Chapter 13:
Phil Wood
2025-08-01
Figure 13.1 Repeated Measures Null Model
Load Children’s Vocabulary Data
(cf. (McArdle & Epstein (1987)))
<a https://docs.google.com/document/d/1IA6_7hM-3pb3-m46wFaMEXO9YQpm8K1q0n050QxwtAg/edit?usp=sharing
Path Diagram via Common Manifest Intercepts
Repeated Measures Null Model via Common
Intercepts
R Program
mcardle <- read.table("C:/Users/woodph/OneDrive - University of Missouri/Documents/SembookCRCPress/Data/Mcardle/mcardle.txt", header=TRUE)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ggplot2)
require(lavaan)## Loading required package: lavaan## This is lavaan 0.6-19
## lavaan is FREE software! Please report any bugs.Fig13.1AModel<-"
! regressions
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e1*wisc_3
wisc_4 ~~ V_e1*wisc_4
RI ~~ V_RI*RI
! means
wisc_1~t1*1
wisc_2~t1*1
wisc_3~t1*1
wisc_4~t1*1
RI~0*1;
"
model<-"
! regressions
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e1*wisc_3
wisc_4 ~~ V_e1*wisc_4
RI ~~ V_RI*RI
! means
wisc_1~t1*1
wisc_2~t1*1
wisc_3~t1*1
wisc_4~t1*1
RI~0*1;
"
Fig13.1AResult<-lavaan(Fig13.1AModel, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.1AResult, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 20 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 6
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 1705.425
## Degrees of freedom 11
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.000
## Tucker-Lewis Index (TLI) -0.121
##
## Robust Comparative Fit Index (CFI) 0.000
## Robust Tucker-Lewis Index (TLI) -0.121
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3271.092
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 6548.183
## Bayesian (BIC) 6558.138
## Sample-size adjusted Bayesian (SABIC) 6548.633
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.869
## 90 Percent confidence interval - lower 0.834
## 90 Percent confidence interval - upper 0.904
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.869
## 90 Percent confidence interval - lower 0.834
## 90 Percent confidence interval - upper 0.904
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 1.501
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 1.000 3.648 0.273
## wisc_2 1.000 3.648 0.273
## wisc_3 1.000 3.648 0.273
## wisc_4 1.000 3.648 0.273
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (t1) 32.164 0.518 62.105 0.000 32.164 2.404
## .wisc_2 (t1) 32.164 0.518 62.105 0.000 32.164 2.404
## .wisc_3 (t1) 32.164 0.518 62.105 0.000 32.164 2.404
## .wisc_4 (t1) 32.164 0.518 62.105 0.000 32.164 2.404
## RI 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_2 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_3 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_4 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## RI (V_RI) 13.307 5.913 2.251 0.024 1.000 1.000require(lavaangui)## Loading required package: lavaangui## Warning: package 'lavaangui' was built under R version 4.5.1## This is lavaangui 0.2.5
## lavaangui is BETA software! Please report any bugs at https://github.com/karchjd/lavaangui/issues#plot_lavaan(Fig13.1AResult)Repeated Measures Null Model
Path Diagram via Random Intercept Mean
Repeated Measures Null Model via Random
Intercept Mean
R Program
require(lavaan)
Fig13.1BModel<-"
! regressions
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e1*wisc_3
wisc_4 ~~ V_e1*wisc_4
RI ~~ V_RI*RI
! means
RI~a1*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.1BResult<-lavaan(Fig13.1BModel, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.1BResult, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 34 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 6
## Number of equality constraints 3
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 1705.425
## Degrees of freedom 11
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.000
## Tucker-Lewis Index (TLI) -0.121
##
## Robust Comparative Fit Index (CFI) 0.000
## Robust Tucker-Lewis Index (TLI) -0.121
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3271.092
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 6548.183
## Bayesian (BIC) 6558.138
## Sample-size adjusted Bayesian (SABIC) 6548.633
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.869
## 90 Percent confidence interval - lower 0.834
## 90 Percent confidence interval - upper 0.904
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.869
## 90 Percent confidence interval - lower 0.834
## 90 Percent confidence interval - upper 0.904
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 1.501
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 1.000 3.648 0.273
## wisc_2 1.000 3.648 0.273
## wisc_3 1.000 3.648 0.273
## wisc_4 1.000 3.648 0.273
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI (a1) 32.164 0.518 62.105 0.000 8.817 8.817
