Skip to main content
SpringerLink
Log in

Empirical Standard Errors for Longitudinal Data Mixed Linear Models

  • Published:
Computational Statistics Aims and scope Submit manuscript

Summary

In longitudinal data analysis, as with cross-sectional regression, correction for heteroscedasticity is important for accurate hypothesis tests and confidence intervals. Using empirical standard errors to adjust for heteroscedasticity is common practice. In this article, we investigate alternative empirical standard errors that have better finite sample properties than those commonly used. We review the properties of an estimator that is based on the deletion, or jackknife, principle. We also consider an estimator based on standardized residuals that is unbiased in many circumstances. Our Monte Carlo simulations show that both alternatives have better small-sample properties than the usual empirical standard errors, regardless as to whether heteroscedasticity is present. Both alternatives behave well in terms of size and power even when the covariance matrix is completely misspecified. We recommend the use of the alternative empirical standard errors whenever there is a reason to suspect that heteroscedasticity is present.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4

Similar content being viewed by others

References

  • Bannerjee, M. and E.W. Frees (1997). Influence diagnostics for longitudinal models. Journal of the American Statistical Association 92, 999–1005.

    Article  MathSciNet  Google Scholar 

  • Carroll, R. J. and Ruppert, D. (1988). Transformation and weighting in regression, Chapman-Hall.

  • Eicker, F. (1967). Limit theorems for regressions with unequal and dependent errors. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1, LeCam, L. M. and Neyman, J. editors, University of California Press, pp, 59–82.

  • Feng, Z., McLerran, D. and J. Grizzle (1996). A comparison of statistical methods for clustered data analysis with Gaussian error. Statistics in Medicine 15, 1793–1806.

    Article  Google Scholar 

  • Gourieroux, C., Monfort, A., and Trognon, A. (1984). Pseudo-maximum likelihood methods: theory. Econometrica, 52, 681–700.

    Article  MathSciNet  Google Scholar 

  • Huber, P. J. (1967). The behaviour of maximum likelihood estimators under non-standard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1, LeCam, L.M. and Neyman, J. editors, University of California Press, pp, 221–33.

  • Kauermann, G. and R. J. Carroll (2001). A note on the efficiency of sandwich covariance matrix estimation. Journal of the American Statistical Association 96, 1387–1396.

    Article  MathSciNet  Google Scholar 

  • Liang, K.-Y. and S.L. Zeger (1986). Longitudinal data analysis using generalized linear models. Biometrika 73, 12–22.

    Article  MathSciNet  Google Scholar 

  • Long, J.S. and L.H. Ervin (2000). Using heteroscedasticity consistent standard errors in the linear regression model. American Statistician 54, 217–224.

    Google Scholar 

  • MacKinnon, J.G. and H. White (1985). Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics 29, 53–57.

    Article  Google Scholar 

  • Manel, L.A. and T.A. DeRouen (2001). A covariance estimator for GEE with improved small-sample properties. Biometrics 57, 126–134.

    Article  MathSciNet  Google Scholar 

  • Pan, W. (2001). On the robust variance estimator in generalized estimating equations. Biometrica 88, 901–906.

    Article  Google Scholar 

  • White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817–38.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank two anonymous reviewers for comments on the paper. The Fortis Health Insurance Professorship and the National Science Foundation Grant Number SES- 0095343 provided funding to support the research of Edward Frees.

Author information

Authors and Affiliations

  1. School of Business, University of Wisconsin-Madison, 975 University Ave, Madison, WI, 53706, USA

    Edward W. Frees

  2. Department of Statistics, University of Wisconsin-Madison, 1210 West Dayton Ave, Madison, WI, 53706, USA

    Chunfang Jin

Authors
  1. Edward W. Frees
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunfang Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Appendix

Appendix

From a standard result on the inverse of partitioned matrices, we have

$$(\text{M}-\text{X}^\prime_i\text{W}_i\text{X}_i)^{-1}=\text{M}^{-1}+\text{M}^{-1}\text{X}^\prime_i\text{W}_i(\text{I}_i-\text{H}_{ii})^{-1}\text{X}_i\text{M}^{-1}$$

With this and the definition of the delete i WLS estimator, we have

$$\text{b}_{(i),WLS}=(\text{M}-\text{X}^\prime_i\text{W}_i\text{X}_i)^{-1}\sum_{k\neq{1}}\text{X}^\prime_k\text{W}_k\text{y}_k\\=\text{M}^{-1}\left(\sum_{k\neq{1}}\text{X}^\prime_k\text{W}_k\text{y}_k\right)+\text{M}^{-1}\text{X}^\prime_i\text{W}_i(\text{I}_i-\text{H}_{ii})^{-1}\text{X}_i\text{M}^{-1}\left(\sum_{k\neq{1}}\text{X}^\prime_k\text{W}_k\text{y}_k\right)\\=\text{b}_{WLS}-\text{M}^{-1}\text{X}^\prime_i\text{W}_i\text{y}_i+\text{M}^{-1}\text{X}^\prime_i\text{W}_i(\text{I}_i-\text{H}_{ii})^{-1}\text{X}_i\left(\text{b}_{WLS}-\text{M}^{-1}\text{X}^\prime_i\text{W}_i\text{y}_i\right)\\=\text{b}_{WLS}-\text{M}^{-1}\text{X}^\prime_i\text{W}_i\text{y}_i+\text{M}^{-1}\text{X}^\prime_i\text{W}_i(\text{I}_i-\text{H}_{ii})^{-1}\left(\text{X}_i\text{b}_{WLS}-\text{H}_{ii}\text{y}_i\right)\\=\text{b}_{WLS}+\text{M}^{-1}\text{X}^\prime_i\text{W}_i\left(\text{I}_i-\text{H}_{ii}\right)^{-1}\left(\left(\text{X}_i\text{b}_{WLS}-\text{H}_{ii}\text{y}_i\right)-(\text{I}_i-\text{H}_{ii})\text{y}_i\right)\\=\text{b}_{WLS}+\text{M}^{-1}\text{X}^\prime_i\text{W}_i(\text{I}_i-\text{H}_{ii})^{-1}\left(\text{X}_i\text{b}_{WLS}-\text{y}_{i}\right)\\=\text{b}_{WLS}-\text{M}^{-1}\text{X}^\prime_i\text{W}_i\left(\text{I}_i-\text{H}_{ii}\right)^{-1}\text{e}_i.$$

This establishes equation (4).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frees, E.W., Jin, C. Empirical Standard Errors for Longitudinal Data Mixed Linear Models. CompStat 19, 455–475 (2004). https://doi.org/10.1007/BF03372107

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03372107

Keywords

Search

Navigation