas.4Wave.Boys.dat) (L12.Panas.4Wave.Girls.
dat)
72
Comparing groups on shapes of growth curves
a1
a2
y21
Boys
Intercept
Slope
0
1
1
Addresses question of whether basis weights --
i.e., loadings of observed variables on to latent
slope variable(s) -- are equal across groups.
1
1
l32
l22
1
Time 1
Time 2
Time 3
Time 4
q11
q22
q33
q44
a1
a2
y21
Intercept
Slope
Girls
0
1
1
1
1
l32
l22
1
Time 1
Time 2
Time 3
Time 4
q11
q22
q33
q44
(L12.1.2GroupGC.LevelShape.4Wave.NegAFF)
73
Comparing groups on shapes of growth curves
a1
2.13
-.34
a2
-.16
Boys
Intercept
Slope
.29
.10
0
1
1
With loadings freely estimated across
groups ?2(12, N1684) 52.18 RMSEA
.065(.048.083) NNFI .965 CFI .965
1
1
1
.52
.28
Time 1
Time 2
Time 3
Time 4
.31
.31
.31
.31
-.26
a1
2.05
a2
-.15
Intercept
Slope
Girls
.31
.13
0
1
1
1
1
1
.96
.39
Time 1
Time 2
Time 3
Time 4
.29
.29
.29
.29
(L12.1.2GroupGC.LevelShape.4Wave.NegAFF)
74
Comparing groups on shapes of growth curves
a1
2.14
-.31
a2
-.18
Boys
Intercept
Slope
.31
.13
0
1
1
With loadings constrained to be equal across
groups ?2(14, N1684) 74.21 Change in
fit ??2(2) 22.03, plt.001 Conclusion Boys and
girls differ in the shape of their growth over
time.
1
1
1
.73
.35
Time 1
Time 2
Time 3
Time 4
.31
.31
.31
.31
-.28
a1
2.05
a2
-.15
Intercept
Slope
Girls
.29
.16
0
1
1
1
1
1
.73
.35
Time 1
Time 2
Time 3
Time 4
.29
.29
.29
.29
(L12.2GroupGC.EqualShape.4Wave.NegAFF)
75
Comparing groups on shapes of Positive Affect
a1
3.59
.16
a2
-.16
Boys
Intercept
Slope
.41
.18
0
1
1
With loadings freely estimated across
groups ?2(12, N1684) 45.10 RMSEA
.057(.040.075) NNFI .976 CFI .976
1
1
1
.84
1.21
Time 1
Time 2
Time 3
Time 4
.39
.39
.39
.39
.02
a1
3.58
a2
-.06
Intercept
Slope
Girls
.36
.05
0
1
1
1
1
1
2.08
1.63
Time 1
Time 2
Time 3
Time 4
.44
.44
.44
.44
(L12.3.2GroupGC.LevelShape.4Wave.PosAFF)
76
Comparing groups on shapes of Positive Affect
a1
3.60
.14
a2
-.16
Boys
Intercept
Slope
.41
.17
0
1
1
With loadings constrained to be equal across
groups ?2(14, N1684) 49.93 Change in
fit ??2(2) 4.83, ns Conclusion Boys and girls
are not sig dif in the shape of their growth over
time.
1
1
1
1.31
.96
Time 1
Time 2
Time 3
Time 4
.39
. 39
. 39
.39
.03
a1
3.58
a2
-.09
Intercept
Slope
Girls
.36
.12
0
1
1
1
1
1
1.31
.96
Time 1
Time 2
Time 3
Time 4
.45
.45
.45
.45
Note Higher absolute loadings among girls are
represented in latent variance when loadings
constrained.
(L12.4.2GroupGC.EqualShape.4Wave.PosAFF)
77
Linear trend in Positive Affect
a1
3.71
.033
a2
-.01
Boys
Intercept
Slope
.25
.003
-3
1
1
1
3
1
1
-1
Time 1
Time 2
Time 3
Time 4
.43
.43
.43
.43
?2(16, N1684) 85.17 RMSEA .072(.057.087) NNF
I .962 CFI .949 Comparison to level
shape ??2(2) 35.24, plt.001
.00ns
a1
3.60
a2
.00ns
Intercept
Slope
Girls
.28
.002ns
-3
1
1
1
3
1
1
-1
Time 1
Time 2
Time 3
Time 4
.47
.47
.47
.47
(L12.5.2GroupGC.Linear.4Wave.PosAFF)
78
Linear Quadratic trend in Positive Affect
-.01ns
a2
a1
3.71
a3
-.02ns
.03
Boys
-.01
-.01
Linear
Quad
Intercept
.01
.03
.27
1
1
-3
3
-1
1
1
1
-1
1
-1
1
?2(8, N1684) 17.64 RMSEA .038(.013.062) NNFI
.989 CFI .993 Improvement from linear
only ??2(2) 35.24, plt.001
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.00ns
a2
a1
3.60
a3
-.02ns
.00ns
.00ns
-.004
Girls
Linear
Quad
Intercept
.01
.04
.30
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.6.2GroupGC.Linear.4Wave.PosAFF)
79
Testing differences in growth parameter means
Group dummy variable
One option (if only interested in means)
a2
a1
a3
Intercept
Quad -ratic
Linear
1
1
1
1
1
1
-1
-1
-3
3
1
-1
Time 1
Time 2
Time 3
Time 4
80
Testing differences in growth parameter means
-.01ns
An equivalent option (but also allows for
comparison of variances/covariances)
a2
a1
3.71
a3
-.02ns
.03
Boys
-.01
-.01
Linear
Quad
Intercept
.01
.03
.27
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.00ns
a2
a1
3.60
a3
-.02ns
.00ns
.00ns
-.004
Girls
Linear
Quad
Intercept
.01
.04
.30
