introduction to growth modeling in mplus - ehe rmc...growth modeling in mplus linear growth factors...

Growth Modeling in Mplus

Introduction to Growth Modeling in Mplus

Xin Feng, Ph.D.

Department of Human Sciences

Human Development & Family Science Program


Overview

Longitudinal research

Foundations of latent growth curve modeling

Data preparation

Growth modeling:

Linear growth models

Nonlinear growth models

Multivariate growth models

Multiple-group growth models


Longitudinal research

Studies of change:

Intra-individual variationHow does an individual change over time?

Predictors/causes of intra-individual change

Inter-individual variationDo individual change in different amounts or in different directions?

Predictors/causes of inter-individual differences in intra-individual change

Interrelations among changes in multiple constructs


Features of a longitudinal study

Multiple waves of data (repeated measures)Same individuals measured at different times

Change as a process vs. an increment

An outcome that change systematically over time

Measures with good psychometric properties

Longitudinal equivalence of the scores

A sensible metric of time


Conceptualizing time (time metrics)

Age in years, months, weeks

Experiential time: amount of time something is experienced

Years of schooling, length of relationship, amount of practice

Episodic time: time of onset of a life eventPuberty, birth of a child, retirement


Example

NLSY97 (https://www.nlsinfo.org/content/cohorts/nlsy97)

Age of cohort: born between Jan. 1, 1980 and Dec. 31, 1984; 12-17 years old in 1997

Survey years: 1997-2002 (first 5 yrs.)

Time of measurement

Cohort 97-98 98-99 99-00 00-01 01-02

1984 12-13 13-14 14-15 15-16 16-17

1983 13-14 14-15 15-16 16-17 17-18

1982 14-15 15-16 16-17 17-18 18-19

1981 15-16 16-17 17-18 18-19 20-21

1980 16-17 17-18 18-19 20-21 21-22

How would you model change? What is the time metric?

https://www.nlsinfo.org/content/cohorts/nlsy97


Interval of measurement

How fast is the developmental process?Intervals must be equal to or less than expected processes of change

Cyclical processesStudies of school achievement at yearly intervals vs. half-year intervals

Measurement occasions must span the expected period of change

Effect of change may take time to unfoldWhat’s the optimal lag of the effect

Measuring multiple lags


Little (2013). Longitudinal structural equation modeling.

Time span of the study


Number of assessments

More time points

Provide better description of the pattern of change

Can examine more complex trajectories of change

Provide more reliable estimate of individual growth parameters

DV

Time


A conceptual framework for longitudinal research

Longitudinal research involves:

A theoretical model of change

A design that can capture the process of change

A statistical model that tests the theoretical model


Foundations of latent growth curve modeling


Individual change over time

30

35

40

45

1 2 3 4 5 6

y

t

The intercept The slope

3i 4i

5i

1i

6i

titiiitiy 10


t

y

titiiitiy 10

iii

iii

rx

rx

111101

001000


Latent growth models

Structral equation modeling approach for individual change

Rooted in the factor analysis literature

An application of CFA

Unobserved latent construct cannot be measured directly but are indicated by responses to a number of observed variables (indicators)

η

Y1 Y2 Y3

E1 E2 E3


Latent growth models

Latent factors: intercept and slope(s)

Intercept factorFree mean and variance

All measures have loadings set to one

Slope factor Free mean and variance

Loadings define the time metric

Time scores is determined by the shape of the growth curve

y1 y2 y3 y4

η0 η1

ε1 ε2 ε3 ε4

ii

ii

111

000

titiitiy 10

1 11

1


The scaling of time

Equidistant time scores

Non-equidistant time scores

Slope loading

2 yrs. 3 yrs. 4 yrs. 5 yrs. 6yrs.

0 1 2 3 4

0 .1 .2 .3 .4

Slope loading

Grade 1 Grade 3 Grade 4 Grade 5 Grade 6

0 2 3 4 5

0 .2 .3 .4 .5


The scaling of time

The intercept is defined at time zero0 loading at time 1 intercept = initial status

0 loading at last time point intercept = final status

Change in scaling of time Will not change model fit, slope mean and variance, and error variances

Will change intercept mean and variance and slope-intercept covariance

Slope loading

Grade 1 Grade 3 Grade 4 Grade 5 Grade 6

0 2 3 4 5

-5 -3 -2 -1 0

-2 0 1 2 3


Linear growth factors

For both the intercept and slope growth factors there is a mean and a variance

Intercept (initial status)

Mean

Average of the outcome across individual at the time point where the time score is zero

When the first time score is zero, it represent the initial status

Variance

how much individuals differ in their intercepts


Linear slopeMean: average growth rate across individuals

Variance: individual variation of the growth rate

Covariance between intercept and slope

Outcome (observed variable) parametersIntercepts: fixed at zero (not estimated)

Residual variance: time specific and measurement error variance

Residual covariance: relations between time-specific and measurement error variation across time


Model fit test statistics

2: statistical fit indexBasis of all other fit statistics

Logic of 2 Model TestsH0: Implied and sample matrices are the same (i.e., statistically equal)

H1: Implied and sample matrices are different

Goal: Fail to reject the null hypothesis


Relative fit indices (distance from worst fit)

Comparative Fit Index (CFI)

Tucker-Lewis Index (TLI)

