The Equivalence of the Maximum Likelihood and a Modified Least Squares for a Case of Generalized Linear Model
Applied and Computational Mathematics
Volume 3, Issue 5, October 2014, Pages: 268-272
Received: Oct. 11, 2014; Accepted: Nov. 3, 2014; Published: Nov. 10, 2014
Views 2637      Downloads 125
Author
Ahsene Lanani, MAM Laboratory, Department of Mathematics, University of Constantine 1. Constantine 25000, Algeria
Article Tools
Follow on us
Abstract
During the analysis of statistical data, one of the most important steps is the estimation of the considered parameters model. The most common estimation methods are the maximum likelihood and the least squares. When the data are considered normal, there is equivalence between the two methods, so there is no privilege for one or the other method. However, if the data are not Gaussian, this equivalence is no longer valid. Also, if the normal equations are not linear, we make use of iterative methods (Newton-Raphson algorithm, Fisher, etc ...). In this work, we consider a particular case where the data are not normal and solving equations are not linear and that it leads to the equivalence of the maximum likelihood method at least squares but modified. At the end of the work, we concluded by referring to the application of this modified method for solving the equations of Liang and Zeger.
Keywords
Maximum Likelihood, Linear Mixed Model, Newton-Raphson Algorithm, Weighted Least Squares
To cite this article
Ahsene Lanani, The Equivalence of the Maximum Likelihood and a Modified Least Squares for a Case of Generalized Linear Model, Applied and Computational Mathematics. Vol. 3, No. 5, 2014, pp. 268-272. doi: 10.11648/j.acm.20140305.22
References
[1]
E. M. Chi, and G. C. Reinsel, “Models for longitudinal data with random effects and AR(1) errors,’’ JASA, Theory and Methods, Vol.84, No.406. pp. 452-459, 1989.
[2]
M. Crowder, “On the use of a working matrix correlation in using GLM for repeated measures,’’ Biometrika, 82, vol. 2, pp. 407-410, 1995.
[3]
M. Crowder, “On repeated measures analysis with misspecified covariance structure,’’ JRSS, B 63, part1, pp. 55-62, 2001.
[4]
A. P. Dempster, N. M. Laird, and R. B. Rubin, “Maximum likelihood with incomplete data via the E-M algorithm,’’ JRSS, B39, pp. 1-38, 1977.
[5]
A. P. Dempster, D. B. Rubin, and R. K. Tsutakawa, “Estimation covariance component models,’’ JASA, vol. 76, pp. 341-353, 1981.
[6]
P. J. Diggle, K. Y. Liang, and S. L. Zeger, “Analysis of longitudinal data,’’ Oxford science publications, 1994.
[7]
W. J. Dixon, “BMDP statistical software,’’ manual volume 2, University of California press: Berkley, CA, 1988.
[8]
G. M. Fitzmaurice, and N.M. Laird, “A likelihood based method for analysing longitudinal binary responses,’’ Biometrika, vol. 80, 1, pp. 141-151, 1993.
[9]
J. L. Foulley, F. Jaffrézic, and C. R. Granié, “EM-REML estimation of covariance parameters in gaussian mixed models for longitudinal data analysis,’’ Genet. Sel. Evol. Vol. 32, 129-141, 2000.
[10]
A. Galecki and T. Burzykowski, ‘’Linear Mixed-effects Models using R,’’ Springer, 2012.
[11]
D. A. Harville, “Maximum likelihood approaches to variance components estimation and to related problem,’’ JASA, Vol.72, No.358, pp. 320-329, 1977.
[12]
R. I. Jennrich, and M. D. Scluchter, “Unbalanced Repeated-Measure Models with Structured Covariance Matrices,’’ Biometrics, vol. 42, pp. 805-820, 1986.
[13]
J. Jiang, “Linear and Generalized Linear Mixed models and their applications,’’ Springer, 2007.
[14]
H. R. Jones, and Boadi-Boateng, “Unequally spaced longitudinal data with AR(1) serial correlation,’’ Biometrics, vol. 47, pp. 161-175, 1991.
[15]
N. M. Laird, and J. H. Ware, “Random effect models for longitudinal data,’’ Biometrics, 38, pp. 963-974, 1982.
[16]
K. Y. Liang, and S. Zeger, “Longitudinal data analysis using GLM,’’ Biometrika, vol. 73, 1, pp. 13-22, 1986.
[17]
M. J. Lindstrom and D. M. Bates, “Newton-Raphson and E-M algorithm for linear mixed effect models for repeated -measures data,’’ JASA, V83, N 404, pp. 1014-1022, 1988.
[18]
R. C. Littell, G. A. Milliken, W. W. Stroup, and R. D. Wolfinger, “System for mixed models,’’ SAS institute inc: Cary, N.C, 1996.
[19]
R. C. Littell, J. Pendergast, and R. Natarajan, “Modelling covariance Structure in the analysis of repeated measures data,’’ Statist. Med. V19, pp. 1793-1819, 2000.
[20]
P. McCullagh, and J. A. Nelder, “Generalized linear model,’’ 2nd edition. Chapman and Hall London, 1989.
[21]
C. E. McCulloch, and S. R. Searle, “Generalized Linear Mixed Models’’, Wiley 2001.
[22]
T. A. Park, “A comparison of the GEE approach with the maximum likelihood approach for repeated measurements,’’ Stat. Med. V12, pp. 1723-32, 1993.
[23]
T. Park, and Y. J. Lee, “Covariance models for nested measures data: analysis of ovarian steroid secretion data,’’ Statist. Med. V21, pp. 143-164, 2002.
[24]
H. D. Patterson, and R. Thompson, “Recovery of inter-block information when block sizes are unequal,’’ Biometrika, vol. 58, pp. 545-554, 1971.
[25]
J. C. Pinheiro, and D. M. Bates, “Mixed Effect Models in S and S-Plus,’’ Springer 2000.
[26]
R. L. Prentice, and L. P. Zhao, “Estimating equations for parameters in means and covariance of multivariate discrete and continuous responses,’’ Biometrics 47, pp. 825-839, 1991.
[27]
G. Verbeke, and G. Molenberghs, “Linear Mixed Models for longitudinal data. New York: Springer, 2000.
[28]
R. W. M. Wedderburn, “Quasi-likelihood function; GLM and the Gauss-Newton method,’’ Biometrika 61, 3, pp. 439-447, 1974.
[29]
V. Witkovsky, “Matlab algorithm mixed.m for solving Henderson's mixed model equations,’’ Institute of Measurement Science Slovak Academy of Sciences, 2002.
[30]
L. P. Zhao, and R. L. Prentice, “Correlated binary regression using a quadratic exponential model,’’ Biometrika 77, pp. 642-648, 1990.
[31]
S. L. Zeger, K. Y. Liang, and P. S. Albert, “Models for longitudinal data: a generalized equation approach,’’ Biometrics 44, pp. 1049-1060, 1988.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186