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Relationships between the velocity of the ice hockey wrist shot and selected human factors Moyls, Peter William 1981

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RELATIONSHIPS BETWEEN THE VELOCITY OF THE ICE HOCKEY WRIST SHOT AND SELECTED HUMAN FACTORS by PETER WILLIAM MOYLS B.P.E., The University of British Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENTOOF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF PHYSICAL EDUCATION in THE FACULTY OF GRADUATE STUDIES (PHYSICAL EDUCATION) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA FEBRUARY .1981 ('cVeter Williatn'Moyls,"'1981 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physical Education Graduate Studies The University of Bri t ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date February 16,1981. i i ABSTRACT The main purpuse of this study was to investigate the relation-ships between the velocity of the ice hockey wrist shot and selected human factors in order to be able to predict the velocity of the wrist shot. A second objective was to identify the modifiable human factors most highly related with puck velocity in order to guide further research investigating the causal factors of a high speed wrist shot. Thirty-five subjects were selected from junior, intermediate, collegiate, senior and professional leagues to take part in the study. Their ages ranged from seventeen to twenty-nine. Forty-eight- strength, f l e x i b i l i t y , muscle•ratio, and anthropo-metric variables were measured in the Buchanan Fitness Centre at U.B.C. Puck velocity was measured by the use of a radar gun at the Thunderbird arena. Each subject shot a minimum of five shots and the top three scores were averaged to give the puck velocity score. With this data, Stepwise Regression and A l l Possible Subsets Regression techniques were used to find the best possible regression equation for predicting puck velocity. The product-moment correlations of puck velocity with each variable were tested for significance at the .05 level. The following correlations were significant: arm adduction at the shoulder of the lower arm, forearm supination of the upper arm, wrist strength in exten-sion of the upper arm, wrist strength in flexion of the lower arm, and diagonal arm strength of the lower arm. The best linear regression equation found for predicting puck i i i velocity was: Variable group Y(puck velocity) = 73.0946 + X^(arm girth upper arm)(.7544) A + X2(wrist f l e x i b i l i t y upper arm)(.1374) F + X^forearm rotation upper arm) (-.1779) F + X^(trunk rotation)(-.0574) F + X 5(trunk flexion)(.1645) S + Xg(arm abduction upper arm)(-1.1586) S + X^(arm adduction lower arm)(.8949) S + X (diagonal motion upper arm)(-.2990) S o + Xg(forearm supination upper arm)(1.7023) S + X-^ gCf lexion/extension elbow lower arm) (-7 .5737) R Where: A is anthropometric, F i s f l e x i b i l i t y , S i s strength, R i s Ratio. This equation had a multiple R value of .813 and a standard error of estimate of 4.52. Several transformations were performed on the data in an attempt to reduce the amount of uncertainty in the prediction equation. It was found that cubing the strength variables led to a new regression equation which accounted for more of the uncertainty and used fewer variables. The best regression equation using transformed data was: Y(puck velocity) = 103.1292 + X ^ s t i c k length) (.3405) + (shoulder rotation lower arm)(.0626) F + X 3(trunk rotation)(-.0972) F + X^(hip extension glide leg)(-.2334) S + X,. (elbow flexion upper arm)(-.1918) S 3 3 + X,(arm adduction lower arm) (.0001) S o 3 3 + X^(forearm supination upper arm) (.0116) S 3 3 + Xg(wrist extension upper arm) (.0001) S 3 3 + Xg(elbow flexion upper arm) (.0003) S iv This equation had a multiple R of .871 and a standard error of estimate of 3.74. The transformed data equation reduced the uncertainty by 10% with one fewer variable being required. It is recommended that the variables with a significant correlation with puck velocity and the variables in the untransformed data equation be investigated further by implementing them into a training program for hockey players. In this way i t might be empirically verified that these factors are the causes of a fast shot. V CONTENTS Page LIST OF TABLES v i i LIST OF FIGURES v i i i Chapter 1. INTRODUCTION 1 Statement of the Problem 4 2. REVIEW OF LITERATURE 7 Shooting Technique 7 Off-Ice Training 9 Biomechanical Characteristics Related to Shooting . . 10 Physiological Characteristics Related to Shooting . . 14 . Other Sports 20 Authorities Census 21 Summary 22 3. PROCEDURES 24 Sample . . 24 Design 25 Selection of Variables 25 Instruments and their Reliability 29 Testing Procedures 31 Analysis of Data • 34 4. RESULTS AND DISCUSSION 37 Results: Multiple Regression Using Raw Scores . . . . 37 v i Page Discussion: Multiple Regression Using Raw Scores . . . 46 Results: Multiple Regression Using Non Linear Transformations . . 52 Discussion: Non Linear Versus Linear Equations . . . 56 5. SUMMARY AND CONCLUSIONS 64 REFERENCES 67 APPENDIX A. VARIABLE TESTING PROCEDURES 70 B. FLEXIBILITY WARMUP 76 C. PRODUCT-MOMENT CORRELATION MATRIX 78 v i i TABLES Table Page 2.1 Census of Authorities 22 3.1 Independent Variables Selected 27 4.1 Descriptive Statistics for A l l Variables 38 4.2 Product-Moment Correlation Table of Test Variables and Puck Velocity 40 4.3 Order of Entering of Variables Using Stepwise Program . 41 4.4 Order of Entering of Variables Using A l l Possible Subsets Program 43 4.5 Anthropometric Variables Order of Entering 46 4.6 F l e x i b i l i t y Variables Order of Entering 47 4.7 Strength Variables Order of Entering 49 4.8 Muscle Ratio Variables Order of Entering 50 4.9 Product-Moment Correlation Table of Cubed Strength Variables and Puck Velocity 53 4.10 Order of Entering of Variables Using Stepwise Program for Transformed Data 54 4.11 Order of Entering of Variables Using A l l Possible Subsets Program for Transformed Data 55 4.12 Observed and Predicted Scores with 95% Confidence Intervals 59 4.13 Observed and Predicted Scores with 95% Confidence Intervals for Transformed Data 60 v i i i FIGURES Figure Page 1. Standard Error of Estimate Versus Number of Variables . . 45 2. Adjusted R Squared Versus Number of Variables 45 3. Multiple R Versus Number of Variables 45 4. Standard Error of Estimate Versus Number of Variables for Transformed Data 57 5. Adjusted R Squared Versus Number of Variables for Transformed Data 57 6. Multiple R Versus Number of Variables for Transformed Data 57 7. Residuals Plotted in Order of Shot Velocity from Lowest to Highest . \. 62 8. Residuals Plotted in Order of Shot Velocity from Lowest to Highest for Transformed Data 63 ix ACKNOWLEDGEMENTS In writing this thesis I was fortunate to be guided by some exceptional people. At this time I would like to acknowledge them. A very special thanks goes to Dr. Robert Schutz who chaired my thesis committee and gave me excellent guidance in the development and completion of this thesis. At this time I would like to thank him for his time and continuous interest in me as a student through-out my years at university. I also wish to thank the rest of my thesis committee, Dr. Stan Brown, Mr. Bert Halliwell, and Dr. Bob Hindmarch whose help and guidance were greatly appreciated. I sincerely wish to thank my parents for their encouragement and help through my years at university. A very special thanks goes to Margot McCullough who without her help with the testing and editing this thesis would not have been completed. Her continuous encouragement throughout the writing.;of this thesis is .greatly. Appreciated. 