Acceptable fit: .90-.95

Close fit: ≥ .95

Absolute fit indices (distance from perfect fit) Root Mean Square Error of Approximation (RMSEA)

Standardized Root Mean Residual (SRMR)

Acceptable fit: .05-.08

Close fit: ≤ .05


Comparing growth model with HLM/MLM

Advantages of HLM & MLMMany time points

Individually-varying times of observation with missingness

Flexibility in accommodating more than two hierarchical levels


Advantages of latent growth modelingCan test model fit

Can specify measurement models and test measurement invariance

Can model multiple processes simultaneously

Model parameters (i.e., latent intercept and slope) can serve as predictors of other variables


Data preparation


Data sets

CLSNCNFL.csvMaternal report on child-parent relationship

Two subscales: closeness with child & conflict with child

5 time points: Grade 1, 3, 4, 5, & 6

Demographic variablesChild gender and race, maternal education and race

mathreading.csvStandardized test scores

5 time points: Kindergarten, Grade 1, 3, 5, & 8

Math and reading

Child gender


Data structure

Person-level (wide) format

Person-period (long) format

ID CLSNG1 CLSNG3 CLSNG4

52 39 39 38

56 38 32 32

61 35 28 35

ID Grade CLSN

52 1 39

52 3 39

52 4 38

56 1 38

56 3 32

56 4 32

61 1 35

61 3 28

61 4 35


Data-related assumptions

Measures

All repeatedly measured variables are continuous

Measures are equivalent over time (measurement invariance)

Distribution

Endogenous variables are multivariate normal

Missing data

Data are missing at random (MAR)


Steps in growth modeling

Preliminary descriptive statisticsMeans, variances, correlations, distributions, outliers…

Determine the shape of the growth curve from theory and/or data

Individual plots

Mean plots

Fit unconditional growth model

Modify models as needed

Add covariates—the conditional model


BASIC analysis

TITLE: BASIC analysis

DATA: FILE = CLSNCNFL.csv;

VARIABLE: NAMES = ID CSEX CRACE MRACE MAGE MED

CLSNG1 CLSNG3 CLSNG4 CLSNG5 CLSNG6

CNFLG1 CNFLG3 CNFLG4 CNFLG5 CNFLG6;

USEVARIABLES = CLSNG1 CLSNG3 CLSNG4 CLSNG5 CLSNG6;

MISSING = ALL (-99);

ANALYSIS: TYPE = BASIC;

PLOT: TYPE = PLOT3;

!SERIES = CLSNG1 CLSNG3 CLSNG4 CLSNG5 CLSNG6(*);

SERIES = CLSNG1(0) CLSNG3(2) CLSNG4(3) CLSNG5(4) CLSNG6(5);


Closeness with child


Conflict with child


Unconditional growth model – Linear growth


Mplus language

y1 y2 y3 y4

i s

ε1 ε2 ε3 ε4

MODEL:

i s| y1@0 y2@1 y3@2 y4@3;

! Alternative language;

MODEL:

i BY y1-y4@1

s BY y1@0 y2@1 y3@2 y4@3;

[y1-y4@0]; !Fix the intercepts of y1-y4 at 0;

[i s]; !Freely estimate the latent means of intercept & slope;


Fitting a linear growth model – closenessDATA:

VARIABLE:NAMES = ID … CLSNG1-CNFLG6;USEVARIABLES = CLSNG1-CLSNG6;MISSING = ALL (-99);

ANALYSIS: ESTIMATOR = ML;

MODEL:i s | CLSNG1@0 CLSNG3@2 CLSNG4@3 CLSNG5@4 CLSNG6@5;

OUTPUT:

STANDARDIZED;

PLOT:TYPE = PLOT3;SERIES = CLSNG1(0) CLSNG3(2) CLSNG4(3)

CLSNG5(4) CLSNG6(5);

CLSN G1

CLSN G4

CLSN G5

CLSN G6

i s

CLSN G3

02 3 4

5


Sample and estimated means – closeness


MODEL FIT INFORMATION

Chi-Square Test of Model Fit

Value 60.101Degrees of Freedom 10P-Value 0.0000

RMSEA (Root Mean Square Error Of Approximation)

Estimate 0.06790 Percent C.I. 0.051 0.083Probability RMSEA <= .05 0.040

CFI/TLI

CFI 0.977TLI 0.977

Mplus output


MODEL RESULTSTwo-Tailed

Estimate S.E. Est./S.E. P-ValueI |

CLSNG1 1.000 0.000 999.000 999.000CLSNG3 1.000 0.000 999.000 999.000CLSNG4 1.000 0.000 999.000 999.000CLSNG5 1.000 0.000 999.000 999.000CLSNG6 1.000 0.000 999.000 999.000

S |CLSNG1 0.000 0.000 999.000 999.000CLSNG3 2.000 0.000 999.000 999.000CLSNG4 3.000 0.000 999.000 999.000CLSNG5 4.000 0.000 999.000 999.000CLSNG6 5.000 0.000 999.000 999.000

