Computational Statistics

, Volume 32, Issue 4, pp 1747–1765 | Cite as

Weighted least squares estimation for exchangeable binary data

  • Dale Bowman
  • E. Olusegun George
Original Paper


Parametric models of discrete data with exchangeable dependence structure present substantial computational challenges for maximum likelihood estimation. Coordinate descent algorithms such as the Newton’s method are usually unstable, becoming a hit or miss adventure on initialization with a good starting value. We propose a method for computing maximum likelihood estimates of parametric models for finitely exchangeable binary data, formalized as an iterative weighted least squares algorithm.


Exchangeability Completely monotonic function Weighted least squares Excess risk Benchmark dose 


  1. Bowman D (1999) A parametric independence test for clustered binary data. Stat Probab Lett 41:1–7CrossRefMATHGoogle Scholar
  2. Bowman D, Chen J, George EO (1995) Estimating variance functions in developmental toxicity studies. Biometrics 51:1523–1528CrossRefMATHGoogle Scholar
  3. Bowman D, George EO (1995) A saturated model for analyzing exchangeable binary data: applications to clinical and developmental toxicity studies. J Am Stat Assoc 90:871–879CrossRefMATHGoogle Scholar
  4. Bowman D, George EO (2015) Likelihood estimation for exchangeable multinomial data. Commun Stat Theory Methods. doi: 10.1080/03610926.2015.1053934
  5. Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16:11901208CrossRefMATHMathSciNetGoogle Scholar
  6. Carter RE, Zhang X, Woolson RF, Apfel CC (2009) Statistical analysis of correlated relative risks. J Data Sci 7:397–407Google Scholar
  7. Dang X, Keeton SL, Peng H (2009) A unified approach for analyzing exchangeable binary data with applications to developmental toxicity studies. Stat Med 28:2580–2604CrossRefMathSciNetGoogle Scholar
  8. de Finetti B (1974) Theory of probability. Wiley, New YorkMATHGoogle Scholar
  9. Diaconis P, Freedman D (1082) De Finetti’s theorem for symmetric location families. Ann Stat 10(1):84–189MATHMathSciNetGoogle Scholar
  10. George EO, Bowman D (1995) A full likelihood procedure for analysing exchangeable binary data. Biometrics 51:512–523CrossRefMATHMathSciNetGoogle Scholar
  11. George EO, Bowman D, An Q (2013) Regression models for analyzing clustered binary and continuous outcomes under the assumption of exchangeability. In: deLeon A, Chough K (eds) Analysis of mixed data: methods and applications. CRC Press, New York, pp 93–107Google Scholar
  12. Holston JF, Gaines TB, Nelson CJ, LaBorde JB, Gaylor DW, Sheehan DM, Young JF (1991) Developmental toxicity of 2,4,5-trichlorophenoxiacetic acid:I, multireplicated dose response studies in four inbred strains and one outbred stock of mice. Fund Appl Toxic 19:286–297CrossRefGoogle Scholar
  13. Knight K (2000) Mathematical statistics. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar
  14. Kuk A (2004) A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions. Appl Stat 53:369–386MATHMathSciNetGoogle Scholar
  15. Lipsitz SR, Laird NM, Harrington DP (1991) Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association. Biometrika 78:153–160CrossRefMathSciNetGoogle Scholar
  16. Miller ME, Davis CS, Landis JR (1993) The analysis of longitudinal polytomous data: generalized estimating equations and connections with weighted least squares. Biometrics 49:1033–1044CrossRefMATHGoogle Scholar
  17. Parsons N, Edmondson R, Gilmour S (2006) A generalied estimating equation method for fitting autocorrelated ordinal score data with an application in horticultural research. J R Stat Soc C 55:507–524CrossRefMATHGoogle Scholar
  18. Rosen M (1999) A weighted least squares approach to estimating the per unit agreement rate in clustered binary data. In: Proceedings of the biometrics section of the American Statistical Association, pp 65–70Google Scholar
  19. Touloumis A, Agresti A, Kateri M (2013) GEE for multinomial responses using a local odds ratios parameterization. Biometrics 69:633–640. doi: 10.1111/biom.12054 CrossRefMATHMathSciNetGoogle Scholar
  20. Xu P, Fu W, Zhu L (2013) Shrinkage estimation analysis of correlated binary data with a diverging number of parameters. Sci China Math 56:359–377CrossRefMATHMathSciNetGoogle Scholar
  21. Yu C, Zelterman D (2002) Statistical inference for familial disease clusters. Biometrics 58:481–491CrossRefMATHMathSciNetGoogle Scholar
  22. Yu C, Zelterman D (2008) Sums of exchangeable Bernoulli random variables for family and litter frequency data. Comput Stat Data Anal 52:1636–1649CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical ScienceThe University of MemphisMemphisUSA

Personalised recommendations