The Annals of Statistics

Generalized fiducial inference for normal linear mixed models

Abstract

While linear mixed modeling methods are foundational concepts introduced in any statistical education, adequate general methods for interval estimation involving models with more than a few variance components are lacking, especially in the unbalanced setting. Generalized fiducial inference provides a possible framework that accommodates this absence of methodology. Under the fabric of generalized fiducial inference along with sequential Monte Carlo methods, we present an approach for interval estimation for both balanced and unbalanced Gaussian linear mixed models. We compare the proposed method to classical and Bayesian results in the literature in a simulation study of two-fold nested models and two-factor crossed designs with an interaction term. The proposed method is found to be competitive or better when evaluated based on frequentist criteria of empirical coverage and average length of confidence intervals for small sample sizes. A MATLAB implementation of the proposed algorithm is available from the authors.

Article information

Source
Ann. Statist., Volume 40, Number 4 (2012), 2102-2127.

Dates
First available in Project Euclid: 30 October 2012

https://projecteuclid.org/euclid.aos/1351602538

Digital Object Identifier
doi:10.1214/12-AOS1030

Mathematical Reviews number (MathSciNet)
MR3059078

Zentralblatt MATH identifier
1257.62075

Citation

Cisewski, Jessi; Hannig, Jan. Generalized fiducial inference for normal linear mixed models. Ann. Statist. 40 (2012), no. 4, 2102--2127. doi:10.1214/12-AOS1030. https://projecteuclid.org/euclid.aos/1351602538