S WITHI 0.204 0.067 3.051 0.002

MeansI 37.958 0.074 513.747 0.000S -0.352 0.020 -17.973 0.000

VariancesI 3.298 0.300 11.010 0.000S 0.126 0.022 5.687 0.000

covariance


STDYX StandardizationTwo-Tailed

Estimate S.E. Est./S.E. P-ValueS WITHI 0.316 0.133 2.381 0.017

MeansI 20.902 0.951 21.989 0.000S -0.990 0.103 -9.587 0.000

Residual VariancesCLSNG1 0.500 0.037 13.527 0.000CLSNG3 0.416 0.019 21.638 0.000CLSNG4 0.428 0.019 22.858 0.000CLSNG5 0.348 0.019 18.578 0.000CLSNG6 0.371 0.021 18.011 0.000

R-SQUAREObserved Two-TailedVariable Estimate S.E. Est./S.E. P-ValueCLSNG1 0.500 0.037 13.527 0.000CLSNG3 0.584 0.019 30.367 0.000CLSNG4 0.572 0.019 30.610 0.000CLSNG5 0.652 0.019 34.824 0.000CLSNG6 0.629 0.021 30.539 0.000

correlation


Correlations between slope and intercept

+r = high with high (or low with low)

-r = high with low (or low with high)

Little (2013)


Fit a linear growth model using conflict G1-G6


Linear growth model – conflict

DATA:

VARIABLE:

MODEL:i s | CNFLG1@0 CNFLG3@2 CNFLG4@3 CNFLG5@4 CNFLG6@5;

OUTPUT:STANDARDIZED MODINDICES (3.84)

RESIDUAL;

PLOT:TYPE = PLOT3;SERIES = CNFLG1(0) CNFLG3(2) CNFLG4(3)

CNFLG5(4) CNFLG6(5);

CNFL G1

CNFL G4

CNFL G5

CNFL G6

i s

CNFL G3

02 3 4

5







CFI/TLI

CFI 0.985TLI 0.985

Mplus output


Modification indices

M.I. E.P.C. Std E.P.C. StdYX E.P.C.BY StatementsI BY CNFLG1 4.939 -0.029 -0.142 -0.024I BY CNFLG3 12.975 0.027 0.136 0.023S BY CNFLG3 5.743 0.831 0.428 0.072S BY CNFLG4 4.898 -0.630 -0.324 -0.056

ON/BY StatementsS ON I /I BY S 999.000 0.000 0.000 0.000

WITH StatementsCNFLG4 WITH CNFLG1 4.207 -1.041 -1.041 -0.106CNFLG4 WITH CNFLG3 36.109 2.695 2.695 0.281CNFLG5 WITH CNFLG3 8.814 -1.412 -1.412 -0.135CNFLG6 WITH CNFLG4 10.824 -1.645 -1.645 -0.162CNFLG6 WITH CNFLG5 7.302 1.868 1.868 0.169

Means/Intercepts/Thresholds[ CNFLG3 ] 10.503 0.391 0.391 0.066[ CNFLG4 ] 5.744 -0.252 -0.252 -0.044

Source of misfit

Time scores

Residual covariances


Model modifications

RecommendedTime scores for slope growth factor

Residual covariances for outcomes

Not recommendedOutcome variable intercepts

Loadings for intercept growth factor


RESIDUAL OUTPUTESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED)

Model Estimated Means/Intercepts/ThresholdsCNFLG1 CNFLG3 CNFLG4 CNFLG5 CNFLG6________ ________ ________ ________ ________

1 15.271 15.840 16.125 16.409 16.694

Residuals for Means/Intercepts/ThresholdsCNFLG1 CNFLG3 CNFLG4 CNFLG5 CNFLG6________ ________ ________ ________ ________

1 -0.074 0.298 -0.184 -0.031 0.071

Standardized Residuals (z-scores) for Means/Intercepts/ThresholdsCNFLG1 CNFLG3 CNFLG4 CNFLG5 CNFLG6________ ________ ________ ________ ________

1 -1.767 3.033 -2.097 -0.381 1.055


Sample and estimated means – conflict


Unconditional growth model – Nonlinear growth


Polynomial trajectories

CLSN G3

CLSN G4

CLSN G5

CLSN G6

i s

20 35

CLSN G1

4

q

0

4 9 16 25

Quadratic:i s q | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6@5;

Or alternatively,

i BY CLSNG1-CLSNG6@1;

s BY y1@0 y2@1 y3@2 y4@3 y5@4;

q BY CLSNG1@0 CLSNG3@4 CLSNG4@9 CLSNG5@16 CLSNG6@25;

Cubic:i s q c| CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6@5;


Fitting a quadratic growth model to the closeness data

WARNING: THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IS NOT POSITIVE DEFINITE. THIS COULD INDICATE A NEGATIVE VARIANCE/RESIDUAL VARIANCE FOR A LATENT VARIABLE, A CORRELATION GREATER OR EQUAL TO ONE BETWEEN TWO LATENT VARIABLES, OR A LINEAR DEPENDENCY AMONG MORE THAN TWO LATENT VARIABLES. CHECK THE TECH4 OUTPUT FOR MORE INFORMATION. PROBLEM INVOLVING VARIABLE S.