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.7.2GroupGC.EqualGrowthMeans.4Wave.PosAFF)
81
Differences in group intercepts
82
Differences in group linear slopes
83
Differences in group quadratic slopes
84
Testing differences in growth parameter means
-.01ns
a2
a1
3.71
a3
-.02ns
.03
Boys
-.01
-.01
Linear
Quad
Intercept
.01
.03
.27
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.00ns
a2
a1
3.60
a3
-.02ns
.00ns
.00ns
-.004
Girls
Linear
Quad
Intercept
.01
.04
.30
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.7.2GroupGC.EqualGrowthMeans.4Wave.PosAFF)
85
Testing differences in growth parameter means
Results Omnibus test of sex differences in
growth parameter means (?2(11) 50.09) -
(?2(8) 17.64) ??2(3) 32.45, p lt .001 Test
of sex differences in each growth parameter mean
individually Intercept (?2(9) 31.31) -
(?2(8) 17.64) ??2(1) 13.67, p lt .001 Boys
have higher levels of positive affect than girls
at the intercept point (i.e., at the time between
waves 2 3). Linear slope (?2(9)
35.02) - (?2(8) 17.64) ??2(1) 17.38, p lt
.001 Boys have (on average) greater increases in
positive affect over time than do
girls. Quadratic slope (?2(9) 17.67) -
(?2(8) 17.64) ??2(1) 0.03, ns There
is not a significant sex difference in
(quadratic) curvature
(L12.7.2GroupGC.EqualGrowthMeans.4Wave.PosAFF)
86
Testing differences in growth parameter variances
-.01ns
a2
a1
3.71
a3
-.02ns
.03
Boys
-.01
-.01
Linear
Quad
Intercept
.01
.03
.27
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.00ns
a2
a1
3.60
a3
-.02ns
.00ns
.00ns
-.004
Girls
Linear
Quad
Intercept
.01
.04
.30
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.8.2GroupGC.EqualGrowthVariances.4Wave.PosAFF)
87
Testing differences in growth parameter variances
Results Omnibus test of sex differences in
growth parameter variances (?2(11) 18.70) -
(?2(8) 17.64) ??2(3) 1.06, ns Test of sex
differences in each growth parameter variance
individually Intercept (?2(9) 18.48) -
(?2(8) 17.64) ??2(1) 0.84, ns Linear
slope (?2(9) 17.65) - (?2(8) 17.64)
??2(1) 0.01, ns Quadratic slope (?2(9)
17.83) - (?2(8) 17.64) ??2(1) 0.19,
ns Conclusion There is no evidence that either
boys or girls are more variable in their levels
(at intercept between waves 2 3), linear
change, or 9quadratic) curvature of change in
positive affect over time.
(L12.8.2GroupGC.EqualGrowthVariances.4Wave.PosAFF)
88
Testing differences in growth parameter
interrelations
-.01ns
a2
a1
3.71
a3
-.02ns
.03
Boys
-.01
-.01
Linear
Quad
Intercept
.01
.03
.27
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.00ns
a2
a1
3.60
a3
-.02ns
.00ns
.00ns
-.004
Girls
Linear
Quad
Intercept
.01
.04
.30
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.9.2GroupGC.PhantomGrowthCorrelations.4Wave.Po
sAFF)
89
Testing differences in growth parameter
interrelations
-.08ns
-.22
-.60
Boys
Intercept
Linear
Quad
a2
.52
.08
.18
a1
3.71
a3
-.02ns
.03
Linear
Quad
Intercept
0
0
0
1
1
-3
3
-1
Note identical model fit with phantom
variables ?2(8, N1684) 17.64 RMSEA
.038 NNFI .99
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.37
.37
.37
.37
.01ns
-.29
.01ns
Intercept
Linear
Quad
Girls
.55
.07
.19
a2
a1
3.60
-.02ns
a3
.00ns
Linear
Quad
Intercept
0
0
0
1
1
-3
3
-1
1
1
1
-1
1
-1
1
Time 1
Time 2
Time 3
Time 4
.40
.40
.40
.40
(L12.9.2GroupGC.PhantomGrowthCorrelations.4Wave.Po
sAFF)
90
Testing differences in growth parameter
interrelations
Results Omnibus test of sex differences in
growth parameter correlations (?2(11) 23.71)
- (?2(8) 17.64) ??2(3) 6.07, ns Test of
sex differences in each growth parameter variance
individually Intercept with Linear slope
(?2(9) 21.15) - (?2(8) 17.64) ??2(1)
3.51, p .06 Intercept with Quadratic slope
(?2(9) 18.28) - (?2(8) 17.64) ??2(1)
0.64, ns Linear with Quadratic slope
(?2(9) 19.47) - (?2(8) 17.64) ??2(1)
1.83, ns Conclusion There is no solid evidence
that the intercorrelations among growth
parameters differ by sex.
(L12.10.2GroupGC.EqualGrowthCorrelations.4Wave.Pos
AFF)