MCStd Function Use Case 4: Difference of Standardized Regression Coefficients in Multiple Groups

Ivan Jacob Agaloos Pesigan

2024-10-22

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

library(semmcci)
library(lavaan)

Data

In this example, we use data from Kwan & Chan (2014) with three groups (Hong Kong, Japan, and Korea) 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)

knitr::kable(
  x = covs_hongkong, digits = 4,
  caption = "Covariance Matrix for Hong Kong"
)

Covariance Matrix for Hong Kong

	Y1	X1	X2	X3
Y1	8176.0021	27.3990	28.2320	31.2722
X1	27.3990	0.9451	0.6006	0.4326
X2	28.2320	0.6006	0.7977	0.3779
X3	31.2722	0.4326	0.3779	0.8956

nobs_hongkong
#> [1] 4625

knitr::kable(
  x = covs_japan, digits = 4,
  caption = "Covariance Matrix for Japan"
)

Covariance Matrix for Japan

	Y1	X1	X2	X3
Y1	9666.8658	34.2501	35.2189	30.6472
X1	34.2501	1.0453	0.6926	0.5027
X2	35.2189	0.6926	1.0777	0.4524
X3	30.6472	0.5027	0.4524	0.9583

nobs_japan
#> [1] 5943

knitr::kable(
  x = covs_korea, digits = 4,
  caption = "Covariance Matrix for Korea"
)

Covariance Matrix for Korea

	Y1	X1	X2	X3
Y1	8187.6921	31.6266	37.3062	30.9021
X1	31.6266	0.9271	0.6338	0.4088
X2	37.3062	0.6338	1.0007	0.3902
X3	30.9021	0.4088	0.3902	0.8031

nobs_korea
#> [1] 5151

Model Specification

We regress Y1 on X1, X2, and X3. We label the regression coefficient $\gamma_{1}$ for the three groups as gamma1.g1, gamma1.g2, and gamma1.g3, $\gamma_{2}$ for the three groups as gamma2.g1, gamma2.g2, and gamma2.g3, and $\gamma_{3}$ for the three groups as gamma3.g1, gamma3.g2, and gamma3.g3.

model <- "
  Y1 ~ c(gamma1.g1, gamma1.g2, gamma1.g3) * X1
  Y1 ~ c(gamma2.g1, gamma2.g2, gamma2.g3) * X2
  Y1 ~ c(gamma3.g1, gamma3.g2, gamma3.g3) * X3
  gamma1.g12 := gamma1.g1 - gamma1.g2
  gamma1.g13 := gamma1.g1 - gamma1.g3
  gamma1.g23 := gamma1.g2 - gamma1.g3
  gamma2.g12 := gamma2.g1 - gamma2.g2
  gamma2.g13 := gamma2.g1 - gamma2.g3
  gamma2.g23 := gamma2.g2 - gamma2.g3
  gamma3.g12 := gamma3.g1 - gamma3.g2
  gamma3.g13 := gamma3.g1 - gamma3.g3
  gamma3.g23 := gamma3.g2 - gamma3.g3
"

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 = list(covs_hongkong, covs_japan, covs_korea),
  sample.nobs = list(nobs_hongkong, nobs_japan, nobs_korea)
)

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.g1   0.0568 0.0191 20000  0.0194  0.0943
#> gamma2.g1   0.1985 0.0186 20000  0.1615  0.2353
#> gamma3.g1   0.2500 0.0150 20000  0.2207  0.2793
#> Y1~~Y1      0.8215 0.0103 20000  0.8005  0.8409
#> X1~~X1      1.0000 0.0000 20000  1.0000  1.0000
#> X1~~X2      0.6917 0.0076 20000  0.6765  0.7064
#> X1~~X3      0.4702 0.0114 20000  0.4478  0.4924
#> X2~~X2      1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3      0.4471 0.0118 20000  0.4239  0.4699
#> X3~~X3      1.0000 0.0000 20000  1.0000  1.0000
#> gamma1.g2   0.1390 0.0164 20000  0.1068  0.1712
#> gamma2.g2   0.1792 0.0158 20000  0.1480  0.2098
#> gamma3.g2   0.1688 0.0139 20000  0.1416  0.1961
#> Y1~~Y1.g2   0.8371 0.0088 20000  0.8196  0.8539
#> X1~~X1.g2   1.0000 0.0000 20000  1.0000  1.0000
#> X1~~X2.g2   0.6525 0.0075 20000  0.6374  0.6669
#> X1~~X3.g2   0.5023 0.0096 20000  0.4833  0.5209
#> X2~~X2.g2   1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3.g2   0.4452 0.0103 20000  0.4249  0.4650
#> X3~~X3.g2   1.0000 0.0000 20000  1.0000  1.0000
#> gamma1.g3   0.0863 0.0170 20000  0.0526  0.1193
#> gamma2.g3   0.2557 0.0164 20000  0.2235  0.2878
#> gamma3.g3   0.2289 0.0139 20000  0.2016  0.2565
#> Y1~~Y1.g3   0.7761 0.0103 20000  0.7556  0.7959
#> X1~~X1.g3   1.0000 0.0000 20000  1.0000  1.0000
#> X1~~X2.g3   0.6580 0.0079 20000  0.6422  0.6731
#> X1~~X3.g3   0.4738 0.0108 20000  0.4527  0.4948
#> X2~~X2.g3   1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3.g3   0.4353 0.0113 20000  0.4131  0.4570
#> X3~~X3.g3   1.0000 0.0000 20000  1.0000  1.0000
#> gamma1.g12 -0.0821 0.0251 20000 -0.1311 -0.0336
#> gamma1.g13 -0.0294 0.0257 20000 -0.0801  0.0208
#> gamma1.g23  0.0527 0.0236 20000  0.0070  0.0993
#> gamma2.g12  0.0193 0.0244 20000 -0.0288  0.0674
#> gamma2.g13 -0.0572 0.0248 20000 -0.1056 -0.0080
#> gamma2.g23 -0.0765 0.0228 20000 -0.1214 -0.0318
#> gamma3.g12  0.0811 0.0204 20000  0.0413  0.1208
#> gamma3.g13  0.0211 0.0204 20000 -0.0188  0.0613
#> gamma3.g23 -0.0601 0.0196 20000 -0.0987 -0.0216

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

Kwan, J. L. Y., & Chan, W. (2014). Comparing squared multiple correlation coefficients using structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 21(2), 225–238. https://doi.org/10.1080/10705511.2014.882673

Pesigan, I. J. A., & Cheung, S. F. (2023). Monte Carlo confidence intervals for the indirect effect with missing data. Behavior Research Methods, 56(3), 1678–1696. https://doi.org/10.3758/s13428-023-02114-4