1 Chapter 1 INTRODUCTION The o f f i c i a l game of ice hockey started in North America in the early 1900's, and was characterized by skilled stickhandling and passing. However, over the years the emphasis of the game gradually changed from stickhandling and passing to shooting. The development of the slapshot in the late f i f t i e s and early sixties led to goals being scored with greater frequency from distances further away than before. With an increasing emphasis on faster shots the high velocity wrist shot became more and more prominent. The wrist shot has a major advantage over the slapshot in that i t requires a shorter backswing and can be executed in a significantly shorter time period. Alexander, Haddow, and Schultz (1963) found that although the slapshot is capable of producing a higher puck velocity, i t is not, on the average, as accurate as the wrist shot. Thus i t is important that players develop a high speed wrist shot as well as a slapshot. Hockey stars such as Phil Esposito and Mike Bossy are two splendid examples of players who have developed a highly effective high speed wrist shot. In recent years articles by well known hockey players have been written explaining how to perform a wrist shot and, although most of these athletes explain how to perform a wrist shot, l i t t l e is said about what human characteristics are important to develop in order to execute the shot well. Successful players Phil Esposito (1975), Rod Gilbert (1972), and Ken Hodge (1967) maintain that the way of improving one's shot velocity and accuracy is by hours of practise. In :the early stages of one's development as a hockey player, i t is probable that routine practise plays a significant role in the improvement of one's shot, but i t seems reasonable to conjecture that, later, specific exercises undertaken in conjunction with disciplined practise can produce a faster shot. A question often raised when discussing shooting pertains to why one person is able to shoot the puck faster than another. Indeed this is a central concern of coaches, sports scientists, players, and others interested in developing superior performance in hockey players . In an attempt to resolve this problem one may begin by looking at what human factors are required to produce a high velocity wrist shot. Brown (1980) suggests that when analyzing a motor s k i l l such as the wrist shot the human factors can be broken down into five basic categories: genetic makeup, body type, psychomotor a b i l i t i e s , physical characteristics, and technical execution of the s k i l l . These categories consist of either modifiable or inherent factors. One might i n i t i a l l y hypothesize that inherent growth factors such as size may be limiting factors in obtaining high puck velocity. However upon investigation this does not seem to be true. The Russian coach Tarasov (1973) states that a player doesn't have to be abnormally strong to produce a hard shot. Subjectively one observes that some of the smaller lighter players are among the hardest shooters, examples being: Danny Gare 5'9" 175 lbs, Guy Lafleur 6'0" 180 lbs, and Bobby Schmautz 5'9" 176 lbs. It appears that body size and possible inherent body characteristics such as somatotype may not be as important as one 3 might suppose. Many authors have attempted to isolate the key factors of a high velocity wrist shot. Hull (1967) attributed puck speed to arm and wrist strength. He also claims that much of the speed with which a puck moves is a combination of the technical perfection of the mechanics of the shot and the shooter's strength. Tarasov (1973) believes that i t results from the proper technique of using the thumb of the upper hand and the strength of the lower hand on the shaft to lock the blade while executing the shot. He also maintains that shifting the weight of the supporting leg at the proper instant is a crucial element in determining the puck's velocity. Hodge (1967) states that timing,"coordination and strength are the key factors. Chambers (1979) states that the speed and the power of the wrist shot depend on the strength in the arms, shoulders and wrists, along with coordination of the shooting movement and upper body rotation. The beliefs presented by these athletes and coaches are not based on empirical data but rather on subjective opinions. However, these thoughts should not be discounted because they are subjective in nature but research should be conducted in an attempt to substantiate these beliefs. Research in the mechanics of shooting has just begun, and there i s no concensus as to what the most important factors are in a high speed wrist shot. It is hoped that this study w i l l shed further light on this subject. 4 Statement of the Problem The aim of this study is to investigate the relationships between the velocity of the ice hockey wrist shot and selected human factors. Subproblems The main problem can be broken down into the following subproblems. 1. To investigate and select the human factors with the highest linear relationships with puck velocity. 2. To identify the modifiable human factors most highly related linearly to the puck velocity of the wrist shot. 3. To find the best possible regression equation for predicting puck velocity. Definition of Terms Wrist shot (Sweep Shot): A shot in ice hockey that is characterized by a sweeping motion of the blade of the stick on the ice and a weight transfer from one leg to the other. The puck remains in contact from the beginning of the execution of the shot to the point of release. Upper Arm and Hand: Refers to the limb in contact with the top of the shaft of the stick. Lower Arm and Hand: Refers to the limb in contact with the middle of the shaft of the stick. 5 Drive Leg and Foot: Refers to the leg and foot that l i e on the same side of the midline as the lower arm. It is characterized by extension of the knee during the execution of the wrist shot. Glide Leg and Foot: Refers to the leg and foot that l i e to the same side of the midline as the upper arm. It is characterized by flexion of the knee during execution of the wrist shot. Delimitations Relationships between puck velocity - and human factors can only be applied when speaking of players who have developed a pattern of execution consistent with those required in this study. These relationships w i l l have no bearing on players who are in the develop-mental stages of shooting or below the age of seventeen. The results of this study are delimited to players who participate at the junior, university, senior, intermediate and professional levels. Assumption and Limitations It is assumed that the subject w i l l use his game style while executing the shot and that he w i l l not alter his shooting pattern throughout the test. One tester w i l l be solely concerned with observing the pattern demonstrated and w i l l watch to be certain the pattern remains consistent throughout the test. It is also assumed that what the subject does prior to the testing sessions w i l l not affect the results. The subjects w i l l be instructed to avoid strenuous exercise prior to testing. The subject w i l l be required to provide his own stick for the study. Kingston (1980) states that stick flex, height, weight, curve 6 and l i e may limit the validity of the findings. However, requiring the subjects to use sticks to which they are not accustomed would li k e l y upset the shooting motion. Thus having the subjects use their own stick seems to be the best solution. The sample may be biased in that only volunteer subjects directly available to the researcher w i l l be used. There is no evidence to suggest that this bias w i l l be large. Significance of the Study Whether the individual is an athlete or a coach, there is a need to know the human factors that are important in achieving a high speed wrist shot. It is hoped that this study w i l l provide information beyond that given by most hockey books, and that i t w i l l also give a good basis for further research. By using the regression equation developed in this study one should be able to predict the velocity of a person's wrist shot. The relationships of the selected human factors to puck velocity cannot be looked at in terms of causality but only in degree of line a r i t y . One can hypothesize that the factors that have significant relationships with puck velocity may indicate aspects to be concentrated on during on-and off-ice training periods. Using the framework lai d by this study one could investigate the causes of a faster shot by analyzing a training program incorporating the factors proven to be related to puck velocity in this research. Thus in this proposed future study one could speak of the "causes" of a faster shot rather than just the characteristics related to a fast shot. 7 Chapter 2 REVIEW OF LITERATURE This chapter presents a general review of the literature which has a bearing on factors which may be involved in the execution of an effective ice hockey wrist shot. Books, journals, research papers, and expert opinion (from personal contact) form the basis of literature reviewed in this chapter. The content of this chapter is divided into six main categories: shooting technique, off-ice training, biomechanical characteristics, physiological characteristics, other sports and authorities census. Shooting Technique Many articles have been written