This tutorial walks through the fitting of growth models with nonlinearity in random coefficients in several different frameworks (e.g., multilevel modeling framework, structural equation modeling framework), and demonstrates these models using different R packages (knowing how to fit the models in different packages can be helpful when trying to fit more complex models as each packages as its own advantages and limitations). In this tutorial, we will be using a sample data set that includes repeated measures of individuals’ height during infancy and early childhood.
The code and example provided in this tutorial are from Chapter 12 of Grimm, Ram, and Estabrook (2016), with a few additions in code and commentary. Please refer to the chapter for further interpretations and insights about the analyses.
This tutorial provides line-by-line code to examine growth models with nonlineary in parameters using the:
Load required libraries and read in the data
library(nlme)
library(lavaan)## This is lavaan 0.5-20## lavaan is BETA software! Please report any bugs.#set filepath for long data file
filepath.long<-"http://quant-d.ssri.psu.edu/sites/qdev/files/bgs_height_long_0.csv"
#set filepath for wide data file
filepath.wide<-"http://quant-d.ssri.psu.edu/sites/qdev/files/bgs_height_wide.csv"
#read in the .csv files using the url() function
hght_long <- read.csv(file=url(filepath.long),header=TRUE,na.string='.')
head(hght_long)## id age hght
## 1 1 1 53.5
## 2 1 3 60.4
## 3 1 6 67.2
## 4 1 9 70.9
## 5 1 12 76.2
## 6 1 15 81.1hght_wide <- read.csv(file=url(filepath.wide),header=TRUE,na.string='.')
head(hght_wide)## id hght01 hght03 hght06 hght09 hght12 hght15 hght18 hght24 hght36
## 1 1 53.5 60.4 67.2 70.9 76.2 81.1 NA 89.5 95.9
## 2 2 54.5 58.8 66.0 69.1 74.5 78.2 81.2 86.7 NA
## 3 3 49.5 56.0 61.9 64.1 67.9 69.6 73.8 NA 85.1
## 4 4 56.9 64.6 70.0 72.5 76.1 80.4 82.3 87.9 94.3
## 5 5 56.7 63.1 NA 73.1 77.3 81.8 83.5 88.9 99.1
## 6 6 56.1 61.9 67.9 72.3 77.3 79.9 82.3 88.3 97.5nlme library.hght.jb.nlme <- nlme(hght~b_1i + b_2i * (age/12) + b_3i * (exp(b_4i*(age/12))-1),
data = hght_long,
fixed = b_1i + b_2i + b_3i + b_4i ~ 1,
random = b_1i + b_2i + b_3i + b_4i ~ 1,
groups = ~ id,
start = c(50, 10, -18, -2),
na.action = na.exclude)
summary(hght.jb.nlme)## Nonlinear mixed-effects model fit by maximum likelihood
