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

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

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

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

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

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

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

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

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

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

[11]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

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

[13]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[14]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

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

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

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

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

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

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

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

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

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

[24]  David R. Cox.  The analysis of binary data , 1970 .

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

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

[27]  S. Kullback,et al.  On the Analysis of Multidimensional Contingency Tables , 1971 .

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

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

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

[31]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

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

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

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

[35]  George A. F. Seber,et al.  Linear regression analysis , 1977 .

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

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

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

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

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

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