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_2 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_3 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## .wisc_4 (V_e1) 165.645 9.469 17.493 0.000 165.645 0.926
## RI (V_RI) 13.307 5.912 2.251 0.024 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.1BResult)Repeated Measures Alternate Model
Path Diagram via Repeated Measures Alternate Model via Variant Intercepts
Repeated Measures Alternate Model via Variant
Intercepts
R Program Individual Means
require(lavaan)
Fig13.2AModel<-"
! regressions
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e1*wisc_3
wisc_4 ~~ V_e1*wisc_4
RI ~~ V_RI*RI
! means
wisc_1~t1*1
wisc_2~t2*1
wisc_3~t3*1
wisc_4~t4*1
RI~0*1;
"
Fig13.2AResult<-lavaan(Fig13.2AModel, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.2AResult, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 3
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 184.942
## Degrees of freedom 8
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.785
## Tucker-Lewis Index (TLI) 0.839
##
## Robust Comparative Fit Index (CFI) 0.785
## Robust Tucker-Lewis Index (TLI) 0.839
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2510.850
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 5033.701
## Bayesian (BIC) 5053.609
## Sample-size adjusted Bayesian (SABIC) 5034.599
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.329
## 90 Percent confidence interval - lower 0.289
## 90 Percent confidence interval - upper 0.371
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.329
## 90 Percent confidence interval - lower 0.289
## 90 Percent confidence interval - upper 0.371
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.241
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 1.000 7.160 0.888
## wisc_2 1.000 7.160 0.888
## wisc_3 1.000 7.160 0.888
## wisc_4 1.000 7.160 0.888
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (t1) 18.781 0.565 33.253 0.000 18.781 2.328
## .wisc_2 (t2) 26.553 0.565 47.013 0.000 26.553 3.292
## .wisc_3 (t3) 35.982 0.565 63.708 0.000 35.982 4.460
## .wisc_4 (t4) 47.341 0.565 83.819 0.000 47.341 5.869
## RI 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_2 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_3 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_4 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## RI (V_RI) 51.265 5.421 9.456 0.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.2AResult)Path Diagram Mean Differences Relative to Initial Time
Repeated Measures Alternate Model Mean
Differences Relative to Time 1
R Program Mean Differences Relative to Time 1
require(lavaan)
Fig13.2BModel<-"
! regressions
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e1*wisc_3
wisc_4 ~~ V_e1*wisc_4
RI ~~ V_RI*RI
! means
wisc_1~t1*1
wisc_2~t2*1
wisc_3~t3*1
wisc_4~t4*1
RI~0*1;
"
Fig13.2BResult<-lavaan(Fig13.2BModel, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.2BResult, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 27 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 3
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 184.942
## Degrees of freedom 8
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.785
## Tucker-Lewis Index (TLI) 0.839
##
## Robust Comparative Fit Index (CFI) 0.785
## Robust Tucker-Lewis Index (TLI) 0.839
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2510.850
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 5033.701
## Bayesian (BIC) 5053.609
## Sample-size adjusted Bayesian (SABIC) 5034.599
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.329
## 90 Percent confidence interval - lower 0.289
## 90 Percent confidence interval - upper 0.371
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.329
## 90 Percent confidence interval - lower 0.289
## 90 Percent confidence interval - upper 0.371
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.241
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 1.000 7.160 0.888
## wisc_2 1.000 7.160 0.888
## wisc_3 1.000 7.160 0.888
## wisc_4 1.000 7.160 0.888
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (t1) 18.781 0.565 33.253 0.000 18.781 2.328
## .wisc_2 (t2) 26.553 0.565 47.013 0.000 26.553 3.292
## .wisc_3 (t3) 35.982 0.565 63.708 0.000 35.982 4.460
## .wisc_4 (t4) 47.341 0.565 83.819 0.000 47.341 5.869
## RI 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_2 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_3 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## .wisc_4 (V_e1) 13.810 0.789 17.493 0.000 13.810 0.212
## RI (V_RI) 51.265 5.421 9.456 0.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.2BResult)Factor Mean (FM) Growth Model
McArdle and Epstein (1987) called this the CURVE model (p. 118) (McArdle & Epstein (1987)).