## Model: hght ~ b_1i + b_2i * (age/12) + b_3i * (exp(b_4i * (age/12)) - 1)
## Data: hght_long
## AIC BIC logLik
## 2005.178 2070.649 -987.589
##
## Random effects:
## Formula: list(b_1i ~ 1, b_2i ~ 1, b_3i ~ 1, b_4i ~ 1)
## Level: id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## b_1i 2.6774685 b_1i b_2i b_3i
## b_2i 0.8717526 0.319
## b_3i 3.4099947 0.491 0.449
## b_4i 0.2746913 -0.120 -0.275 -0.642
## Residual 0.8041717
##
## Fixed effects: b_1i + b_2i + b_3i + b_4i ~ 1
## Value Std.Error DF t-value p-value
## b_1i 50.99063 0.3422353 495 148.99288 0
## b_2i 9.31946 0.1607672 495 57.96865 0
## b_3i -17.84870 0.4814401 495 -37.07357 0
## b_4i -2.13255 0.0769100 495 -27.72791 0
## Correlation:
## b_1i b_2i b_3i
## b_2i 0.022
## b_3i 0.376 0.607
## b_4i 0.263 -0.642 -0.536
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -3.482492379 -0.518214899 0.001140808 0.517854147 2.745161193
##
## Number of Observations: 581
## Number of Groups: 83lavaan library.jb.hght.lavaan <- '
#latent variable definitions
#defining the intercept
eta_1 =~ 1*hght01 +
1*hght03 +
1*hght06 +
1*hght09 +
1*hght12 +
1*hght15 +
1*hght18 +
1*hght24 +
1*hght36
#defining the slope
eta_2 =~ .0833*hght01 +
.25*hght03 +
.5*hght06 +
.75*hght09 +
1*hght12 +
1.25*hght15 +
1.5*hght18 +
2*hght24 +
3*hght36
#defining eta_3
eta_3 =~ start(-.15)*L13*hght01 +
start(-.40)*L23*hght03 +
start(-.64)*L33*hght06 +
start(-.78)*L43*hght09 +
start(-.87)*L53*hght12 +
start(-.92)*L63*hght15 +
start(-.95)*L73*hght18 +
start(-.98)*L83*hght24 +
start(-.99)*L93*hght36
#defining eta_4
eta_4 =~ start(-1.25)*L14*hght01 +
start(-2.65)*L24*hght03 +
start(-3.16)*L34*hght06 +
start(-2.82)*L44*hght09 +
start(-2.24)*L54*hght12 +
start(-1.66)*L64*hght15 +
start(-1.19)*L74*hght18 +
start(-0.56)*L84*hght24 +
start(-0.11)*L94*hght36
#latent means
eta_1 ~ start(51)*1
eta_2 ~ start(9.25)*1
eta_3 ~ start(-17.8)*alpha3*1
eta_4 ~ 0 # start(0.07)*1
#introducing phantom factor, setting all parameters to 0 and mean to alpha4
phantom =~ 0*hght01
phantom ~~ 0*phantom
phantom ~~ 0*eta_1
phantom ~~ 0*eta_2
phantom ~~ 0*eta_3
phantom ~~ 0*eta_4
phantom ~ start(-2)*alpha4*1
#contraints to define factor loadings
alpha4 < -2 #add this constraint in so that it finds the viable solution
L13 == exp(alpha4*0.0833)-1
L23 == exp(alpha4*0.25)-1
L33 == exp(alpha4*0.50)-1
L43 == exp(alpha4*0.75)-1
L53 == exp(alpha4*1.00)-1
L63 == exp(alpha4*1.25)-1
L73 == exp(alpha4*1.50)-1
L83 == exp(alpha4*2.00)-1
L93 == exp(alpha4*3.00)-1
L14 == alpha3 * 0.0833 * exp(alpha4*0.0833)-1
L24 == alpha3 * 0.25 * exp(alpha4*0.25)-1
L34 == alpha3 * 0.50 * exp(alpha4*0.50)-1
L44 == alpha3 * 0.75 * exp(alpha4*0.75)-1
L54 == alpha3 * 1.00 * exp(alpha4*1.00)-1
L64 == alpha3 * 1.25 * exp(alpha4*1.25)-1
L74 == alpha3 * 1.50 * exp(alpha4*1.50)-1
L84 == alpha3 * 2.00 * exp(alpha4*2.00)-1
L94 == alpha3 * 3.00 * exp(alpha4*3.00)-1
#factor variances
eta_1 ~~ start(7.4)*eta_1
eta_2 ~~ start(0.81)*eta_2
eta_3 ~~ start(11.63)*eta_3
eta_4 ~~ start(0.08)*eta_4
