Generalized Linear Models

The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

[1]  Stephen E. Fienberg,et al.  The Analysis of Multidimensional Contingency Tables , 1970 .

[2]  Y. Bishop,et al.  Full Contingency Tables, Logits, and Split Contingency Tables , 1969 .

[3]  G. Patil,et al.  On Certain Properties of the Exponential-Type Families , 1965 .

[4]  P. McCullagh.  Quasi-Likelihood Functions , 1983 .

[5]  H. O. Lancaster.  The combination of probabilities arising from data in discrete distributions. , 1949, Biometrika.

[6]  Calyampudi R. Rao,et al.  Linear Statistical Inference and Its Applications. , 1975 .

[7]  G. Seber,et al.  Linear Regression Analysis , 1980 .

[8]  J. Gaddum.  Probit Analysis , 1948, Nature.

[9]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[10]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[11]  H. D. Patterson,et al.  Analysis of Factorial Arrangements when the Data are Proportions , 1952 .

[12]  F. Yates.  The analysis of contingency tables with groupings based on quantitative characters. , 1948, Biometrika.

[13]  Yoo Yoon Jae,et al.  On the Analysis , 2009 .

[14]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[15]  J. Mandel.  A New Analysis of Variance Model for Non-additive Data , 1971 .

[16]  Leo A. Goodman,et al.  Association Models and Canonical Correlation in the Analysis of Cross-Classifications Having Ordered Categories , 1981 .

[17]  S. Kullback.  Information Theory and Statistics , 1959 .

[18]  J. Nelder.  Weighted Regression, Quantal Response Data, and Inverse Polynomials , 1968 .

[19]  J. Nelder,et al.  The GLIM System Release 3. , 1979 .

[20]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[21]  J. O. Irwin.  A Note on the Subdividsion of ? 2 into Components , 1949 .

[22]  J. O. Irwin,et al.  A note on the subdivision of chi2 into components. , 1949, Biometrika.

[23]  J. A. Nelder,et al.  Inverse Polynomials, a Useful Group of Multi-Factor Response Functions , 1966 .

[24]  D. Cox,et al.  The analysis of binary data , 1971 .

[25]  H. O. Lancaster,et al.  The derivation and partition of chi2 in certain discrete distributions. , 1949, Biometrika.

[26]  F. Yates.  THE RECOVERY OF INTER-BLOCK INFORMATION IN BALANCED INCOMPLETE BLOCK DESIGNS , 1940 .

[27]  A. Dempster.  An overview of multivariate data analysis , 1971 .

[28]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[29]  F. Yates.  Experimental Design. Selected Papers. , 1972 .

[30]  A. E. Maxwell,et al.  Analysing Qualitative Data , 1962 .

[31]  R. A. Fisher.  A biological assay of tuberculins. , 1949, Biometrics.

[32]  D. Cox,et al.  Notes on Some Aspects of Regression Analysis , 1968 .

[33]  A. Kimball.  Short-Cut Formulas for the Exact Partition of χ 2 in Contingency Tables , 1954 .

[34]  S. Ziebland,et al.  Analysing qualitative data , 2000, BMJ : British Medical Journal.

[35]  H. O. Lancaster.  The exact partition of chi 2 and its application to the problem of the pooling of small expectations. , 1950, Biometrika.

[36]  Shelby J. Haberman,et al.  Tests for Independence in Two-Way Contingency Tables Based on Canonical Correlation and on Linear-By-Linear Interaction , 1981 .

[37]  Robin Thompson,et al.  Composite Link Functions in Generalized Linear Models , 1981 .

[38]  M. W. Birch.  Maximum Likelihood in Three-Way Contingency Tables , 1963 .

[39]  E. H. Simpson,et al.  The Interpretation of Interaction in Contingency Tables , 1951 .

[40]  W. C. Guenther,et al.  Analysis of variance , 1968, The Mathematical Gazette.

[41]  S. R. Searle.  Linear Models , 1971 .

[42]  A. Linder,et al.  Experimental Design: Selected Papers , 1971 .

[43]  S. Kullback,et al.  Contingency tables with given marginals. , 1968, Biometrika.

[44]  K. Pearson,et al.  Biometrika , 1902, The American Naturalist.

[45]  J. Nelder.  A Reformulation of Linear Models , 1977 .

[46]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[47]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[48]  Evgueni A. Haroutunian,et al.  Information Theory and Statistics , 2011, International Encyclopedia of Statistical Science.

[49]  J. Mandel.  The Analysis of Latin Squares with a Certain Type of Row-Column Interaction , 1959 .