Comparison of naïve, Kenward–Roger, and parametric bootstrap interval approaches to small-sample inference in linear mixed models

References

• Aitken, A. C. (1935). On least squares and linear combination of observations. Proceedings of the Royal Society of Edinburgh 55:42–48.
   PubMedGoogle Scholar
• Butar, F. B., Lahiri, P. (2003). On measures of uncertainty of empirical Bayes small-area estimators. Journal of Statistical Planning and Inference 112(1–2):63–76.
   Web of Science ®Google Scholar
• Catellier, D. J., Muller, K. E. (2000). Tests for Gaussian repeated measures with missing data in small samples. Statistics in Medicine 19:1101–1114.
   PubMed Web of Science ®Google Scholar
• Chernick, M. C. (2008). Bootstrap Methods: A Guide for Practitioners and Researchers. 2nd ed. Hoboken, NJ: Wiley.
   Google Scholar
• Costa, P. T., Jr., McCrae, R. R. (1992). Revised NEO Personality Inventory (NEO-PI-R) and NEO Five-Factor Inventory (NEO-FFI) Professional Manual. Odessa, FL: Psychological Assessment Resources.
   Google Scholar
• Das, K., Jiang, J. M., Rao, J. N. K. (2004). Mean squared error of empirical predictor. Annals of Statistics 32(2):818–840.
   Web of Science ®Google Scholar
• Datta, G. S., Lahiri, P. (2000). A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems. Statistica Sinica 10(2):613–627.
   Web of Science ®Google Scholar
• Datta, G. S., Rao, J. N. K., Smith, D. D. (2005). On measuring the variability of small area estimators under a basic area level model. Biometrika 92(1):183–196.
   Google Scholar
• Demidenko, E. (2004). Mixed Models: Theory and Applications. New Jersey: John Wiley & Sons, Inc.
   Google Scholar
• Efron, B. (1979). Another look at the jackknife. The Annals of Statistics 7(1):1–26.
   Web of Science ®Google Scholar
• Efron, B. (1981). Nonparametric standard errors and confidence intervals. The Canadian Journal of Statistics 9(2):139–158.
   Google Scholar
• Efron, B., Tibshirani, R. J. (1993). An Introduction to the Bootstrap. London: Chapman & Hall.
   Google Scholar
• Fai, A. H. T., Cornelius, P. L. (1996). Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments. Journal of Statistical Computation and Simulation 54(4):363–378.
   Web of Science ®Google Scholar
• Fisher, R. A. (1925). Statistical Methods for Research Workers. Edinburgh and London: Oliver & Boyd.
   Google Scholar
• Fuller, W. A., Harter, R. M. (1987). The multivariate components of variance model for small area estimation. In: Platek, R., Rao, J. N. K., Sarndal, C. E., Singh, M. P., eds. Small Area Statistics: An International Symposium. New York: Wiley, pp. 103–123.
   Google Scholar
• Giesbrecht, F. G., Burns, J. C. (1985). Two-stage analysis based on a mixed model: Large-sample asymptotic theory and small-sample simulation results. Biometrics 41(2):477–486.
   Web of Science ®Google Scholar
• Goldberger, A. S. (1962). Best linear unbiased prediction in generalized linear-regression model. Journal of the American Statistical Association 57(298):369–375.
   Web of Science ®Google Scholar
• Gomez, E. V., Schaalje G. B., Fellingham, G. W. (2005). Performance of the Kenward-Roger method when the covariance structure is selected using AIC and BIC. Communications in Statistics—Simulation and Computation 34:377–392.
   Web of Science ®Google Scholar
• Gonzalez-Manteiga, W., Lombardia, M. J., Molina, I., Morales, D., Santamaria, L. (2008). Bootstrap mean squared error of a small-area EBLUP. Journal of Statistical Computation and Simulation 78(5):443–462.
   Web of Science ®Google Scholar
• Halekoh, U., Højsgaard, S. (2012). A Kenward-Roger approximation and parametric bootstrap methods for tests in linear mixed models–the R package pbkrtest. Available at: http://people.math.aau.dk/∼sorenh/software/pbkrtest/doc/pbkrtest-JSS.pdf
   Google Scholar
• Hall, P. (1992). The Bootstrap and Edgeworth Expansion. New York: Springer.
   Google Scholar
• Hall, P., Maiti, T. (2006). On parametric bootstrap methods for small area prediction. Journal of the Royal Statistical Society Series B-Statistical Methodology 68:221–238.
   Web of Science ®Google Scholar
• Harville, D. A., Jeske, D. R. (1992). Mean squared error of estimation or prediction under a general linear model. Journal of the American Statistical Association 87(419):724–731.
   Web of Science ®Google Scholar
• Henderson, C. R. (1963). Selection index and expected genetic advance. In: Statistical Genetics and Plant Breeding. National Academy of Sciences, National Research Council, Pub. 982, Washington, DC: U. S. Government Printing Office, pp. 141–163.
