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Modelling Rankings in R: the Plackettluce Package
Computational Statistics (2020) 35:1027–1057 https://doi.org/10.1007/s00180-020-00959-3 ORIGINAL PAPER Modelling rankings in R: the PlackettLuce package Heather L. Turner1 · Jacob van Etten2 · David Firth1,3 · Ioannis Kosmidis1,3 Received: 30 October 2018 / Accepted: 18 January 2020 / Published online: 12 February 2020 © The Author(s) 2020 Abstract This paper presents the R package PlackettLuce, which implements a generalization of the Plackett–Luce model for rankings data. The generalization accommodates both ties (of arbitrary order) and partial rankings (complete rankings of subsets of items). By default, the implementation adds a set of pseudo-comparisons with a hypothetical item, ensuring that the underlying network of wins and losses between items is always strongly connected. In this way, the worth of each item always has a finite maximum likelihood estimate, with finite standard error. The use of pseudo-comparisons also has a regularization effect, shrinking the estimated parameters towards equal item worth. In addition to standard methods for model summary, PlackettLuce provides a method to compute quasi standard errors for the item parameters. This provides the basis for comparison intervals that do not change with the choice of identifiability constraint placed on the item parameters. Finally, the package provides a method for model- based partitioning using covariates whose values vary between rankings, enabling the identification of subgroups of judges or settings with different item worths. The features of the package are demonstrated through application to classic and novel data sets. This research is part of cooperative agreement AID-OAA-F-14-00035, which was made possible by the generous support of the American people through the United States Agency for International Development (USAID). -
Points Report Talladega Superspeedway UAW-Ford 500 UNOFFICIAL Provided by NASCAR Statistical Services - Sun, Oct 2, 2005 @ 08:09 PM Eastern UNOFFICIAL
Points Report Talladega Superspeedway UAW-Ford 500 UNOFFICIAL Provided by NASCAR Statistical Services - Sun, Oct 2, 2005 @ 08:09 PM Eastern UNOFFICIAL Pos Driver BPts Points -Ldr -Nxt Starts Poles Wins T5s T10s DNFs Money Won PPos G/L 1 Tony Stewart 120 5519 0 0 29 2 5 14 20 1 $5,829,335 5 4 2 Ryan Newman 80 5515 -4 -4 29 6 1 8 13 3 $4,590,956 3 1 3 Rusty Wallace 35 5443 -76 -72 29 0 0 8 16 0 $4,048,697 2 -1 4 Jimmie Johnson 80 5437 -82 -6 29 1 3 10 17 4 $5,622,476 1 -3 5 Greg Biffle 115 5421 -98 -16 29 0 5 11 16 1 $4,563,902 6 1 6 Carl Edwards 40 5419 -100 -2 29 1 2 9 12 1 $3,512,909 8 2 7 Matt Kenseth 75 5408 -111 -11 29 1 1 8 13 4 $4,595,182 9 2 8 Jeremy Mayfield 50 5407 -112 -1 29 0 1 4 8 1 $3,831,461 7 -1 9 Mark Martin 50 5381 -138 -26 29 0 