A Reformulation of Linear Models

SUMMARY Dissatisfaction is expressed with aspects of the current exposition of linear models, including the neglect of marginality, unnecessary differences between models for finite and infinite populations, failure to distinguish different kinds of random terms, impositon of unnecessary and inconsistent constraints on parameters, and lack of an adequate notation for negative components of variance. The reformulation, exemplified for crossed and nested classifications of balanced data, and for simple orthogonal designed experiments, is designed to integrate finite and infinite populations, random and fixed effects, excess and deficit of variance, to avoid unnecessary constraints on parameters, and to lead naturally to interesting hypotheses about the model terms.

[1]  A. Dawid.  Invariant distributions and analysis of variance models , 1977 .

[2]  R. R. Hocking,et al.  The Use of the R( )-Notation with Unbalanced Data , 1976 .

[3]  J. Nelder.  Hypothesis testing in linear-models , 1976 .

[4]  Virgil L. Anderson,et al.  Restriction Errors: Another Dimension in Teaching Experimental Statistics , 1974 .

[5]  M. Kutner.  Hypothesis Testing in Linear Models (Eisenhart Model I) , 1974 .

[6]  Murray Aitkin,et al.  Simultaneous Inference and the Choice of Variable Subsets in Multiple Regression , 1974 .

[7]  J. Kleffe,et al.  Bates and best quadratic unbiased estimators for variance components and heteroscedastie variances in linear models , 1974 .

[8]  Ivor Francis,et al.  A Comparison of Several Analysis of Variance Programs , 1973 .

[9]  C. E. Rogers,et al.  Symbolic Description of Factorial Models for Analysis of Variance , 1973 .

[10]  Oscar Kempthorne,et al.  Probability, Statistics, and data analysis , 1973 .

[11]  H. D. Patterson,et al.  Recovery of inter-block information when block sizes are unequal , 1971 .

[12]  C. R. Rao,et al.  Estimation of variance and covariance components--MINQUE theory , 1971 .

[13]  D. S. Tracy,et al.  Generalized Inverse Matrices: With Applications to Statistics , 1971 .

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

[15]  H. Herne,et al.  Biomedical Computer Programs. X‐Series Supplement , 1970 .

[16]  S. R. Searle,et al.  A Discontinuity in Mixed Model Analyses , 1969 .

[17]  F. Yates.  A fresh look at the basic principles of the design and analysis of experiments , 1967 .

[18]  John A. Nelder,et al.  The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance , 1965, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[19]  John A. Nelder,et al.  The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance , 1965, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[20]  F. Graybill,et al.  Theorems Concerning Eisenhart's Model II , 1961 .

[21]  R. Plackett.  Models in the Analysis of Variance , 1960 .

[22]  D. Cox.  THE INTERPRETATION OF THE EFFECTS OF NON-ADDITIVITY IN THE LATIN SQUARE , 1958 .

[23]  J. Tukey,et al.  AVERAGE VALUES OF MEAN SQUARES IN FACTORIALS , 1956 .

[24]  M. B. Wilk,et al.  Some Aspects of the Analysis of Factorial Experiments in a Completely Randomized Design , 1956 .

[25]  P. Erdös,et al.  A limit theorem for the maximum of normalized sums of independent random variables , 1956 .

[26]  H. Smith.  Variance components, finite populations, and experimental inference , 1955 .

[27]  N. L. Franklin,et al.  Statistical Analysis in Chemistry and the Chemical Industry , 1955 .

[28]  John A. Nelder,et al.  The interpretation of negative components of variance , 1954 .

[29]  Oscar Kempthorne,et al.  The Design and Analysis of Experiments , 1952 .

[30]  M. Healy,et al.  Restricted randomization and quasi-Latin squares , 1950 .

[31]  C. Eisenhart.  The assumptions underlying the analysis of variance. , 1947, Biometrics.

[32]  H. E. Daniels.  The Estimation of Components of Variance , 1939 .

[33]  E. Pitman.  SIGNIFICANCE TESTS WHICH MAY BE APPLIED TO SAMPLES FROM ANY POPULATIONS III. THE ANALYSIS OF VARIANCE TEST , 1938 .

[34]  B. L. Welch.  ON THE z-TEST IN RANDOMIZED BLOCKS AND LATIN SQUARES , 1937 .

[35]  E. Pitman.  Significance Tests Which May be Applied to Samples from Any Populations , 1937 .

[36]  J. Neyman,et al.  Statistical Problems in Agricultural Experimentation , 1935 .

[37]  Frank Yates,et al.  The Analysis of Multiple Classifications with Unequal Numbers in the Different Classes , 1934 .