John W. Tukey

Simultaneous conjoint measurement: A new type of fundamental measurement

The essential character of what is classically considered, e.g., by N. R. Campbell, the fundamental measurement of extensive quantities is described by an axiomatization for the comparision of effects of (or responses to) arbitrary combinations of “quantities” of a single specified kind. For example, the effect of placing one arbitrary combination of masses on a pan of a beam balance is compared with another arbitrary combination on the other pan. Measurement on a ratio scale follows from such axioms. In this paper, the essential character of simultaneous conjoint measurement is described by an axiomatization for the comparision of effects of (or responses to) pairs formed from two specified kinds of “quantities”. The axioms apply when, for example, the effect of a pair consisting of one mass and one difference in gravitational potential on a device that responds to momentum is compared with the effect of another such pair. Measurement on interval scales which have a common unit follows from these axioms; usually these scales can be converted in a natural way into ratio scales. A close relation exists between conjoint measurement and the establishment of response measures in a two-way table, or other analysis-of-variance situations, for which the “effects of columns” and the “effects of rows” are additive. Indeed, the discovery of such measures, which are well known to have important practical advantages, may be viewed as the discovery, via conjoint measurement, of fundamental measures of the row and column variables. From this point of view it is natural to regard conjoint measurement as factorial measurement.

Variations of Box Plots

Abstract Box plots display batches of data. Five values from a set of data are conventionally used; the extremes, the upper and lower hinges (quartiles), and the median. Such plots are becoming a widely used tool in exploratory data analysis and in preparing visual summaries for statisticians and nonstatisticians alike. Three variants of the basic display, devised by the authors, are described. The first visually incorporates a measure of group size; the second incorporates an indication of rough significance of differences between medians; the third combines the features of the first two. These techniques are displayed by examples.

A Projection Pursuit Algorithm for Exploratory Data Analysis

An algorithm for the analysis of multivariate data is presented and is discussed in terms of specific examples. The algorithm seeks to find one-and two-dimensional linear projections of multivariate data that are relatively highly revealing.

The Fitting of Power Series, Meaning Polynomials, Illustrated on Band-Spectroscopic Data

The prototype of fitting polynomials to equally-spaced data—in which the equalspacing is theoretically precise and the data is accurate to many decimal places—arises in the analysis of band spectra. A hard look at such examples forces us to reexamine our thinking on such diverse issues as: How to formulate such problems, the use of robust/resistant techniques in polynomial regression, which coordinates to use and why, the basic properties of linear least squares, choices in stopping a fit, and improved ways to describe our answers. Our results and attitudes apply rather directly to other situations where we are fitting a sum of functions of a single variable. When two or more different variables, subject to error, blunder, or omission, underlie the carriers to be considered, regression/fitting problems are likely to need not only the considerations presented here, but others as well. To a varying extent, the same will be true of nonlinear fitting/regression problems.

Testing the statistical certainty of a response to increasing doses of a drug.

Experiments in which the treatments are composed of a series of doses of a compound and a zero dose control are often used in animal toxicity studies. A test procedure is proposed to assess trends in the response variable. The notion of a no-statistical-significance-of-trend (NOSTASOT) dose is introduced, and questions of multiplicity of statistical tests in this context are addressed.

The Examination and Analysis of Residuals

A number of methods for examining the residuals remaining after a conventional analysis of variance or least-squares fitting have been explored during the past few years. These give information on various questions of interest, and in particular, aid in assessing the validity or appropriateness of the conventional analysis. The purpose of this paper is to make a variety of these techniques more easily available, so that they can be tried out more widely. Techniques of analysis, some graphical, some wholly numerical, and others mixed, are discussed in terms of the residuals that result from fitting row and column means to entries in a two-way array (or in several two-way arrays). Extensions to more complex situations, and some of the uses of the results of examination, are indicated.

Performance of Some Resistant Rules for Outlier Labeling

Abstract The techniques of exploratory data analysis include a resistant rule for identifying possible outliers in univariate data. Using the lower and upper fourths, FL and FU (approximate quartiles), it labels as “outside” any observations below FL − 1.5(FU — FL ) or above FU + 1.5(FU — FL ). For example, in the ordered sample −5, −2, 0, 1, 8, FL = −2 and FU = 1, so any observation below −6.5 or above 5.5 is outside. Thus the rule labels 8 as outside. Some related rules also use cutoffs of the form FL — k(FU — FL ) and FU + k(FU — FL ). This approach avoids the need to specify the number of possible outliers in advance; as long as they are not too numerous, any outliers do not affect the location of the cutoffs. To describe the performance of these rules, we define the some-outside rate per sample as the probability that a sample will contain one or more outside observations. Its complement is the all-inside rate per sample. We also define the outside rate per observation as the average fraction of outs...

The Bagplot: A Bivariate Boxplot

Abstract We propose the bagplot, a bivariate generalization of the univariate boxplot. The key notion is the half space location depth of a point relative to a bivariate dataset, which extends the univariate concept of rank. The “depth median” is the deepest location, and it is surrounded by a “bag” containing the n/2 observations with largest depth. Magnifying the bag by a factor 3 yields the “fence” (which is not plotted). Observations between the bag and the fence are marked by a light gray loop, whereas observations outside the fence are flagged as outliers. The bagplot visualizes the location, spread, correlation, skewness, and tails of the data. It is equivariant for linear transformations, and not limited to elliptical distributions. Software for drawing the bagplot is made available for the S-Plus and MATLAB environments. The bagplot is illustrated on several datasets—for example, in a scatterplot matrix of multivariate data.

We Need Both Exploratory and Confirmatory

Abstract We often forget how science and engineering function. Ideas come from previous exploration more often than from lightning strokes. Important questions can demand the most careful planning for confirmatory analysis. Broad general inquiries are also important. Finding the question is often more important than finding the answer. Exploratory data analysis is an attitude, a flexibility, and a reliance on display, NOT a bundle of techniques, and should be so taught. Confirmatory data analysis, by contrast, is easier to teach and easier to computerize. We need to teach both; to think about science and engineering more broadly; to be prepared to randomize and avoid multiplicity.

A Nonparametric Sum of Ranks Procedure for Relative Spread in Unpaired Samples

Abstract A nonparametric procedure is presented to test the null hypothesis that two independent samples come from the same population against the alternative hypothesis that the samples come from populations differing in variability or “spread.” Extensive tables of critical values are included for n 1≤n 2≤20. Large sample procedures are presented which include a correction for tied observations. The test is entirely distribution-free under the usual randomization procedures against the null hypothesis that the two distributions are identical. The absence of any normality assumption is a particularly important feature of the test, because its parametric alternative, the F test for variance differences, is quite sensitive to departures from normality. The test has the additional advantage of being directly applicable to non-numerical ordinal data.