Lecture 09: Path Analysis

Homework 2

Author
Affiliation

Jihong Zhang*, Ph.D

Educational Statistics and Research Methods (ESRM) Program*

University of Arkansas

Published

October 9, 2024

Modified

October 11, 2024

Homework 2B

library(ESRM64503)
library(kableExtra)
library(tidyverse)
library(DescTools) # Desc() allows you to quick screen data
library(lavaan) # Desc() allows you to quick screen data
# options(digits = 3)
head(dataMath)
  id hsl cc use msc mas mse perf female
1  1  NA  9  44  55  39  NA   14      1
2  2   3  2  77  70  42  71   12      0
3  3  NA 12  64  52  31  NA   NA      1
4  4   6 20  71  65  39  84   19      0
5  5   2 15  48  19   2  60   12      0
6  6  NA 15  61  62  42  87   18      0
dim(dataMath)
[1] 350   9

Homework 2B: Setup

Homework 2B: Model 1 - Correlation Model with MSE, HSL, and CC

model01.syntax = "
# Variances:
mse ~~ mse 
hsl ~~ hsl   
cc ~~ cc

# Covariance:
mse ~~ hsl
mse ~~ cc
hsl ~~ cc

# Means:
mse ~ 1  
hsl ~ 1   
cc ~ 1   
"

## Estimation for model01
model01.fit <- cfa(model01.syntax, data=dataMath, mimic="MPLUS", fixed.x = TRUE, estimator = "MLR") 

## Print output
standardizedsolution(model01.fit)
  lhs op rhs est.std    se      z pvalue ci.lower ci.upper
1 mse ~~ mse   1.000 0.000     NA     NA    1.000    1.000
2 hsl ~~ hsl   1.000 0.000     NA     NA    1.000    1.000
3  cc ~~  cc   1.000 0.000     NA     NA    1.000    1.000
4 mse ~~ hsl   0.490 0.044 11.096  0.000    0.403    0.576
5 mse ~~  cc   0.252 0.054  4.646  0.000    0.146    0.358
6 hsl ~~  cc   0.156 0.055  2.860  0.004    0.049    0.263
7 mse ~1       6.178 0.235 26.283  0.000    5.718    6.639
8 hsl ~1       3.721 0.155 23.933  0.000    3.416    4.025
9  cc ~1       1.754 0.078 22.426  0.000    1.600    1.907
## filter correlations among varaibles
output_mod1 <- standardizedsolution(model01.fit) |> filter(op == "~~", lhs != rhs)
output_mod1
  lhs op rhs est.std    se      z pvalue ci.lower ci.upper
1 mse ~~ hsl   0.490 0.044 11.096  0.000    0.403    0.576
2 mse ~~  cc   0.252 0.054  4.646  0.000    0.146    0.358
3 hsl ~~  cc   0.156 0.055  2.860  0.004    0.049    0.263

  • For Mathematics Self-Efficacy (MSE) and High School Math Experience (HSL), there is a moderate but significantly positive correlation with r = 0.49 (SE = 0.044), p < .001

  • For Mathematics Self-Efficacy (MSE) and College Math Experience (CC), there is a strong and significantly positive correlation with r = 0.252 (SE = 0.054), p < .001

  • For High School Math Experience (HSL) and College Math Experience (CC), there is a weak and significantly positive correlation with r = 0.156 (SE = 0.055), p = .004

Homework 2B: Model 2 - constraining the correlation of HSL and CC to 0

model02.syntax = "
# Variances:
mse ~~ mse 
hsl ~~ hsl   
cc ~~ cc

# Covariance:
mse ~~ hsl
mse ~~ cc
hsl ~~ 0*cc

# Means:
mse ~ 1  
hsl ~ 1   
cc ~ 1   
"
model02.fit <- cfa(model02.syntax, data=dataMath, mimic="MPLUS", fixed.x = TRUE, estimator = "MLR") 

## Print output
standardizedsolution(model02.fit)
  lhs op rhs est.std    se      z pvalue ci.lower ci.upper
1 mse ~~ mse   1.000 0.000     NA     NA    1.000    1.000
2 hsl ~~ hsl   1.000 0.000     NA     NA    1.000    1.000
3  cc ~~  cc   1.000 0.000     NA     NA    1.000    1.000
4 mse ~~ hsl   0.469 0.045 10.359      0    0.380    0.557
5 mse ~~  cc   0.190 0.054  3.512      0    0.084    0.296
6 hsl ~~  cc   0.000 0.000     NA     NA    0.000    0.000
7 mse ~1       6.251 0.233 26.791      0    5.794    6.708
8 hsl ~1       3.719 0.156 23.886      0    3.414    4.024
9  cc ~1       1.750 0.078 22.328      0    1.597    1.904
output_mod2 <- standardizedsolution(model02.fit) |> filter(op == "~~", lhs != rhs)
output_mod2
  lhs op rhs est.std    se      z pvalue ci.lower ci.upper
1 mse ~~ hsl   0.469 0.045 10.359      0    0.380    0.557
2 mse ~~  cc   0.190 0.054  3.512      0    0.084    0.296
3 hsl ~~  cc   0.000 0.000     NA     NA    0.000    0.000

  • For Mathematics Self-Efficacy (MSE) and High School Math Experience (HSL), there is a moderate but significantly positive correlation with r = 0.469 (SE = 0.045), p < .001

  • For Mathematics Self-Efficacy (MSE) and College Math Experience (CC), there is a strong and significantly positive correlation with r = 0.19 (SE = 0.054), p < .001

  • For High School Math Experience (HSL) and College Math Experience (CC), there is a weak and significantly positive correlation with r = 0 (SE = 0), p = NA

Homework 2B: AIC and BIC

anova_output <- anova(model01.fit, model02.fit)
anova_output

Scaled Chi-Squared Difference Test (method = "satorra.bentler.2001")

lavaan->lavTestLRT():  
   lavaan NOTE: The "Chisq" column contains standard test statistics, not the 
   robust test that should be reported per model. A robust difference test is 
   a function of two standard (not robust) statistics.
            Df    AIC    BIC  Chisq Chisq diff Df diff Pr(>Chisq)   
model01.fit  0 5445.7 5480.4 0.0000                                 
model02.fit  1 5450.8 5481.7 7.1812     7.8104       1   0.005195 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

  • For Model 1, AIC = 5445.7 and BIC = 5480.4

  • For Model 2, AIC = 5450.8 and BIC = 5481.7

  • According to the criteria of lower AIC/BIC values indicating better model fit and the result of Likelihood ratio test (\chi^2(0, 1)= 7.8104, p < .001), we prefer model 1 as it has significantly better model fit than model 2.

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