CFI/TLI

CFI 0.992TLI 0.987

SRMR (Standardized Root Mean Square Residual)

Value 0.128


STANDARDIZED MODEL RESULTS

STDYX StandardizationTwo-Tailed

Estimate S.E. Est./S.E. P-Value

S WITHI 1.124 1.170 0.961 0.336

Q WITHI -0.572 0.499 -1.147 0.251S -0.905 0.040 -22.402 0.000


Fit a quadratic model to math scores from kindergarten to grade 8 using the “MathReading.csv” data set

MODEL:i s q| mathK@0 mathG1@1 mathG3@3 mathG5@5 mathG8@8;


Quadratic model of math scores







CFI/TLI

CFI 0.995TLI 0.992


Value 0.051

Quadratic model of math scores



Estimate S.E. Est./S.E. P-ValueS WITH

I 33.424 5.969 5.600 0.000

Q WITHI -3.474 0.611 -5.689 0.000S -4.661 0.618 -7.547 0.000

MeansI 34.590 0.552 62.656 0.000S 24.704 0.446 55.445 0.000Q -1.494 0.047 -31.588 0.000

VariancesI 114.294 10.799 10.583 0.000S 50.657 5.715 8.863 0.000Q 0.462 0.078 5.890 0.000

R-SQUAREMATHK 0.912 0.056 16.424 0.000MATHG1 0.707 0.025 28.472 0.000MATHG3 0.882 0.021 41.751 0.000MATHG5 0.883 0.019 45.430 0.000MATHG8 0.982 0.081 12.098 0.000


Piecewise growth models

One way of capturing nonlinear growth

Can be used to represent different phases of development

Each piece has its own growth factors

Example: one intercept factor, two slope growth factors

S1: 0 1 2 2 2 2S2: 0 0 0 1 2 3

0

2

4

6

8

10

12

1 2 3 4 5 6


Fitting the closeness data

CLSN G3

CLSN G4

CLSN G5

CLSN G6

i s1

20 33

CLSN G1

3

s2

0

0 0 1 2

MODEL:i s1 | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@3 CLSNG6@3;

i s2 | CLSNG1@0 CLSNG3@0 LSNG4@0 CLSNG5@1 CLSNG6@2;


Mplus outputMODEL FIT INFORMATION





CFI/TLI

CFI 0.997TLI 0.994


Value 0.101




I |CLSNG1 1.000 0.000 999.000 999.000CLSNG3 1.000 0.000 999.000 999.000CLSNG4 1.000 0.000 999.000 999.000CLSNG5 1.000 0.000 999.000 999.000CLSNG6 1.000 0.000 999.000 999.000

S1 |CLSNG1 0.000 0.000 999.000 999.000CLSNG3 2.000 0.000 999.000 999.000CLSNG4 3.000 0.000 999.000 999.000CLSNG5 3.000 0.000 999.000 999.000CLSNG6 3.000 0.000 999.000 999.000

S2 |CLSNG1 0.000 0.000 999.000 999.000CLSNG3 0.000 0.000 999.000 999.000CLSNG4 0.000 0.000 999.000 999.000CLSNG5 1.000 0.000 999.000 999.000CLSNG6 2.000 0.000 999.000 999.000



Estimate S.E. Est./S.E. P-ValueMeans

I 37.941 0.075 507.369 0.000S1 -0.339 0.031 -11.109 0.000S2 -0.373 0.047 -7.980 0.000

VariancesI 2.231 0.492 4.535 0.000S1 0.173 0.080 2.168 0.030S2 0.635 0.144 4.411 0.000

S1 WITHI 0.576 0.178 3.240 0.001

S2 WITHI -0.129 0.128 -1.010 0.312S1 -0.082 0.064 -1.283 0.199

R-SQUARECLSNG1 0.359 0.076 4.727 0.000CLSNG3 0.636 0.019 32.710 0.000CLSNG4 0.681 0.024 28.295 0.000CLSNG5 0.649 0.018 36.444 0.000CLSNG6 0.668 0.034 19.736 0.000


Growth model with free time scores

Model identification:

One time score must be fixed to zero and at least one time score must be fixed to a non-zero value (usually 1)

Common approach: fixing the time score following the centering point at 1

0

2

4

6

8

10

1 2 3 4

*

*

*

i s | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6*5;

i s | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6;




I |CLSNG1 1.000 0.000 999.000 999.000CLSNG3 1.000 0.000 999.000 999.000CLSNG4 1.000 0.000 999.000 999.000CLSNG5 1.000 0.000 999.000 999.000CLSNG6 1.000 0.000 999.000 999.000

S |CLSNG1 0.000 0.000 999.000 999.000CLSNG3 2.000 0.000 999.000 999.000CLSNG4 3.000 0.000 999.000 999.000CLSNG5 4.000 0.000 999.000 999.000CLSNG6 4.721 0.222 21.254 0.000

Mplus output


Conditional growth models


Conditional growth models

Adding covariates to the growth model

Types of covariatesTime-invariant covariates: vary across individuals not time

Time-varying covariates: vary across time and individual


Time-invariant covariates

CLSN G1

CLSN G3

CLSN G4

CLSN G5

i s

MODEL:i s | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6@5;

i s ON CSEX;

OUTPUT:

STANDARDIZED TECH4;

CLSN G6

CSEX


Gender as time-invariant covariate – output



Value 65.865

Degrees of Freedom 13

P-Value 0.0000


Estimate 0.060

90 Percent C.I. 0.046 0.075

Probability RMSEA <= .05 0.113

CFI/TLI

CFI 0.976

TLI 0.972


MODEL RESULTS (CSEX: 0=female, 1=male)

