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ORIGINAL RESEARCH

Injuries in Little League Baseball From 1987 Through 1996

Implications for Prevention

Frederick O. Mueller, PhD; Stephen W. Marshall, PhD; Daniel P. Kirby

THE PHYSICIAN AND SPORTSMEDICINE - VOL 29 - NO.7 - JULY 2001


ABSTRACT

BACKGROUND: Baseball is a very popular team sport in the United States, but there has been concern with the number of injuries and fatalities.

OBJECTIVE: This study describes the incidence of injuries to Little League Baseball, Inc, players from 5 to 12 years old from 1987 through 1996.

METHODS: Descriptive statistical analyses of injuries derived from Little League Baseball, Inc, insurance data from 17,221,210 player-years of follow-up in the 10-year study period. An average of 1,722,121 children participated every year.

RESULTS: During the study, there were 29,038 injuries and an injury rate of 1.69 injuries per 1,000 participants per season. Ball-related injuries totaled 15,266, and batters had the greatest number of ball injuries. There were 12,306 facial and teeth injuries, mostly in defensive players. Sliding was associated with 60% of the injuries to base runners. Approximately 25% of the injuries were considered severe (fractures, dislocations, and concussions), and 13 players died.

CONCLUSION: Based on the data, youth baseball appears to be a very safe sport, but there are areas where injury prevention is possible. Data and additional attention are needed concerning face mask use for players in the field, modified balls for minimizing contact injuries, education on sliding technique, and use of safety bases for runners.

Baseball is the second most commonly played team sport in the country (after basketball), with approximately 8.6 million players from 6 to 17 years old participating each year (1). In 1995 more than 4 million boys and girls played in organized amateur youth baseball leagues, with Little League Baseball, Inc, (LLB) participants making up more than 50% of the total. Baseball has been cited as having a high number of injuries. The Consumer Product Safety Commission (CPSC) recorded 88 deaths in baseball from 1973 to 1995, more than any other sport (2). Sixty-eight deaths were caused by ball impact, and 13 resulted from bat impact. Yen and Metzl (3) note that baseball is played by millions of children in the United States and is considered a safe sport, but they estimate that there are more than 100,000 acute baseball injuries every year in 5- to 14-year-old children. They state that baseball is the leading cause of sports-related eye injury (3).

We present the incidence of injuries to LLB players between the ages of 5 and 12 for the 10-year period, 1987 through 1996, and also discuss controversies associated with youth baseball and equipment. These issues include impact-reduced balls, batter chest protectors, face masks, and safety bases. We present arguments for protective equipment and the potential impact its use might have on the distribution of LLB injuries.

Methods

LLB represents only players in that organization, though many other youth baseball organizations exist. LLB has been collecting insurance data for many years from all its leagues (about 6,200) in the United States. That database was used for the current analysis.

All injuries not directly related to baseball were deleted from the insurance database, and thus not considered in the analysis. Analysis was restricted to players 5 to 12 years old, since this is the predominant age-group in LLB. Data on all participants collected annually by LLB were used to calculate incidence rates per 100,000 participants; the number of total participants in a given year was used as the denominator. Because opportunities for injury prevention strategies are sought for reducing injury incidence, data presentation is organized by the underlying cause of injury (injury contact).

Results

LLB had an average of 1,722,121 boy and girl participants from 1987 through 1996, yielding 17,221,210 athlete-years of follow-up. There were 29,038 injuries (including warm-up and batting circle injuries) and an injury rate of 1.69 injuries per 1,000 participants per season in 5- to 12-year-olds.

Comparison of these numbers with those from other sports is difficult because of the lack of national studies for this age-group. The CPSC has sports injury data from emergency departments, but it does not classify the data according to organized and unorganized play, and does not have the total number of participants of each sport (denominator). The authors contacted a number of organizations, but nationwide injury and participation data are not available for any youth sports programs. LLB has the only national data available on both numbers of injuries and participants.

Overall injuries. The greatest number of injuries (6,137) was associated with base runners (21.1% of all injuries). On defense, infielders had the most injuries: 6,012 (20.7% of all injuries). Batters were third with 5,567 injuries (19.2%). When combined, the data for runners and batters totaled 11,704 injuries, or 40.3% of all injuries. By position, outfielders accounted for 4,119 injuries (14.2%) and were second to infielders among defensive players. Catchers, with 3,649 injuries (12.6%), and pitchers, with 2,080 injuries (7.2%), had the fewest injuries among active players.

