Neil C. Schwertman

Is the Property of Being Positively Correlated Transitive

Suppose that X, Y, and Z are random variables and that X and Y are positively correlated and that Y and Z are likewise positively correlated. Does it follow that X and Z must be positively correlated? As we shall see by example, the answer is (perhaps surprisingly) “no.” We prove, though, that if the correlations are sufficiently close to 1, thenX and Z must be positively correlated. We also prove a general inequality that relates the three correlations. The ideas should be accessible to students in a first (postcalculus) course in probability and statistics.

OPTIMAL LIMITS FOR ATTRIBUTES CONTROL CHARTS

Attributes control charts have historically been used with 3-sigma limits. When such an approach is used there is the implicit assumption that the normal approximation to the binomial and Poisson distributions will be adequate. Control chart properties ..

More Probability Models for the NCAA Regional Basketball Tournaments

Abstract In this department The American Statistician publishes articles, reviews, and notes of interest to teachers of the first mathematical statistics course and of applied statistics courses. The department includes the Accent on Teaching Materials section; suitable contents for the section are described under the section heading. Articles and notes for the department, but not intended specifically for the section, should be useful to a substantial number of teachers of the indicated types of courses or should have the potential for fundamentally affecting the way in which a course is taught.

A simple more general boxplot method for identifying outliers

Abstract The boxplot method (Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977) is a graphically-based method of identifying outliers which is appealing not only in its simplicity but also because it does not use the extreme potential outliers in computing a measure of dispersion. The inner and outer fences are defined in terms of the hinges (or fourths), and therefore are not distorted by a few extreme values. Such distortion could lead to failing to detect some outliers, a problem known as “masking”. A method for determining the probability associated with any fence or observation is proposed based on the cumulative distribution function of the order statistics. This allows the statistician to easily assess, in a probability sense, the degree to which an observation is dissimilar to the majority of the observations. In addition, an adaptation for approximately normal but somewhat asymmetric distributions is suggested.

Smoothing an indefinite variance-covariance matrix

The usual estimator of the dispersion matrix has a distinct advantage over other estimation procedures since it is computationally feasible for a data set with a substantial number of missing observations. However, this estimator, when the data vectors have some missing elements, may not have the required property of being at least positive semidefinite. The smoothing procedure suggested in this paper rectifies this deficiency. In addition, com-putational procedures are proposed. The smoothing procedure is illustrated by an example and a Monte Carlo experiment shows that smoothing substantially increases the power of the test proposed by Kleinbaum(1973).

Can the NCAA Basketball Tournament Seeding be Used to Predict Margin of Victory

Abstract Following the announcement by the NCAA of the seeding and placement of mens basketball teams in the regional tournaments there is often much discussion among basketball afficionados of the fairness. A statistical analysis of simple regression models for the tournament games shows that indeed there is a strong association between the seed positions of the teams and the actual margin of victory; in fact, fairly reliable prediction models of actual margin of victory in tournament games can be achieved based primarily on the seed numbers alone.

A Note on the Geisser-Greenhouse Correction for Incomplete Data Split-Plot Analysis

Abstract Growth curve, wear curve, and repeated measurement experiments frequently are analyzed as split-plot designs with time used as the subplot treatment. This method is particularly convenient when the data have missing values in the repeated measurements since multivariate methods are not easily adapted to data with different dispersion matrices. This article establishes that the dispersion structure which guarantees that the subplot test ratio has an F distribution for complete data applies to incomplete data as well.

Implementing Optimal Attributes Control Charts

We present a FORTRAN program that can be used for determining control limits for a p, np, c, or u chart such that, assuming known parameter values, the absolute deviation between the average run length and a specified nomial value is minimized for each ..

A Simple Noncalculus Proof That the Median Minimizes the Sum of the Absolute Deviations

It is widely known among statisticians that the median minimizes the sum of the absolute deviation about any point for a set of x’s, xl, x2, x3, . . . , x,. Some authors (e.g., David 1970) point out that when n is even the minimizing point is not necessarily unique. The proof that the median minimizes the sum of the absolute deviations is omitted in many mathematical statistics textbooks. Other textbooks (e.g., Bickel and Dobsum 1977, p. 54; Cramer 1946, p. 179; De Groot 1975, p. 170; Dwass 1970, p. 341; Von Mises 1964, pp. 373-374) suggest or prove the result using expectation and integral calculus for continuous data. Wasan (1970, p. 119) used a similar expectation argument for discrete distributions only. Sposito, Smith, and McCormick (1978) provided a somewhat involved proof using summations. Bloomfield and Steiger (1983) provided a rather difficult and more general investigation of the minimization of the general Lp norm that, when p = 1, proves the median minimizes the sum of the absolute derivations. A somewhat simplified calculus proof was given by Shad (1969). Aitken (1952, pp. 32-34) provided a clever proof with only minor use of calculus but did not provide a convenient computational procedure. Some authors, such as Gentle, Sposito, and Kennedy (1977), have used the absolute deviation for various applications, whereas the general L, norm has been used by others. For example, Sielken and Hartley (1973) and Sposito (1982) showed how one can obtain unbiased L , and Lp estimators. First consider some difficulties that must be overcome when using differentiation in the proof. From any set of x’s construct the ordered set x(’), x ( ~ ) , . . . , x(,), where x ( ~ ) 5 X(2) 5 * * * 5 X(,), and define the sum of the absolute deviations about any point, say a, as D ( a ) = I x ( ~ ) a]. A proof using differentiation of D ( a ) requires considerable care, since D(a) is nondifferentiable at a = x(~) (i = 1, 2, . . . , n). In addition, D ( a ) = Xf=l ( a x(~)) + (x(~) a) = (2k n)a Cf=, x0) + x(~), where x ( ~ ) I a 5 x ( ~ + I ) . The fact that k and each of the two summations are functions of a must be considered in all differentiations. The following noncalculus proof is a simple alternative, based on sets, that is appropriate for continuous or discrete populations, readily demonstrates the nonuniqueness for even n, and provides a convenient method of computation. Consequently, this proof should be easier for most students to understand.

A monte carlo study of alternative procedures for testing the hypothesis of parallelism for complete and incomplete growth curve data

Data collected over time on the same experimental unit, frequently called growth curve data, is typical of many clinical, biological, medical and agricultural studies. Such data is usually highly correlated and may be difficult to analyze if there are missing observations. Monte Carlo simulations using a broad spectrum of dispersion structures are used to compare for significance level and power of various procedures for testing the parallelism of the response curves for both complete and incomplete growth curve data. The various analysis methods used are (1) the split-plot; (2) Hotellings T-square; (3) analysis of the estimated regression coefficients for each experimental unit by Hotellings T-square; (4) successive differences; and (5) estimation of missing data then using the procedures 1 through 4; and (6), adjusting these procedures using the Geisser-Greenhouse correction as appropriate. Of these methods, for complete data the split-plot analysis using the Geisser-Greenhouse correction was most sat...