coeff.robust: Heteroscedasticity-Consistent and Cluster-Robust Standard...

In misty: Miscellaneous Functions 'T. Yanagida'

Heteroscedasticity-Consistent and Cluster-Robust Standard Errors

Description

This function computes (1) heteroscedasticity-consistent or cluster-robust standard errors standard errors and significance values for (generalized) linear models estimated by using the lm() or the glm() function and (2) cluster-robust standard errors for multilevel and linear mixed-effects models estimated by using the lmer() function from the lme4 package that are robust to the violation of the homoscedasticity assumption. For linear models the heteroscedasticity-robust F-test is computed as well. By default, the function uses the HC4 estimator for (generalized) linear models and the heteroscedastic-robust CR2 estimator for multilevel and linear mixed-effects models. Note that cluster-robust standard errors are available only for two-level models.

Usage

coeff.robust(model, cluster = NULL,
             type = c("HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5",
                      "CR0", "CR1", "CR1p", "CR1S", "CR2", "CR3"),
             digits = 2, p.digits = 3, write = NULL, append = TRUE,
             check = TRUE, output = TRUE)

Arguments

model	a fitted model of class lm, glm, lmerMod, or lmerModLmerTest.
cluster	a vector representing the nested grouping structure (i.e., group or cluster variable). This argument is used only when requesting cluster-robust standard errors for (generalized) linear models estimated by using the lm() or glm() function. Note that the length of the vector needs to match the number of rows of the data frame used to estimate the (generalized) linear model. In the presence of missing data, the length of the vector needs to match the number of rows of the data frame after listwise deletion of missing data.
type	a character string specifying the estimation type for (generalized) linear models estimated by using the lm() or glm() function, where "H0" gives White's estimator and "H1" to "H5" are refinement of this estimator. See help page of the vcovHC() function in the R package sandwich for more details. Alternatively, a character string specifying the estimation type for multilevel and linear mixed-effects model estimated by using the lmer() function from the lme4 package, where "CR0" is the original form of the sandwich estimator (Liang & Zeger, 1986), "CR1" multiplies CR0 by m / (m - 1), where m is the number of clusters, "CR1p" multiplies CR0 by m / (m - p), where p is the number of covariates, "CR1S" multiplies CR0 by (m (N-1)) / [(m - 1)(N - p)], where N is the total number of observations, "CR2" (default) is the "bias-reduced linearization" adjustment proposed by Bell and McCaffrey (2002) and further developed in Pustejovsky and Tipton (2017), and "CR3" approximates the leave-one-cluster-out jackknife variance estimator (Bell & McCaffrey, 2002). See help page of the vcovCR() function in the R package clubSandwich for more details.
digits	an integer value indicating the number of decimal places to be used for displaying results. Note that information criteria and chi-square test statistic are printed with digits minus 1 decimal places.
p.digits	an integer value indicating the number of decimal places
write	a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append	logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check	logical: if TRUE (default), argument specification is checked.
output	logical: if TRUE (default), output is shown.

Details

Heteroscedasticity-Consistent Standard Errors

The family of heteroscedasticity-consistent (HC) standard errors estimator for the model parameters of a regression model is based on an HC covariance matrix of the parameter estimates and does not require the assumption of homoscedasticity. HC estimators approach the correct value with increasing sample size, even in the presence of heteroscedasticity. On the other hand, the OLS standard error estimator is biased and does not converge to the proper value when the assumption of homoscedasticity is violated (Darlington & Hayes, 2017). White (1980) introduced the idea of HC covariance matrix to econometricians and derived the asymptotically justified form of the HC covariance matrix known as HC0 (Long & Ervin, 2000). Simulation studies have shown that the HC0 estimator tends to underestimate the true variance in small to moderately large samples (N \leq 250) and in the presence of leverage observations, which leads to an inflated type I error risk (e.g., Cribari-Neto & Lima, 2014). The alternative estimators HC1 to HC5 are asymptotically equivalent to HC0 but include finite-sample corrections, which results in superior small sample properties compared to the HC0 estimator. Long and Ervin (2000) recommended routinely using the HC3 estimator regardless of a heteroscedasticity test. However, the HC3 estimator can be unreliable when the data contains leverage observations. The HC4 estimator, on the other hand, performs well with small samples, in the presence of high leverage observations, and when errors are not normally distributed (Cribari-Neto, 2004). In summary, it appears that the HC4 estimator performs the best in terms of controlling the type I and type II error risk (Rosopa, 2013). As opposed to the findings of Cribari-Neto et al. (2007), the HC5 estimator did not show any substantial advantages over HC4. Both HC5 and HC4 performed similarly across all the simulation conditions considered in the study (Ng & Wilcox, 2009). Note that the F-test of significance on the multiple correlation coefficient R also assumes homoscedasticity of the errors. Violations of this assumption can result in a hypothesis test that is either liberal or conservative, depending on the form and severity of the heteroscedasticity. Hayes (2007) argued that using a HC estimator instead of assuming homoscedasticity provides researchers with more confidence in the validity and statistical power of inferential tests in regression analysis. Hence, the HC3 or HC4 estimator should be used routinely when estimating regression models. If a HC estimator is not used as the default method of standard error estimation, researchers are advised to at least double-check the results by using an HC estimator to ensure that conclusions are not compromised by heteroscedasticity. However, the presence of heteroscedasticity suggests that the data is not adequately explained by the statistical model of estimated conditional means. Unless heteroscedasticity is believed to be solely caused by measurement error associated with the predictor variable(s), it should serve as warning to the researcher regarding the adequacy of the estimated model.