One interesting note was that warm-up activities and being in the on-deck circle accounted for 5.1% of the total injuries. LLB eliminated the on-deck circle for its Little League Baseball division in 1996, but Senior and Big League divisions continue to use one.

Ball-related injuries by position. Ball injuries accounted for 15,266 injuries (52.6% of all injuries, table 1). The batted ball accounted for 5,882, the pitched ball accounted for 5,609, and the thrown ball accounted for 3,775 injuries (20.2%, 19.3%, and 13% of all injuries, respectively). Batters experienced the most ball-related injuries, followed in order by infielders, outfielders, catchers, pitchers, and base runners. As would be expected, pitched balls during games caused 95% of the ball injuries to the batters.


TABLE 1. Ball-Related Injuries to Little League Baseball Players by Position and Anatomic Site
Batted Ball Thrown Ball Pitched Ball Total

Position No. of Injuries (% of Row)

Batter 235 (4.7) 0 4,757 (95.3) 4,992 (100)
Infielder 2,135 (53.5) 1,856 (46.5) 0 3,991 (100)
Catcher 313 (19.0) 480 (29.2) 852 (51.8) 1,645 (100)
Outfielder 2,142 (77.2) 631 (22.8) 0 2,773 (100)
Pitcher 1,022 (75.0) 341 (25.0) 0 1,363 (100)
Base runner 35 (7.0) 467 (93.0) 0 502 (100)

Total (% of row) 5,882 (38.6) 3,775 (24.7) 5,609 (36.7) 15,226 (100)

Batted Ball Thrown Ball Pitched Ball Hit by Ball Total

Anatomic Site No. of Injuries (% of Row)

Face 3,566 (38.4) 3,581 (38.6) 1,193 (12.9) 941 (10.1) 9,281 (100)
Teeth 1,304 (43.1) 1,399 (46.2) 322 (10.7) 0 3,025 (100)
Head 440 (39.6) 437 (39.3) 235 (21.1) 0 1,112 (100)
Knee or ankle 275 (46.4) 0 318 (53.6) 0 593 (100)
Chest 70 (32.4) 35 (16.2) 111 (51.4) 0 216 (100)

Total (% of row) 5,655 (39.8) 5,452 (38.3) 2,179 (15.3) 941 (6.6) 14,227 (100)

Batters most frequently injured their hand or finger, followed by the face; arm, wrist, or elbow; knee or ankle; head or neck; chest; and leg. The three leading injury types for batters were contusions (46.0%), fractures (29.7%), and sprains (6.4%).

Infielders received most of their injuries in games, and ball injuries accounted for 3,991 (66.4%) of their injuries. The batted ball accounted for more injuries to infielders than the thrown ball (53.5% vs 46.5%). The face was injured most often and accounted for 2,253 (37%) of the total injuries to infielders. Other injuries in order were to the teeth; hand or fingers; arm, wrist, or elbow; knee or ankle; and head or neck. Fractures accounted for one third of the injuries to infielders, with contusions a close second. Other injuries in order were dental injuries, lacerations, and sprains. Concussions made up only 1.2% of the total injuries to infielders.

Outfielders compared with players at other positions experienced more injuries during practice than in games. The baseball was responsible for two thirds of the total injuries to outfielders. Approximately 77% of the ball-related injuries were caused by the batted ball. Facial injuries (eye, face, mouth, nose, and lips) accounted for 40.7% of the injuries. Tooth injuries were second, making up 18.4% of the total injuries. Fractures were most prevalent (30.3%), followed by contusions (25.6%), dental injuries (18.5%), lacerations (11.3%), sprains (6.9%), and concussions (1.5%). One outfielder died from injuries.

Catchers received 3,649 total injuries, 1,645 of them from being hit by a baseball. A pitched ball caused 51.8% of the ball injuries. Two thirds of the injuries to catchers took place in games. The body part most injured was the hand or fingers (36.0% of total injuries), followed by the arm, wrist, or elbow (16.1%); face (14.7%); knee or ankle (9.5%); teeth (6.7%); and head or neck (5.6%). As was true for outfielders, infielders, and base runners, fractures were the most common injury observed in catchers (35.4%). Other injuries in order were contusions (32.0%), sprains (13.8%), dental (6.8%), lacerations (5.0%), and concussions (1.0%). There was one fatality among catchers.

