A Geometric Interpretation of Inferences Based on Ranks in the Linear Model

Abstract Four different approaches, based on ranks, to testing hypotheses are unified through the geometry of the linear model. The various tests are identified with different but algebraically equivalent forms of the classical F test. Small sample differences are investigated via a Monte Carlo study using both rank and signed rank tests.

[1]  Peter J. Bickel,et al.  On Some Alternative Estimates for Shift in the $P$-Variate One Sample Problem , 1964 .

[2]  Thomas P. Hettmansperger,et al.  A Robust Alternative Based on Ranks to Least Squares in Analyzing Linear Models , 1977 .

[3]  E. Lehmann,et al.  Nonparametrics: Statistical Methods Based on Ranks , 1976 .

[4]  J. L. Hodges,et al.  Rank Methods for Combination of Independent Experiments in Analysis of Variance , 1962 .

[5]  Herman Chernoff,et al.  ASYMPTOTIC NORMALITY AND EFFICIENCY OF CERTAIN NONPARAMETRIC TEST STATISTICS , 1958 .

[6]  Pranab Kumar Sen,et al.  Asymptotically distribution-free aligned rank order tests for composite hypotheses for general multivariate linear models , 1977 .

[7]  Gary Simon.  Computer Simulation Swindles, with Applications to Estimates of Location and Dispersion , 1976 .

[8]  J. L. Hodges,et al.  The Efficiency of Some Nonparametric Competitors of the t-Test , 1956 .

[9]  J. L. Hodges,et al.  Estimates of Location Based on Rank Tests , 1963 .

[10]  T. A. Ryan,et al.  Algorithm 516: An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two-Sample Location Problem [G1] , 1977, TOMS.

[11]  Ian Barrodale,et al.  Algorithm 478: Solution of an Overdetermined System of Equations in the l1 Norm [F4] , 1974, Commun. ACM.

[12]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[13]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[14]  J. McKean,et al.  The Geometry of Robust Procedures in Linear Models , 1980 .

[15]  H. Toutenburg,et al.  Lehmann, E. L., Nonparametrics: Statistical Methods Based on Ranks, San Francisco. Holden‐Day, Inc., 1975. 480 S., $ 22.95 . , 1977 .

[16]  R. Welsch,et al.  The Hat Matrix in Regression and ANOVA , 1978 .

[17]  J. Jurecková,et al.  Nonparametric Estimate of Regression Coefficients , 1971 .

[18]  Louis A. Jaeckel.  Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals , 1972 .

[19]  T. Hettmansperger,et al.  Robust analysis of variance based upon a likelihood ratio criterion , 1980 .

[20]  T. A. Bray,et al.  A Convenient Method for Generating Normal Variables , 1964 .

[21]  P. Sen.  On a Class of Rank Order Tests for the Parallelism of Several Regression Lines , 1969 .

[22]  Richard H. Bartels,et al.  Linearly Constrained Discrete I1 Problems , 1980, TOMS.

[23]  Paul W. Holland,et al.  Two Robust Alternatives to Least-Squares Regression , 1977 .

[24]  J. N. Adichie.  Rank Score Comparison of Several Regression Parameters , 1974 .

[25]  C. Kraft,et al.  Linearized Rank Estimates and Signed-Rank Estimates for the General Linear Hypothesis , 1972 .

[26]  R. Worthley,et al.  A Small-Sample Comparison of Rank Score Tests for Parallelism of Several Regression Lines , 1978 .

[27]  S. R. Searle,et al.  The theory of linear models and multivariate analysis , 1981 .

[28]  Thomas P. Hettmansperger,et al.  Statistical inference based on ranks , 1985 .

[29]  C. F. Smit.  An empirical comparison of several tests for parallelism of regression lines , 1979 .

[30]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[31]  Thomas P. Hettmansperger,et al.  Tests of hypotheses based on ranks in the general linear model , 1976 .

[32]  J. N. Adichie.  Rank Tests of Sub-Hypotheses in the General Linear Regression , 1978 .

[33]  Thomas P. Hettmansperger,et al.  A robust analysis of the general linear model based on one step R-estimates , 1978 .

[34]  Joseph W. McKean,et al.  THE USE AND INTERPRETATION OF ROBUST ANALYSIS OF VARIANCE , 1982 .