Ch. 3 Path Notation
Phil Wood
2025-07-18
Read in Data
setwd(dirname(rstudioapi::getActiveDocumentContext()$path))
#Load lavaan (Install if you haven't already)
#Install lavaan if needed
if (!require("lavaan", character.only = TRUE)) {
install.packages("lavaan")
library(lavaan, character.only = TRUE)
}## Loading required package: lavaan## This is lavaan 0.6-19
## lavaan is FREE software! Please report any bugs.library(lavaan)
#Install HistData if needed
if (!require("HistData", character.only = TRUE)) {
install.packages("HistData")
library(HistData, character.only = TRUE)
}## Loading required package: HistDatalibrary(HistData)
# Load the Galton data
data(GaltonFamilies)
# Confirm variable names
names(GaltonFamilies)## [1] "family" "father" "mother" "midparentHeight"
## [5] "children" "childNum" "gender" "childHeight"Figure 3.3 a, b, and c (Also Figure 3.4a): Regression model with named parameters
model <- '
# Regression path with named parameter
childHeight ~ b1*midparentHeight
# Variance of predictor
midparentHeight ~~ var_mph*midparentHeight
# Residual (error) variance of outcome
childHeight ~~ var_e*childHeight
# Intercept of outcome
childHeight ~ b0*1
# Mean of predictor
midparentHeight ~ m*1
'
# Fit the model
fit <- sem(model, data = GaltonFamilies)
# Show results
summary(fit, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6-19 ended normally after 8 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 5
##
## Number of observations 934
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Model Test Baseline Model:
##
## Test statistic 101.534
## Degrees of freedom 1
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.000
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -4340.025
## Loglikelihood unrestricted model (H1) -4340.025
##
## Akaike (AIC) 8690.049
## Bayesian (BIC) 8714.247
## Sample-size adjusted Bayesian (SABIC) 8698.367
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
## P-value H_0: RMSEA <= 0.050 NA
## P-value H_0: RMSEA >= 0.080 NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## childHeight ~
## mdprntHgh (b1) 0.637 0.062 10.357 0.000 0.637 0.321
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .childHght (b0) 22.636 4.261 5.313 0.000 22.636 6.328
## mdprntHgh (m) 69.207 0.059 1174.115 0.000 69.207 38.418
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## mdprntH (vr_m) 3.245 0.150 21.610 0.000 3.245 1.000
## .chldHgh (var_) 11.479 0.531 21.610 0.000 11.479 0.897Make a Path Diagram
#Make a Path Diagram
#lavaan Shiny GUI is a graphical user interface (GUI) built with Shiny,
#It allows users to specify and run SEM models without writing full R code.
#How to Launch lavaan GUI
#1. Install required packages
#Install shiny if you haven't already:
#Install lavaangui if needed
if (!require("shiny", character.only = TRUE)) {
install.packages("shiny")
library(shiny, character.only = TRUE)
}## Loading required package: shinyif (!require("lavaangui", character.only = TRUE)) {
install.packages("lavaangui")
library(lavaangui, character.only = TRUE)
}## Loading required package: lavaangui## This is lavaangui 0.2.4
## lavaangui is BETA software! Please report any bugs at https://github.com/karchjd/lavaangui/issuesrequire(lavaangui)
#After executing the following line, you will see a window open. You can then
#move path elements around, save as an SVG or other format and export to the
#graphics package of your choice for further processing
#You'll have to run this interactively. The plot looks as follows:
#plot_lavaan(fit)
Figure 3.3a One Predictor Standardized
Model
Figure3.3b Unstandardized One Predictor
Regression Model
Figure3.3c Unstandardized One Predictor
Regression Model with Intercept
Figure 3.4b .Model with Error Standard Deviation
modelErrSD <- '
# Regression path from predictor to outcome
childHeight ~ b1*midparentHeight