on the "proper technique" to be used in the execution of the wrist shot, and, overall, most authorities agree on what are the important technical characteristics of the shot. However, there is a disagreement as to the relative importance of these characteristics. Hayward (1978), Johnston (1976), Jones (1979), MacDonald (1978), and Patrick and Monahan (1957) a l l l i s t the following as important ingredients of a good shot: I n i t i a l phase- At the start of the shot the puck is well back behind the drive leg, the wrists are cocked and the weight is on the drive leg; the puck should be towards the heel of the blade. Transfer phase- The puck is swept forward. At the same time 8 there i s a transfer of weight from the drive leg to the glide leg. As the stick comes across the midline of the body the wrists uncock. Follow through phase- The weight now becomes directly over the glide leg, and the stick follows through until i t points to the target, completing the shot. These authors differ somewhat as to what they believe are the most important aspects of each phase, but they a l l agree that these components comprise an effective technique. Phase 1 and 3 are primarily style and form of execution and are easy to monitor. Phase 2 is more concerned with the power of the legs and the strength and f l e x i b i l i t y of the wrists and arms. This phase is of major concern to the development of the test variables in this study. Being more or less in agreement on technique, the authorities are unanimous in stating that the technique must be practised over and over again. Professional hockey players Esposito (1975), Hull (1967), and Hodge (1967) a l l credit hours of practise with improving their shots. Hull (p.126) states that "no one whether in the N.H.L. or the Kiwanis Atom League gets enough practise shooting". Robb'(1972), a motor learning researcher, concurs that many hours of practise are necessary before a skilled movement is executed smoothly and efficiently. She suggests that there are three phases of learning: the plan formation phase, wherein the learner must formulate an executive plan and understand the sequential organization of the s k i l l , the practise phase, in which the learner must practise in order to f i x the perform-ance sequence in the human system, and the autonomous phase, wherein the learner is now able to perform the total executive program almost without conscious effort. This study deals only with subjects who 9 are in the third phase of learning. Off-Ice Training In recent years off-ice training and practise has become more popular with many coaches; e.g., Tremble (1978) maintains that off-ice practise is valuable in conditioning the athletes and maximizing the use of ice time. He states that a l l shots can be practised:off the ice. The literature on this form of training is related to the present investigation, in that the advocates of off-ice training invariably speculate on what factors produce a good fast shot. For example, in Sports Talk magazine (1978) i t is recommended that four off-ice exercises'"should'berconsidered beneficial to shooting. Forward and reverse wrist curls are recommended as exercises to develop the wrist flexors and extensors both of which are important to the wrist shot. In the second exercise, the wrist r o l l , the arms are extended straight out from the body and grasp a short stick. Attached to the stick i s a rope with weight attached to i t . The player winds the cord around the stick l i f t i n g the weight towards the stick. This strengthens the wrist flexors and extensors as well as some of the shoulder and forearm muscles. The third exercise consists of rotating a small weight in'ithe hand in c i r c l e s . The fourth exercise involves a method of "muscle fighting" or eccentric contractions. One hand pushes against the other in front of the chest and the hands move back and forth across the body. This exercise works the triceps, biceps and pectoral muscles of the body. The author of the a r t i c l e f e l t that the exercises would help a shooter develop the strength needed to execute an effective wrist shot. 10 Biomechanical Characteristics Related to Shooting Physiological Characteristics Halliwell, Gropel and Ward (1977) performed a kinematic analysis of the snap shot using professional hockey players as subjects. Their research concentrated on the motion of the hands required to s k i l l f u l l y execute the stationary snap: shot. Seven professional hockey players were filmed while executing the snap shot, with linear hand displacement and shooting accuracy being recorded for each subject. They found that linear hand displacement for the upper hand was : similar for both the high and low snap shot. The same was found to be true for the lower hand. The movement patterns for the upper and lower . hands were parallel although certain spatial and temporal differences were evident. For both the high and the low snap shot the upper hand started from a position behind the puck and moved toward the target. However, just prior to impact the upper hand reversed direction and moved away from the target back towards the body. Throughout the shot the lower hand moves toward the target. From these observations they inferred that, in both the high and low snap shot, when the upper hand was producing a pulling force away from the target, the lower hand was executing a pushing force towards the target. In this case the shaft acts as a f i r s t class lever with the lower hand acting as a fulcrum. One concludes from this that both arms are instrumental in force application. Although the snap shot and the wrist shot differ slightly in execution, . i t is'•.reasonable to assume that this same pattern takes place in the wrist shot. The wrist shot differs from the snap shot in that there is a longer sweeping motion and a distinct transfer of weight ( 11 from one side of the body to the other. The shots are similar in that they both require a similar cocking and uncocking of the wrists, although in the snap shot this tends to be more dynamic in nature. MacGillivary and Watson (1964) performed a kinesiological analysis, of the skating wrist shot. Through the use of film analysis they identified the muscles responsible for the primary movements of action for each of the three phases of movement of the shot. They liste d the movements of each body segment and the musc'les responsible for their action. No indication of the relative importance of the various movements or muscle groups was given. From their mechanical analysis they came up with six main points: 1". The mass should be moving in the direction of the shot. 2. Additional force is given by the torque transmitted from the feet on the ice, through the rotation of the thighs at the hips and the rotation of the trunk around the spine and f i n a l l y through the shoulders, arms and wrists. 3. The sum of the forces involved is increased by delaying arm and shoulder muscle movements. 4. The individual should drive through the puck so as to lessen the possible decrease in energy and momentum due to premature diminution of force. 5. The subject should follow through'..towards the target. 6. The mechanical advantage is increased when the lower hand moves down the shaft of the stick. A l l of these points relate directly to the proper technique mentioned earlier. Points 2, 3, and 4.also involve human factors such as muscle balance and strength, and body f l e x i b i l i t y . The other points primarily 12 deal with learned style and form. Stick Characteristics Kingston (1980) maintains that the interaction of stick characteristics (i.e., weight, height, and flex) with technique and arm strength are important factors that should be considered when discussing puck velocity. Perhaps golf is the most obvious other sport in which a "stick" is used, with the presence of considerable flex. In looking at the effect of shaft flex in the golf swing, Daish (1972) found that the characteristics of the shaft do not affect the drive to any appreciable extent. He suggested that this was due to the short time of contact between the club head and the b a l l . Since the hockey wrist shot has a longer contact period then the golf swing, the shaft plays a more important role in the wrist shot and could conceivably have an effect on puck speed. Based on these principles the well known sports manufacturer C.C.M. developed a method of determining the most effective stick flex for a player based on his wrist strength. Unfortunately the concept was not publicly supported and is no longer marketed. Jensen and Schultz (1970) explain that flex is a.'temporarily stored counterforce. If a surface or implement used in a performance has elasticity, then an applied force produces bend or compression which represents stored energy. This stored energy depends on the amount of disfigurement of the object combined with i t s a b i l i t y to spring back. This disfigurement of the shaft occurs in the wrist shot and results in stored energy which i s , in turn, converted into kinetic 13 energy of the puck. It is well known from Physics that the mass of the implement striking an object w i l l affect the velocity of that object. Daish (1972) found that in the golf swing increasing the club mass to infi n i t y would increase the bal l velocity by at most one seventh. He deduced the club mass M, the bal l mass m, and the velocity v are related by the formula v= kM M + m for some constant k. In the limit as M-*.