I ONCSEX -0.395 0.147 -2.682 0.007

S ONCSEX -0.017 0.039 -0.438 0.661

S WITHI 0.201 0.067 3.009 0.003

InterceptsI 38.157 0.104 365.387 0.000S -0.343 0.028 -12.443 0.000

Residual VariancesI 3.263 0.298 10.948 0.000S 0.127 0.022 5.696 0.000


MODEL RESULTS

S WITHI 0.204 0.067 3.051 0.002

MeansI 37.958 0.074 513.747 0.000S -0.352 0.020 -17.973 0.000

VariancesI 3.298 0.300 11.010 0.000S 0.126 0.022 5.687 0.000

Results of unconditional model


R-SQUARE

Observed Two-TailedVariable Estimate S.E. Est./S.E. P-Value

CLSNG1 0.501 0.037 13.533 0.000CLSNG3 0.583 0.019 30.334 0.000CLSNG4 0.573 0.019 30.632 0.000CLSNG5 0.652 0.019 34.826 0.000CLSNG6 0.629 0.021 30.548 0.000

Latent Two-TailedVariable Estimate S.E. Est./S.E. P-Value

I 0.012 0.009 1.346 0.178S 0.001 0.003 0.219 0.827


TECHNICAL 4 OUTPUT

ESTIMATED MEANS FOR THE LATENT VARIABLESI S CSEX________ ________ ________

1 37.957 -0.352 0.507

ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLESI S CSEX________ ________ ________

I 3.302S 0.202 0.127CSEX -0.099 -0.004 0.250

ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLESI S CSEX________ ________ ________

I 1.000S 0.313 1.000CSEX -0.109 -0.024 1.000


Conditional growth model for CNFL

CNFL G1

CNFL G3

CNFL G4

CNFL G5

i s

Exercise: Estimate a conditional model

with CNFLG1-CNFLG6 and include child gender and maternal education as time-invariant covariates

MODEL:

i s | CNFLG1@0 CNFLG3@2 CNFLG4@3 CNFLG5@4 CNFLG6@5;

i s ON CSEX MED;

CNFL G6

CSEX MED







CFI/TLI

CFI 0.985TLI 0.982


Value 0.030

Mplus output



I ONCSEX -0.079 0.349 -0.226 0.821MED -0.074 0.071 -1.049 0.294

S ONCSEX -0.132 0.065 -2.032 0.042MED -0.039 0.013 -2.901 0.004

S WITHI -0.342 0.240 -1.427 0.154

InterceptsI 16.381 1.058 15.484 0.000S 0.907 0.199 4.564 0.000

R-SQUARELatent Two-TailedVariable Estimate S.E. Est./S.E. P-Value

I 0.001 0.003 0.533 0.594S 0.049 0.029 1.678 0.093


Time-varying covariates

CLSN G1

CLSN G3

CLSN G4

CLSN G5

i

s

CLSN G6

CNFL G1

CNFL G3

CNFL G4

CNFL G5

CNFL G6

MODEL:

i s | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6@5;

CLSNG1 ON CNFLG1;

CLSNG3 ON CNFLG3;

CLSNG4 ON CNFLG4;

CLSNG5 ON CNFLG5;

CLSNG6 ON CNFLG6;


Mplus output



Value 99.036


P-Value 0.0000


Estimate 0.052

90 Percent C.I. 0.041 0.064


CFI/TLI

CFI 0.970

TLI 0.965


MODEL RESULTS

Two-Tailed


CLSNG1 ON

CNFLG1 -0.123 0.011 -11.207 0.000

CLSNG3 ON

CNFLG3 -0.140 0.008 -16.505 0.000

CLSNG4 ON

CNFLG4 -0.150 0.009 -16.035 0.000

CLSNG5 ON

CNFLG5 -0.165 0.010 -15.856 0.000

CLSNG6 ON

CNFLG6 -0.181 0.012 -14.855 0.000


MODEL RESULTS

Two-Tailed


S WITH

I 0.163 0.059 2.752 0.006

Means

I 39.827 0.181 220.153 0.000

S -0.135 0.055 -2.450 0.014

Variances

I 2.611 0.264 9.902 0.000

S 0.120 0.021 5.832 0.000

R-SQUARE

CLSNG1 0.521 0.037 14.207 0.000

CLSNG3 0.601 0.019 30.981 0.000

CLSNG4 0.579 0.019 29.923 0.000

CLSNG5 0.670 0.019 35.467 0.000

CLSNG6 0.672 0.021 32.246 0.000


Time-varying and time-invariant covariates

CLSN G1

CLSN G3

CLSN G4

CLSN G5

i

s

CLSN G6

CNFL G1

CNFL G3

CNFL G4

CNFL G5

CNFL G6

CSEX MED

MODEL:

i s | CLSNG1@0 CLSNG3@2 LSNG4@3 CLSNG5@4 CLSNG6@5;

i s ON CSEX MED;

CLSNG1 ON CNFLG1;

CLSNG3 ON CNFLG3;

CLSNG4 ON CNFLG4;

CLSNG5 ON CNFLG5;

CLSNG6 ON CNFLG6;


