• Bernoulli
  • Volume 20, Number 4 (2014), 1979-1998.

Restricted likelihood representation and decision-theoretic aspects of meta-analysis

Andrew L. Rukhin

Full-text: Open access


In the random-effects model of meta-analysis a canonical representation of the restricted likelihood function is obtained. This representation relates the mean effect and the heterogeneity variance estimation problems. An explicit form of the variance of weighted means statistics determined by means of a quadratic form is found. The behavior of the mean squared error for large heterogeneity variance is elucidated. It is noted that the sample mean is not admissible nor minimax under a natural risk function for the number of studies exceeding three.

Article information

Bernoulli, Volume 20, Number 4 (2014), 1979-1998.

First available in Project Euclid: 19 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

DerSimonian–Laird estimator Hedges estimator Mandel–Paule procedure minimaxity quadratic forms random-effects model Stein phenomenon


Rukhin, Andrew L. Restricted likelihood representation and decision-theoretic aspects of meta-analysis. Bernoulli 20 (2014), no. 4, 1979--1998. doi:10.3150/13-BEJ547.

Export citation


  • [1] Borenstein, M., Hedges, L., Higgins, J. and Rothstein, H. (2009). Introduction to Meta-Analysis. New York: Wiley.
  • [2] Brown, L.D. (1988). The differential inequality of a statistical estimation problem. In Statistical Decision Theory and Related Topics, IV, Vol. 1 (West Lafayette, Ind., 1986) (S.S. Gupta and J.O. Berger, eds.) 299–324. New York: Springer.
  • [3] DerSimonian, R. and Laird, N. (1986). Meta-analysis in clinical trials. Control. Clin. Trials 7 177–188.
  • [4] Efron, B. and Morris, C. (1973). Stein’s estimation rule and its competitors – an empirical Bayes approach. J. Amer. Statist. Assoc. 68 117–130.
  • [5] Harville, D.A. (1985). Decomposition of prediction error. J. Amer. Statist. Assoc. 80 132–138.
  • [6] Jackson, D., Bowden, J. and Baker, R. (2010). How does the DerSimonian and Laird procedure for random effects meta-analysis compare with its more efficient but harder to compute counterparts? J. Statist. Plann. Inference 140 961–970.
  • [7] Maatta, J.M. and Casella, G. (1990). Developments in decision-theoretic variance estimation. Statist. Sci. 5 90–120. With comments and a rejoinder by the authors.
  • [8] Marchand, É. and Strawderman, W.E. (2005). On improving on the minimum risk equivariant estimator of a scale parameter under a lower-bound constraint. J. Statist. Plann. Inference 134 90–101.
  • [9] Marchand, É. and Strawderman, W.E. (2012). A unified minimax result for restricted parameter spaces. Bernoulli 18 635–643.
  • [10] Marshall, A.W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering 143. New York: Academic Press.
  • [11] Morris, C.N. and Normand, S.L. (1992). Hierarchical models for combining information and for meta-analyses. In Bayesian Statistics, Vol. 4 (Peñíscola, 1991) (J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, eds.) 321–344. New York: Oxford Univ. Press.
  • [12] Paule, R.C. and Mandel, J. (1982). Consensus values and weighting factors. J. Res. Natl. Bur. Stand. 87 377–385.
  • [13] Rukhin, A.L. (1995). Admissibility: Survey of a concept in progress. Int. Stat. Rev. 63 95–115.
  • [14] Rukhin, A.L. (2012). Estimating common mean and heterogeneity variance in two study case meta-analysis. Statist. Probab. Lett. 82 1318–1325.
  • [15] Rukhin, A.L. (2013). Estimating heterogeneity variance in meta-analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 451–469.
  • [16] Rukhin, A.L. (2014). Supplement to “Restricted likelihood representation and decision-theoretic aspects of meta-analysis.” DOI:10.3150/13-BEJ543SUPP.
  • [17] Searle, S.R., Casella, G. and McCulloch, C.E. (1992). Variance Components. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.

Supplemental materials

  • Supplementary material: Restricted likelihood representation and decision-theoretic aspects of meta-analysis: Electronic supplement. The supplement contains the proof of Theorem 2.1.