Pitchers had 2,080 total injuries, and 1,363 (65.5%) were from being hit by the ball—1,022 of them from the batted ball. The body parts injured were, in order, the face (32.3%); arm, wrist, or elbow (13.4%); knee or ankle (10.1%); teeth (10.1%); hand or fingers (9.7%); and head or neck (7.8%). The ranking of participant injury type in pitchers was contusions (36.5%), fractures (25.4%), sprains (11.2%), dental injury (10.1%), lacerations (6.9%), and concussions (2.0%).

Base runners experienced only 502 ball-related injuries, and 467 (93.0%) resulted from a thrown ball. Nearly three quarters of the injuries to runners (73.7%) occurred in games. The body part most injured in runners was the ankle, followed by the arm, wrist, or elbow, and the knee. Ankle and knee injuries accounted for about 40% of runner injuries.

Ball-related injuries by anatomic site. Of the 15,266 ball-related injuries, 14,227 (93.4%) were to the body parts listed in table 1. Most (80.6%) of all ball-related injuries were to the face ( 9,281 injuries) and teeth (3,025).

Ball injuries to the chest have received much attention over the years because of deaths caused by the batter being hit in the chest with a pitched ball. (See "Commotio Cordis: A Deadly Consequence of Chest Trauma," November 2000, page 31.) But in this study, only 216 ball-related chest injuries (1.4% of all ball-related injuries) were seen, and none resulted in death.

Sliding, running, and tagging injuries. Base runner sliding accounted for 3,703 injuries (12.8% of all injuries), and an additional 418 injuries were related to base running (1.4% of all injuries, table 2). Fractures, sprains, and contusions accounted for 85% of the sliding and running injuries. Forty percent of the injuries were fractures, 30% sprains, and 14% contusions. The knee and ankle were by far the most often injured body parts in both sliding and running. Sliding made up approximately 80% of the injuries and running 19%.


TABLE 2. Sliding, Running, and Tagging Injuries in Little League Players by Position and Anatomic Site
Sliding Running Tagging Total

Position No. of Injuries (% of Row)

Base runner 3,703 (89.9) 418 (10.1) 0 4,121 (100)
Infielder 0 0 179 (100) 179 (100)
Pitcher 0 0 39 (100) 39 (100)

No. (% of row) 3,703 (85.3) 418 (9.6) 218 (5.1) 4,339 (100)

Sliding Running Tagging Total

Anatomic Site No. of Injuries (% of Row)

Knee or ankle 1,761 (80.6) 423 (19.4) 0 2,184 (100)
Face or head 0 0 0 0
Teeth 17 (60.7) 1 (3.6) 10 (35.7) 28 (100)

No. (% of row) 1,778 (80.4) 424 (19.2) 10 (0.4) 2,212 (100)

The few tagging injuries reported were mainly to infielders (179 injuries) and pitchers (39). One surprising finding was the absence of tagging injuries to catchers, but this might be a classification artifact. The more violent tagging collisions take place at home plate.

Colliding, falling, and hit-by-bat injuries. Collisions and falls accounted for 4,637 injuries (16.0% of all injuries), and most of those injuries were associated with infielders and runners (table 3). Outfielders had 924 injuries and catchers 730 from collisions and falls (19.9% and 15.7% of all collisions and falls, respectively). Most collisions in baseball happen either when two outfielders or an outfielder and an infielder collide while trying to catch a fly ball. The second type of collision involves a base runner colliding with either an infielder or catcher. Falls happen for diverse reasons, and the number of injuries from this cause is approximately half that of collisions. The catcher was the only position player injured by being hit with a bat, and this was rare (565 injuries) in the study period. The knee and ankle top the list of body parts injured (see table 3), but if the facial, head, and teeth injuries were combined into a single category, their number would exceed knee and ankle injuries.