Mplus output



Value 108.730


P-Value 0.0000


Estimate 0.049

90 Percent C.I. 0.038 0.059


CFI/TLI

CFI 0.969

TLI 0.961


MODEL RESULTS

Two-Tailed


I ON

CSEX -0.274 0.150 -1.823 0.068

MED 0.037 0.031 1.191 0.234

S ON

CSEX -0.084 0.040 -2.090 0.037

MED 0.012 0.008 1.425 0.154

CLSNG1 ON

CNFLG1 -0.123 0.011 -11.211 0.000

CLSNG3 ON

CNFLG3 -0.140 0.008 -16.534 0.000

CLSNG4 ON

CNFLG4 -0.150 0.009 -16.070 0.000

CLSNG5 ON

CNFLG5 -0.165 0.010 -15.877 0.000

CLSNG6 ON

CNFLG6 -0.181 0.012 -14.863 0.000


R-SQUARE

Observed Two-Tailed

Variable Estimate S.E. Est./S.E. P-Value

CLSNG1 0.521 0.037 14.190 0.000

CLSNG3 0.600 0.019 30.888 0.000

CLSNG4 0.581 0.019 29.951 0.000

CLSNG5 0.670 0.019 35.430 0.000

CLSNG6 0.673 0.021 32.287 0.000

Latent Two-Tailed


I 0.011 0.010 1.111 0.266

S 0.022 0.017 1.281 0.200


Run a linear growth model on CNFLG1-CNFLG6, with CLSNG1-CLSNG6 as time-varying covariates and CSEX and MED as time-invariant covariates.


Time-varying & time-invariant covariates



Value 150.554


P-Value 0.0000


Estimate 0.061

90 Percent C.I. 0.051 0.071


CFI/TLI

CFI 0.966

TLI 0.958


Value 0.082


MODEL RESULTS

Two-Tailed


I ON

CSEX -0.396 0.373 -1.063 0.288

MED -0.028 0.078 -0.362 0.717

S ON

CSEX -0.154 0.068 -2.253 0.024

MED -0.040 0.014 -2.769 0.006

CNFLG1 ON

CLSNG1 -0.516 0.047 -10.999 0.000

CNFLG3 ON

CLSNG3 -0.458 0.030 -15.115 0.000

CNFLG4 ON

CLSNG4 -0.444 0.026 -16.977 0.000

CNFLG5 ON

CLSNG5 -0.418 0.028 -15.154 0.000

CNFLG6 ON

CLSNG6 -0.391 0.034 -11.541 0.000


S WITH

I -0.347 0.232 -1.494 0.135

Intercepts

CNFLG1 0.000 0.000 999.000 999.000

CNFLG3 0.000 0.000 999.000 999.000

CNFLG4 0.000 0.000 999.000 999.000

CNFLG5 0.000 0.000 999.000 999.000

CNFLG6 0.000 0.000 999.000 999.000

I 35.332 2.103 16.801 0.000

S -0.117 0.492 -0.238 0.812

R-SQUARE

Latent Two-Tailed


I 0.011 0.010 1.111 0.266

S 0.022 0.017 1.281 0.200



Parallel processes

Analytical proceduresEstimate a growth model for each process separately

Fit the unconditional model and determine the shape of the growth curve

Modify the model if necessary

Joint analysis of multiple processes

Add covariates


Parallel process model

CLSN G1

CLSN G3

CLSN G4

CLSN G5

CLi CLs

CLSN G6

MODEL:cli cls| CLSNG1@0 CLSNG3@2 CLSNG4@3 CLSNG5@4 CLSNG6@5;

cni cns| CNFLG1@0 CNFLG3@2 CNFLG4@3 CNFLG5@4 CNFLG6@5;

OUTPUT:STANDARDIZED MODINDICES;

CNFL G1

CNFL G3

CNFL G4

CNFL G5

CNi CNs

CNFL G6







CFI/TLI

CFI 0.957TLI 0.953


Value 0.105

Mplus output


CLS WITHCLI 0.185 0.067 2.760 0.006

CNI WITHCLI -4.879 0.461 -10.593 0.000CLS -0.140 0.115 -1.225 0.221

CNS WITHCLI 0.365 0.082 4.480 0.000CLS -0.148 0.021 -6.908 0.000CNI -0.366 0.241 -1.522 0.128

MeansCLI 37.956 0.074 513.261 0.000CLS -0.351 0.020 -17.939 0.000CNI 15.279 0.175 87.458 0.000CNS 0.283 0.033 8.647 0.000

VariancesCLI 3.373 0.301 11.216 0.000CLS 0.132 0.022 5.928 0.000CNI 24.696 1.505 16.404 0.000CNS 0.280 0.062 4.507 0.000


MODEL MODIFICATION INDICESM.I. E.P.C. Std E.P.C. StdYX E.P.C.