Path Diagram
Factor Mean (FM) Growth Model
R Program
require(lavaan)
Fig13.4Model<-"
! regressions
RI=~L1*wisc_1
RI=~L2*wisc_2
RI=~L3*wisc_3
RI=~L4*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
RI ~~ 1.0*RI
! means
RI~a1*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.4Result<-lavaan(Fig13.4Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.4Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 64.823
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.929
## Tucker-Lewis Index (TLI) 0.929
##
## Robust Comparative Fit Index (CFI) 0.929
## Robust Tucker-Lewis Index (TLI) 0.929
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2450.791
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4917.581
## Bayesian (BIC) 4944.126
## Sample-size adjusted Bayesian (SABIC) 4918.780
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.219
## 90 Percent confidence interval - lower 0.173
## 90 Percent confidence interval - upper 0.269
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.219
## 90 Percent confidence interval - lower 0.173
## 90 Percent confidence interval - upper 0.269
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.142
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 (L1) 4.079 0.219 18.636 0.000 4.079 0.758
## wisc_2 (L2) 5.719 0.302 18.911 0.000 5.719 0.852
## wisc_3 (L3) 7.689 0.402 19.142 0.000 7.689 0.954
## wisc_4 (L4) 10.108 0.530 19.085 0.000 10.108 0.925
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI (a1) 4.671 0.254 18.384 0.000 4.671 4.671
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 12.302 0.994 12.380 0.000 12.302 0.425
## .wisc_2 (V_e1) 12.302 0.994 12.380 0.000 12.302 0.273
## .wisc_3 (V_e3) 5.808 1.247 4.656 0.000 5.808 0.089
## .wisc_4 (V_e4) 17.233 2.629 6.554 0.000 17.233 0.144
## RI 1.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.4Result)Factor Mean Shift (FM-Shift) Growth Model
Path Diagram
Factor Mean Shift (FM-Shift) Growth Model
R Program
require(lavaan)
Fig13.6Model<-"
! regressions
RI=~L1*wisc_1
RI=~L2*wisc_2
RI=~L3*wisc_3
RI=~L4*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
RI ~~ 1.0*RI
! means
RI~a1*1
wisc_1~t1*1
wisc_2~t1*1
wisc_3~t1*1
wisc_4~t1*1
"
Fig13.6Result<-lavaan(Fig13.6Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.6Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 98 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 13
## Number of equality constraints 4
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 13.744
## Degrees of freedom 5
## P-value (Chi-square) 0.017
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.989
## Tucker-Lewis Index (TLI) 0.987
##
## Robust Comparative Fit Index (CFI) 0.989
## Robust Tucker-Lewis Index (TLI) 0.987
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2425.251
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4868.503
## Bayesian (BIC) 4898.366
## Sample-size adjusted Bayesian (SABIC) 4869.851
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.093
## 90 Percent confidence interval - lower 0.036
## 90 Percent confidence interval - upper 0.153
## P-value H_0: RMSEA <= 0.050 0.096
## P-value H_0: RMSEA >= 0.080 0.691
##
## Robust RMSEA 0.093
## 90 Percent confidence interval - lower 0.036
## 90 Percent confidence interval - upper 0.153
## P-value H_0: Robust RMSEA <= 0.050 0.096
## P-value H_0: Robust RMSEA >= 0.080 0.691
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.038
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI =~
## wisc_1 (L1) 5.444 0.330 16.496 0.000 5.444 0.868
## wisc_2 (L2) 6.419 0.342 18.761 0.000 6.419 0.900
## wisc_3 (L3) 7.593 0.398 19.092 0.000 7.593 0.948
## wisc_4 (L4) 9.026 0.507 17.811 0.000 9.026 0.901
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## RI (a1) 7.986 0.907 8.807 0.000 7.986 7.986