TABLE 3. Colliding, Falling, Hit-by-Bat, and Other Injuries Among Little League Baseball Players by Position and Anatomic Site
Colliding Falling Hit by Bat Other Total

Position No. of Injuries (% of Row)

Catcher 730 (37.2) 0 565 (28.8) 669 (34.0) 1,964 (100)
Infielder 1,025 (58.8) 311 (17.9) 0 406 (23.3) 1,742 (100)
Base runner 662 (43.7) 643 (42.5) 0 209 (13.8) 1,514 (100)
Outfielder 489 (38.0) 435 (33.8) 0 363 (28.2) 1,287 (100)
Pitcher 274 (42.2) 68 (10.5) 0 307 (47.3) 649 (100)
Batter 0 0 0 557 (100) 557 (100)

Total (% of row) 3,180 (41.2) 1,457 (18.9) 565 (7.3) 2,511 (32.6) 7,713 (100)

Colliding Falling Hit by Bat Other Unknown Total

Anatomic Site No. of Injuries (% of Row)

Knee or ankle 567 (35.4) 467 (29.2) 0 522 (32.6) 45 (2.8) 1,601 (100)
Face 417 (36.7) 0 0 530 (46.7) 188 (16.6) 1,135 (100)
Teeth 187 (19.0) 29 (2.9) 500 (50.8) 201 (20.4) 67 (6.8) 984 (100)
Head 202 (23.3) 0 366 (42.3) 278 (32.1) 20 (2.3) 866 (100)
Chest 53 (41.7) 0 23 (18.1) 49 (38.6) 2 (1.6) 127 (100)

Total (% of row) 1,426 (30.2) 496 (10.5) 889 (18.9) 1,580 (33.5) 322 (6.8) 4,713 (100)

Injury severity. The only measure of severity available from the data was the type of injury associated with particular body parts. We selected injuries believed to be most severe and compared them with the body parts most injured (table 4). Almost one fourth of the head injuries were concussions. Contusions were the predominant facial injury (42.4%), but fractures accounted for 2,689 of 8,972 (30%) facial injuries. Contusions can be severe injuries, but in most cases they are considered minor. Facial fractures are severe injuries. Fractures and dislocations (severe injuries) accounted for approximately half of the injuries to the hand, arm, and elbow and for 30% of the injuries to the knee, leg, and ankle. Although all injuries in table 4 can be considered severe, if fractures, dislocations, and concussions are considered the most severe, these accounted for 7,204 (27.7%). Fatalities are the most severe injury, and their distribution by cause is shown in table 5. Nearly half (6 of 13) resulted from being struck by a ball.


TABLE 4. Severity and Anatomic Site of Baseball Injuries Among Little League Baseball Players
Head Face Teeth Knee, Ankle, and Leg Chest Hand, Arm, and Elbow Total

Type of Injury No. (% of Row)
Higher Severity
Fracture 55
(2.8)
2,689
(30.0)
0 1,618
(27.8)
5
(1.5)
2,262
(46.8)
6,629
(25.5)
Dislocation 0 0 0 124 (2.1) 0 25
(0.5)
149
(0.6)
Concussion 426
(21.6)
0 0 0 0 0 426
(1.6)
Lower Severity
Laceration 392
(19.8)
1,834
(20.4)
0 232
(3.9)
0 132
(2.7)
2,590
(10.0)
Contusion 861
(43.6)
3,803
(42.4)
0 1,293
(22.2)
294
(85.7)
1,728
(35.8)
7,979
(30.7)
Dental 0 0 4,037
(100)
0 0 0 4,037
(15.5)
Other 242
(12.2)
646
(7.2)
0 2,563
(44.0)
44
(12.8)
685
(14.2)
4,180
(16.1)

Total 1,976
(100)
8,972
(100)
4,037
(100)
5,830
(100)
343
(100)
4,832
(100)
25,990
(100)


TABLE 5. Causes of Fatalities in Little League Baseball Players, 1987 Through 1996
Cause No. Percent

Pitched ball 3 23
Sliding 2 15
Thrown ball 2 15
Batted ball 1 8
Collisions 1 8
Falls 1 8
Other 3 23

Total 13 100

Discussion

These data present a descriptive analysis of the injuries in the nation's largest organized youth baseball league and are the only national injury information available for this age-group. The size of the population followed (2.5 million) and the fact that 10 years of data are available for analysis have allowed us to describe youth baseball injury in great detail. Epidemiologic analysis of these data is important in guiding future research in youth baseball safety.