0 7 14 2 $4,593,193 4 -5 10 Kurt Busch 105 5339 -180 -42 29 0 3 8 15 3 $5,721,366 10 0 11 Kevin Harvick 45 3321 0 0 29 2 1 3 9 1 $4,098,812 12 1 12 Jamie McMurray 10 3307 -14 -14 29 1 0 4 8 3 $3,309,124 13 1 13 Elliott Sadler 55 3288 -33 -19 29 3 0 1 10 2 $4,077,999 11 -2 14 Dale Jarrett 15 3270 -51 -18 29 1 1 4 6 1 $3,859,097 15 1 15 Joe Nemechek 50 3222 -99 -48 29 1 0 1 8 2 $3,437,038 16 1 16 Jeff Gordon 65 3202 -119 -20 29 2 3 5 9 8 $5,686,421 14 -2 17 Brian Vickers 55 3187 -134 -15 29 1 0 5 10 3 $3,363,819 18 1 18 Dale Earnhardt Jr. -
2, Jack Sprague, Mike Skinner Most Poles: 3, Jack Sprague, Mike Skinner Most Top 5S: 7, Jack Sprague Most Top 10S: 9, Ron Hornaday, Matt Crafton
Miscellaneous NCWTS Records at LVMS Most wins: 2, Jack Sprague, Mike Skinner Most poles: 3, Jack Sprague, Mike Skinner Most Top 5s: 7, Jack Sprague Most Top 10s: 9, Ron Hornaday, Matt Crafton. Most laps led (career): 318, Jack Sprague Most laps led: (race): 114, Todd Bodine (9/24/05), Mike Skinner (9/23/06) Most laps led (winner): 114, Todd Bodine (9/24/05), Mike Skinner (9/23/06) Fewest laps led (winner): 2, Shane Hmiel (9/25/04) Most laps led (non-winner): 104, Jack Sprague (10/14/01) Most NCWTS starts LVMS: 16, Matt Crafton Best start for winner: 1st, Sprague (11/8/98), Starr (10/13/02), Gaughan (9/27/03), Skinner (9/23/06), Kvapil (9/22/07), Dillon (9/25/10), Hornaday (10/15/11) Worst start for winner: 21st, Shane Hmiel (9/25/04) Youngest winner: Erik Jones (2014) 18 years, 3 months, 29 days Oldest winner: Ron Hornaday (2011) 55 years, 3 months, 25 days Qualifying record: Mike Skinner (2006) 30.326 seconds, 178.065 mph Best finish for rookie: 1, Johnny Sauter (9/26/09), Austin Dillon (9/25/10) Best start for rookie: 1, Bryan Reffner (11/3/96), Austin Dillon (9/25/10) Most rookies in field: 18, 9/25/10 (A. Dillon, Lofton, Mayhew, Bowles, Jackson, J. Earnhardt, Gosselin, Piquet, Cobb, C. Long, Greenfield, Karthikeyan, Raymer, Fenton, Pursley, Garvey, Hobgood, J. Long) Most running at finish: 32, 11/8/98, 9/25/04 Fewest running at finish: 18, 9/29/12 Slowest race (speed): 101.070 mph (9/20/08) Fastest race (speed): 143.163 (10/1/16) Most cautions: 12 (9/20/08) Fewest cautions: 2 (10/13/02) Driver Records Driver Starts Wins Poles Top -
Club Racing Media Guide and Record Book
PLAYGROUND EARTH BEGINS WHERE YOUR DRIVEWAY ENDS. © 2013 Michelin North America, Inc. BFGoodrich® g-ForceTM tires bring track-proven grip to the street. They have crisp steering response, sharp handling and predictable feedback that bring out the fun of every road. They’re your ticket to Playground EarthTM. Find yours at bfgoodrichtires.com. Hawk Performance brake pads are the most popular pads used in the Sports Car Club of America (SSCA) paddock. For more, visit us at www.hawkperformance.com. WHAT’SW STOPPING YOU? Dear SCCA Media Partners, Welcome