## .wisc_1 (t1) -24.692 5.730 -4.309 0.000 -24.692 -3.936
## .wisc_2 (t1) -24.692 5.730 -4.309 0.000 -24.692 -3.461
## .wisc_3 (t1) -24.692 5.730 -4.309 0.000 -24.692 -3.082
## .wisc_4 (t1) -24.692 5.730 -4.309 0.000 -24.692 -2.465
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 9.704 0.885 10.962 0.000 9.704 0.247
## .wisc_2 (V_e1) 9.704 0.885 10.962 0.000 9.704 0.191
## .wisc_3 (V_e3) 6.544 1.161 5.639 0.000 6.544 0.102
## .wisc_4 (V_e4) 18.857 2.419 7.795 0.000 18.857 0.188
## RI 1.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.6Result)Orthogonal Free Curve Slope Intercept (FCSI) Model (a.k.a. “Latent Basis”)
Path Diagram
Orthogonal Free Curve Slope Intercept (FCSI)
Growth Model
R Program
require(lavaan)
Fig13.8Model<-"
! regressions
S=~L1*wisc_1
S=~L2*wisc_2
S=~L3*wisc_3
S=~L4*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ 1.0*S
RI ~~ V_RI*RI
S ~~ 0.0*RI
! means
S~aS*1
RI~aI*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.8Result<-lavaan(Fig13.8Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.8Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 64 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 11
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 7.718
## Degrees of freedom 4
## P-value (Chi-square) 0.102
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.995
## Tucker-Lewis Index (TLI) 0.993
##
## Robust Comparative Fit Index (CFI) 0.995
## Robust Tucker-Lewis Index (TLI) 0.993
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2422.238
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4864.477
## Bayesian (BIC) 4897.658
## Sample-size adjusted Bayesian (SABIC) 4865.975
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.068
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.139
## P-value H_0: RMSEA <= 0.050 0.278
## P-value H_0: RMSEA >= 0.080 0.454
##
## Robust RMSEA 0.068
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.139
## P-value H_0: Robust RMSEA <= 0.050 0.278
## P-value H_0: Robust RMSEA >= 0.080 0.454
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.040
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_1 (L1) 3.612 0.689 5.246 0.000 3.612 0.572
## wisc_2 (L2) 4.839 0.644 7.518 0.000 4.839 0.683
## wisc_3 (L3) 6.315 0.626 10.093 0.000 6.315 0.781
## wisc_4 (L4) 8.118 0.662 12.262 0.000 8.118 0.810
## RI =~
## wisc_1 1.000 4.306 0.682
## wisc_2 1.000 4.306 0.607
## wisc_3 1.000 4.306 0.533
## wisc_4 1.000 4.306 0.430
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI 0.000 0.000 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 6.348 0.715 8.884 0.000 6.348 6.348
## RI (aI) -4.147 5.934 -0.699 0.485 -0.963 -0.963
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 8.297 0.872 9.517 0.000 8.297 0.208
## .wisc_2 (V_e1) 8.297 0.872 9.517 0.000 8.297 0.165
## .wisc_3 (V_e3) 6.898 1.171 5.891 0.000 6.898 0.106
## .wisc_4 (V_e4) 15.894 2.673 5.946 0.000 15.894 0.158
## S 1.000 1.000 1.000
## RI (V_RI) 18.545 5.694 3.257 0.001 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.8Result)Correlated Free Curve Slope Intercept (FCSI) Model (a.k.a. “Latent Basis”)
Path Diagram
Correlated Free Curve Slope Intercept (FCSI)
Growth Model
R Program
require(lavaan)
Fig13.10Model<-"
! regressions
S=~L2*wisc_2
S=~L3*wisc_3
S=~L4*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ 1.0*S
RI ~~ V_RI*RI
RI ~~ CRIS*S
! means
S~aS*1
RI~aI*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.10Result<-lavaan(Fig13.10Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.10Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 120 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 11
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 7.718
## Degrees of freedom 4
## P-value (Chi-square) 0.102
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.995