#factor covariances
eta_1 ~~ start(0.64)*eta_2
eta_1 ~~ start(4.45)*eta_3
eta_1 ~~ start(0.03)*eta_4
eta_2 ~~ start(1.35)*eta_3
eta_2 ~~ start(-0.08)*eta_4
eta_3 ~~ start(-0.49)*eta_4
#manifest variances
hght01 ~~ start(.64)*theta*hght01
hght03 ~~ theta*hght03
hght06 ~~ theta*hght06
hght09 ~~ theta*hght09
hght12 ~~ theta*hght12
hght15 ~~ theta*hght15
hght18 ~~ theta*hght18
hght24 ~~ theta*hght24
hght36 ~~ theta*hght36
#manifest means (fixed to zero)
hght01 ~ 0*1
hght03 ~ 0*1
hght06 ~ 0*1
hght09 ~ 0*1
hght12 ~ 0*1
hght15 ~ 0*1
hght18 ~ 0*1
hght24 ~ 0*1
hght36 ~ 0*1'
lavaan.fit <- lavaan(jb.hght.lavaan,
data = hght_wide,
meanstructure = TRUE,
estimator = "ML",
missing = "fiml",
fixed.x = FALSE,
mimic="mplus",
control=list(iter.max=500),
verbose=FALSE)
summary(lavaan.fit, fit.measures=TRUE)## lavaan (0.5-20) converged normally after 1829 iterations
##
## Number of observations 83
##
## Number of missing patterns 31
##
## Estimator ML
## Minimum Function Test Statistic 103.452
## Degrees of freedom 39
## P-value (Chi-square) 0.000
##
## Model test baseline model:
##
## Minimum Function Test Statistic 955.844
## Degrees of freedom 36
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.930
## Tucker-Lewis Index (TLI) 0.935
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -987.467
## Loglikelihood unrestricted model (H1) -935.741
##
## Number of free parameters 15
## Akaike (AIC) 2004.935
## Bayesian (BIC) 2041.217
## Sample-size adjusted Bayesian (BIC) 1993.904
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.141
## 90 Percent Confidence Interval 0.108 0.174
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.306
##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 =~
## hght01 1.000
## hght03 1.000
## hght06 1.000
## hght09 1.000
## hght12 1.000
## hght15 1.000
## hght18 1.000
## hght24 1.000
## hght36 1.000
## eta_2 =~
## hght01 0.083
## hght03 0.250
## hght06 0.500
## hght09 0.750
## hght12 1.000
## hght15 1.250
## hght18 1.500
## hght24 2.000
## hght36 3.000
## eta_3 =~
## hght01 (L13) -0.159 0.005 -28.938 0.000
## hght03 (L23) -0.405 0.012 -34.751 0.000
## hght06 (L33) -0.646 0.014 -46.575 0.000
## hght09 (L43) -0.789 0.012 -63.764 0.000
## hght12 (L53) -0.875 0.010 -89.055 0.000
## hght15 (L63) -0.925 0.007 -126.678 0.000
## hght18 (L73) -0.956 0.005 -183.196 0.000
## hght24 (L83) -0.984 0.002 -399.676 0.000
## hght36 (L93) -0.998 0.000 -2154.873 0.000
## eta_4 =~
## hght01 (L14) -2.250 0.038 -58.536 0.000
## hght03 (L24) -3.653 0.107 -34.037 0.000
## hght06 (L34) -4.158 0.182 -22.881 0.000
## hght09 (L44) -3.819 0.214 -17.862 0.000
## hght12 (L54) -3.236 0.212 -15.287 0.000
## hght15 (L64) -2.663 0.189 -14.077 0.000
## hght18 (L74) -2.188 0.158 -13.852 0.000
## hght24 (L84) -1.561 0.096 -16.214 0.000
## hght36 (L94) -1.105 0.026 -42.016 0.000
## phantom =~
## hght01 0.000
##
## Covariances:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 ~~
## phantom 0.000
## eta_2 ~~
## phantom 0.000
## eta_3 ~~
## phantom 0.000
## eta_4 ~~
## phantom 0.000
## eta_1 ~~
## eta_2 0.561 0.585 0.960 0.337
## eta_3 3.960 1.641 2.413 0.016
## eta_4 0.105 0.304 0.347 0.729
## eta_2 ~~
## eta_3 1.346 0.849 1.586 0.113
## eta_4 -0.078 0.143 -0.542 0.588
## eta_3 ~~
## eta_4 -0.488 0.383 -1.274 0.203
##
## Intercepts:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 51.076 0.349 146.465 0.000
## eta_2 9.296 0.168 55.402 0.000
## eta_3 (alp3) -17.836 0.481 -37.051 0.000
## eta_4 0.000
## phantom (alp4) -2.076 0.078 -26.508 0.000
## hght01 0.000
## hght03 0.000
## hght06 0.000
## hght09 0.000
## hght12 0.000
## hght15 0.000
## hght18 0.000
## hght24 0.000
## hght36 0.000
##
## Variances:
## Estimate Std.Err Z-value P(>|z|)
## phantom 0.000
## eta_1 7.545 1.828 4.126 0.000
## eta_2 0.809 0.340 2.380 0.017
## eta_3 11.626 3.006 3.868 0.000
## eta_4 0.076 0.079 0.959 0.338
## hght01 (thet) 0.644 0.053 12.175 0.000
## hght03 (thet) 0.644 0.053 12.175 0.000
## hght06 (thet) 0.644 0.053 12.175 0.000
## hght09 (thet) 0.644 0.053 12.175 0.000
## hght12 (thet) 0.644 0.053 12.175 0.000
## hght15 (thet) 0.644 0.053 12.175 0.000
## hght18 (thet) 0.644 0.053 12.175 0.000
## hght24 (thet) 0.644 0.053 12.175 0.000
## hght36 (thet) 0.644 0.053 12.175 0.000
##
## Constraints:
## |Slack|
## -2 - (alpha4) 0.076
## L13 - (exp(alpha4*0.0833)-1) 0.000
## L23 - (exp(alpha4*0.25)-1) 0.000
## L33 - (exp(alpha4*0.50)-1) 0.000
## L43 - (exp(alpha4*0.75)-1) 0.000
## L53 - (exp(alpha4*1.00)-1) 0.000
## L63 - (exp(alpha4*1.25)-1) 0.000
## L73 - (exp(alpha4*1.50)-1) 0.000
## L83 - (exp(alpha4*2.00)-1) 0.000
## L93 - (exp(alpha4*3.00)-1) 0.000
## L14-(alpha3*0.0833*exp(alpha4*0.0833)-1) 0.000
## L24 - (alpha3*0.25*exp(alpha4*0.25)-1) 0.000
## L34 - (alpha3*0.50*exp(alpha4*0.50)-1) 0.000
## L44 - (alpha3*0.75*exp(alpha4*0.75)-1) 0.000
## L54 - (alpha3*1.00*exp(alpha4*1.00)-1) 0.000
## L64 - (alpha3*1.25*exp(alpha4*1.25)-1) 0.000
## L74 - (alpha3*1.50*exp(alpha4*1.50)-1) 0.000
## L84 - (alpha3*2.00*exp(alpha4*2.00)-1) 0.000
## L94 - (alpha3*3.00*exp(alpha4*3.00)-1) 0.000nlme library.hght.jb.nlme.fo <- nlme(hght ~ (b_1 + d_1i) + (b_2 + d_2i) * (age/12) +
(b_3 + d_3i) * (exp(b_4 * (age/12))-1) +
d_4i * (b_3 * (age/12) * exp(b_4 * (age/12))),
data = hght_long,
fixed = b_1 + b_2 + b_3 + b_4 ~ 1,
random = d_1i + d_2i + d_3i + d_4i ~ 1,
groups =~ id,
