Nonparametric Bootstrap Confidence Intervals for Variance Components Applied to Interlaboratory Comparisons

Exact confidence intervals for variance components in linear mixed models rely heavily on normal distribution assumptions. If the random effects in the model are not normally distributed, then the true coverage probabilities of these conventional intervals may be erratic. In this paper we examine the performance of nonparametric bootstrap confidence intervals based on restricted maximum likelihood (REML) estimators. Asymptotic theory suggests that these intervals will achieve the nominal coverage value as the sample size increases. Incorporating a small-sample adjustment term in the bootstrap confidence interval construction process improves the performance of these intervals for small to intermediate sample sizes. Simulation studies suggest that the bootstrap standard method (with a transformation) and the bootstrap bias-corrected and accelerated (BCa) method produce confidence intervals that have good coverage probabilities under a variety of distribution assumptions. For an interlaboratory comparison of mercury concentration in oyster tissue, a balanced one-way random effects model is used to quantify the proportion of the variation in mercury concentration that can be attributed to the laboratories. In this application the exact confidence interval using normal distribution theory produces misleading results and inferences based on nonparametric bootstrap procedures are more appropriate.

[1]  Robert Tibshirani,et al.  Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy , 1986 .

[2]  Bootstrapping Robust Estimates for Clustered Data , 2010 .

[3]  Jiming Jiang,et al.  Partially observed information and inference about non-Gaussian mixed linear models , 2005, math/0603073.

[4]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[5]  D. Harville,et al.  Confidence intervals for a variance ratio, or for heritability, in an unbalanced mixed linear model. , 1985, Biometrics.

[6]  M. Chernick,et al.  A Bootstrap Test for Comparing Two Variances: Simulation of Size and Power in Small Samples , 2011, Journal of biopharmaceutical statistics.

[7]  Brent D. Burch.  Generalized confidence intervals for proportions of total variance in mixed linear models , 2007 .

[8]  Hari Iyer,et al.  EXACT CONFIDENCE INTERVALS FOR A VARIANCE RATIO (OR HERITABILITY) IN A MIXED LINEAR MODEL , 1997 .

[9]  M. Chernick,et al.  Revisiting Qualms about Bootstrap Confidence Intervals , 2009 .

[10]  Hong Lai,et al.  Nonparametric bootstrapping for hierarchical data , 2010 .

[11]  Peter Hall,et al.  Adaptive inference for the two-sample scale problem , 1997 .

[12]  Phillip I. Good,et al.  A Practitioner's Guide to Resampling for Data Analysis, Data Mining, and Modeling , 2011 .

[13]  Nathaniel Schenker,et al.  Qualms about Bootstrap Confidence Intervals , 1985 .

[14]  C. Field,et al.  Bootstrapping clustered data , 2007 .

[15]  D. Joanes,et al.  Comparing measures of sample skewness and kurtosis , 1998 .

[16]  Peter H. Westfall,et al.  A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions , 1987 .

[17]  Brent D. Burch.  Assessing the performance of normal-based and REML-based confidence intervals for the intraclass correlation coefficient , 2011, Comput. Stat. Data Anal..

[18]  Closed‐Form Approximations to the REML Estimator of a Variance Ratio (or Heritability) in a Mixed Linear Model , 2001, Biometrics.

[19]  D. Harville.  Making REML computationally feasible for large data sets: use of the Gibbs sampler , 2004 .

[20]  Jiming Jiang.  REML estimation: asymptotic behavior and related topics , 1996 .

[21]  B. Efron.  Better Bootstrap Confidence Intervals , 1987 .

[22]  Brady T. West,et al.  Linear Mixed Models: A Practical Guide Using Statistical Software , 2006 .

[23]  B. Efron,et al.  The Jackknife: The Bootstrap and Other Resampling Plans. , 1983 .

[24]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[25]  Alan M. Polansky,et al.  Stabilizing bootstrap‐t confidence intervals for small samples , 2000 .

[26]  A. Davison,et al.  Non‐parametric bootstrap confidence intervals for the intraclass correlation coefficient , 2003, Statistics in medicine.

[27]  J. Shao,et al.  The jackknife and bootstrap , 1996 .

[28]  Michael R Chernick,et al.  Bootstrap Methods: A Guide for Practitioners and Researchers , 2007 .

[29]  B. Efron,et al.  Bootstrap confidence intervals , 1996 .

[30]  S. Willie,et al.  NOAA National Status and Trends Program : thirteenth round intercomparison exercise results for trace metals in marine sediments and biological tissues , 1995 .