to what is truly a new era of the Sports Car Club of America, as the SCCA National Champi- onship Runoffs heads west for the first time in 46 years to Mazda Raceway Laguna Seca. This event, made possible with the help of our friends and partners at Mazda and the Sports Car Rac- ing Association of the Monterey Peninsula (SCRAMP), was met with questions at its announcement that have been answered in a big way, with more than 530 drivers on the entry list and a rejunvenaton of the west coast program all season long. While we haven’t been west of the Rockies since River- side International Raceway in 1968, it’s hard to believe it will be that long before we return again. The question on everyone’s mind, even more than usual, is who is going to win? Are there hidden gems on the west coast who may be making their first Runoffs appearance that will make a name for themselves on the national stage, or will the traditional contenders learn a new track quickly enough to hold their titles? My guess is that we’ll see some of each. -
Atlantic News
This Page © 2004 Connelly Communications, LLC, PO Box 592 Hampton, NH 03843- Contributed items and logos are © and ™ their respective owners Unauthorized reproduction 2 of this page or its contents for republication in whole or in part is strictly prohibited • For permission, call (603) 926-4557 • AN-Mark 9A-EVEN- Rev 12-16-2004 PAGE 2A | ATLANTIC NEWS | JUNE 30, 2006 | VOL 31, NO 25 ATLANTICNEWS.COM . WEATHER THE WEEKEND FRIDAY, JUNE 30 SATURDAY, JULY 1 SUNDAY, JULY 2 MONDAY, JULY 3 CHANCE OF LIGHTNING CHANCE OF THUNDER CHANCE OF CLOUDS PARTLY CLOUDY HIGH: 77° LOW: 63° HIGH: 75° LOW: 61° HIGH: 70S°LOW: 60S° HIGH: 70’S°LOW: 60’S° Cyan NEWBURYPORT’S HAMPTON HARBOR’S PORTSMOUTH HARBOR’S DOVER POINT’S Magenta 10 DAY TIDES 10 DAY TIDES 10 DAY TIDES 10 DAY TIDES HIGH TIDE LOW TIDE HIGH TIDE LOW TIDE HIGH TIDE LOW TIDE HIGH TIDE LOW TIDE Tides provided by DATE AM PM AM PM . DATE AM PM AM PM . DATE AM PM AM PM . DATE AM PM AM PM . Yellow www.MaineHarbors.com 30 FRI 3:18 3:58 10:23 10:32 30 FRI 3:09 3:49 9:29 9:38 30 FRI 3:01 3:41 9:44 9:53 30 FRI 4:20 5:00 10:39 10:48 1SAT 3:51 4:29 10:17 10:32 1SAT 3:45 4:23 10:25 10:40 1SAT 3:53 4:31 10:10 10:25 1SAT 5:04 5:42 11:20 11:35 Black 2SUN 4:36 5:12 10:58 11:21 2SUN 4:30 5:06 11:06 11:29 2SUN 4:38 5:14 10:51 11:14 2SUN 5:49 6:25 - 12:01 3MON 5:24 5:56 11:40 - 3MON 5:18 5:50 11:48 - 3MON 5:26 5:58 11:33 - 3MON 6:37 7:09 12:24 12:43 4TUE 6:14 6:42 12:12 12:25 4TUE 6:08 6:36 12:20 12:33 4TUE 6:16 6:44 12:05 12:18 4TUE 7:27 7:55 1:15 1:28 IRST UARTER 5WED 7:08 7:30 1:06 1:12 5WED 7:02 7:24 1:14 1:20 -
Points Report Martinsville Speedway Subway 500 UNOFFICIAL Provided by NASCAR Statistical Services - Sun, Oct 22, 2006 @ 05:07 PM Eastern UNOFFICIAL
