Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

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14-36 CHAPTER 14 Bootstrap Methods and Permutation Tests page 14-19) is so far from normal that we are reluctant to use the bootstrap t or percentile confidence intervals. Now we will bootstrap the correlation coefficient. This is our first use of the bootstrap for a statistic that depends on two related variables. As with the difference of means, we must pay attention to how we should resample. Major League Baseball (MLB) owners claim they need limitations on EXAMPLE 14.9 player salaries to maintain competitiveness among richer and poorer teams. This argument assumes that higher salaries attract better players. Is there a relationship between an MLB player’s salary and his performance? Table 14.2 contains the names, 2002 salaries, and career batting averages of 50 randomly selected MLB players (excluding pitchers). 9 The scatterplot in Figure 14.15 suggests that the relationship between salary and batting average is weak. The sample correlation is r = 0.107. Is this small correlation significantly different from 0? To find out, we can calculate a 95% confidence interval for the population correlation and see whether or not it covers 0. If the confidence interval does not cover 0, the observed correlation is significant at the 5% level. TABLE 14.2 Major League Baseball salaries and batting averages Name Salary Average Name Salary Average Matt Williams $9,500,000 0.269 Greg Colbrunn $1,800,000 0.307 Jim Thome $8,000,000 0.282 Dave Martinez $1,500,000 0.276 Jim Edmonds $7,333,333 0.327 Einar Diaz $1,087,500 0.216 Fred McGriff $7,250,000 0.259 Brian L. Hunter $1,000,000 0.289 Jermaine Dye $7,166,667 0.240 David Ortiz $950,000 0.237 Edgar Martinez $7,086,668 0.270 Luis Alicea $800,000 0.202 Jeff Cirillo $6,375,000 0.253 Ron Coomer $750,000 0.344 Rey Ordonez $6,250,000 0.238 Enrique Wilson $720,000 0.185 Edgardo Alfonzo $6,200,000 0.300 Dave Hansen $675,000 0.234 Moises Alou $6,000,000 0.247 Alfonso Soriano $630,000 0.324 Travis Fryman $5,825,000 0.213 Keith Lockhart $600,000 0.200 Kevin Young $5,625,000 0.238 Mike Mordecai $500,000 0.214 M. Grudzielanek $5,000,000 0.245 Julio Lugo $325,000 0.262 Tony Batista $4,900,000 0.276 Mark L. Johnson $320,000 0.207 Fernando Tatis $4,500,000 0.268 Jason LaRue $305,000 0.233 Doug Glanville $4,000,000 0.221 Doug Mientkiewicz $285,000 0.259 Miguel Tejada $3,625,000 0.301 Jay Gibbons $232,500 0.250 Bill Mueller $3,450,000 0.242 Corey Patterson $227,500 0.278 Mark McLemore $3,150,000 0.273 Felipe Lopez $221,000 0.237 Vinny Castilla $3,000,000 0.250 Nick Johnson $220,650 0.235 Brook Fordyce $2,500,000 0.208 Thomas Wilson $220,000 0.243 Torii Hunter $2,400,000 0.306 Dave Roberts $217,500 0.297 Michael Tucker $2,250,000 0.235 Pablo Ozuna $202,000 0.333 Eric Chavez $2,125,000 0.277 Alexis Sanchez $202,000 0.301 Aaron Boone $2,100,000 0.227 Abraham Nunez $200,000 0.224
14.4 Bootstrap Confidence Intervals 14-37 Salary (in millions of dollars) 8 6 4 2 0 .200 .250 .300 .350 Batting average FIGURE 14.15 Career batting average and 2002 salary for a random sample of 50 Major League Baseball players. How shall we resample from Table 14.2? Because each observation consists of the batting average and salary for one player, we resample players (that is, observations). Resampling batting averages and salaries separately would lose the tie between a player’s batting average and his salary. Software such as S-PLUS automates proper resampling. Once we have produced a bootstrap distribution by resampling, we can examine the distribution and form a confidence interval in the usual way. We need no special formulas or procedures to handle the correlation. Figure 14.16 shows the bootstrap distribution and normal quantile plot for the sample correlation for 1000 resamples from the 50 players in our sample. The bootstrap distribution is close to normal and has small bias, so a 95% bootstrap t confidence interval appears reasonable. The bootstrap standard error is SE boot = 0.125. The t interval using the bootstrap standard error is r ± t ∗ SE boot = 0.107 ± (2.009)(0.125) = 0.107 ± 0.251 = (−0.144, 0.358) The 95% bootstrap percentile interval is (2.5% percentile, 97.5% percentile) = (−0.128, 0.356) = (0.107 − 0.235, 0.107 + 0.249) The two confidence intervals are in reasonable agreement. The confidence intervals give a wide range for the population correlation, and both include 0. These data do not provide significant evidence that there is a relationship between salary and batting average. A larger sample might result in a significant relationship, but the evidence from this sample suggests
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