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Quantile regression in random effects meta-analysis model

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Abstract

In meta-analysis model, due to the appearance of publication bias or outliers, as well as the small sample size, the normal assumption is usually unreliable. Therefore, the exploration of more robust estimation, such quantile regression (QR) method, is extremely important in meta-analysis area. This paper studies the QR estimation method in random-effects meta-analysis model based on the reformulation by asymmetric Laplace distribution (ALD). The maximum likelihood estimation using Monte Carlo Expectation Maximization algorithm and the Bayesian estimation using Markov chain Monte Carlo (MCMC) algorithm are proposed for computation of the QR estimates. The significance tests of regression coefficients are suggested using likelihood ratio statistics. For MCMC algorithm, a simple and efficient Gibbs sampling algorithm is employed based on a location-scale mixture representation of the ALD, and information criterions are considered for choosing the hyper-parameters. Monte Carlo simulations are conducted to study the finite sample performance of the proposed methodology and analysis of two real data sets are presented for illustrations. Our results show that QR estimation methods perform very well, especially in case of non-normal assumption in meta-regression models. The detailed algorithms and software code are available for easy use in applications.

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References

  • Aghamohammadi A, Mohammadi S (2017) Bayesian analysis of penalized quantile regression for longitudinal data. Stat Pap 58:1035–1053

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B (Stat Methodol) 63:167–241

    MathSciNet  MATH  Google Scholar 

  • Booth JG, Hobert JP (1999) Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J R Stat Soc Ser B (Stat Methodol) 61:265–285

    MATH  Google Scholar 

  • Chen Y, Hong C, Riley RD (2015) An alternative pseudolikelihood method for multivariate random-effects meta-analysis. Stat Med 34(3):361–380

    MathSciNet  Google Scholar 

  • Colditz GA, Brewer TF, Berkey CS, Wilson ME, Burdick E, Fineberg HV, Mosteller F (1994) Efficacy of BCG vaccine in the prevention of tuberculosis: meta-analysis of the published literature. J Am Med Assoc 271:698–702

    Google Scholar 

  • Dagpunar JS (1989) An easily implemented generalised inverse Gaussian generator. Commun Stat Simul Comput 18(2):703–710

    MathSciNet  Google Scholar 

  • Dai X, Jin L, Tian M, Shi L (2019) Bayesian local influence for spatial autoregressive models with heteroscedasticity. Stat Pap 60:1423–1446

    MathSciNet  MATH  Google Scholar 

  • Devroye L (2014) Random variate generation for the generalized inverse Gaussian distribution. Stat Comput 24(2):239–246

    MathSciNet  MATH  Google Scholar 

  • Galarza CE, Castro LM, Louzada F, Lachos VH (2020) Quantile regression for nonlinear mixed effects models: a likelihood based perspective. Stat Pap 61:1281–1307

    MathSciNet  MATH  Google Scholar 

  • Galvao AF, Poirier A (2019) Quantile regression random effects. Ann Econ Stat 134:109–148

    Google Scholar 

  • Geraci M, Bottai M (2007) Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8(1):140–154

    MATH  Google Scholar 

  • Hardy RJ, Thompson SG (1996) A likelihood approach to meta-analysis with random effects. Stat Med 15(6):619–629

    Google Scholar 

  • Higgins JP, Thompson SG, Spiegelhalter DJ (2009) A re-evaluation of random-effects meta-analysis. J R Stat Soc A Stat Soc 172(1):137–159

    MathSciNet  Google Scholar 

  • Higgins JP, Whitehead A, Simmonds M (2011) Sequential methods for random-effects meta-analysis. Stat Med 30(9):903–921

    MathSciNet  Google Scholar 

  • Hobert JP, Casella G (1996) The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. J Am Stat Assoc 91(436):1461–1473

    MathSciNet  MATH  Google Scholar 

  • Hörmann W, Leydold J, Derflinger G (2004) Automatic nonuniform random variate generation. Springer, Berlin

    MATH  Google Scholar 

  • Hörmann W, Leydold J (2014) Generating generalized inverse Gaussian random variates. Stat Comput 24(4):547–557

    MathSciNet  MATH  Google Scholar 

  • Koenker R, Bassett G Jr (1978) Regression quantiles. Econometrica 46(1):33–50

    MathSciNet  MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Econometric society monograph series. Cambridge University Press, Cambridge

    Google Scholar 

  • Kozumi H, Kobayashi G (2011) Gibbs sampling methods for Bayesian quantile regression. J Stat Comput Simul 81(11):1565–1578

    MathSciNet  MATH  Google Scholar 

  • Levine RA, Casella G (2001) Implementations of the Monte Carlo EM algorithm. J Comput Graph Stat 10(3):422–439

    MathSciNet  Google Scholar 

  • Leydold J, Hörmann W (2011) Generating generalized inverse Gaussian random variates by fast inversion. Comput Stat Data Anal 55(1):213–217