start = c(50, 10, -18, -2),
na.action = na.exclude)
summary(hght.jb.nlme.fo)## Nonlinear mixed-effects model fit by maximum likelihood
## Model: hght ~ (b_1 + d_1i) + (b_2 + d_2i) * (age/12) + (b_3 + d_3i) * (exp(b_4 * (age/12)) - 1) + d_4i * (b_3 * (age/12) * exp(b_4 * (age/12)))
## Data: hght_long
## AIC BIC logLik
## 2005.213 2070.684 -987.6063
##
## Random effects:
## Formula: list(d_1i ~ 1, d_2i ~ 1, d_3i ~ 1, d_4i ~ 1)
## Level: id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## d_1i 2.7167272 d_1i d_2i d_3i
## d_2i 0.8928535 0.267
## d_3i 3.3840109 0.482 0.433
## d_4i 0.2775674 0.027 -0.299 -0.525
## Residual 0.8027907
##
## Fixed effects: b_1 + b_2 + b_3 + b_4 ~ 1
## Value Std.Error DF t-value p-value
## b_1 50.96848 0.3478280 495 146.53357 0
## b_2 9.33387 0.1602774 495 58.23568 0
## b_3 -17.83233 0.4667397 495 -38.20615 0
## b_4 -2.11293 0.0752805 495 -28.06740 0
## Correlation:
## b_1 b_2 b_3
## b_2 -0.006
## b_3 0.385 0.578
## b_4 0.326 -0.634 -0.462
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -3.43001311 -0.51726171 -0.02358487 0.52150038 2.78477882
##
## Number of Observations: 581
## Number of Groups: 83nlme library.hght.spline.nlme <- nlme(hght ~ b_1i + b_2i * (pmin(0, age - b_4i)) +
b_3i * (pmax(0, age - b_4i)),
data = hght_long,
fixed = b_1i + b_2i + b_3i + b_4i ~ 1,
random = b_1i + b_2i + b_3i + b_4i ~ 1,
groups =~ id,
start = c(60, 5, 2, 8),
na.action = na.omit)
summary(hght.spline.nlme) ## Nonlinear mixed-effects model fit by maximum likelihood
## Model: hght ~ b_1i + b_2i * (pmin(0, age - b_4i)) + b_3i * (pmax(0, age - b_4i))
## Data: hght_long
## AIC BIC logLik
## 2293.053 2358.524 -1131.526
##
## Random effects:
## Formula: list(b_1i ~ 1, b_2i ~ 1, b_3i ~ 1, b_4i ~ 1)
## Level: id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## b_1i 2.70810506 b_1i b_2i b_3i
## b_2i 0.28603962 0.107
## b_3i 0.05567495 0.475 0.444
## b_4i 0.29364264 0.792 -0.506 0.235
## Residual 1.21258074
##
## Fixed effects: b_1i + b_2i + b_3i + b_4i ~ 1
## Value Std.Error DF t-value p-value
## b_1i 71.92542 0.3602927 495 199.63051 0
## b_2i 2.51072 0.0540711 495 46.43362 0
## b_3i 0.90033 0.0098067 495 91.80777 0
## b_4i 7.72086 0.1427460 495 54.08808 0
## Correlation:
## b_1i b_2i b_3i
## b_2i -0.153
## b_3i 0.028 0.168
## b_4i 0.617 -0.685 -0.190
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.639119082 -0.529769751 0.009075796 0.574138864 2.623652491
##
## Number of Observations: 581
## Number of Groups: 83lavaan library.Note: this model results in estimation issues (e.g., variance of the knot point was estimated to be negative), thus the model results should not be interpreted.