°°, v becomes v«° = k. Since the average golf club is about seven times as heavy as the b a l l , v= 7km = 7k . Hence y<o= 8^  > and the velocity of the b a l l can be 7m+m 8 v 7 increased by at most 1 . In hockey the puck and stick weight average 7 0.15 kg and .69 kg respectively. Hence Vo<>_ k _ 0.84 1.22 . v ~ 0.69k ~ 0.69 0.84 Thus one might increase the puck velocity in the wrist shot by. at most 22%, i f the muscles could manage to sweep an i n f i n i t e l y heavy stick. Doublxng the stick weight results in a ratio voo_ 1.11. Unfortunately v doubling the weight of the stick would require the subject to be. porportionally stronger in order to produce the same stick motion and velocity and this is not easily achieved. Thus far the research cited indicates that technique, stick characteristics and biomechanical principles of the body are integral components of a good wrist shot,. 14 Physiological •Characteristics Related'to Shooting This section deals with the anthropometric and physiological characteristics of a hockey player, and discusses how they may relate to the velocity of the wrist shot. Somatotypes Three somatotypes analyses have been performed on U.B.C. hockey teams in the last twenty years. Selder (1964), found that a l l the players he tested were in the dominant mesomorph category. Ennos (1976) repeated the analysis on the 1976 team and found that a l l but one subject were in the dominant mesomorph category. Comparing the 1964 and 1976 teams he found that there was no significant difference in somatotypes between the two teams. On the other hand J Ennos found that the 1976 team was significantly - t a l l e r and heavier then the 1964 team. He also concluded that body size but not body structure had changed in the;-selection process of ice hockey players in the past twelve years. Moyls and Nobbs (1979) repeated the procedure on the 1979 hockey team. Grouping the players according to the position played, they found that defensemen were significantly heavier and and higher in endomorphic ratings than the forwards. Again no significant differences insomatotype ratings were found between the • three teams. The somatotypes of successful hockey players does not appear to have changed over the fifteen year period, 1964-79, but size has,especially weight. These three studies indicate that successful hockey players are in a highly selective somatotype category. Success in the game of hockey requires a dominant mesomorphic somatotype. S k i l l in shooting 15 is of course one small part of the competent hockey player's arsenal. At this point there is l i t t l e evidence that shooting s k i l l is related to a specific somatotype. A l l good shooters seem to be in the dominant mesomorph category (in spite of considerable physical variation) . But this is probably due to the fact that they are good hockey players, rather .than specifically good shooters. Strength Several investigators have attempted to isolate selected strength of arm movements affecting puck velocity in the wrist shot. Alexander, Drake, Reichenbach, and Haddow (1964) investigated the effect of isometric strength development on the shot velocity of varsity ice hockey players. Eighteen members of the University of Alberta varsity hockey team were used in this study, with nine of these subjects used as a control group. Four of the remaining nine were analyzed on film to determine the arm movements thought to be important in shooting. From this analysis, the subjects to be tested were placed on a program consisting of eight isometric exercises specific-ally designed to develop these arm movements. The exercises for the upper arm were designed to increase strength for arm adduction at shoulder, arm adduction at shoulder in hyperextension, wrist supination and wrist extension. The lower arm exercises were designed to increase strength in forearm extension, arm flexion at shoulder, wrist pronation and wrist flexion. During the training period, which lasted for five weeks, a l l subjects also participated in regular daily practises. The investigators found that the speed of shooting in both the skating slap and wrist shots for the experimental group improved significantly 16 (p.<.05) as a result of the training program. For the experimental group, the skating wrist shot mean increased from 65.5 M.P.H. to 70.8 M.P.H. while the control group improved from 67.5 M.P ."H. :to'69v4" M.PVH. For the skating slap shot the experimental group mean improved from 71.1 MoP.H. to 75.3 M.P.H. while the contol group improved from 70.6 M.P.H. to 70.7 M.P.H. Reed, Cotton, Hansen, and Gauthier (1979) investigated upper body strength and handedness shooting characteristics of junior and professional hockey players. Seventy-nine juniors and forty-three professionals were classified into l e f t - and right- groups. These two groups were in turn broken down into l e f t and." right shots giving a total of four groups. Left and right grip strength, shoulder strength in horizontal abduction and forearm strength in pronation were measured on each subject. In measuring grip strength, the results indicated a trend towards greater right thanle'ft scores in a l l subgroups. This trend was more pronounced for professionals than i t was for the juniors. The horizontal abduction showed a similar trend in that the means were higher on the right side than on the l e f t , for a l l groups. However, only the professional group demonstrated significant right strength dominance. One infers from this study that both sides of the body should be tested. Moyls (1979) investigated the effect, of dominant hand placement on accuracy and velocity of the standing snap shot. Subjects from the University of British Columbia varsity hockey team were classified by dominant hand and shooting characteristics. Shooting accuracy, velocity, and l e f t and right grip strength were measured on each subject. Placement of the strongest hand in either of the two 17 positions did not result in significant differences in either accuracy or velocity. Furthermore, the analysis showed no significant difference between the group with the dominant hand placed on the shaft of the stick and the group with the dominant hand placed on the top position on the stick as far as accuracy and velocity were concerned. Hand position had a correlation of -.013 with accuracy and a correlation of .268 with puck velocity. Muscle Balance The a b i l i t y of the agonist and antagonist muscles to work together in opposition is an important component of human movement. In order to generate a force, prime movers act without interference from antagonistic muscle groups. The efficiency and smoothness of the movement w i l l depend, to a large measure, on the strength, balance and f l e x i b i l i t y of agonist-antagonist (and synergistic) pairs about joints. Jensen and Schultz (1970) state that an inadequate range of motion in certain joints may restrict one's ab i l i t y to perform. In f l e x i b i l i t y acts as a resistance or brake to both speed and strength of movement. One's a b i l i t y to reduce resistance by relaxing the antagonistic muscles is a key factor in the speed of the movement. This concept of muscle balance appears to be an important factor to be considered in a movement such as the wrist shot. Jensen and Schultz maintain that the overdevelopment of one set of muscles can result in a person becoming musclebound. This decreases the speed and force of the movement by upsetting the agonist-antagonistic balance essential to a f l u i d smooth movement. Jesse (1977) emphasizes the importance of minimizing muscular imbalance through adequate testing and properly designed . 