William Kruskal

Use of Ranks in One-Criterion Variance Analysis

Abstract Given C samples, with n i observations in the ith sample, a test of the hypothesis that the samples are from the same population may be made by ranking the observations from from 1 to Σn i (giving each observation in a group of ties the mean of the ranks tied for), finding the C sums of ranks, and computing a statistic H. Under the stated hypothesis, H is distributed approximately as χ2(C – 1), unless the samples are too small, in which case special approximations or exact tables are provided. One of the most important applications of the test is in detecting differences among the population means.* * Based in part on research supported by the Office of Naval Research at the Statistical Research Center, University of Chicago.

Measures of Association for Cross Classifications. II: Further Discussion and References

Abstract Our earlier discussion of measures of association for cross classifications [66] is extended in two ways. First, a number of supplementary remarks to [66] are made, including the presentation of some new measures. Second, historical and bibliographical material beyond that in [66] is critically surveyed; this includes discussion of early work in America by Doolittle and Peirce, early work in Europe by Korosy, Benini, Lipps, Deuchler and Gini, more recent work based on Shannon-Wiener information, association measures based on latent structure, and relevant material in the literatures of meteorology, ecology, sociology, and anthropology. New expressions are given for some of the earlier measures of association. * This research was supported in part by the Army, Navy, and Air Force through the Joint Services Advisory Committee for Research Groups in Applied Mathematics and Statistics, Contract No. N6ori-02035; and in part by the Office of Naval Research. This paper, in whole or in part, may be repro...

Relative importance by averaging over orderings

Abstract Many ways have been suggested for explicating the ambiguous concept of relative importance for independent variables in a multiple regression setting. There are drawbacks to all the explications, but a relatively acceptable one is available when the independent variables have a relevant, known ordering: consider the proportion of variance of the dependent variable linearly accounted for by the first independent variable; then consider the proportion of remaining variance linearly accounted for by the second independent variable; and so on. When, however, the independent variables do not have a relevant ordering, that approach fails. The primary suggestion of this article is to rescue the idea by averaging relative importance over all orderings of the independent variables. Variations and extensions of the idea are described.

Measures of Association for Cross Classifications, IV: Simplification of Asymptotic Variances

Abstract The asymptotic sampling theory discussed in our 1963 article [3] for measures of association presented in earlier articles [1, 2] turns on the derivation of asymptotic variances that may be complex and tedious in specific cases. In the present article, we simplify and unify these derivations by exploiting the expression of measures of association as ratios. Comments on the use of asymptotic variances, and on a trap in their calculation, are also given.

Concepts of Relative Importance in Recent Scientific Literature

Abstract How is the ambiguous concept of relative importance for independent variables handled in the scientific literature? We sampled from a population of recent papers with relative importance (or the equivalent) in their titles. We found widespread apparent desire to make relative-importance statements, but little self-conscious interest in interpretation or in looking at more than one specification of the concept. We were unhappy to find that a substantial fraction (one-fifth) of the papers used statistical significance to measure relative importance.

An Approximate Statistical Test for Correlations between Proportions

The observed means and variances of data occurring as proportions or percentages may be used to estimate analogous parameters of a theoretical open array, X, which, on closure, yields a new array, Y, whose means and variances are exactly those of the observed data, but in which the covariances have been generated entirely by closure. The correlations in Y, found directly from the means and variances of X, are appropriate null values against which the observed correlations may be tested, A testing procedure is outlined, and a practical example is given.

Miracles and Statistics: The Casual Assumption of Independence

Abstract The primary theme of this address is cautionary: Statistical independence is far too often assumed casually, without serious concern for how common is dependence and how difficult it can be to achieve independence (or related structures). After initial discussion of statistics and religion, the address turns to miracles, especially Humes critique and Babbages reply. Stress is given the often tacit or unexamined assumption of independence among witnesses of a putative miracle. Other contexts of multiple testimony are treated, and the address ends with contemporary casual assumptions of independence: nuclear reactor safety, repeated measurements, and so forth. Other topics include prayer, circularity of argument, and the tension between skepticism about testimony and the pragmatic need to accept most of it provisionally.

The Significance of Fisher: A Review of R.A. Fisher: The Life of a Scientist

Abstract A number of colleagues have made helpful criticism and comments. They certainly do not uniformly agree with my judgments and emphases, but my warm appreciation goes to Keith Baker, Albert Biderman, Richard Brown, K. Alexander Brownlee, Donald T. Campbell, William G. Cochran, Lee J. Cronbach, Cuthbert Daniel, F.N. David, Arthur P. Dempster, Churchill Eisenhart, Stephen E. Fienberg, David Finney, Milton Friedman, I.J. Good, Bernard G. Greenberg, N.T. Gridgeman, William Jaffe, Oscar Kempthorne, Erich L. Lehmann, Richard C. Lewontin, Donald MacKenzie, William G. Madow, Margaret E. Martin, Frederick Mosteller, Jerzy Neyman, John W. Pratt, Donald B. Rubin, I.R. Savage, Hilary L. Seal, Hanan Selvin, Oscar B. Sheynin, David L. Sills, Theodor D. Sterling, George Stigler, Stephen M. Stigler, Fred L. Strodtbeck, Alan Stuart, Judith M. Tanur, Ronald Thisted, Howard Wainer, Frank Yates, Arnold Zellner and Harriet Zuckerman. Joan Fisher Boxs biography of R.A. Fisher presents a lively and detailed description ...

Statistics in Society: Problems Unsolved and Unformulated

Abstract Many problems important to society have major statistical components, yet are not wholly statistical in the usual sense. Examples of such problems are discussed, and a tentative classification is proposed. The first examples are those of inconsistencies and the like in large data sets, for example, census information showing a number of women, aged 15 through 19, with 12 or more children. The second group of examples deals with ambiguity of classification, for example, of ethnic origin. Other problems discussed briefly include clarity of statistical graphics, and confidentiality. Stress is laid on novel theoretical issues posed by these problems, and also on the social or political forces that generate the problems. A professional response is proposed in which resources are sought to study with care the behavior of relevant statistical activities.

A statistical model : Frederick Mosteller's contributions to statistics, science, and public policy

Contents: Biography.- Bibliography.- Contributions as a Scientific Generalist.- Contributions to Mathematical Statistics.- Contributions to Methodology and Applications.- Fred as Educator.- Fred at Harvard.- Reviews of Book Contributions.