Points Report Martinsville Speedway Subway 500 UNOFFICIAL Provided by NASCAR Statistical Services - Sun, Oct 22, 2006 @ 05:07 PM Eastern UNOFFICIAL Pos Driver BPts Points -Ldr -Nxt Starts Poles Wins T5s T10s DNFs Money Won PPos G/L 1 Matt Kenseth 125 5848 0 0 32 0 4 14 19 1 $6,016,155 2 1 2 Kevin Harvick 70 5812 -36 -36 32 1 4 12 17 1 $5,420,509 3 1 3 Jimmie Johnson 80 5807 -41 -5 32 1 5 10 20 1 $7,961,024 7 4 4 Denny Hamlin # 75 5801 -47 -6 32 3 2 6 16 1 $3,839,097 6 2 5 Jeff Burton 85 5800 -48 -1 32 4 1 7 19 2 $4,562,952 1 -4 6 Dale Earnhardt Jr. 85 5754 -94 -46 32 0 1 9 14 3 $4,834,715 5 -1 7 Mark Martin 60 5752 -96 -2 32 0 0 7 14 1 $3,884,723 4 -3 8 Kasey Kahne 80 5749 -99 -3 32 5 6 11 17 4 $5,648,016 8 0 9 Jeff Gordon 125 5707 -141 -42 32 1 2 13 15 7 $5,372,879 10 1 10 Kyle Busch 100 5677 -171 -30 32 1 1 9 17 2 $4,359,843 9 -1 11 Tony Stewart 105 4108 0 0 32 0 3 13 17 4 $6,149,737 11 0 12 Carl Edwards 65 3862 -246 -246 32 0 0 9 17 3 $4,116,401 12 0 13 Greg Biffle 110 3616 -492 -246 32 2 1 6 13 5 $3,947,866 13 0 14 Casey Mears 10 3537 -571 -79 32 0 0 2 7 1 $4,928,481 15 1 15 Brian Vickers 45 3483 -625 -54 32 0 1 5 9 2 $3,535,676 15 0 16 Kurt Busch 75 3456 -652 -27 32 6 1 7 10 2 $4,513,382 16 0 17 Ryan Newman 60 3402 -706 -54 32 2 0 2 7 3 $4,881,165 18 1 18 Clint Bowyer # 35 3387 -721 -15 32 0 0 3 9 3 $3,497,787 17 -1 19 Scott Riggs 20 3204 -904 -183 31 2 0 1 7 1 $2,898,216 20 1 20 Bobby Labonte 35 3203 -905 -1 32 0 0 3 8 7 $4,178,676 23 3 21 Martin Truex Jr. -
Points Report Watkins Glen International Sirius Satellite Radio
Points Report Watkins Glen International Sirius Satellite Radio at the Glen UNOFFICIAL Provided by NASCAR Statistical Services - Mon, Aug 15, 2005 @ 11:58 AM Eastern UNOFFICIAL Pos Driver BPts Points -Ldr -Nxt Starts Poles Wins T5s T10s DNFs Money Won PPos G/L 1 Tony Stewart 95 3113 0 0 22 1 5 10 14 1 $4,741,476 1 0 2 Jimmie Johnson 70 3008 -105 -105 22 1 2 9 14 3 $4,624,235 2 0 3 Greg Biffle 85 2861 -252 -147 22 0 5 7 11 1 $3,677,698 3 0 4 Rusty Wallace 30 2855 -258 -6 22 0 0 5 12 0 $3,176,469 4 0 5 Mark Martin 30 2782 -331 -73 22 0 0 6 12 1 $3,896,003 6 1 6 Kurt Busch 65 2692 -421 -90 22 0 2 7 11 3 $4,673,753 5 -1 7 Jeremy Mayfield 25 2684 -429 -8 22 0 0 3 5 1 $2,987,013 8 1 8 Ryan Newman 65 2646 -467 -38 22 5 0 5 10 2 $3,573,815 7 -1 9 Jamie McMurray 5 2599 -514 -47 22 1 0 4 7 2 $2,655,488 11 2 10 Carl Edwards 25 2593 -520 -6 22 0 2 6 8 1 $2,748,798 10 0 11 Elliott Sadler 35 2590 -523 -3 22 2 0 1 9 2 $3,267,615 12 1 12 Dale Jarrett 10 2590 -523 0 22 1 0 3 5 0 $3,029,074 9 -3 13 Jeff Gordon 55 2526 -587 -64 22 2 3 5 8 6 $4,860,850 14 1 14 Kevin Harvick 25 2523 -590 -3 22 1 1 3 6 0 $3,213,445 13 -1 15 Dale Earnhardt Jr. -
Post-Race Report