18 conditioning programs. An exact balance of muscles cannot be obtained because of their differences in anatomical size and length. However, he states that participation in some specific sports can actually aggravate the situation, resulting in increased muscular imbalance. He cites two studies in the Soviet Union which claim to have proven conclusively that participation in competitive sport creates muscular imbalance in the body. Jesse adds that many" conditioning programs devised by coaches also add to muscle imbalance. He points out that the goal of training and conditioning, at least from the standpoint of injury prevention, should reduce the ratio of imbalance to as small a differential as possible. F l e x i b i l i t y is d i f f i c u l t to aquire in a joint when there is a great disparity in the ratio of strength between the muscles on both sides of the joint. Constantly overstretched muscles on one side of a joint', caused by an imbalance created by stronger muscles on the opposite side, result in weakness and loss of power in the stretched muscles. A weak atrophic muscle is easily fatigued and is exposed to loss of elas t i c i t y sooner than a strong non-fatigued muscle. A tired muscle looses some of i t s a b i l i t y to relax. But these antagonistic muscles must relax completely while the prime mover muscles move the joint i f optimum speed and coordinat-ion are to be obtained. Jesse strongly maintains that an evaluation of muscle balance is required when analyzing the characteristics of skilled movement and injury prevention. Important to any skilled movement is the efficient summation of forces by muscles or muscle groups. Jensen and Schultz (1970) maintain that few performances result from the upper extremities alone, and in almost a l l cases these movements must be closely coordinated 19 and assisted by the movements in the neck, trunk, and lower extremities. In the ice hockey wrist shot the proper sequential summation of muscle groups is v i t a l to the success of the shot. Coaches King (1980), Kingston (1980) and Watt (1980) maintain that optimum-synchron-ization and summation of the forces in the shot w i l l increase the velocity of the wrist shot and may offset the effect of lack of strength. The force.must be transferred from:'the'.contact with the ice through the legs to the body, arms and stick. Without a proper summation of forces the shooter w i l l attain l i t t l e effective power in his shot. Eye Reflexes Specific to the hockey wrist shot is the importance of eye movement. Johnston (1976) feels that the head should l i f t to look at the target during the execution of the shot to ensure accuracy of the shot. On the other hand, Brown (1980) maintains that i f the eyes l i f t during the execution of the shot there is a change in the organization of the muscles due to eye and neck reflexes affecting extensor muscles. Consequently, by keeping the eyes fixed, one avoids interupting the organization of the muscle sequence in the shot. Research by Gardiner (1969) indicates that there are two main reflexes occurring when the head moves. The f i r s t originates in the labyrinthine receptors located in the inner ear. The second is located in the neck. In connection with these reflexes Gardiner points out that the head exerts important influences upon positions of movement of the trunk and limbs. The neck reflexes ensure that the body follows the head movements. Reflexes generated by the labyrinthine receptors ensure that-the centre of gravity of the body f a l l s over the base of 20 support. Gardiner speculates that simple labyrinthine and neck reflexes are concerned at a l l times in the coordination of limb movements and total body patterns. She cites work by Litmer which has shown that neck reflexes act more efficiently upon the upper limbs while labyrinthine reflexes have more impact on the lower limbs. Positions and movement of the head may be used to reinforce contractions of the arm muscles by involving tonic neck reflexes . Various positions of the head favor specific movements of the trunk and limbs. Fixing the head by focussing the eyes on a point stabilizes the gravitational effect on the labyrinthine receptors. The neck reflexes can make any necessary adjustments of the body to keep i t aligned with a correctly positioned head. Gardiner also notes that many motor patterns require learning to inhibit the responses. In summary, the work of Gardiner shows that both in learning to shoot and in'.''the execution of a good shot, positioning of the head can play a significant role. Other Sports Research done on other act i v i t i e s and sports related to this topic is minimal. Of particular interest, however, is Wiren's (1968) investigation of the human factors influencing the golf drive for distance. Various strength, f l e x i b i l i t y , anthropometric and distance tests were performed on fifty-one golfers. This data was entered into a stepwise regression program for predicting drive distance. Eighteen physical variables entered the equation as contributing predictors. Wiren then also performed a film analysis on a few of the golfers. From the film analysis he developed a second regression equation with four variables entering into the equation. He then combined the four 21 variables from the film analysis with the eight best physical predictors to give an overall regression equation. This overall equation had a multiple correlation coefficient of .953. He concluded that these models were reasonably accurate tools for predicting the distance of a golf drive. The variables having the highest correlation with drive distance were speed of downswing .789, length of backswing .719, clubhead speed prior to impact .680, handicap .612, right wrist palmar flexion strength .586, age .522, wrist cock retention (delayed hit) .523, thigh girth .494, right ankle plantar flexion strength .491, and l e f t shoulder horizontal extension strength .484. Although the golf swing is different from the wrist shot in execution several of the principles such as summation of forces and body f l e x i b i l i t y are similar. Thus Wiren's work can help in the understanding of the problems faced in this study. Authorities Census Table 2.1 represents a summary of the characteristics various authorities feel are important in the execution of the wrist shot. The authority.'.y name is listed in the category which he feels is the most important factor in the execution of the high speed wrist shot. Some names f a l l in more than one column; in these cases they feel that more than one major factor is contributing to the shot. There is a remarkable similarity in the authorities responses. It appears that the coordination of the movement (ideal summation of forces) is thought to be about as important as muscular strength. King (1980) states "after analyzing many good shooters that:on .occaision I have observed shooters who generate great velocity yet don't appear 22 to have great physical strength, I believe this is because "the synchronization of the mechanics are refined and exact". Watt (1980) and Kingston (1980) concur that coordination of movement can offset lack of strength. It appears that ideally, coordination and strength w i l l together produce the best results. Table 2.1. Census of Authorities Strength Coordination (muscle summation) Fl e x i b i l i t y * Bobby Hull (arms, wrists) Ken Hodge * Ken Hodge Dave Chambers (arms,wrists) shoulders * Dave Chambers Dave Chambers Clare Drake Clare Drake George Kingston (arms) George Kingston * Marshall Johnston * Marshall Johnston Tom Watt (forearms, wrists) Tom Watt Dave King (forearms, trunk) Dave King * Anatoli Tarasov Howie Wenger (elbows, knees), shoulders,hips Bob Hindmarch (legs) Bob Hindmarch *- Indicates information found in authority-Is book or a r t i c l e . Other information is based on personal contact. Summary The literature reviewed in this chapter indicates that many hours of practising the proper technique are necessary i f one is going to be able to shoot the puck at a high velocity. If optimum results 23 are to occur then the exact technique is essential. Strength i s known to play a role in determining the velocity of the wrist shot, but the exact nature of this role and the muscle groups involved has not been determined. It i s thought that muscular balance and optimum synchronization of muscle forces are v i t a l components related to a high speed shot. By l i f t i n g the head during the execution of the shot, the synchronization of muscle forces is altered. It i s s t i l l unknown whether this effect i s important to the outcome of the shot in.terms of puck velocity. L i t t l e researc'h has been done in the area of f l e x i b i l i t y and the role i t plays in executing a high velocity wrist shot. At the present time i t s effects remain unknown. 