    MathSciNet  MATH  Google Scholar 

  • Liu Y, DeSantis SM, Chen Y (2018) Bayesian mixed treatment comparisons meta-analysis for correlated outcomes subject to reporting bias. J R Stat Soc Ser C (Appl Stat) 67(1):127–144

    MathSciNet  Google Scholar 

  • Luo Y, Lian H, Tian M (2012) Bayesian quantile regression for longitudinal data models. J Stat Comput Simul 82(11):1635–1649

    MathSciNet  MATH  Google Scholar 

  • Marino MF, Farcomeni A (2015) Linear quantile regression models for longitudinal experiments: an overview. Metron 73:229–247

    MathSciNet  MATH  Google Scholar 

  • Martino L, Míguez J (2011) A generalization of the adaptive rejection sampling algorithm. Stat Comput 21(4):633–647

    MathSciNet  MATH  Google Scholar 

  • Park JS, Qian GQ, Jun Y (2008) Monte Carlo EM algorithm in logistic linear models involving non-ignorable missing data. Appl Math Comput 197(1):440–450

    MathSciNet  MATH  Google Scholar 

  • Partlett C, Riley RD (2017) Random effects meta-analysis: coverage performance of 95% confidence and prediction intervals following REML estimation. Stat Med 36(2):301–317

    MathSciNet  Google Scholar 

  • Quintana FA, Liu JS, del Pino GE (1999) Monte Carlo EM with importance reweighting and its applications in random effects models. Comput Stat Data Anal 29(4):429–444

    MATH  Google Scholar 

  • Robert CP, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Searle SR, Casella G, McCulloch CE (2009) Variance components. Wiley, Hoboken

    MATH  Google Scholar 

  • Shi L, Zuo S, Yu D, Zhou X (2017) Influence diagnostics in meta-regression model. Res Synth Methods 8(3):343–354

    Google Scholar 

  • Sidik K, Jonkman JN (2007) A comparison of heterogeneity variance estimators in combining results of studies. Stat Med 26(9):1964–1981

    MathSciNet  Google Scholar 

  • Sutton AJ, Higgins JP (2008) Recent developments in meta-analysis. Stat Med 27(5):625–650

    MathSciNet  Google Scholar 

  • Thompson SG, Sharp SJ (1999) Explaining heterogeneity in meta-analysis: a comparison of methods. Stat Med 18(20):2693–2708

    Google Scholar 

  • Tian Y, Lian H, Tian M (2017) Bayesian composite quantile regression for linear mixed-effects models. Commun Stat Theory Methods 46(15):7717–7731

    MathSciNet  MATH  Google Scholar 

  • Van Houwelingen HC, Arends LR, Stijnen T (2002) Advanced methods in meta-analysis: multivariate approach and meta-regression. Stat Med 21(4):589–624

    Google Scholar 

  • Viechtbauer W, Cheung MWL (2010) Outlier and influence diagnostics for meta-analysis. Res Synth Methods 1(2):112–125

    Google Scholar 

  • Wang J (2012) Bayesian quantile regression for parametric nonlinear mixed effects models. Stat Methods Appl 21(3):279–295

    MathSciNet  MATH  Google Scholar 

  • Wang RW, Dunn DW, Luo J, He JZ, Shi L (2015) The importance of spatial heterogeneity and self-restraint on mutualism stability: a quantitative review. Sci Rep 5:14826

    Google Scholar 

  • Whitehead A, Whitehead J (1991) A general parametric approach to the meta-analysis of randomized clinical trials. Stat Med 10(11):1665–1677

    Google Scholar 

  • Yu K, Moyeed RA (2001) Bayesian quantile regression. Stat Probab Lett 54(4):437–447

    MathSciNet  MATH  Google Scholar 

  • Yuan Y, Yin G (2010) Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics 66(1):105–114

    MathSciNet  MATH  Google Scholar 

  • Yue YR, Rue H (2011) Bayesian inference for additive mixed quantile regression models. Comput Stat Data Anal 55(1):84–96

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research were supported by key project of National Natural Science Foundation (NSFC) of China (No. 11931015), and NSFC project (No. 11671348, No. 11801370, No. 12271471) and Shanghai Natural Science Foundation (No. 18ZR1427200).

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Authors and Affiliations

  1. School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, China

    Xiaowen Dai & Libin Jin

  2. Interdisciplinary Research Institute of Data Science, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, China

    Xiaowen Dai & Libin Jin

  3. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China

    Lei Shi

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  1. Xiaowen Dai
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  2. Libin Jin
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Correspondence to Lei Shi.

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Dai, X., Jin, L. & Shi, L. Quantile regression in random effects meta-analysis model. Stat Methods Appl 32, 469–492 (2023). https://doi.org/10.1007/s10260-022-00660-3

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