spline.hght.lavaan <- '
#latent variable definitions
#defining the intercept
eta_1 =~ 1*hght01 +
1*hght03 +
1*hght06 +
1*hght09 +
1*hght12 +
1*hght15 +
1*hght18 +
1*hght24 +
1*hght36
#defining the slope
eta_2 =~ 1*L12*hght01 +
3*L22*hght03 +
6*L32*hght06 +
9*L42*hght09 +
12*L52*hght12 +
15*L62*hght15 +
18*L72*hght18 +
24*L82*hght24 +
36*L92*hght36
#defining eta_3
eta_3 =~ NA*L13*hght01 +
L23*hght03 +
L33*hght06 +
L43*hght09 +
L53*hght12 +
L63*hght15 +
L73*hght18 +
L83*hght24 +
L93*hght36
#defining eta_4
eta_4 =~ start(-0.90)*L14*hght01 +
start(-0.90)*L24*hght03 +
start(-0.90)*L34*hght06 +
start(-2.57)*L44*hght09 +
start(-2.57)*L54*hght12 +
start(-2.57)*L64*hght15 +
start(-2.57)*L74*hght18 +
start(-2.57)*L84*hght24 +
start(-2.57)*L94*hght36
#latent means
eta_1 ~ start(71)*1
eta_2 ~ start(2.5)*alpha2*1
eta_3 ~ start(.09)*alpha3*1
eta_4 ~ start(7.7)*alpha4*1
#factor variances
#fixing variances to zero b/c getting negative estimates
eta_1 ~~ start(7.3)*eta_1
eta_2 ~~ 0*eta_2
eta_3 ~~ start(0)*eta_3
eta_4 ~~ start(0.08)*eta_4
#factor covariances
eta_1 ~~ start(.08)*eta_2
eta_1 ~~ 0*eta_3
eta_1 ~~ start(.03)*eta_4
eta_2 ~~ 0*eta_3
eta_2 ~~ start(.40)*eta_4
eta_3 ~~ 0*eta_4
#introducing phantom factors, setting all parameters to 0 and mean to alpha4
phantom2 =~ 0*hght01
phantom2 ~~ 0*phantom2
phantom2 ~~ 0*eta_1
phantom2 ~~ 0*eta_2
phantom2 ~~ 0*eta_3
phantom2 ~~ 0*eta_4
phantom2 ~ start(.29)*alpha2*1
phantom3 =~ 0*hght01
phantom3 ~~ 0*phantom3
phantom3 ~~ 0*phantom2
phantom3 ~~ 0*eta_1
phantom3 ~~ 0*eta_2
phantom3 ~~ 0*eta_3
phantom3 ~~ 0*eta_4
phantom3 ~ start(.29)*alpha3*1
phantom4 =~ 0*hght01
phantom4 ~~ 0*phantom4
phantom4 ~~ 0*phantom3
phantom4 ~~ 0*phantom2
phantom4 ~~ 0*eta_1
phantom4 ~~ 0*eta_2
phantom4 ~~ 0*eta_3
phantom4 ~~ 0*eta_4
phantom4 ~ start(.29)*alpha4*1
#contraints to define factor loadings
L12 == 1 - alpha4
L22 == 3 - alpha4
L32 == 6 - alpha4
L42 == 9 - alpha4
L52 == 12 - alpha4
L62 == 15 - alpha4
L72 == 18 - alpha4
L82 == 24 - alpha4
L92 == 36 - alpha4
L13 == sqrt(( 1 - alpha4)^2)
L23 == sqrt(( 3 - alpha4)^2)
L33 == sqrt(( 6 - alpha4)^2)
L43 == sqrt(( 9 - alpha4)^2)
L53 == sqrt((12 - alpha4)^2)
L63 == sqrt((15 - alpha4)^2)
L73 == sqrt((18 - alpha4)^2)
L83 == sqrt((24 - alpha4)^2)
L93 == sqrt((36 - alpha4)^2)
L14 == (-alpha2 * sqrt(( 1 - alpha4)^2) + alpha3 * 1 - alpha3 * alpha4) / sqrt(( 1 - alpha4)^2)
L24 == (-alpha2 * sqrt(( 3 - alpha4)^2) + alpha3 * 3 - alpha3 * alpha4) / sqrt(( 3 - alpha4)^2)
L34 == (-alpha2 * sqrt(( 6 - alpha4)^2) + alpha3 * 6 - alpha3 * alpha4) / sqrt(( 6 - alpha4)^2)