24 Chapter 3 PROCEDURES This chapter reviews the testing procedures used in measuring the human factors thought to be important in predicting the velocity of the wrist shot. Of primary interest are the rationales for variable selection, variable measurement methods, research design, and the st a t i s t i c a l techniques selected for the analysis of the data. The detailed step-by-step procedures for measuring the test variables are found in Appendix A. Sample The sample was made up of thirty-five active male hockey players drawn from five a b i l i t y levels of hockey. These levels were: junior, intermediate, senior, university, and professional. The subjects varied in age from seventeen to twenty-nine. It may be noted that there is an overlap in calibre at a l l five levels. Some of the subjects have participated in four of the five levels within a time span of three years. Criteria for selection were that the subject was currently participating in organized hockey at one of these five levels, and that he_ had the ab i l i t y to execute a well-developed wrist shot. The features looked for in determining whether a player had a well developed wrist shot were the following; (i) sweeping motion, of the stick through the shot; ( i i ) weight transfer when puck passes midline 25 of body (right handed shot— from right leg to l e f t leg); ( i i i ) extended follow through; (iv) and weight ending up directly over supporting leg. I n i t i a l l y a l i s t was drawn up of sixty-five players who met the required c r i t e r i a . From this l i s t , players were contacted un t i l a sample of thirty-five players had agreed to participate in the study. It was hoped that by meeting the cr i t e r i a the sample would be consistent with regard to method of execution, but f a i r l y broad in range in shooting a b i l i t y . Design This descriptive f i e l d study was based on a multiple correlat-ion design and ut i l i z e d Forward Stepwise Regression and A l l Possible Subsets Regression techniques in analyzing the data. Selection of Variables A major portion of this study deals with the selection of the independent variables to be measured. When analyzing a motor s k i l l there are several conventional approaches available. The characteristics which can be analyzed f a l l naturally into two categories: modifiable and non-modifiable. It has been the intent of this study to look at characteristics that are subject to modifications. This automatically eliminated such characteristics as genetic traits and body somatotypes, since neither of these can be changed to any great extent. Therefore this study concentrated onranalyzing the wrist shot through an investigation of physical characteristics; this appears to be the simplest means of analyzing primarily modifiable characteristics such as strength, f l e x i b i l i t y and muscle balance yet also incorporates some 26 non-modifiable characteristics in the area of anthropometry. Having decided on the general components of the motor s k i l l to be analyzed, the next task was to select the specific physical variables to be investigated. Several letters were written to authorities in the sport of hockey. They were asked to l i s t the specific muscle groups, joints and body dimensions that they deemed to be most important in terms of strength, f l e x i b i l i t y and composition in determining or restricting the puck velocity achieved in the execution of the wrist shot. It may be noted that several of the authorities specified the optimum summation of forces as an important element in determining puck velocity. No attempt 'was made in this study to try to alter the coordination of the shooting motion other than to ensure the execution was consistent with that already specified. The optimum summation of forces was not determined in this study. The test battery was selected on the basis of the responses received, related literature, and the author's subjective opinion. The physical variables chosen can be grouped into four categories: 1. Anthropometric Measures 2. F l e x i b i l i t y _.. . 3. Muscular 'Strength 4. Muscular Balance Table 3.1 l i s t s the independent variables used in this study. It may be noted that the anthropometric variables were investigative in nature. Very: . l i t t l e : information was found in related literature that predicted what relationship these anthropometric variables would have with a motor s k i l l of this nature. 27 Table 3.1. Independent Variables Selected for Testing"*" Group Variable Abbreviation Anthropometric Age AGE Variables Height HT Weight WT Arm Length Upper Arm UARM L Arm Length Lower Arm LARM L Arm Length/Stick Length ARM/STI Wrist Girth Upper Arm UWRISTG Wrist Girth Lower Arm LWRISTG Forearm Girth Upper Arm UFORARMG Forearm Girth Lower Arm LFORARMG Arm Girth Upper Arm UARM G Arm Girth Lower Arm LARM G Hand Dominance HAND Fl e x i b i l i t y Wrist Flexion Extension Upper Arm UWRISTFX Variables Wrist Flexion Extension Lower Arm LWRISTFX Shoulder Rotation Upper Arm USHROT Shoulder Rotation Lower Arm LSHROT Forearm Rotation Upper Arm UFORROT Forearm Rotation Lower Arm LFORROT Trunk Rotation TRUNKROT Strength Trunk Flexion TRUNKFLE Variables Trunk Extension TRUNKEXT Hip Flexion Glide Leg GLHIPFLE Hip Extension Glide Leg GLHIPEXT Hip Flexion Drive Leg DRHIPFLE Hip Extension Drive Leg DRHIPEXT Arm Abduction at Shoulder Upper Arm USHABD Arm Adduction at Shoulder Lower Arm LSHADD Diagonal Motion Upper Arm UARMDIAG Diagonal Motion Lower Arm LARMDIAG Forearm Supination Upper Arm UFORSUP . .. Forearm Pronation Lower Arm- LFORPRO Wrist Extension Upper Arm UWRISTST Wrist Flexion Lower Arm LWRISTST' Elbow Flexion Upper Arm. UELBFLEX Elbow Extension Upper Arm UELBEXT Elbow Flexion Lower Arm LELBFLEX Muscle Ratio Flexion/Extension Glide Hip F/E GHIP Variables Flexion/Extension Drive Hip F/E DHIP Flexion/Extension Trunk T F/E Drive/Glide Leg Flexion D/G FLEX Drive/Glide Leg Extension D/G EXT Flexion/Extension Elbow Upper Arm U F/E EL Flexion/Extension Elbow Lower Arm L F/E EL Upper/Lower Elbow Flexion U/L F EL Upper/Lower Elbow Extension U/L E EL Stick Variable Stick Length STICK L I. Upper and Lower Arm refer to the limb that grasps the stick in the upper and lower shaft positions (see upper and lower arm definitions, p.4) 2 8 The relaxed arm girths were selected because they are primarily a measure of bone development; the flexed forearm girths because they are predominantly a measure of muscle development; and the wrist girths because they are a combination of bone and muscular tendinous tissue. These measurements appear to cover the bone and muscle development of the arm. Stick length was the only characteristic of the stick to be measured. It was believed that the relationship of the arm to stick length was important when considering the mechanical advantage in the execution of the wrist shot. MacGillivary and Watson (1964) showed that by lowering the bottom hand on the shaft, one increases the mechanical advantage in the shot by decreasing the length of the resistance arm and increasing the length of the force arm. The resistance arm i s the distance from the lower hand on the stick to the puck. The force arm is from the top hand of the stick to the lower hand. Thus by changing the position of the lower hand one can alter the mechanical advantage. On the other hand, this process of lowering the bottom hand on the stick does decrease the speed advantage of the stick. The ratio of arm length to stick length affects the lower hand position. The muscular strength variables were selected by analyzing the shooting motion and choosing those elements of the motion deemed most important by the authorities consulted. An attempt was made to include variables which covered the major areas of the body, such as arms, legs and trunk; however, a more concentrated emphasis was placed on the arms. The f l e x i b i l i t y variables were selected for similar reasons, 29 again with the most concentrated area being the arms and shoulders. The muscular balance ratios were selected on the basis of considerations presented