## Tucker-Lewis Index (TLI) 0.993
##
## Robust Comparative Fit Index (CFI) 0.995
## Robust Tucker-Lewis Index (TLI) 0.993
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2422.238
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4864.477
## Bayesian (BIC) 4897.658
## Sample-size adjusted Bayesian (SABIC) 4865.975
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.068
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.139
## P-value H_0: RMSEA <= 0.050 0.278
## P-value H_0: RMSEA >= 0.080 0.454
##
## Robust RMSEA 0.068
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.139
## P-value H_0: Robust RMSEA <= 0.050 0.278
## P-value H_0: Robust RMSEA >= 0.080 0.454
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.040
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_2 (L2) 1.227 0.143 8.562 0.000 1.227 0.173
## wisc_3 (L3) 2.702 0.302 8.961 0.000 2.702 0.334
## wisc_4 (L4) 4.505 0.503 8.954 0.000 4.505 0.450
## RI =~
## wisc_1 1.000 5.621 0.890
## wisc_2 1.000 5.621 0.793
## wisc_3 1.000 5.621 0.695
## wisc_4 1.000 5.621 0.561
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI (CRIS) 3.612 0.689 5.246 0.000 0.643 0.643
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 6.348 0.715 8.884 0.000 6.348 6.348
## RI (aI) 18.786 0.442 42.503 0.000 3.342 3.342
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 8.297 0.872 9.517 0.000 8.297 0.208
## .wisc_2 (V_e1) 8.297 0.872 9.517 0.000 8.297 0.165
## .wisc_3 (V_e3) 6.898 1.171 5.891 0.000 6.898 0.106
## .wisc_4 (V_e4) 15.894 2.673 5.946 0.000 15.894 0.158
## S 1.000 1.000 1.000
## RI (V_RI) 31.594 3.740 8.447 0.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.10Result)Linear Slope Intercept Model
Path Diagram
Linear Slope Intercept Growth Model
R Program
require(lavaan)
Fig13.11Model<-"
! regressions
S=~0.88*wisc_2
S=~2.73*wisc_3
S=~4.72*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ V_S*S
RI ~~ V_RI*RI
RI ~~ CRIS*S
! means
S~aS*1
RI~aI*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.11Result<-lavaan(Fig13.11Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.11Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 70 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 79.486
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.911
## Tucker-Lewis Index (TLI) 0.911
##
## Robust Comparative Fit Index (CFI) 0.911
## Robust Tucker-Lewis Index (TLI) 0.911
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2458.122
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4932.244
## Bayesian (BIC) 4958.789
## Sample-size adjusted Bayesian (SABIC) 4933.442
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.294
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.294
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.084
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_2 0.880 0.699 0.099
## wisc_3 2.730 2.169 0.273
## wisc_4 4.720 3.751 0.372
## RI =~
## wisc_1 1.000 5.511 0.845
## wisc_2 1.000 5.511 0.777
## wisc_3 1.000 5.511 0.692
## wisc_4 1.000 5.511 0.546
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI (CRIS) 4.131 0.611 6.760 0.000 0.943 0.943
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 5.885 0.092 63.917 0.000 7.405 7.405
## RI (aI) 19.936 0.429 46.506 0.000 3.617 3.617
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 12.198 1.245 9.797 0.000 12.198 0.287
## .wisc_2 (V_e1) 12.198 1.245 9.797 0.000 12.198 0.242
## .wisc_3 (V_e3) 5.722 1.176 4.867 0.000 5.722 0.090
## .wisc_4 (V_e4) 18.346 2.792 6.570 0.000 18.346 0.180
## S (V_S) 0.632 0.196 3.215 0.001 1.000 1.000
## RI (V_RI) 30.376 3.792 8.011 0.000 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.11Result)Age as Time Model
Path Diagram
Age as Time Growth Model
R Program
require(lavaan)
Fig13.13Model<-"
! regressions
S=~6.95*wisc_2
S=~8.8*wisc_3
S=~10.79*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
S=~6.07*wisc_1