References

• Burch, B. D. (2011). Assessing the performance of normal-based and REML-based confidence intervals for the intraclass correlation coefficient. Comput. Statist. Data Anal. 55 1018–1028.
• Burch, B. D. and Iyer, H. K. (1997). Exact confidence intervals for a variance ratio (or heritability) in a mixed linear model. Biometrics 53 1318–1333.
• Burdick, R. K. and Graybill, F. A. (1992). Confidence Intervals on Variance Components. Statistics: Textbooks and Monographs 127. Dekker, New York.
• Casella, G. and Berger, R. L. (2002). Statistical Inference, 2nd ed. Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, CA.
• Chopin, N. (2002). A sequential particle filter method for static models. Biometrika 89 539–551.
• Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.
• Cisewski, J. and Hannig, J. (2012). Supplement to “Generalized fiducial inference for normal linear mixed models.” DOI:10.1214/12-AOS1030SUPP.
• Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411–436.
• Diaconis, P. and Freedman, D. (1986a). On inconsistent Bayes estimates of location. Ann. Statist. 14 68–87.
• Diaconis, P. and Freedman, D. (1986b). On the consistency of Bayes estimates (with discussion). Ann. Statist. 14 1–67.
• Douc, R., Cappé, O. and Moulines, E. (2005). Comparison of resampling schemes for particle filtering. In 4th International Symposium on Image and Signal Processing and Analysis 64–69.
• Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 2344–2376.
• Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
• E, L., Hannig, J. and Iyer, H. (2008). Fiducial intervals for variance components in an unbalanced two-component normal mixed linear model. J. Amer. Statist. Assoc. 103 854–865.
• Elster, C. (2000). Evaluation of measurement uncertainty in the presence of combined random and analogue-to-digital conversion errors. Measurement Science and Technology 11 1359–1363.
• Fisher, R. A. (1930). Inverse probability. Math. Proc. Cambridge Philos. Soc. xxvi 528–535.
• Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to unknown parameters. Proc. R. Soc. Lond. Ser. A 139 343–348.
• Fisher, R. A. (1935). The fiducial argument in statistical inference. Annals of Eugenics VI 91–98.
• Fraser, D. A. S. (1961a). The fiducial method and invariance. Biometrika 48 261–280.
• Fraser, D. A. S. (1961b). On fiducial inference. Ann. Math. Statist. 32 661–676.
• Fraser, D. A. S. (1966). Structural probability and a generalization. Biometrika 53 1–9.
• Fraser, D. A. S. (1968). The Structure of Inference. Wiley, New York.
• Frenkel, R. B. and Kirkup, L. (2005). Monte Carlo-based estimation of uncertainty owing to limited resolution of digital instruments. Metrologia 42 L27–L30.
• Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal. 1 515–533 (electronic).
• Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian Data Analysis, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL.
• Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500–531.
• GUM (1995). Guide to the Expression of Uncertainty in Measurement. International Organization for Standardization (ISO), Geneva, Switzerland.
• Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica 19 491–544.
• Hannig, J. (2012). Generalized fiducial inference via discretizations. Statist. Sinica. To appear.
• Hannig, J., Iyer, H. and Patterson, P. (2006). Fiducial generalized confidence intervals. J. Amer. Statist. Assoc. 101 254–269.
• Hannig, J., Iyer, H. K. and Wang, J. C. M. (2007). Fiducial approach to uncertainty assessment: Accounting for error due to instrument resolution. Metrologia 44 476–483.
• Hannig, J. and Lee, T. C. M. (2009). Generalized fiducial inference for wavelet regression. Biometrika 96 847–860.
• Hartley, H. O. and Rao, J. N. K. (1967). Maximum-likelihood estimation for the mixed analysis of variance model. Biometrika 54 93–108.
• Hernandez, R. P., Burdick, R. K. and Birch, N. J. (1992). Confidence intervals and tests of hypotheses on variance components in an unbalanced two-fold nested design. Biom. J. 34 387–402.
• Hernandez, R. P. and Burdick, R. K. (1993). Confidence intervals and tests of hypotheses on variance components in an unbalanced two-factor crossed design with interactions. J. Stat. Comput. Simul. 47 67–77.
• Jasra, A., Stephens, D. A. and Holmes, C. C. (2007). On population-based simulation for static inference. Stat. Comput. 17 263–279.
• Jeyaratnam, S. and Graybill, F. A. (1980). Confidence intervals on variance components in three-factor cross-classification models. Technometrics 22 375–380.
• Jiang, J. (2007). Linear and Generalized Linear Mixed Models and Their Applications. Springer, New York.
• Khuri, A. I. (1987). Measures of imbalance for unbalanced models. Biom. J. 29 383–396.
• Khuri, A. I. and Sahai, H. (1985). Variance components analysis: A selective literature survey. Internat. Statist. Rev. 53 279–300.
• Kong, A., Liu, J. S. and Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89 278–288.
• Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics 38 963–974.
• Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
• Lee, Y. and Nelder, J. A. (1996). Hierarchical generalized linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 58 619–678.
• Lee, Y., Nelder, J. A. and Pawitan, Y. (2006). Generalized Linear Models with Random Effects: Unified Analysis via $H$-Likelihood. Monographs on Statistics and Applied Probability 106. Chapman & Hall/CRC, Boca Raton, FL.
• Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. R. Stat. Soc. Ser. B Stat. Methodol. 20 102–107.
• Lira, I. H. and Woger, W. (1997). The evaluation of standard uncertainty in the presence of limited resolution of indicating devices. Measurement Science and Technology 8 441–443.
• Liu, J. S. and Chen, R. (1995). Blind deconvolution via sequential imputations. J. Amer. Statist. Assoc. 90 567–576.
• Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
• Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice 197–223. Springer, New York.
• Mossman, D. and Berger, J. O. (2001). Intervals for posttest probabilities: A comparison of 5 methods. Medical Decision Making 21 498–507.
• O’Connell, A. A. and McCoach, D. B., eds. (2008). Multilevel Modeling of Educational Data. Information Age Publishing, Charlotte, NC.
• Schweder, T. and Hjort, N. L. (2002). Confidence and likelihood. Scand. J. Stat. 29 309–332.
• Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.
• Taraldsen, G. (2006). Instrument resolution and measurement accuracy. Metrologia 43 539–544.
• Ting, N., Burdick, R. K., Graybill, F. A., Jeyaratnam, S. and Lu, T.-F. C. (1990). Confidence intervals on linear combinations of variance components that are unrestricted in sign. J. Stat. Comput. Simul. 35 135–143.
• Tsui, K.-W. and Weerahandi, S. (1989). Generalized $p$-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 602–607.
• van der Vaart, A. W. (2007). Asympotitc Statistics. Statististical and Probabilistic Mathematics 8. Cambridge Univ. Press, New York.
• Wandler, D. V. and Hannig, J. (2012). Generalized fiducial confidence intervals for extremes. Extremes 15 67–87.
• Weerahandi, S. (1993). Generalized confidence intervals. J. Amer. Statist. Assoc. 88 899–905.
• Willink, R. (2007). On the uncertainty of the mean of digitized measurements. Metrologia 44 73–81.
• Wolfinger, R. D. and Kass, R. E. (2000). Nonconjugate Bayesian analysis of variance component models. Biometrics 56 768–774.
• Xie, M., Singh, K. and Strawderman, W. E. (2011). Confidence distributions and a unifying framework for meta-analysis. J. Amer. Statist. Assoc. 106 320–333.
• Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci. 7 369–387.

Supplemental materials

• Supplementary material: Additional simulation results. The asymptotic stability of the algorithm, with respect to the sample size and the particle sample size, was tested, and the simulation results are included in this document. The raw results for the simulation study in Section 3 are also displayed, along with additional summary figures.