Mark Schilling

Multivariate Two-Sample Tests Based on Nearest Neighbors

Abstract A new class of simple tests is proposed for the general multivariate two-sample problem based on the (possibly weighted) proportion of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample. Large values of the test statistics give evidence against the hypothesis H of equality of the two underlying distributions. Asymptotic null distributions are explicitly determined and shown to involve certain nearest neighbor interaction probabilities. Simple infinite-dimensional approximations are supplied. The unweighted version yields a distribution-free test that is consistent against all alternatives; optimally weighted statistics are also obtained and asymptotic efficiencies are calculated. Each of the tests considered is easily adapted to a permutation procedure that conditions on the pooled sample. Power performance for finite sample sizes is assessed in simulations.

Is Human Height Bimodal

The combined distribution of heights of men and women has become the canonical illustration of bimodality when teaching introductory statistics. But is this example appropriate? This article investigates the conditions under which a mixture of two normal distributions is bimodal. A simple justification is presented that a mixture of equally weighted normal distributions with common standard deviation σ is bimodal if and only if the difference between the means of the distributions is greater than 2σ. More generally, a mixture of two normal distributions with similar variability cannot be bimodal unless their means differ by more than approximately the sum oftheirstandard deviations. Examination of national survey data on young adults shows that the separation between the distributions of mens and womens heights is not wide enough to produce bimodality. We suggest reasons why histograms of height nevertheless often appear bimodal.

The Longest Run of Heads.

Mark F. Schilling is Associate Professor at California State University, Northridge. He received his BA and M.A. in mathe? matics at the University of California at San Diego and his doctorate was earned in statistics at the University of California at Berkeley in 1979 under the supervision of Peter J. Bickel. Schilling was employed at the University of Southern California prior to his appointment at C.S.U. Northridge. Dr. Schillings research interests include statistical methods for multidimensional data and the probabilistic behavior of repet? itive sequences. His hobbies include sports (and statistics), boomerang flying, music, and hiking.

Spatial designs when the observations are correlated

Suppose n observations are to be taken within a compact region, where the objective is to estimate the mean level of a multidimensional stationary process using the ordinary sample mean as the estimator. Simulated annealing is used to search for optimal (variance minimizing) designs for the case when the observations are correlated. The results give insight into the sensitivity of optimal designs to the strength and nature of the correlation present, extending and reinforcing previous results for the one-dimensional case. An important outcome is that designs which space out the sampling locations evenly are optimal if the correlation is low.

The Surprising Predictability of Long Runs

Summary When data arise from a situation that can be modeled as a collection of n independent Bernoulli trials with success probability p, a simple rule of thumb predicts the approximate length that the longest run of successes will have, often with remarkable accuracy. The distribution of this longest run is well approximated by an extreme value distribution. In some cases we can practically guarantee the length that the longest run will have. Applications to coin and die tossing, roulette, state lotteries and the digits of π are given.

A Coverage Probability Approach to Finding an Optimal Binomial Confidence Procedure

The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sternes procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.

Effect of age, parity, and race on the incidence of pregnancy associated hypertension and eclampsia in the United States.

PURPOSE To describe the incidence of pregnancy associated hypertension and eclampsia from adolescence through the fifth decade of life, including the effect of parity and race, in the United States. METHODS Data were evaluated from the National Center for Health Statistics (vital statistics section). The data were stratified by maternal age group, parity (G1, first pregnancy; G2+, second or higher pregnancy), and racial group. RESULTS The incidence of pregnancy associated hypertension (PAH) decreased with increased age in late adolescence in the G2+ group but not the G1 group (total and all racial groups). The incidence of PAH was significantly greater for non-Hispanic black or non-Hispanic white than Hispanic groups for all age groups (P⩽.02) except age ⩽15years (G2+ group) and 45-54years (both G1 and G2+ groups). The incidence of eclampsia decreased with increased age in late adolescence in the G2+ group (total and all racial groups) and the G1 group (total and non-Hispanic black groups). The incidence of eclampsia was significantly greater for non-Hispanic black than non-Hispanic white and for non-Hispanic white than Hispanic groups for all age groups except age ⩽15years in the G2+ group. The incidence of PAH and eclampsia increased substantially in both G1 and G2+ groups in the fifth decade of life (total and all racial groups). CONCLUSIONS The incidence of PAH (G2+ group) and eclampsia (G1 and G2+ groups) decreased with increased age during adolescence and increased in the fifth decade (G1 and G2+ groups).