BY Statements

CLI BY CNFLG3 10.007 0.010 0.018 0.003CNI BY CNFLG3 12.143 0.026 0.132 0.022

WITH Statements

CLSNG3 WITH CLSNG1 10.449 -0.806 -0.806 -0.248CLSNG4 WITH CLSNG3 34.403 0.923 0.923 0.246CNFLG1 WITH CLSNG1 17.644 -1.483 -1.483 -0.243CNFLG1 WITH CLSNG3 19.870 1.219 1.219 0.196CNFLG1 WITH CLSNG6 17.032 -1.540 -1.540 -0.202CNFLG3 WITH CLSNG3 26.050 -1.206 -1.206 -0.198CNFLG4 WITH CLSNG4 21.477 -1.086 -1.086 -0.183CNFLG4 WITH CNFLG3 38.643 2.753 2.753 0.285CNFLG5 WITH CLSNG5 41.691 -1.631 -1.631 -0.270CNFLG5 WITH CLSNG6 40.380 1.906 1.906 0.271CNFLG6 WITH CLSNG6 47.896 -2.357 -2.357 -0.304CNFLG6 WITH CNFLG4 12.141 -1.702 -1.702 -0.170


Parallel process model

CLSN G1

CLSN G3

CLSN G4

CLSN G5

CLi CLs

CLSN G6

MODEL:cli cls| CLSNG1@0 CLSNG3@2 CLSNG4@3 CLSNG5@4 CLSNG6@5;


CLSNG1-CLSNG6 PWITH CNFLG1-CNFLG6;

CNFL G1

CNFL G3

CNFL G4

CNFL G5

CNi CNs

CNFL G6







CFI/TLI

CFI 0.982TLI 0.977


Value 0.106

Mplus output


MODEL RESULTS

CLSNG1 WITHCNFLG1 -1.807 0.359 -5.027 0.000

CLSNG3 WITHCNFLG3 -1.015 0.245 -4.142 0.000

CLSNG4 WITHCNFLG4 -0.801 0.241 -3.325 0.001

CLSNG5 WITHCNFLG5 -1.533 0.265 -5.779 0.000

CLSNG6 WITHCNFLG6 -2.417 0.366 -6.603 0.000


MODEL RESULTS

CLS WITHCLI 0.198 0.067 2.950 0.003

CNI WITHCLI -3.584 0.492 -7.278 0.000CLS -0.513 0.126 -4.077 0.000

CNS WITHCLI -0.008 0.097 -0.080 0.936CLS -0.012 0.027 -0.458 0.647CNI -0.336 0.241 -1.390 0.164

MeansCLI 37.954 0.074 513.122 0.000CLS -0.350 0.020 -17.875 0.000CNI 15.273 0.175 87.499 0.000CNS 0.284 0.033 8.693 0.000


Parallel process model with covariates

CLSN G1

CLSN G3

CLSN G4

CLSN G5

CLiCLs

CLSN G6

MODEL:

CLi CLs| CLSNG1@0 CLSNG3@2 CLSNG4@3 CLSNG5@4 CLSNG6@5;


CLSNG1-CLSNG6 PWITH CNFLG1-CNFLG6;

CLi CLs CNi CNs ON CSEX MED;

CNFL G1

CNFL G3

CNFL G4

CNFL G5

CNiCNs

CNFL G6

CSEX

MED


Mplus output



Value 157.880


P-Value 0.0000


Estimate 0.045

90 Percent C.I. 0.037 0.053


CFI/TLI

CFI 0.981

TLI 0.975


MODEL RESULTS

CLI ON

CSEX -0.383 0.147 -2.603 0.009

MED 0.083 0.030 2.794 0.005

CLS ON

CSEX -0.014 0.039 -0.357 0.721

MED 0.011 0.008 1.380 0.168

CNI ON

CSEX -0.089 0.349 -0.254 0.799

MED -0.075 0.071 -1.066 0.287

CNS ON

CSEX -0.129 0.065 -1.984 0.047

MED -0.039 0.013 -2.943 0.003


Multiple group models


Comparing (Nested) Models

Multiple group growth modelsTesting group differences in parameters

Nested 2 tests (or 2 tests)

Used when one model (A) is exactly the same as another (B), except one or more parameters in (A):

have been removed (constrained to zero)

constrained to a fixed (non-zero) value

constrained to be equal to another parameter


Steps in fitting multi-group models

Y1 Y2 Y3 Y4

i s

Y5

Y1 Y2 Y3 Y4

i s

Y5

Group 1

Group 2

1. Multi-group model without constrains

2. Set the intercept and slope means equal across groups

3. Set the intercept and slope variances and covariance equal across groups

4. Set residual variances of the outcome variables equal across groups


The unconditional model for multiple groups

VARIABLES:

USEVARIABLES = CLSNG1 CLSNG3 CLSNG4 CLSNG5 CLSNG6 CSEX;

GROUPING = CSEX (0=FEMALE 1=MALE);

MODEL:

i s| CLSNG1@0 CLSNG3@2 CLSNG4@3 CLSNG5@4 CLSNG6@5;