# Latent error factor with fixed variance
e =~ stddev_e*childHeight + NA*childHeight
e ~~ 1*e
# Residual variance removed from observed childHeight
childHeight ~~ 0*childHeight
# Variance of predictor
midparentHeight ~~ var_mph*midparentHeight
# Intercept of outcome
childHeight ~ b0*1
# Mean of predictor
midparentHeight ~ m*1
'
# Fit the model
fitErrSD <- sem(modelErrSD, data = GaltonFamilies)
# Show results
summary(fitErrSD, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6-19 ended normally after 28 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 5
##
## Number of observations 934
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Model Test Baseline Model:
##
## Test statistic 101.534
## Degrees of freedom 1
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.000
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -4340.025
## Loglikelihood unrestricted model (H1) -4340.025
##
## Akaike (AIC) 8690.049
## Bayesian (BIC) 8714.247
## Sample-size adjusted Bayesian (SABIC) 8698.367
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
## P-value H_0: RMSEA <= 0.050 NA
## P-value H_0: RMSEA >= 0.080 NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## e =~
## chldHgh (std_) 3.388 0.078 43.220 0.000 3.388 0.947
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## childHeight ~
## mdprntHgh (b1) 0.637 0.062 10.357 0.000 0.637 0.321
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .childHght (b0) 22.636 4.261 5.313 0.000 22.636 6.328
## mdprntHgh (m) 69.207 0.059 1174.115 0.000 69.207 38.418
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## e 1.000 1.000 1.000
## .chldHgh 0.000 0.000 0.000
## mdprntH (vr_m) 3.245 0.150 21.610 0.000 3.245 1.000#plot_lavaan(fitErrSD)
Figure 3.4b One Predictor Model with Intercept
and Error Standard Deviation
Path Diagram for Correlation Coefficient
#Path Diagram for Correlation Coefficient (Z Score Factors)
modelCorr <- '
# Latent factors with unit variance
ChildZ =~ NA*childHeight+SDChild*childHeight
ChildZ ~~ 1*ChildZ
ParentZ =~ NA*midparentHeight+SDParent*midparentHeight
ParentZ ~~ 1*ParentZ
# Residual variance removed from observed childHeight
childHeight ~~ 0*childHeight
midparentHeight ~~ 0*midparentHeight
# Correlation of Z Score Child and Z Score Parent
ChildZ ~~ R*ParentZ
# Mean of Child and Parent
childHeight ~ b0*1
midparentHeight ~ m*1
'
# Fit the model
fitCorr <- sem(modelCorr, data = GaltonFamilies)
# Show results
summary(fitCorr, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6-19 ended normally after 15 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 5
##
## Number of observations 934
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Model Test Baseline Model:
##
## Test statistic 101.534
## Degrees of freedom 1
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.000
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -4340.025
## Loglikelihood unrestricted model (H1) -4340.025
##
## Akaike (AIC) 8690.049
## Bayesian (BIC) 8714.247
## Sample-size adjusted Bayesian (SABIC) 8698.367
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
## P-value H_0: RMSEA <= 0.050 NA
## P-value H_0: RMSEA >= 0.080 NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ChildZ =~
## chldHgh (SDCh) 3.577 0.083 43.220 0.000 3.577 1.000
## ParentZ =~
## mdprntH (SDPr) 1.801 0.042 43.220 0.000 1.801 1.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ChildZ ~~
## ParentZ (R) 0.321 0.029 10.935 0.000 0.321 0.321
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .childHght (b0) 66.746 0.117 570.215 0.000 66.746 18.658
## .mdprntHgh (m) 69.207 0.059 1174.115 0.000 69.207 38.418
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## ChildZ 1.000 1.000 1.000
## ParentZ 1.000 1.000 1.000
## .childHeight 0.000 0.000 0.000
## .midparentHeght 0.000 0.000 0.000#plot_lavaan(fitCorr)
Figure3.4c Path Diagram of a Correlation
Coefficient based on Adjoined Matrix