A Suggestion for Sunflower Plots

Abstract Although sunflower plots are highly effective for displaying bivariate data with coincident observations, they possess certain disadvantages involving graphical perception of data. Moreover, sunflower plots for data that arrive “on line” can be updated only by completely redrawing the affected sunflowers. We propose a variation of the traditional sunflower plot that addresses these issues.

Understanding Probability: Chance Rules in Everyday Life

Baseball is fertile ground for statistical analysis. Not only does the ASA have a section on Statistics in Sports, but there is an entire organization, the Society for American Baseball Research (www.sabr.org), dedicated to the analysis of baseball data. This interest of statisticians in baseball is reciprocated, and the recent success of the Oakland A’s (chronicled by Lewis 2003) is emblematic of the trend toward employing statistics in franchise operations. In Baseball’s All-Time Best Hitters, Michael Schell applies his statistical knowledge to determine the best hitters in the sport. Although it is not intended as a “teaching statistics with baseball” book, the author succeeds in presenting statistical ideas which are subtle yet easily understood by a fan with little or no statistical acumen. For example, Schell’s research shows that left-handed batters hit for higher average than right-handed hitters (p. 19). But after accounting for position (lefties are disproportionately placed at first base or outfield—positions which are less demanding defensively and are expected to produce more offensively), there seems to be little difference (p. 23). The book can be divided into two main sections. The first half focuses on the statistical methodology used to “level the playing field.” The author’s underlying assumption is that the population of baseball talent is roughly normal, and thus individual performances can be standardized by comparing a batter’s performance to those of his contemporaries. This approach is markedly different than the one chosen by Berry, Reese, and Larkey (1999), who used a Bayesian bridge to compare athletes to contemporaries and contemporaries’ contemporaries, and so on. In the book’s many analyses, Schell considers only players who achieved a minimum of 8,000 at-bats (4,000 for retired players), a decision he explains in Chapter 1. Subsequent chapters detail the methods used to adjust the performance of qualifying players. Factors considered are late-career declines (chapter 2), era of play (chapter 3), league-wide talent (chapter 4), and ballpark effects (chapter 5). Although a statistical novice may be tempted to skip these chapters, doing so would be a mistake, as they contain a meticulously detailed history of baseball’s evolution. In particular, the discussion of various rule changes (chapter 3) is fascinating. For instance, I was surprised to learn that in 1876, batters could request pitches in a given location, and pitchers were required to oblige them—underhanded (p. 47)! The second half of the book discusses the author’s findings in great detail. Schell presents player rankings based on career performance (chapter 6), performance by defensive position (chapter 7), and single-season performance (chapter 8). In addition to batter-specific results, he examines the effect of various ballparks on hitting (chapter 9), offers his thoughts on the Hall of Fame (chapter 11), and compares current (as of 1999) players to all-time greats (chapter 12). The closing chapter is particularly interesting, as enough time has passed to reexamine those on the list. For example, Mike Piazza (6th) and Frank Thomas (12th), two of the game’s most feared hitters in 1999, have suffered dramatic declines in their production during the intervening years. Baseball’s All-Time Best Sluggers picks up where its predecessor left off, finetuning the adjustments in Baseball’s All-Time Best Hitters and examining other offensive events. To complement the adjustments used in the first book, Schell introduces a power transformation to correct for skewness and kurtosis (p. 24). But man does not live by power transformations alone, and the real reason to read this book is the expanded list of offensive categories. While Hitters focused on base-hits and tended to favor notable singles hitters like Tony Gwynn (in Schell’s opinion, the best ever), Sluggers considers all kinds of baseball output—batting average, doubles-plus-triples, triples, home runs, runs scored, runs batted in (RBIs), walks, strikeouts, stolen bases, and hit-by-pitches (p. 2). Like Hitters, this book is roughly split between sections describing methods (Chapters 2 through 6) and findings (Chapters 7 through 12). By and large, the methods used to normalize sluggers are the same as those introduced for hitters. Schell again accounts for aging, era, talent pool, and home ballpark. Readers will certainly gravitate toward Schell’s ranking of home run hitters. However, it is the breadth of analyses that makes Sluggers so enjoyable. Though stolen bases are not as common in the modern game as in days past, Schell painstakingly lists all-time base-stealers despite conceding the difficulty in doing so. “The key complication is the nonnormality of stolen base data, especially since players have failed to steal a single base in more than 10% of regular player seasons since 1947” (p. 163). However, this difficulty does not stop Schell from examining the stolen base credentials of the game’s greats. While few would dispute Rickey Henderson, Luis Aparicio, and Vince Coleman being among the best base-stealers, there are a few surprises in the top-10 list (p. 166). For instance, journeymen Otis Nixon (6th place) and Tom Goodwin (8th) were more productive base-stealers over their careers than Hall of Famers Ty Cobb (9th) and Lou Brock (10th), even though both Cobb and Brock were known for their stolen bases. Although it is difficult to criticize Schell’s completeness, it would be nice to see a follow-on to Table 11.3 on p. 152 listing easy strikeout victims. Though not as compelling, perhaps, as debating home-run prowess, some (I confess, I’m one) might find a discussion of prodigious strikeout totals interesting. As an example, in 2002 Milwaukee manager, Jerry Royster, benched Jose Hernandez (known affectionately in my household as “the human windmill”) during the season’s final week to prevent him from breaking Bobby Bonds’s single-season strikeout mark. (Bonds had 189 strikeouts in 1970; Hernandez finished with 188.) Adam Dunn subsequently broke the record, striking out 195 times in 2004. Whose season was worse? (I suspect Hernandez’s since Dunn garnered many more walks and pounded more home-runs.) In fairness to Schell, he lists the best strikeout performances (fewest from a hitter’s perspective) and the title of the book implies that the focus is on good performances, not bad ones. So one can hardly fault him for this omission. While statistically inclined readers will enjoy debating Schell’s methods, the books’ raison d’etre is their ranking of great single-season and career performances. Even those who disagree with the author’s methods or conclusions (or both) will no doubt concede that these books are thoroughly researched and thoughtfully written. Far from dry, Schell sprinkles anecdotes throughout which give the reader an appreciation for both the difficulty in quantifying performance and the greatness of many of baseball’s best. For example, Schell notes that Ted Williams’s plate-discipline and high walk total cost him the batting title in 1954 (Sluggers, p. 155). With its lucid prose and detailed presentations, statisticians and casual baseball fans alike will enjoy Baseball’s All-Time Best Hitters and Baseball’s All-Time Best Sluggers.

Measuring Diversity in the United States

D iversity is a word we hear frequently these days. But what does it actually mean? Sociologically, the word is used in reference to the number and degree of representation of racial and ethnic groups in a university, a city neighborhood, and so forth. Still, the notion is somewhat vague. What do people really have in mind when they say, for example, that the United States is more diverse than it has been in the past? In order to come up with some sort of mathematical definition of diversity, consider a population of individuals comprised of k groups that are represented in the population in proportions P;. i = l, 2, ... , k. A reasonable objective is to come up with some function of the p;s that measures the extent to which the population is spread across these groups. If the groups had an ordinal relationship where we could assign values to them corresponding to a numerical scale, then one possible measure would be the variance of the distribution, or equivalently its square root, the standard deviation. But data on race and ethnicity are not ordinal, so these measures make no sense here. The definitions of race and ethnicity used by the Census Bureau are influenced greatly by self-identification and do not represent any clear-cut scientific defmition of biological stock. In fact, the in the United States