   Google Scholar
• Hulting, F. L., Harville, D. A. (1991). Some Bayesian and non-Bayesian procedures for the analysis of comparative experiments and for small-area estimation: Computational aspects, frequentist properties, and relationships. Journal of the American Statistical Association 86(415):557–568.
   Web of Science ®Google Scholar
• Jiang, J. M., Lahiri, P., Wan, S. M. (2002). A unified jackknife theory for empirical best prediction with M-estimation. Annals of Statistics 30(6):1782–1810.
   Web of Science ®Google Scholar
• Kackar, R. N., Harville, D. A. (1981). Unbiasedness of two-stage estimation and prediction procedures for mixed linear-models. Communications in Statistics—Theory and Methods 10(13):1249–1261.
   Web of Science ®Google Scholar
• Kackar, R. N., Harville, D. A. (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear-models. Journal of the American Statistical Association 79(388):853–862.
   Web of Science ®Google Scholar
• Kenward, M. G., Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997.
   PubMed Web of Science ®Google Scholar
• Kenward, M. G., Roger, J. H. (2009). An improved approximation to the precision of fixed effects from restricted maximum likelihood. Computational Statistics & Data Analysis 53(7):2583–2595.
   Web of Science ®Google Scholar
• Lahiri, P., Rao, J. N. K. (1995). Robust estimation of mean squared error of small-area estimators. Journal of the American Statistical Association 90(430):758–766.
   Web of Science ®Google Scholar
• Laird, N. M., Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics 38(4):963–974.
   PubMed Web of Science ®Google Scholar
• Littell, R. C. (2002). Analysis of unbalanced mixed model data: A case study comparison of ANOVA versus REML/GLS. Journal of Agricultural, Biological, and Environmental Statistics 7(4):472–490.
   Web of Science ®Google Scholar
• Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D. (1996). SAS System for Mixed Models. Cary, NC: SAS Institute Inc.
   Google Scholar
• McLean, R. A., Sanders, W. L. (1988). Approximating degrees of freedom for standard errors in mixed linear models. In: Proceedings of the Statistical Computing Section, American Statistical Association, pp. 50–59.
   Google Scholar
• Pereira, L. N., Coelho, P. S. (2010). Assessing different uncertainty measures of EBLUP: A resampling-based approach. Journal of Statistical Computation and Simulation 80(7):713–727.
   Web of Science ®Google Scholar
• Pfefferman, D., Glickman, H. (2004). Mean square error approximation in small area estimation by use of parametric and nonparametric bootstrap. In: Proceedings of the Section on Survey Research Methods, American Statistical Association, pp. 4167–4178.
   Google Scholar
• Prasad, N. G. N., Rao, J. N. K. (1990). The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association 85(409):163–171.
   Web of Science ®Google Scholar
• SAS Institute Inc. (2014). SAS/STAT® 13.2 User's Guide. Cary, NC: SAS Institute Inc.
   Google Scholar
• Satterthwaite, F. E. (1941). Synthesis of variance. Psychometrika 6:309–316.
   Google Scholar
• Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin 2(6):110–114.
   Google Scholar
• Savin, A., Wimmer, G., Witkovský (2003). On Kenward-Roger confidence intervals for common mean in interlaboratory trials. Measurement Science Review 3(1):53–56.
   Google Scholar
• Schaalje, G. B., McBride, J. B., Fellingham, G.W. (2002). Adequacy of approximations to distributions of test statistics in complex mixed linear models. Journal of Agricultural, Biological & Environmental Statistics 7(4):512–524.
   Web of Science ®Google Scholar
• Searle, S. R., Casella, G., McCulloch, C. E. (1992). Variance Components. New York: Wiley.
   Google Scholar
• Skene, S. S., Kenward, M. G. (2010). The analysis of very small samples of repeated measurements I: An adjusted sandwich estimator. Statistics in medicine 29(27):2825–2837.
   PubMed Web of Science ®Google Scholar
• Terracciano, A., McCrae, R. R., Brant, L. J., Costa, P. T., Jr. (2005). Hierarchical linear modeling analyses of the NEO-PI-R scales in the Baltimore Longitudinal Study of Aging. Psychology and Aging 20(3):493–506.
   PubMed Web of Science ®Google Scholar
• Van der Leeden, R., Meijer, E., Busing, F. M. T. A. (2008). Resampling multilevel models. In: van der Leeden, R., Meijer, E., eds. Handbook of Multilevel Analysis. New York: Springer, pp. 401–433.
   Google Scholar
• Wang, J. Y., Fuller, W. A. (2003). The mean squared error of small area predictors constructed with estimated area variances. Journal of the American Statistical Association 98(463):716–723.
   Web of Science ®Google Scholar