Study limitations. One of the limitations of the LLB research project was that all injuries were identified using compensated insurance claims. Some injury claims may not be compensated because of coverage by another party. However, a comparison of claims that result in compensation with claims that do not indicates that the patterns of injury type, mechanism, and age distribution are very similar in both types of claims. Also, claims in which LLB was the sole provider have very similar characteristics to claims in which LLB was the secondary provider. Finally, injury rates and injury claim patterns recorded in the system are very stable from year to year. All these facts, taken together, indicate that these insurance claims data provide a valid and comprehensive picture of injury in LLB.

Implications for safety. The CPSC stated in 1996 that the National Electronic Injury Surveillance System estimated that 168,000 emergency-department visits occurred annually from baseball, softball, and T-ball injuries among 5- to 14-year-olds (2). The CPSC concluded that about one third of these injuries could be prevented or reduced in severity if equipment such as reduced-impact balls, safety bases, and face guards were universally used. These conclusions made headlines in many national newspapers and caused concern among baseball administrators and parents.

We agree that injury prevention is important for baseball and support any intervention that has proven effectiveness. It is also important to frame each intervention within the overall spectrum of injury in this sport. In the context of the data presented here, the potential benefits of safety bases, reduced-impact balls, and face masks are large. However, the benefits of chest protectors for batters appear to be of lesser importance.

Safety bases. Sliding was associated with approximately 60% of the injuries to base runners. Sliding has always been associated with a large percentage of youth baseball injuries and will remain at the top of the list until proper sliding techniques are universally taught and the debate over the use of safety bases is settled. Janda et al (4) studied college and minor league teams for 2 years and found an 80% reduction in the number of sliding injuries when breakaway bases were used. In our study, if a similar magnitude reduction had occurred by using the breakaway bases, approximately 3,000 sliding injuries would have been prevented.

Other researchers have noted similar results for other types of safety bases. Sendre et al (5) found players using the compression (Hollywood) base had fewer sliding injuries. Hosey and Puffer (6) studied sliding injuries in collegiate baseball and softball players and found that in baseball players, the injury rate was higher for feet-first slides than for head-first slides. They recorded 37 injuries in 3,889 slides in 637 games.

Modified balls and chest protectors. Modified balls, designed to generate lower-impact forces, also hold considerable promise for baseball, a sport in which more than half of all injuries are from ball contact. Although the impact dynamics of modified balls clearly differ from traditional baseballs, there is still a need for a large-scale epidemiologic study examining their effectiveness in the field. If a modified ball could prevent or reduce the severity of even a fraction of the roughly 15,000 ball-related injuries seen during this research project, its widespread adoption would be warranted.

Many people believe that all batters should be made to wear protective vests (7-9). However, there is no evidence that a vest will protect the batter, and—as shown in this study—the number of chest injuries is small. In a study assessing the relative risk of fatal cardiac injury from various baseballs, Janda et al (8) found that soft-core baseballs may not differ from standard baseballs with regard to the risk of fatal chest-impact injuries. The authors stated that other techniques, such as preventive coaching, need to be implemented when trying to improve baseball safety. Link et al (10) found that with safety baseballs and regulation balls, the likelihood of ventricular fibrillation was proportional to the hardness of the ball, with the softest balls associated with the lowest risk. They concluded that ventricular fibrillation occurrence depended on the precise timing of the impact. Vincent and McPeak (11) presented data from the United States Commotio Cordis Registry in Minneapolis that documented 40 cases of commotio cordis in baseball as of June 1998. The authors suggested that consideration be given to having automated external defibrillators and personnel trained in their use available for youth injuries.

Face masks. Some have suggested that face masks be required on all helmets to prevent facial injuries to the batter, and that very good shatterproof eye protection also be used (12-15). (See "A Practical Guide for Sports Eye Protection," June 2000, page 49.) While our study documents 9,281 facial injuries, it does not reveal how many batters were wearing face masks when they were injured. We need data to assess the protective value of face masks for batters and whether face masks can reduce severity. The data presented here show that most facial injuries in 5- to 12-year-olds are contusions, but there were also a large number of fractures. Since most of the facial injuries are caused by the batted ball, additional data are needed concerning the use of the face mask while running the bases and playing in the field.

