MCStd Function Use Case 5: Difference of R-Squared in Multiple Groups

The MCStd() function is used to generate Monte Carlo confidence intervals for differences between \(R^{2}\) in multiple groups.

library(semmcci)
library(lavaan)

Data

In this example, we use data from Kwan and 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 (\(X_{3}\))

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

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)

varnames <- c("Y1", "X1", "X2", "X3")
nobs_hongkong <- 4625
covs_hongkong <- matrix(
  data = c(
    8176.0021, 27.3990, 28.2320, 31.2722,
    27.3990, 0.9451, 0.6006, 0.4326,
    28.2320, 0.6006, 0.7977, 0.3779,
    31.2722, 0.4326, 0.3779, 0.8956
  ),
  nrow = 4
)
colnames(covs_hongkong) <- rownames(covs_hongkong) <- varnames
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_japan <- 5943
covs_japan <- matrix(
  data = c(
    9666.8658, 34.2501, 35.2189, 30.6472,
    34.2501, 1.0453, 0.6926, 0.5027,
    35.2189, 0.6926, 1.0777, 0.4524,
    30.6472, 0.5027, 0.4524, 0.9583
  ),
  nrow = 4
)
colnames(covs_japan) <- rownames(covs_japan) <- varnames
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_korea <- 5151
covs_korea <- matrix(
  data = c(
    8187.6921, 31.6266, 37.3062, 30.9021,
    31.6266, 0.9271, 0.6338, 0.4088,
    37.3062, 0.6338, 1.0007, 0.3902,
    30.9021, 0.4088, 0.3902, 0.8031
  ),
  nrow = 4
)
colnames(covs_korea) <- rownames(covs_korea) <- varnames
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

Model Specification

We regress Y1 on X1, X2, and X3. We label the error variance \(\zeta_{1}\) for the three groups as psi1.g1, psi1.g2, and psi1.g3. \(R^{2}\) is defined using the := operator in the lavaan model syntax using the following equation

\[\begin{equation} R^{2} = 1 - \psi^{\ast} \end{equation}\]

where \(\psi^{\ast}\) is the standardized error variance.

model <- "
  Y1 ~ X1 + X2 + X3
  Y1 ~~ c(psi1.g1, psi1.g2, psi1.g3) * Y1
  rsq.g1 := 1 - psi1.g1
  rsq.g2 := 1 - psi1.g2
  rsq.g3 := 1 - psi1.g3
  rsq.g12 := rsq.g1 - rsq.g2
  rsq.g13 := rsq.g1 - rsq.g3
  rsq.g23 := rsq.g2 - rsq.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 and 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.

Note: The parameterization of \(R^{2}\) and above should only be interpreted using the output of the MCStd() function since the input in the functions defined by := require standardized estimates.

unstd <- MC(fit, R = 20000L, alpha = 0.05)
MCStd(unstd)
#> Standardized Monte Carlo Confidence Intervals
#>               est     se     R    2.5%   97.5%
#> Y1~X1      0.0568 0.0190 20000  0.0190  0.0941
#> Y1~X2      0.1985 0.0187 20000  0.1616  0.2351
#> Y1~X3      0.2500 0.0150 20000  0.2207  0.2790
#> psi1.g1    0.8215 0.0103 20000  0.8008  0.8411
#> X1~~X1     1.0000 0.0000 20000  1.0000  1.0000
#> X1~~X2     0.6917 0.0077 20000  0.6764  0.7064
#> X1~~X3     0.4702 0.0115 20000  0.4475  0.4925
#> X2~~X2     1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3     0.4471 0.0118 20000  0.4238  0.4698
#> X3~~X3     1.0000 0.0000 20000  1.0000  1.0000
#> Y1~X1.g2   0.1390 0.0164 20000  0.1071  0.1709
#> Y1~X2.g2   0.1792 0.0159 20000  0.1480  0.2100
#> Y1~X3.g2   0.1688 0.0138 20000  0.1416  0.1961
#> psi1.g2    0.8371 0.0087 20000  0.8193  0.8534
#> X1~~X1.g2  1.0000 0.0000 20000  1.0000  1.0000
#> X1~~X2.g2  0.6525 0.0074 20000  0.6379  0.6669
#> X1~~X3.g2  0.5023 0.0097 20000  0.4832  0.5211
#> X2~~X2.g2  1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3.g2  0.4452 0.0103 20000  0.4245  0.4651
#> X3~~X3.g2  1.0000 0.0000 20000  1.0000  1.0000
#> Y1~X1.g3   0.0863 0.0170 20000  0.0527  0.1194
#> Y1~X2.g3   0.2557 0.0164 20000  0.2235  0.2877
#> Y1~X3.g3   0.2289 0.0139 20000  0.2016  0.2564
#> psi1.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  0.6734
#> X1~~X3.g3  0.4738 0.0108 20000  0.4526  0.4947
#> X2~~X2.g3  1.0000 0.0000 20000  1.0000  1.0000
#> X2~~X3.g3  0.4353 0.0113 20000  0.4132  0.4570
#> X3~~X3.g3  1.0000 0.0000 20000  1.0000  1.0000
#> rsq.g1     0.1785 0.0103 20000  0.1589  0.1992
#> rsq.g2     0.1629 0.0087 20000  0.1466  0.1807
#> rsq.g3     0.2239 0.0103 20000  0.2041  0.2444
#> rsq.g12    0.0155 0.0135 20000 -0.0110  0.0418
#> rsq.g13   -0.0455 0.0146 20000 -0.0742 -0.0169
#> rsq.g23   -0.0610 0.0135 20000 -0.0873 -0.0344

References

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

Ivan Jacob Agaloos Pesigan

2023-03-12

Data

Model Specification

Model Fitting

Standardized Monte Carlo Confidence Intervals

References