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ V_S*S
RI ~~ V_RI*RI
RI ~~ CRIS*S
! means
S~aS*1
RI~aI*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.13Result<-lavaan(Fig13.13Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.13Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 57 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 79.486
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.911
## Tucker-Lewis Index (TLI) 0.911
##
## Robust Comparative Fit Index (CFI) 0.911
## Robust Tucker-Lewis Index (TLI) 0.911
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2458.122
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4932.244
## Bayesian (BIC) 4958.789
## Sample-size adjusted Bayesian (SABIC) 4933.442
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.294
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.294
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.084
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_2 6.950 5.523 0.778
## wisc_3 8.800 6.993 0.879
## wisc_4 10.790 8.575 0.850
## RI =~
## wisc_1 1.000 1.870 0.287
## wisc_2 1.000 1.870 0.264
## wisc_3 1.000 1.870 0.235
## wisc_4 1.000 1.870 0.185
## S =~
## wisc_1 6.070 4.824 0.739
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI (CRIS) 0.297 1.265 0.235 0.814 0.200 0.200
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 5.885 0.092 63.917 0.000 7.405 7.405
## RI (aI) -15.784 0.619 -25.496 0.000 -8.441 -8.441
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 12.198 1.245 9.797 0.000 12.198 0.287
## .wisc_2 (V_e1) 12.198 1.245 9.797 0.000 12.198 0.242
## .wisc_3 (V_e3) 5.722 1.176 4.867 0.000 5.722 0.090
## .wisc_4 (V_e4) 18.346 2.792 6.570 0.000 18.346 0.180
## S (V_S) 0.632 0.196 3.215 0.001 1.000 1.000
## RI (V_RI) 3.497 10.031 0.349 0.727 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.13Result)Orthogonal Linear Growth Model
Path Diagram
Orthogonal Linear Growth Model
R Program
require(lavaan)
Fig13.15Model<-"
# Measurement model
S =~ L1*wisc_1 + L2*wisc_2 + L3*wisc_3 + L4*wisc_4
RI =~ 1*wisc_1 + 1*wisc_2 + 1*wisc_3 + 1*wisc_4
# Residual variances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
# Latent variances and covariances
S ~~ 1*S
RI ~~ V_RI*RI
S ~~ 0*RI
# Latent means
S ~ aS*1
RI ~ aI*1
wisc_1 ~ 0*1
wisc_2 ~ 0*1
wisc_3 ~ 0*1
wisc_4 ~ 0*1
# Nonlinear equality constraints
# (Another Way to think about the loadings falling all on a line)
L2==L1+0.186282811*(L4-L1)
L3==L1+0.578746825*(L4-L1)
"
Fig13.15Result<-lavaan(Fig13.15Model, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.15Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 65 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 11
## Number of equality constraints 3
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 79.738
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.911
## Tucker-Lewis Index (TLI) 0.911
##
## Robust Comparative Fit Index (CFI) 0.911
## Robust Tucker-Lewis Index (TLI) 0.911
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2458.248
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4932.497
## Bayesian (BIC) 4959.041
## Sample-size adjusted Bayesian (SABIC) 4933.695
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.295
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.245
## 90 Percent confidence interval - lower 0.199
## 90 Percent confidence interval - upper 0.295
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.084
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_1 (L1) 5.205 1.040 5.003 0.000 5.205 0.798
## wisc_2 (L2) 5.903 0.970 6.087 0.000 5.903 0.832
## wisc_3 (L3) 7.373 0.848 8.699 0.000 7.373 0.926
## wisc_4 (L4) 8.952 0.773 11.586 0.000 8.952 0.887
## RI =~
## wisc_1 1.000 1.811 0.278
## wisc_2 1.000 1.811 0.255
## wisc_3 1.000 1.811 0.228
## wisc_4 1.000 1.811 0.180
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI 0.000 0.000 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 7.410 1.162 6.378 0.000 7.410 7.410