Loop Data Statistics Post-Race Report Dodge/Save Mart 350 June 25, 2006 Provided by STATS LLC and NASCAR - Tuesday, June 24, 2008 NASCAR Nextel Cup Series Average Running Position Sum of driver position on each lap - divided by the laps run in the race. Dodge/Save Mart 350 Infineon Raceway June 25, 2006 Car Finish Average Rk. Number Driver Team Pos. Place 1 12 Ryan Newman alltel 2 2.945 2 24 Jeff Gordon Dupont 1 3.464 3 2 Kurt Busch Miller Lite 5 3.927 4 16 Greg Biffle National Guard/Subway 4 5.873 5 60 Boris Said SoBe/No Fear 9 7.173 6 20 Tony Stewart The Home Depot 28 8.109 7 48 Jimmie Johnson Lowe's 10 9.291 8 6 Mark Martin AAA 13 10.109 9 29 Kevin Harvick GM Goodwrench 24 11.064 10 31 Jeff Burton Cingular Wireless 7 13.991 11 17 Matt Kenseth DEWALT 17 14.182 12 5 Kyle Busch Kellogg's 11 14.591 13 38 Elliott Sadler Pedigree 8 14.964 14 99 Carl Edwards Office Depot 6 15.418 15 01 Joe Nemechek U.S. Army 25 16.518 16 9 Kasey Kahne Dodge Dealers/UAW 31 16.591 17 96 Terry Labonte DLP HDTV 3 18.436 18 7 Robby Gordon Menards/Energizer 40 18.727 19 25 Brian Vickers GMAC 14 19.518 20 1 Martin Truex Jr. Bass Pro Shops/Tracker Boats 15 21.173 21 8 Dale Earnhardt Jr. Budweiser 26 23.373 22 40 Scott Pruett Coors Light 30 23.845 23 11 Denny Hamlin FedEx Freight 12 24.155 24 19 Jeremy Mayfield Dodge Dealers/UAW 22 24.345 25 26 Jamie McMurray Crown Royal 18 25.500 26 07 Clint Bowyer Jack Daniel's 16 25.955 27 66 Jeff Green Best Buy 19 26.336 28 4 P.J. -
Modelling Rankings in R: the Plackettluce Package
Computational Statistics https://doi.org/10.1007/s00180-020-00959-3 ORIGINAL PAPER Modelling rankings in R: the PlackettLuce package Heather L. Turner1 · Jacob van Etten2 · David Firth1,3 · Ioannis Kosmidis1,3 Received: 30 October 2018 / Accepted: 18 January 2020 © The Author(s) 2020 Abstract This paper presents the R package PlackettLuce, which implements a generalization of the Plackett–Luce model for rankings data. The generalization accommodates both ties (of arbitrary order) and partial rankings (complete rankings of subsets of items). By default, the implementation adds a set of pseudo-comparisons with a hypothetical item, ensuring that the underlying network of wins and losses between items is always strongly connected. In this way, the worth of each item always has a finite maximum likelihood estimate, with finite standard error. The use of pseudo-comparisons also has a regularization effect, shrinking the estimated parameters towards equal item worth. In addition to standard methods for model summary, PlackettLuce provides a method to compute quasi standard errors for the item parameters. This provides the basis for comparison intervals that do not change with the choice of identifiability constraint placed on the item parameters. Finally, the package provides a method for model- based partitioning using covariates whose values vary between rankings, enabling the identification of subgroups of judges or settings with different item worths. The features of the package are demonstrated through application to classic and novel data sets. This research is part of cooperative agreement AID-OAA-F-14-00035, which was made possible by the generous support of the American people through the United States Agency for International Development (USAID). -