In a survey of Little League affiliates in Rochester, New York, Pasternack et al (16) followed 2,861 LLB players between the ages of 7 and 18 for 140,932 player-hours. The authors recorded the mechanism, area injured, and position played at the time of injury. A total of 81 injuries occurred; 81% were acute and 19% were overuse. The most frequent mechanism was being hit by the ball, and 68% of these occurred to defensive players. Defensive players experienced 16 of the 18 ball-related facial injuries. The authors stated that since 86% of the head and facial injuries occurred to players on defense, they would not be prevented by using a face mask. They did state that the more severe ball injuries occur to batters and that a face mask could have prevented them, but not helmets that had only eye guards. Only 11 severe injuries were seen in the study. The authors concluded that face masks could eliminate facial injuries to offensive players, but they would only moderately reduce the incidence of ball-related facial injuries, as most of these injuries are sustained by defensive players.

The Line Score

Based on these data, youth baseball appears to be essentially a very safe activity, but there are areas in which protective equipment use may help to lower the risk of injury. In addition, team sports can be used to promote physical activity, leadership, and teamwork. Given the declining levels of physical activity in the general population, and the ever-increasing proportion of sedentary adolescents, we would do well to continue to promote baseball as America's national pastime.

References

  1. Baseball and Softball Council: Baseball: A Report on Participation in America's National Pastime, 1996. North Palm Beach, FL, Sporting Goods Manufacturers Association, 1996, pp 1-11
  2. Kyle SB: Youth Baseball Protective Equipment Project Final Report. US Consumer Product Safety Commission, Washington, DC, 1996
  3. Yen KL, Metzl JD: Sports-specific concerns in the young athlete: baseball. Pediatr Emerg Care 2000;16(3):215-220
  4. Janda DH, Maguire R, Mackesy D, et al: Sliding injuries in college and professional baseball: a prospective study comparing standard and break-away bases. Clin J Sport Med 1993;3(1):78-81
  5. Sendre RA, Keating TM, Hornak JE, et al: Use of the Hollywood Impact Base and standard stationary base to reduce sliding and base-running injuries in baseball and softball. Am J Sports Med 1994;22(4):450-453
  6. Hosey RG, Puffer JC: Baseball and softball sliding injuries: incidence, and the effect of technique in collegiate baseball and softball players. Am J Sports Med 2000;28(3):360-363
  7. Janda DH, Viano DC, Andrzejak DV, et al: An analysis of preventive methods for baseball-induced chest impact injuries. Clin J Sport Med 1992;2(3):172-179
  8. Janda DH, Bir CA, Viano DC, et al: Blunt chest injuries: assessing the relative risk of fatal cardiac injury from various baseballs. J Trauma 1998;44(2):298-303
  9. Crisco JJ, Hendee SP, Greenwald RM: The influence of baseball modulus and mass on head and chest impacts: a theoretical study. Med Sci Sports Exerc 1997;29(1):26-36
  10. Link MS, Wang PJ, Pandian NG, et al: An experimental model of sudden death due to low-energy chest-wall impact (commotio cordis). N Engl J Med 1998;338(25):1805-1811
  11. Vincent GM, McPeak H: Commotio cordis: a deadly consequence of chest trauma. Phys Sportsmed 2000;28(11):31-39
  12. Caveness LS: Ocular and facial injuries in baseball. Int Ophthalmol Clin 1988;28(3):238-241
  13. Vinger PF, Parver L, Alfaro DV 3rd, et al: Shatter resistance of spectacle lenses. JAMA 1997;277(2):142-144
  14. Vinger PF, Duma SM, Crandall LJ: Baseball hardness as a risk factor for eye injuries. Arch Ophthalmol 1999;117(3):354-358
  15. American Academy of Pediatrics Committee on Sports Medicine and Fitness: Risk of injury from baseball and softball in children 5 to 14 years of age. Pediatrics 1994;93(4):690-692
  16. Pasternack JS, Veenema KR, Callahan CM: Baseball injuries: a Little League survey. Pediatrics 1996;98(3 pt 1):445-448

Dr Mueller is professor and chair of the department of exercise and sport science at the University of North Carolina at Chapel Hill. Dr Marshall is research assistant professor in the department of epidemiology and in the Injury Prevention Research Center at the University of North Carolina at Chapel Hill. Mr Kirby is director of risk management of Little League Baseball, Inc, in Williamsport, Pennsylvania. Address correspondence to Frederick O. Mueller, PhD, Dept of Exercise and Sport Science, University of North Carolina at Chapel Hill, CB#8700, 209 Fetzer Gym, Chapel Hill, NC 27599-8700; e-mail to [email protected].


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