## RI (aI) -18.628 12.615 -1.477 0.140 -10.285 -10.285
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 12.216 1.247 9.797 0.000 12.216 0.287
## .wisc_2 (V_e1) 12.216 1.247 9.797 0.000 12.216 0.243
## .wisc_3 (V_e3) 5.718 1.176 4.863 0.000 5.718 0.090
## .wisc_4 (V_e4) 18.347 2.791 6.574 0.000 18.347 0.180
## S 1.000 1.000 1.000
## RI (V_RI) 3.281 11.178 0.293 0.769 1.000 1.000
##
## Constraints:
## |Slack|
## L2 - (L1+0.186282811*(L4-L1)) 0.000
## L3 - (L1+0.578746825*(L4-L1)) 0.000require(lavaangui)
#plot_lavaan(Fig13.15Result)Quadratic Growth Model
Path Diagram
Quadratic Linear Growth Model
R Program
require(lavaan)
Fig13.17Model<-"
! regressions
S=~0.88*wisc_2
S=~2.73*wisc_3
S=~4.72*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
Q=~0.77*wisc_2
Q=~7.45*wisc_3
Q=~22.28*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ V_S*S
RI ~~ V_RI*RI
RI ~~ CRIS*S
Q ~~ V_Q*Q
S ~~ CSQ*Q
RI ~~ CRIQ*Q
! means
S~aS*1
RI~aI*1
Q~aQ*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.17Result<-lavaan(Fig13.17Model, data=mcardle, fixed.x=FALSE, missing="FIML")## Warning: lavaan->lav_object_post_check():
## some estimated lv variances are negativesummary(Fig13.17Result, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 113 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 13
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 55.455
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.935
## Tucker-Lewis Index (TLI) 0.806
##
## Robust Comparative Fit Index (CFI) 0.935
## Robust Tucker-Lewis Index (TLI) 0.806
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2446.106
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4916.213
## Bayesian (BIC) 4956.030
## Sample-size adjusted Bayesian (SABIC) 4918.011
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.362
## 90 Percent confidence interval - lower 0.283
## 90 Percent confidence interval - upper 0.447
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.362
## 90 Percent confidence interval - lower 0.283
## 90 Percent confidence interval - upper 0.447
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.071
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_2 0.880 NA NA
## wisc_3 2.730 NA NA
## wisc_4 4.720 NA NA
## RI =~
## wisc_1 1.000 5.679 0.860
## wisc_2 1.000 5.679 0.811
## wisc_3 1.000 5.679 0.719
## wisc_4 1.000 5.679 0.556
## Q =~
## wisc_2 0.770 NA NA
## wisc_3 7.450 NA NA
## wisc_4 22.280 NA NA
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI (CRIS) 3.684 1.609 2.290 0.022 0.383 0.383
## Q (CSQ) 0.934 0.394 2.369 0.018 1.068 1.068
## RI ~~
## Q (CRIQ) 0.036 0.334 0.107 0.915 0.012 0.012
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 6.897 0.244 28.262 0.000 NA NA
## RI (aI) 19.451 0.452 43.016 0.000 3.425 3.425
## Q (aQ) -0.239 0.050 -4.742 0.000 NA NA
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 11.353 1.235 9.196 0.000 11.353 0.260
## .wisc_2 (V_e1) 11.353 1.235 9.196 0.000 11.353 0.232
## .wisc_3 (V_e3) 7.756 2.256 3.437 0.001 7.756 0.124
## .wisc_4 (V_e4) 35.548 11.895 2.988 0.003 35.548 0.340
## S (V_S) -2.871 1.753 -1.638 0.101 NA NA
## RI (V_RI) 32.249 4.218 7.645 0.000 1.000 1.000
## Q (V_Q) -0.266 0.115 -2.307 0.021 NA NArequire(lavaangui)
#plot_lavaan(Fig13.17Result)Quadratic Growth Model Scaled as Proportion of Time Elapsed (Not in Book)
Path Diagram
Quadratic Linear Growth Model Scaled as
Proportion of Time
This is the default behavior of programs like Amos in which the last measurement occasion is fixed at 1 and the intermediary loadings are fixed as proportions of time elapsed form the first measurement occasion. The resulting solution has positive values due to the reduced collinearity of the linear and quadratic latent variables.