Starting Line up by Row Infineon Raceway Dodge/Save Mart 350 Provided by NASCAR Statistical Services - Fri, Jun 23, 2006 @ 8:55 PM Eastern
Starting Line Up by Row Infineon Raceway Dodge/Save Mart 350 Provided by NASCAR Statistical Services - Fri, Jun 23, 2006 @ 8:55 PM Eastern Driver Date Time Speed Track Race Record: Ricky Rudd 6/23/02 02:42:08 81.007 Pos Car Driver Team Time Speed Row 1: 1 2 Kurt Busch Miller Lite Dodge 76.987 93.055 2 26 Jamie McMurray Crown Royal Ford 77.075 92.948 Row 2: 3 29 Kevin Harvick GM Goodwrench Chevrolet 77.124 92.889 4 12 Ryan Newman Alltel Dodge 77.188 92.812 Row 3: 5 60 Boris Said Sobe-No Fear Ford 77.312 92.663 6 9 Kasey Kahne Dodge Dealers/UAW Dodge 77.453 92.495 Row 4: 7 16 Greg Biffle National Guard/Subway Ford 77.610 92.308 8 6 Mark Martin AAA Ford 77.621 92.295 Row 5: 9 17 Matt Kenseth DeWalt Ford 77.679 92.226 10 01 Joe Nemechek U.S. Army Chevrolet 77.736 92.158 Row 6: 11 24 Jeff Gordon DuPont Chevrolet 77.797 92.086 12 20 Tony Stewart The Home Depot Chevrolet 77.805 92.076 Row 7: 13 31 Jeff Burton Cingular Wireless Chevrolet 77.847 92.027 14 7 Robby Gordon Menards/Energizer Chevrolet 77.848 92.025 Row 8: 15 4 P.J. Jones AERO Exhaust Chevrolet 77.853 92.020 16 48 Jimmie Johnson Lowe's Chevrolet 77.960 91.893 Row 9: 17 32 Ron Fellows Tide/Downy Chevrolet 77.989 91.859 18 21 Ken Schrader Little Debbie Ford 78.008 91.837 Row 10: 19 5 Kyle Busch Kellogg's Chevrolet 78.021 91.821 20 99 Carl Edwards Office Depot Ford 78.102 91.726 Row 11: 21 43 Bobby Labonte Cheerios/Betty Crocker Dodge 78.138 91.684 22 88 Dale Jarrett UPS Ford 78.172 91.644 Row 12: 23 66 Jeff Green Best Buy Chevrolet 78.222 91.585 24 38 Elliott Sadler Pedigree Ford 78.245 91.559 Row 13: 25 10 Scott Riggs Valvoline/Stanley Tools Dodge 78.310 91.483 26 8 Dale Earnhardt Jr. -
Points Report Indianapolis Motor Speedway Allstate 400 at The
Points Report Indianapolis Motor Speedway Allstate 400 at the Brickyard UNOFFICIAL Provided by NASCAR Statistical Services - Sun, Aug 6, 2006 @ 07:50 PM Eastern UNOFFICIAL Pos Driver BPts Points -Ldr -Nxt Starts Poles Wins T5s T10s DNFs Money Won PPos G/L 1 Jimmie Johnson 40 3124 0 0 21 1 4 8 17 0 $6,412,078 1 0 2 Matt Kenseth 80 3017 -107 -107 21 0 2 11 12 1 $4,283,879 2 0 3 Jeff Burton 50 2749 -375 -268 21 3 0 4 13 0 $3,103,062 3 0 4 Kevin Harvick 50 2733 -391 -16 21 0 1 9 12 0 $3,723,688 5 1 5 Kyle Busch 60 2733 -391 0 21 1 1 7 11 1 $3,083,228 4 -1 6 Mark Martin 45 2712 -412 -21 21 0 0 4 10 0 $2,740,103 6 0 7 Denny Hamlin # 50 2648 -476 -64 21 2 2 4 9 1 $2,831,522 8 1 8 Jeff Gordon 85 2627 -497 -21 21 0 2 7 9 4 $3,797,483 9 1 9 Tony Stewart 90 2619 -505 -8 21 0 2 8 11 3 $4,284,666 10 1 10 Dale Earnhardt Jr. 45 2612 -512 -7 21 0 1 6 9 3 $3,355,114 11 1 11 Kasey Kahne 45 2575 -549 -37 21 4 4 6 11 3 $3,891,387 7 -4 12 Greg Biffle 95 2497 -627 -78 21 2 1 5 10 3 $2,863,681 12 0 13 Kurt Busch 40 2438 -686 -59 21 2 1 5 8 1 $3,153,979 13 0 14 Carl Edwards 35 2380 -744 -58 21 0 0 6 10 3 $2,875,976 14 0 15 Casey Mears 5 2279 -845 -101 21 0 0 1 5 1 $3,574,093 15 0 16 Brian Vickers 40 2251 -873 -28 21 0 0 3 5 2 $2,407,541 16 0 17 Clint Bowyer # 15 2224 -900 -27 21 0 0 2 6 1 $2,478,642 19 2 18 Ryan Newman 40 2204 -920 -20 21 2 0 2 5 3 $3,584,752 18 0 19 Jamie McMurray 25 2173 -951 -31 21 0 0 2 6 2 $3,165,068 17 -2 20 Martin Truex Jr. -
Qualifying Order & Worksheet Infineon Raceway Dodge/Save Mart
Qualifying Order & Worksheet Infineon Raceway Dodge/Save Mart 350 Provided by NASCAR Statistical Services - Fri, Jun 23, 2006 @ 2:12 PM Eastern Driver Date Time Speed Track Qualifying Record: Jeff Gordon 6/24/05 75.950 94.325 Q# Car Driver Team √ Time Speed √ Time Speed Rmks 1 01 Joe Nemechek U.S. Army Chevrolet 2 7 Robby Gordon Menards/Energizer Chevrolet 3 29 Kevin Harvick GM Goodwrench Chevrolet 4 18 J.J. Yeley # Interstate Batteries Chevrolet 5 42 Casey Mears Chevron-Texaco Dodge 6 9 Kasey Kahne Dodge Dealers/UAW Dodge 7 22 Dave Blaney Caterpillar Dodge 8 16 Greg Biffle National Guard/Subway Ford 9 * 61 Brian Simo Front Row Motorsports Chevrolet 10 14 Sterling Marlin Waste Management Chevrolet 11 31 Jeff Burton Cingular Wireless Chevrolet 12 * 60 Boris Said Sobe-No Fear Ford 13 1 Martin Truex Jr. # Bass Pro Shops/Tracker Chevrolet 14 24 Jeff Gordon DuPont Chevrolet 15 * 32 Ron Fellows Tide/Downy Chevrolet 16 11 Denny Hamlin # FedEx Freight Chevrolet 17 * 49 Chris Cook DSI Dodge 18 43 Bobby Labonte Cheerios/Betty Crocker Dodge 19 48 Jimmie Johnson Lowe's Chevrolet 20 10 Scott Riggs Valvoline/Stanley Tools Dodge 21 38 Elliott Sadler Pedigree Ford 22 25 Brian Vickers GMAC Chevrolet 23 2 Kurt Busch Miller Lite Dodge 24 * 95 Stanton Barrett Interush Chevrolet 25 96 Terry Labonte DLP® HDTV Chevrolet 26 66 Jeff Green Best Buy Chevrolet 27 8 Dale Earnhardt Jr. Budweiser Chevrolet 28 * 27 Tom Hubert TBA Chevrolet 29 * 02 Brandon Ash Sprinter Trucking, Inc. Dodge 30 * 78 Travis Kvapil Furniture Row Chevrolet 31 * 55 Michael Waltrip NAPA Dodge 32 * 34 Johnny Miller Oak Gloves Chevrolet 33 99 Carl Edwards Office Depot Ford 34 6 Mark Martin AAA Ford 35 45 Kyle Petty Wells Fargo Dodge 36 20 Tony Stewart The Home Depot Chevrolet 37 26 Jamie McMurray Crown Royal Ford 38 * 40 Scott Pruett Coors Light Dodge 39 5 Kyle Busch Kellogg's Chevrolet 40 17 Matt Kenseth DeWalt Ford 41 21 Ken Schrader Little Debbie Ford 42 12 Ryan Newman Alltel Dodge 43 07 Clint Bowyer # Jack Daniels Chevrolet 44 41 Reed Sorenson # Target Dodge 45 19 Jeremy Mayfield Dodge Dealers/UAW Dodge 46 * 4 P.J.