by Jesse (1977) . This section was coordinated with the strength variables already selected in order that extra strength variables need not to be measured. In addition, i t was felt that the strength variables selected for shooting would be the most important to consider in muscular balance. Instruments and their Reliability K15 Radar Gun A K15 radar gun (Digitar K15 doppler A-06-000011 M.P.H. IND. Inc.) was used to measure the velocity of the puck. The transmitter frequency of the gun has a maximum error rate of .05% and the speed of the radio waves is within .001% of the speed of light. The K15 radar gun has an internal calibration system which indicates whether the gun needs adjustment. If the gun requires calibration this system specifies the exact measures to be calibrated to. Problems occur when the radar gun is not positioned directly behind the flig h t of the object. In this case the problem is solved by placing the tester directly behind the net. The signal is broad enough to give accurate results for this situation;,. There are two types of interference that can influence the speeds recorded. The f i r s t type is a low frequency source (e.g., neon lights) in the testing v i c i n i t y . This source is not additive in nature and, as long as the subject's puck velocity is greater than the source reading, the puck velocity w i l l be displayed on the gun. The subjects are capable of shooting the puck at speeds much faster than this source 30 interference. The second type of interference i s additive in nature and is found in the-presence of ultra sound. This is a more serious problem in that::.'the system w i l l add this constant interference to the velocity of the puck. If the source of the interference is established, the true puck velocity can be obtained. In this study i t was found that the f i r s t type of interference was present but did not interfere with the results obtained in any way. No calibration was required throughout the tests. Cybex II A l l the strength tests were measured on the Cybex II isokinetic exerciser. The machine was calibrated before being used and at the end of the study. No major adjustments were required as the Cybex machine has been found to hold calibration for periods much longer than that required in this study. Leighton Flexometer A l l the f l e x i b i l i t y tests were measured by the use of a flexometer developed by Leighton (1966). The flexometer consists of a weighted 360 degree dial and a weighted pointer mounted in a case. The dial and pointer rotate independently of each other and are responsive to the gravitational pull of the earth. The instrument w i l l record any angle of movement as long as i t is positioned 20 degrees off the horizontal. The instrument requires no calibration and is as reliable as-'the tester; however extensive practise with the instrument is required for valid results to be obtained. The investigator has used this instrument in two studies and has obtained reliable results when measuring in the pilot study. 31 Tape A l l the girth variables were measured with a cloth tape. The tape was checked three times for accuracy. It was found that the tape had not stretched throughout the course of the testing but had stretched by three millimeters over thirty centimeters prior to use. Testing Procedures The testing procedures used were those recommended by the manufacturers and researchers who developed the testing equipment. A detailed description of these testing procedures is found in Appendix A. For consistency, only one investigator and one aide were used throughout the course of the testing. A l l the measurements were performed by the investigator. Each subject was required to read land sign a consent form, outlining the testing and timing requirements , prior to any involvement in the study. The testing was broken down into two sessions. The f i r s t session took place at the University of British Columbia Thunderbird Arena where stick length and puck velocity were measured. The second session took place at the University of British Columbia Buchanan Fitness Center. A l l strength, f l e x i b i l i t y and anthropometric measures were tested there. The order of testing was the same for each subject; anthropometric measures f i r s t , f l e x i b i l i t y measures second and strength measures last. Shooting Velocity The procedures used to measure puck velocity followed the principles outlined in Corbett's (1980) paper on Radar Systems. Five 32 pucks were weighed and frozen prior to testing. The subject was positioned on the blueline directly in front of the goal. The tester was situated directly behind the goal with the K15 radar gun aimed at the puck. The subject was instructed to shoot the puck as hard as possible at the middle of the net. He was allowed as many warm up shots as he wanted. Each subject shot a minimum of five test shots. Any shots that missed the net were not included as test shots. If the subject's fourth or f i f t h shot exceeded by 3 K.P.H. or more the speed of the fastest shot in his f i r s t three attempts, he continued to shoot until he had two consecutive shots after his fastest shot within the 3 K.P.H. li m i t . The fastest;three shots were averaged to give the subject's shooting score. The subject's sequence of head motions was also recorded. No subject required more than nine shots. Anthropometric Measures Girths. Girth measurement procedures are outlined in the MOGAP (1976) paper in Appendix A. Each girth was measured and recorded twice. If these measurements differed by more than 10% a third measurement was taken. The closest two scores were averaged to give the test score. Height, Weight and Arm Lengths. Height and arm length were measured By "the use of'"a s'tadiometef. Each arm length was measured twice. When the difference between the two scores was more than 10% a third measurement was taken. The closest two were averaged to give the arm length score. Weight was measured on a balance beam scale. The subject was 33 weighed wearing shorts only. F l e x i b i l i t y Measures Each subject had a brief ten-motion warmup similar to the f l e x i b i l i t y measures being tested (see Appendix B). The flexometer then strapped to the subject and two test t r i a l s were recorded for each movement. The largest value obtained was recorded as the test score. Muscular Strength Measures Eighteen strength tests were administered to the subject. Each test consisted of two t r i a l s for each test. Before the actual test one practise t r i a l was given at about half the maximum force in order to allow the subject to become accustomed to the resistance given by the Cybex machine. The Cybex was set at a maximum arm speed of 30 degrees per second, for a l l but one of the tests. The trunk extension test was executed as an isometric contraction rather than isokinetic as was the case in the other tests. The lever arm length was recorded for each exercise. The printout from the machine gave the torque applied through the f u l l range of motion. By dividing the maximum torque applied by the lever arm length one obtains the maximum force applied by the subject. The procedures for each exercise follow those specified in the Cybex II manual (1978) . Muscle Balance The muscular balance ratios were calculated for three muscle groups in the body. These were obtained by dividing the force in flexion by the force in extension around a body joint. The ratio 34 of hip flexion/hip extension was calculated for both legs. The drive leg was compared to the glide leg for both flexion and extension. The lower arm was similarly compared to the upper arm. Similar comparisons were made for elbow flexion and extension, in both arms. Trunk flexion/ trunk extension was also calculated. The calculated .ratios were used to analyze the balance of muscle forces of the hip, trunk, and elbow joints and to compare the right to the l e f t limb in flexion and extension. Analysis, .of Data ' , There were two major objectives in analyzing the data: f i r s t , to find the human characteristics most important in predicting puck velocity; and second, to find the best overall regression equation for predicting puck velocity. Multiple regression techniques provide a natural and effective tool for achieving these objectives. In the f i r s t instance Stepwise Regression was the technique chosen, but the A l l Possible Subsets technique was required to verify the cutoff point for variables entering the regression equation. Programs BMD:P2R (Stepwise