R Program
require(lavaan)
Fig13.17BModel<-"
! regressions
S=~0.1864*wisc_2
S=~0.5783*wisc_3
S=~1.0*wisc_4
RI=~1.0*wisc_1
RI=~1.0*wisc_2
RI=~1.0*wisc_3
RI=~1.0*wisc_4
Q=~0.77*wisc_2
Q=~7.45*wisc_3
Q=~0.3345*wisc_4
! residuals, variances and covariances
wisc_1 ~~ V_e1*wisc_1
wisc_2 ~~ V_e1*wisc_2
wisc_3 ~~ V_e3*wisc_3
wisc_4 ~~ V_e4*wisc_4
S ~~ V_S*S
RI ~~ V_RI*RI
RI ~~ CRIS*S
Q ~~ V_Q*Q
S ~~ CSQ*Q
RI ~~ CRIQ*Q
! means
S~aS*1
RI~aI*1
Q~aQ*1
wisc_1~0*1;
wisc_2~0*1;
wisc_3~0*1;
wisc_4~0*1;
"
Fig13.17BResult<-lavaan(Fig13.17BModel, data=mcardle, fixed.x=FALSE, missing="FIML")
summary(Fig13.17BResult, fit.measures=TRUE,standardized=TRUE);## lavaan 0.6-19 ended normally after 135 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 13
## Number of equality constraints 1
##
## Number of observations 204
## Number of missing patterns 1
##
## Model Test User Model:
##
## Test statistic 74.087
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Model Test Baseline Model:
##
## Test statistic 830.699
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.913
## Tucker-Lewis Index (TLI) 0.738
##
## Robust Comparative Fit Index (CFI) 0.913
## Robust Tucker-Lewis Index (TLI) 0.738
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -2455.423
## Loglikelihood unrestricted model (H1) -2418.379
##
## Akaike (AIC) 4934.845
## Bayesian (BIC) 4974.663
## Sample-size adjusted Bayesian (SABIC) 4936.643
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.420
## 90 Percent confidence interval - lower 0.341
## 90 Percent confidence interval - upper 0.505
## P-value H_0: RMSEA <= 0.050 0.000
## P-value H_0: RMSEA >= 0.080 1.000
##
## Robust RMSEA 0.420
## 90 Percent confidence interval - lower 0.341
## 90 Percent confidence interval - upper 0.505
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.079
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S =~
## wisc_2 0.186 1.106 0.154
## wisc_3 0.578 3.430 0.443
## wisc_4 1.000 5.931 0.572
## RI =~
## wisc_1 1.000 5.659 0.854
## wisc_2 1.000 5.659 0.787
## wisc_3 1.000 5.659 0.730
## wisc_4 1.000 5.659 0.546
## Q =~
## wisc_2 0.770 0.297 0.041
## wisc_3 7.450 2.877 0.371
## wisc_4 0.335 0.129 0.012
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S ~~
## RI (CRIS) 20.057 3.365 5.960 0.000 0.598 0.598
## Q (CSQ) -1.562 1.348 -1.159 0.246 -0.682 -0.682
## RI ~~
## Q (CRIQ) -0.355 0.256 -1.387 0.165 -0.162 -0.162
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## S (aS) 27.425 0.544 50.398 0.000 4.624 4.624
## RI (aI) 19.930 0.440 45.251 0.000 3.522 3.522
## Q (aQ) 0.032 0.042 0.761 0.447 0.082 0.082
## .wisc_1 0.000 0.000 0.000
## .wisc_2 0.000 0.000 0.000
## .wisc_3 0.000 0.000 0.000
## .wisc_4 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .wisc_1 (V_e1) 11.879 1.198 9.919 0.000 11.879 0.271
## .wisc_2 (V_e1) 11.879 1.198 9.919 0.000 11.879 0.230
## .wisc_3 (V_e3) 3.550 14.389 0.247 0.805 3.550 0.059
## .wisc_4 (V_e4) 1.500 15.963 0.094 0.925 1.500 0.014
## S (V_S) 35.179 19.051 1.847 0.065 1.000 1.000
## RI (V_RI) 32.020 3.978 8.050 0.000 1.000 1.000
## Q (V_Q) 0.149 0.342 0.436 0.663 1.000 1.000require(lavaangui)
#plot_lavaan(Fig13.17BResult)References
McArdle, J. J., & Epstein, D. (1987). Latent growth curves within
developmental structural equation models. Child Development,
58(1), 110–133. https://doi.org/10.2307/1130295