Cluster-Robust Standard Errors

The family of cluster-robust (CR) standard errors estimator for the model parameters of a multilevel and linear mixed-effects model are based on the heteroscedasticity-consistent (HC) standard errors estimators that have been generalized to clustered data (Zhang & Lai, 2024). The standard errors of the CR0 estimator (Liang and Zeger, 1986) rely on large samples, i.e., the CR0 estimator may result in underestimated standard errors with small number of clusters (Cameron & Miller, 2015; Imbens & Kolesar, 2016). However, there is no consensus about the minimum number of clusters, e.g., at least 100 clusters (Maas & Hox, 2004, p. 439), around 40 (Angrist & Pischke, 2008) or 30 clusters (Huang, 2016). The CR2 estimator, also referred to as Bell and McCaffrey (2002) bias-reduced linearization method, has been shown to be effective when used with a small number of clusters (Hugang & Li, 2022). For example, the CR2 estimator performed well in all conditions of a simulation study involving 20, 50, or 100 clusters regardless if homoskedasticity was violated or not. (Huang, et al, 2023). The CR3 estimator tends to over-correct the bias of the CR0 estiamator, while the CR1 estimator tends to under-correct the bias (Pustejovsky & Tipton, 2018). Note that the cluster-robust SE are only robust to violation of the homoscedasticity assumption, while departure from normality or the presence of outliers can influence its performance (MacKinnon, 2012). Statistical significance testing of the regression coefficients is based on the Satterthwaite approximated degrees of freedom (Bell & McCaffrey (2002).

Value

Returns an object of class misty.object, which is a list with following entries:

call	function call
type	type of analysis
model	model specified in model
args	specification of function arguments
result	list with results, i.e., coef for the unstandardized regression coefficients with heteroscedasticity-consistent or cluster-robust standard errors, F.test for the heteroscedasticity-robust F-Test, and sandwich for the sandwich covariance matrix

Note

The computation of heteroscedasticity-consistent standard errors is based on the vcovHC function from the sandwich package (Zeileis, Köll, & Graham, 2020) and the functions coeftest and waldtest from the lmtest package (Zeileis & Hothorn, 2002), while the computation of cluster-robust standard errors uses the vcovCR and the coef_test function in the clubSandwich package.

Author(s)

Takuya Yanagida [e-mail]

References

Angrist, J. D., & Pischke, J.-S. (2008). Mostly harmless econometrics: An empiricist’s companion. Princeton university press. Bell, R. M., & McCaffrey, D. F. (2002). Bias reduction in standard errors for linear regression with multi-stage samples. Survey Methodology, 28(2), 169-181

Cameron, A. C., & Miller, D. L. (2015). A practitioner’s guide to cluster-robust inference. Journal of Human Resources, 50(2), 317-372. https://doi.org/10.3368/jhr.50.2.317

Darlington, R. B., & Hayes, A. F. (2017). Regression analysis and linear models: Concepts, applications, and implementation. The Guilford Press.

Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 45, 215-233. https://doi.org/10.1016/S0167-9473(02)00366-3

Cribari-Neto, F., & Lima, M. G. (2014). New heteroskedasticity-robust standard errors for the linear regression model. Brazilian Journal of Probability and Statistics, 28, 83-95.

Cribari-Neto, F., Souza, T., & Vasconcellos, K. L. P. (2007). Inference under heteroskedasticity and leveraged data. Communications in Statistics - Theory and Methods, 36, 1877-1888. https://doi.org/10.1080/03610920601126589

Hayes, A.F, & Cai, L. (2007). Using heteroscedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39, 709-722. https://doi.org/10.3758/BF03192961

Huang, F. L., & Li, X. (2022). Using cluster-robust standard errors when analyzing group-randomized trials with few clusters. Behavior Research Methods, 54(3), 1181–1199. https://doi.org/10.3758/s13428-021-01627-0

Kuznetsova, A, Brockhoff, P. B., & Christensen, R. H. B. (2017). lmerTest Package: Tests in linear mixed effects models. Journal of Statistical Software, 82 13, 1-26. https://doi.org/10.18637/jss.v082.i13.