CLSN G1

CLSN G3

CLSN G4

CLSN G5

i s

CLSN G6

CLSN G1

CLSN G3

CLSN G4

CLSN G5

i s

CLSN G6

Girls

Boys

Model 1


Mplus output


Value 98.614


P-Value 0.0000

Chi-Square Contribution From Each Group

FEMALE 48.455

MALE 50.159


Estimate 0.084

90 Percent C.I. 0.067 0.100


CFI/TLI

CFI 0.964

TLI 0.964

df = 10 for single group unconditional model


MODEL RESULTS

Two-Tailed


Group FEMALE

S WITH

I 0.063 0.093 0.681 0.496

Means

I 38.166 0.099 384.417 0.000

S -0.344 0.028 -12.110 0.000

Variances

I 3.017 0.377 8.010 0.000

S 0.178 0.033 5.368 0.000

Group MALE

S WITH

I 0.347 0.095 3.650 0.000

Means

I 37.769 0.107 352.896 0.000

S -0.362 0.027 -13.449 0.000

Variances

I 3.353 0.458 7.322 0.000

S 0.080 0.030 2.691 0.007


MODEL RESULTS - Single Group

Two-Tailed


S WITH

I 0.204 0.067 3.050 0.002

Means

I 37.958 0.074 513.743 0.000

S -0.352 0.020 -17.972 0.000

Intercepts

CLSNG1 0.000 0.000 999.000 999.000

CLSNG3 0.000 0.000 999.000 999.000

CLSNG4 0.000 0.000 999.000 999.000

CLSNG5 0.000 0.000 999.000 999.000

CLSNG6 0.000 0.000 999.000 999.000

Variances

I 3.298 0.300 11.009 0.000

S 0.126 0.022 5.687 0.000


Plot

Girls = Red; Boys = Blue


Example: trajectories of CLSN by gender

Model 2Setting the means of intercept and slope factors equal across groups

MODEL:


MODEL FEMALE:

[i] (a);

[s] (b);

MODEL MALE:

[i] (a);

[s] (b);

Constraining the means of intercept to be equal

Constraining the means of slope to be equal


Mplus output


Value 108.136


P-Value 0.0000

Chi-Square Contribution From Each Group

FEMALE 52.716

MALE 55.420


Estimate 0.083

90 Percent C.I. 0.068 0.099


CFI/TLI

CFI 0.960

TLI 0.964


MODEL RESULTS

Two-Tailed


Group FEMALE

Means

I 37.989 0.073 517.063 0.000

S -0.353 0.020 -17.963 0.000

Variances

I 3.048 0.379 8.044 0.000

S 0.178 0.033 5.361 0.000

Group MALE

Means

I 37.989 0.073 517.063 0.000

S -0.353 0.020 -17.963 0.000

Variances

I 3.396 0.462 7.352 0.000

S 0.080 0.030 2.681 0.007

Model Constraints χ2 df ∆χ2

1 Multiple groups model 98.614 20 --

2 Equal intercepts & slope means 108.136 22 9.522**



Testing equal slopes onlyModel 2a

MODEL FEMALE:

[s] (b);

MODEL MALE:

[s] (b);

MODEL FEMALE:[i];[s] (b);MODEL MALE:[i];[s] (b);


1 Multiple groups model 98.614 20 --

2 Equal intercepts & slope means 108.136 22 9.522**

2a Equal slope means 98.829 21 .215


Model 3Setting the intercept and slope variances and covariance equal across groups

3a: Equal intercept variance onlyMODEL FEMALE:

[i];

[s] (b);

i (c);

MODEL MALE:

[i];

[s] (b);

i (c);


Value 99.162


P-Value 0.0000



2a Equal slope means 98.829 21 --

3a Equal intercept variance 99.162 22 .333

3b Equal slope variance 105.704 23 6.542*

3c Equal intercept slope covariance 109.259 23 10.097**

3b: equal slope variance

MODEL FEMALE:

[i];

[s] (b);

i (c);

s (d);

MODEL MALE:

[i];

[s] (b);

i (c);

s (d);

3c: equal intercept slope covarianceMODEL FEMALE:

[s] (b);

i (c);

i WITH s (r);

MODEL MALE:

[s] (b);

i (c);

i WITH s (r);



Model 4Set the error variances equal across groups

MODEL FEMALE:[i];[s] (b);i (c);s;i WITH s;CLSNG1(1)CLSNG3(2)CLSNG4(3)CLSNG5(4)CLSNG6(5);

MODEL MALE:[i];[s] (b);i (c);s;i WITH s;CLSNG1(1)CLSNG3(2)CLSNG4(3)CLSNG5(4)CLSNG6(5);



3a Equal slope means + intercept variance

99.162 22 --

4 Equal error variances 112.917 27 13.755*

Final unconditional modelMODEL:


MODEL FEMALE:[i];[s] (b);i (c);s;i WITH s;

MODEL MALE:[i];[s] (b);i (c);s;i WITH s;


MODEL RESULTS

Two-Tailed


Group FEMALE

S WITH

I 0.040 0.086 0.470 0.638

Means

I 38.179 0.096 396.580 0.000

S -0.354 0.020 -18.093 0.000

Variances

I 3.160 0.291 10.840 0.000

S 0.183 0.032 5.660 0.000

Group MALE

S WITH

I 0.371 0.083 4.468 0.000

Means

I 37.760 0.104 364.771 0.000

S -0.354 0.020 -18.093 0.000

Variances

I 3.160 0.291 10.840 0.000

S 0.076 0.029 2.650 0.008


Trajectories of CLSN by gender

Summary of results—unconditional growth model for closeness

Boys and girls differed on

Initial status: girls were higher on closeness with mothers

Slope variance: larger individual variation around the slope in girls

Correlation between intercept and slope: positive correlation between intercept and slope for boys, no correlation for girls


Thank you!

Questions? [email protected]

mailto:[email protected]

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