L44 == (-alpha2 * sqrt(( 9 - alpha4)^2) + alpha3 * 9 - alpha3 * alpha4) / sqrt(( 9 - alpha4)^2)
L54 == (-alpha2 * sqrt((12 - alpha4)^2) + alpha3 * 12 - alpha3 * alpha4) / sqrt((12 - alpha4)^2)
L64 == (-alpha2 * sqrt((15 - alpha4)^2) + alpha3 * 15 - alpha3 * alpha4) / sqrt((15 - alpha4)^2)
L74 == (-alpha2 * sqrt((18 - alpha4)^2) + alpha3 * 18 - alpha3 * alpha4) / sqrt((18 - alpha4)^2)
L84 == (-alpha2 * sqrt((24 - alpha4)^2) + alpha3 * 24 - alpha3 * alpha4) / sqrt((24 - alpha4)^2)
L94 == (-alpha2 * sqrt((36 - alpha4)^2) + alpha3 * 36 - alpha3 * alpha4) / sqrt((36 - alpha4)^2)
#manifest variances
hght01 ~~ start(1.2)*theta*hght01
hght03 ~~ theta*hght03
hght06 ~~ theta*hght06
hght09 ~~ theta*hght09
hght12 ~~ theta*hght12
hght15 ~~ theta*hght15
hght18 ~~ theta*hght18
hght24 ~~ theta*hght24
hght36 ~~ theta*hght36
#manifest means (fixed to zero)
hght01 ~ 0*1
hght03 ~ 0*1
hght06 ~ 0*1
hght09 ~ 0*1
hght12 ~ 0*1
hght15 ~ 0*1
hght18 ~ 0*1
hght24 ~ 0*1
hght36 ~ 0*1'
lavaan.fit1 <- lavaan(spline.hght.lavaan,
data = hght_wide,
meanstructure = TRUE,
estimator = "ML",
missing = "fiml",
fixed.x = FALSE)## Warning in lavaan(spline.hght.lavaan, data = hght_wide, meanstructure =
## TRUE, : lavaan WARNING: model has NOT converged!summary(lavaan.fit1, fit.measures=TRUE)## ** WARNING ** lavaan (0.5-20) did NOT converge after 668 iterations
## ** WARNING ** Estimates below are most likely unreliable
##
## Number of observations 83
##
## Number of missing patterns 31
##
## Estimator ML
## Minimum Function Test Statistic NA
## Degrees of freedom NA
## P-value NA## Warning in .local(object, ...): lavaan WARNING: fit measures not available if model did not converge##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 =~
## hght01 1.000
## hght03 1.000
## hght06 1.000
## hght09 1.000
## hght12 1.000
## hght15 1.000
## hght18 1.000
## hght24 1.000
## hght36 1.000
## eta_2 =~
## hght01 (L12) -13.998 NA
## hght03 (L22) -12.000 NA
## hght06 (L32) -9.000 NA
## hght09 (L42) -6.000 NA
## hght12 (L52) -3.000 NA
## hght15 (L62) 0.000 NA
## hght18 (L72) 2.999 NA
## hght24 (L82) 9.000 NA
## hght36 (L92) 21.001 NA
## eta_3 =~
## hght01 (L13) 14.000 NA
## hght03 (L23) 12.000 NA
## hght06 (L33) 9.000 NA
## hght09 (L43) 6.000 NA
## hght12 (L53) 3.000 NA
## hght15 (L63) 0.000 NA
## hght18 (L73) 3.000 NA
## hght24 (L83) 9.000 NA
## hght36 (L93) 21.000 NA
## eta_4 =~
## hght01 (L14) -1.197 NA
## hght03 (L24) -1.186 NA
## hght06 (L34) -1.183 NA
## hght09 (L44) -1.191 NA
## hght12 (L54) -1.199 NA
## hght15 (L64) -1.200 NA
## hght18 (L74) -1.163 NA
## hght24 (L84) -1.152 NA
## hght36 (L94) -1.134 NA
## phantom2 =~