Imbens, G. W., & Kolesar, M. (2016). Robust standard errors in small samples: Some practical advice. Review of Economics and Statistics, 98(4), 701-712. https://doi.org/10.1162/REST_a_00552

Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1), 13-22. https://doi.org/10.1093/biomet/73.1.13

Long, J.S., & Ervin, L.H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54, 217-224. https://doi.org/10.1080/00031305.2000.10474549

Maas, C., & Hox, J. J. (2004). The influence of violations of assumptions on multilevel parameter estimates and their standard errors. Computational Statistics & Data Analysis, 46(3), 427-440. https://doi.org/10.1016/j.csda.2003.08.006

MacKinnon, J. G. (2012). Thirty years of heteroskedasticity-robust inference. In X. Chen & N. R. Swanson (Eds.), Recent advances and future directions in causality, prediction, and specification analysis: Essays in honor of Halbert L. White Jr (pp. 437-461). Springer. https://doi.org/10.1007/978-1-4614-1653-1_17

Ng, M., & Wilcoy, R. R. (2009). Level robust methods based on the least squares regression estimator. Journal of Modern Applied Statistical Methods, 8, 284-395. https://doi.org/10.22237/jmasm/1257033840

Pustejovsky, J. E. & Tipton, E. (2018). Small sample methods for cluster-robust variance estimation and hypothesis testing in fixed effects models. Journal of Business and Economic Statistics, 36(4), 672-683. https://doi.org/10.1080/07350015.2016.1247004

Rosopa, P. J., Schaffer, M. M., & Schroeder, A. N. (2013). Managing heteroscedasticity in general linear models. Psychological Methods, 18(3), 335-351. https://doi.org/10.1037/a0032553

White, H. (1980). A heteroskedastic-consistent covariance matrix estimator and a direct test of heteroskedasticity. Econometrica, 48, 817-838. https://doi.org/10.2307/1912934

Zeileis, A., & Hothorn, T. (2002). Diagnostic checking in regression relationships. R News, 2(3), 7-10. http://CRAN.R-project.org/doc/Rnews/

Zeileis A, Köll S, & Graham N (2020). Various versatile variances: An object-oriented implementation of clustered covariances in R. Journal of Statistical Software, 95(1), 1-36. https://doi.org/10.18637/jss.v095.i01

Zhang, Y., & Lai, M. H. C. (2024). Evaluating two small-sample corrections for fixed-effects standard errors and inferences in multilevel models with heteroscedastic, unbalanced, clustered data. Behavior research methods, 56(6), 5930–5946. https://doi.org/10.3758/s13428-023-02325-9

See Also

coeff.std, write.result

Examples

#----------------------------------------------------------------------------
# Example 1: Linear model

mod.lm <- lm(mpg ~ cyl + disp, data = mtcars)
coeff.robust(mod.lm)

#----------------------------------------------------------------------------
# Example 2: Generalized linear model

mod.glm <- glm(carb ~ cyl + disp, data = mtcars, family = poisson())
coeff.robust(mod.glm)

## Not run: 
#----------------------------------------------------------------------------
# Example 3: Multilevel and Linear Mixed-Effects Model

# Load lme4 and misty package
misty::libraries(lme4, misty)

# Load data set "Demo.twolevel" in the lavaan package
data("Demo.twolevel", package = "lavaan")

# Cluster-mean centering, center() from the misty package
Demo.twolevel <- center(Demo.twolevel, x2, type = "CWC", cluster = "cluster")

# Grand-mean centering, center() from the misty package
Demo.twolevel <- center(Demo.twolevel, w1, type = "CGM", cluster = "cluster")

# Estimate two-level mixed-effects model
mod.lmer <- lmer(y1 ~ x2.c + w1.c + x2.c:w1.c + (1 + x2.c | cluster), data = Demo.twolevel)

# Statistical significance testing based on cluster-robust standard errors
coeff.robust(mod.lmer)

#----------------------------------------------------------------------------
# Write Results

# Example 3a: Write results into a text file
coeff.robust(mod.lm, write = "Robust_Coef.txt", output = FALSE)

# Example 3b: Write results into a Excel file
coeff.robust(mod.lm, write = "Robust_Coef.xlsx", output = FALSE)

## End(Not run)

misty documentation built on Sept. 9, 2025, 5:23 p.m.