Skip to contents

MCStd Function Use Case 2: Difference of Standardized Regression Coefficients

Ivan Jacob Agaloos Pesigan

2024-04-14

Source: vignettes/mcstd-2-difference-regression-coefficients.Rmd
mcstd-2-difference-regression-coefficients.Rmd

The MCStd() function is used to generate Monte Carlo confidence intervals for differences between standardized regression coefficients.

Data

In this example, we use data from Kwan & Chan (2011) where child’s reading ability (\(Y_{1}\)) is regressed on parental occupational status (\(X_{1}\)), parental educational level (\(X_{2}\)), and child’s home possession (\(X_{3}\))

\[ Y_{1} = \alpha_{1} + \gamma_{1} X_{1} + \gamma_{2} X_{2} + \gamma_{3} X_{3} + \zeta_{1} . \]

Note that \(\zeta_{1}\) is the stochastic error term with expected value of zero and finite variance \(\psi_{1}\), \(\alpha_{1}\) is the intercept, and \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\) are regression coefficients.

A Three-Regressor Multiple Regression Model (Covariance Structure)
A Three-Regressor Multiple Regression Model (Covariance Structure)
#>           Y1       X1      X2      X3
#> Y1 6088.8281 271.1429 49.5848 20.0337
#> X1  271.1429 226.2577 29.9232  4.8812
#> X2   49.5848  29.9232  9.0692  1.0312
#> X3   20.0337   4.8812  1.0312  0.8371
covs
#>           Y1       X1      X2      X3
#> Y1 6088.8281 271.1429 49.5848 20.0337
#> X1  271.1429 226.2577 29.9232  4.8812
#> X2   49.5848  29.9232  9.0692  1.0312
#> X3   20.0337   4.8812  1.0312  0.8371
nobs
#> [1] 200

Model Specification

We regress Y1 on X1, X2, and X3. We label the regression coefficients as gamma1, gamma2, and gamma3. We then specify the difference between the coefficients using the := operator.

model <- "
  Y1 ~ gamma1 * X1 + gamma2 * X2 + gamma3 * X3
  gamma12 := gamma1 - gamma2
  gamma13 := gamma1 - gamma3
  gamma23 := gamma2 - gamma3
"

Model Fitting

We can now fit the model using the sem() function from lavaan with mimic = "eqs" to ensure compatibility with results from Kwan & Chan (2011).

Note: We recommend setting fixed.x = FALSE when generating standardized estimates and confidence intervals to model the variances and covariances of the exogenous observed variables if they are assumed to be random. If fixed.x = TRUE, which is the default setting in lavaan, MC() will fix the variances and the covariances of the exogenous observed variables to the sample values.

fit <- sem(
  model = model, mimic = "eqs", fixed.x = FALSE,
  sample.cov = covs, sample.nobs = nobs
)

Standardized Monte Carlo Confidence Intervals

Standardized Monte Carlo Confidence intervals can be generated by passing the result of the MC() function to the MCStd() function.

unstd <- MC(fit, R = 20000L, alpha = 0.05)
MCStd(unstd, alpha = 0.05)
#> Standardized Monte Carlo Confidence Intervals
#>             est     se     R    2.5%  97.5%
#> gamma1   0.1207 0.0900 20000 -0.0589 0.2945
#> gamma2   0.0491 0.0913 20000 -0.1294 0.2286
#> gamma3   0.2194 0.0709 20000  0.0784 0.3562
#> Y1~~Y1   0.9002 0.0405 20000  0.8005 0.9579
#> X1~~X1   1.0000 0.0000 20000  1.0000 1.0000
#> X1~~X2   0.6606 0.0406 20000  0.5745 0.7348
#> X1~~X3   0.3547 0.0627 20000  0.2252 0.4715
#> X2~~X2   1.0000 0.0000 20000  1.0000 1.0000
#> X2~~X3   0.3743 0.0622 20000  0.2480 0.4901
#> X3~~X3   1.0000 0.0000 20000  1.0000 1.0000
#> gamma12  0.0716 0.1631 20000 -0.2469 0.3936
#> gamma13 -0.0987 0.1242 20000 -0.3412 0.1438
#> gamma23 -0.1703 0.1271 20000 -0.4168 0.0827

References

Kwan, J. L. Y., & Chan, W. (2011). Comparing standardized coefficients in structural equation modeling: A model reparameterization approach. Behavior Research Methods, 43(3), 730–745. https://doi.org/10.3758/s13428-011-0088-6
Pesigan, I. J. A., & Cheung, S. F. (2023). Monte Carlo confidence intervals for the indirect effect with missing data. Behavior Research Methods. https://doi.org/10.3758/s13428-023-02114-4