## hght01 0.000
## phantom3 =~
## hght01 0.000
## phantom4 =~
## hght01 0.000
##
## Covariances:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 ~~
## eta_2 0.529 NA
## eta_3 0.000
## eta_4 2.512 NA
## eta_2 ~~
## eta_3 0.000
## eta_4 0.418 NA
## eta_3 ~~
## eta_4 0.000
## eta_1 ~~
## phantom2 0.000
## eta_2 ~~
## phantom2 0.000
## eta_3 ~~
## phantom2 0.000
## eta_4 ~~
## phantom2 0.000
## phantom2 ~~
## phantom3 0.000
## eta_1 ~~
## phantom3 0.000
## eta_2 ~~
## phantom3 0.000
## eta_3 ~~
## phantom3 0.000
## eta_4 ~~
## phantom3 0.000
## phantom3 ~~
## phantom4 0.000
## phantom2 ~~
## phantom4 0.000
## eta_1 ~~
## phantom4 0.000
## eta_2 ~~
## phantom4 0.000
## eta_3 ~~
## phantom4 0.000
## eta_4 ~~
## phantom4 0.000
##
## Intercepts:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 98.002 NA
## eta_2 (alp2) 1.170 NA
## eta_3 (alp3) 0.026 NA
## eta_4 (alp4) 15.000 NA
## phantm2 (alp2) 1.170 NA
## phantm3 (alp3) 0.026 NA
## phantm4 (alp4) 15.000 NA
## hght01 0.000
## hght03 0.000
## hght06 0.000
## hght09 0.000
## hght12 0.000
## hght15 0.000
## hght18 0.000
## hght24 0.000
## hght36 0.000
##
## Variances:
## Estimate Std.Err Z-value P(>|z|)
## eta_1 9.417 NA
## eta_2 0.000
## eta_3 0.215 NA
## eta_4 1.145 NA
## phantm2 0.000
## phantm3 0.000
## phantm4 0.000
## hght01 (thet) 1.531 NA
## hght03 (thet) 1.530 NA
## hght06 (thet) 1.525 NA
## hght09 (thet) 1.533 NA
## hght12 (thet) 1.533 NA
## hght15 (thet) 1.533 NA
## hght18 (thet) 1.530 NA
## hght24 (thet) 1.528 NA
## hght36 (thet) 1.531 NA
##
## Constraints:
## |Slack|
## L12 - (1-alpha4) 0.002
## L22 - (3-alpha4) 0.000
## L32 - (6-alpha4) 0.000
## L42 - (9-alpha4) 0.000
## L52 - (12-alpha4) 0.000
## L62 - (15-alpha4) 0.000
## L72 - (18-alpha4) 0.001
## L82 - (24-alpha4) 0.000
## L92 - (36-alpha4) 0.001
## L13 - (sqrt((1-alpha4)^2)) 0.000
## L23 - (sqrt((3-alpha4)^2)) 0.000
## L33 - (sqrt((6-alpha4)^2)) 0.000
## L43 - (sqrt((9-alpha4)^2)) 0.000
## L53 - (sqrt((12-alpha4)^2)) 0.000
## L63 - (sqrt((15-alpha4)^2)) 0.000
## L73 - (sqrt((18-alpha4)^2)) 0.000
## L83 - (sqrt((24-alpha4)^2)) 0.000
## L93 - (sqrt((36-alpha4)^2)) 0.000
## L14-((-lph2*((1-4)^2)+3*1-3*4)/((1-4)^2)) 0.001
## L24-((-lph2*((3-4)^2)+3*3-3*4)/((3-4)^2)) 0.009
## L34-((-lph2*((6-4)^2)+3*6-3*4)/((6-4)^2)) 0.013
## L44-((-lph2*((9-4)^2)+3*9-3*4)/((9-4)^2)) 0.005
## L54-((-2*((12-4)^2)+3*12-3*4)/((12-4)^2)) 0.003
## L64-((-2*((15-4)^2)+3*15-3*4)/((15-4)^2)) 0.004
## L74-((-2*((18-4)^2)+3*18-3*4)/((18-4)^2)) 0.019
## L84-((-2*((24-4)^2)+3*24-3*4)/((24-4)^2)) 0.008
## L94-((-2*((36-4)^2)+3*36-3*4)/((36-4)^2)) 0.010