Regression) and BMD:P9R (All Possible Subsets Regression) were run to identify the best regression equation. (BMD P Series Manual, 1980) Stepwise Regression starts with no variables in the regression equation and enters f i r s t the variable that has the highest correlation with the dependent variable. It then proceeds to enter the variable having the highest partial correlation with the dependent variable. The partial correlation suppresses the effect of the variable already entered into the equation. The program proceeds in this step-by- step 35 fashion. A significance level must be specified prior to the running of the program. The significance level was set by an F-to-enter value of .500. Any variable that had a partial correlation with the dependent variable that gave a F value of .500 or greater, would enter the equation. Those with an F value of less than -.500 would be removed from the equation. Using such a low F-to-enter value resulted in many variables entering the equation; however, ,by monitoring the standard error of estimate, the best regression equation could be selected. The best possible prediction equation c occurs when the standard error of estimate is at a minimum. As a check and aide in finding the best regression equation, the BMD:P9R A l l Possible Subsets Regression program was also run. This program starts with a single variable and gives the ten best individual variables for predicting puck velocity. It then proceeds to give the ten best two'variable' equations and continues on in this fashion; at each step another variable is added. Next the adjusted R squared criterion was applied to give the best regression equation. The adjusted R squared value takes into account the number of independent variables entering the equation and the sample size. The regression equation with the largest adjusted R squared value is selected as the best. The results from these two programs gave the overall best regression equation. This procedure was carried out f i r s t on the raw data and then later on transformed data. The data was transformed in an attempt to produce a better regression equation u t i l i z i n g the variables that did not appear to be related to puck velocity in a 36 linear fashion. The following transformations were applied to the 2 3 — variables: x , x , log x, and Vx::. „ Running the two regression programs again, another best regression equation was calculated. The two equations were compared in terms of predicting a b i l i t y of puck velocity and are discussed in chapter 4. J 37 Chapter 4 RESULTS AND DISCUSSION In this chapter the data are examined s t a t i s t i c a l l y and an optimal regression equation is established. The Stepwise Program and the A l l Possible Subsets Regression Program are analyzed and discussed with regard to selecting the best possible regression equation. The contribution of each group of variables to the regression equation is analyzed in terms of predictive a b i l i t y and multiple correlation values. Various transformations are then applied to the data, and the effect of this process on the regression equation is observed. Specifically, the equations for the transformed data are compared with that for the untransformed data relative to bias and predictive a b i l i t y . Results: Multiple Regression Using Raw Scores Descriptive Statistics Table 4.1 gives the mean and standard deviation for each variable. Puck velocity had a f a i r l y broad range of scores with a minimum speed of 75.7 K.P.H. and a maximum speed of 105.7 K.P.H. The mean value was 86.6 K.P.H., with a standard deviation of 6.5 K.P.H. In general, the sample turned out to be very homogeneous for the anthropometric variables. Comparing the coefficients of Variation, one can see that the f l e x i b i l i t y measures are slightly more variable then the anthropometric measures, and the strength measures tend to be 38 Table 4.1. Descriptive Statistics for A l l Variables Variable - iMean Standard Coefficient Group No. Name Deviation of Variation Anthropometric 1 AGE 22.571 2.465 0.109 Variables 2 HT 180.860 4.811 0.027 3 WT 80.880 5.900 0.073 5 UARM' L 79.431 3.246 0.041 6 LARM L 79.231 3.114 0.039 7 ARN/STI 0.607 0.024' 0.040 8 UWRISTG 18.283 0.766 0.042 9 LWRISTG 18.230 0.783 0.043 10 UFORARMG 29.368 1.419 0.048 11", LFORARMG 29.237 1.220 0.042 12 UARM G 32.605 2.012 0.062 13 LARM G 33.160 1.883 0.057 Fl e x i b i l i t y 15 UWRISTFX 158.114 15.221 0.096 Variables 16 LWRISTFX 156.543 14.745 0.094 17 USHROT 196.228 18.900 0.096 18 LSHROT 197.114 19.282 0.098 19 UFORROT 192.314 21.678 0.113 20 LFORROT 186.514 21.493 0.115 21 TRUNKROT 160.486 25.988 0.162 Strength 22 TRUNKFLE 91.714 15.315 0.167 Variables 23 TRUNKEXT 111.428 19.850 0.178 24 GLHIPFLE 50.023 .'9.585 0.192 25 GLHIPEXT 73.117 14.595 0.200 26 DRHIPFLE 49.011 9.600 0.196 27 DRHIPEXT 73.134 13.910 0.190 28 USHABD 25.100 4.053 0.161 29 LSHADD 35.688 7.204 0.202 30 UARMDIAG 34.563 8.749 0.253 31 LARMDIAG 31.223 9.346 0.299 32 UFORSUP 7.694 1.720 0.223 33 LFORPRO 8.489 2.118 0.249 34 UWRISTST 31.277 8.903 0.285 35 LWRISST 48,403 11.313 0.234 36 UELBFLEX 41.674 7.988 0.192 371 UELBEXT 33.274 7.853 0.236 38 LELBFLEX 42.954 -.7.785 0.181 39 LELBEXT 32.640 <,'6.723 0.206 Muscle Ratio -40 F/E CHIP - 0;695 - 0.126 0.182 Variables 41 F/E DHIP 0.680 0.121 0.179 42 T F/E 0.849 0.209 0.246 43 D/G FLEX 0.980 0.064 0.066 44 D/G EXT 1.005 0.089 0.089 45 U F/E EL 1.282 0.231 0.180 46 L F/E EL 1.340 0.228 0.170 47 U/L F EL 0.976 0.118 0.121 48 U/L E EL 0.948 0.209 0.221 Stick Variable 4 STICK L 130.800 5.417 0.041 Dependent Var.v ibl4 PUCKVEL 86.560 6.525 0.075 1. See Page for a description of the variables and code identification 39 the most variable of a l l . The muscle ratio group is really hetero-skedast ic in that the values of the coefficient of Variation range from .066 for flexion of the drive leg over glide leg to .246 for flexion over extension for the trunk. On the whole, however, one can conclude from this table that the sample was f a i r l y homogeneous. The product-moment correlation matrix is presented in Appendix G. Table 4.2 shows the correlations of puck velocity with the dependent variables. The shoulder adduction motion for the lower arm had the highest correlation with puck velocity, yielding a value of .451. The next highest correlation was with the forearm supination strength measure for the upper arm; this had a correlation of .431. These two variables had the highest linear relationships with puck velocity. For ; thirty-five degrees of freedom a correlation coefficient of .325 is required for the relationship to be significant at the .05 level. The five variables which qualified under this c r i t e r i a , were; arm-adduction at the shoulder of the lower arm (.451), forearm supination of the upper arm (.431), wrist strength in extension of upper arm (.418), diagonal arm strength of lower arm (.378) and wrist strength in flexion of lower arm (.344). It is interesting to note that a l l of these variables are measures of arm strength. No other variables were significant at the .05 level. Stepwise and A l l Possible Subsets Regression Analysis The P:2R (Stepwise Regression) and P:9R (All Possible Subsets Regression) programs were run on the data. The procedures are described in Chapter 3. Table 4.3 shows the order of variables entering the 2 2 . equation and the F-rvalue, Multiple R, and .the ..increase, in R rafter each step for the Stepwise Regression program. Table 4.4 shows the order of 40 Table 4.2. Product-Moment Correlation Table of Test Variables and Puck Velocity PUCKVEL AGE .0293 HT .0189 WT .2094 STICK L .0318 UARM L -.0435 LARM L .0386 ARM/STI -.0380 UWRISTG .2509 LWRISTG .3047 UFORARMG .2603 LFORARMG .2755 UARM G .3210 LARM G .3128 UWRISTFX .0189 LWRISTFX -.1984 USHROT .1453 LSHROT .0599 UFORROT -.2207 LFORROT -.2587 TRUNKROT -.0424 TRUNKFLE .1920 TRUNKEXT .1432 GLHIPFLE .1530 GLHIPEXT .2874 DRHIPFLE .1951 DRHIPEXT .3110 USHABD .3041 LSHADD .4511 UARMDIAG .2000 LARMDIAG .3778 UFORSUP .4311 LFORPRO .1212 UWRISTST .4176 LWRISTST .3437 UELBFLEX .1807 UELBEXT .1064 LELBFLEX .1972 LELBEXT .2359 F/E GHIP -.1380 F/E DHIP -.1391 T F/E .0046 D/G FLEX .0923 D/G EXT .0928 U F/E EL .0516 L F/E EL -.0993 U/L F EL .0302 U/L E EL -.2873 41 Table 4.3. Order of Entering of Variables Using Stepwise Program No. of Variable R Increase in F-To- F-To Variables Entered Removed Squared R Squared Enter Remove 1 LSHADD .2035 .2035 8.431 2 UFORROT .2910 .0875 3.948 3 USHROT .3656 .0746 3.648 4 UWRISTFX .4148 .0492 2.524 5 L F/E EL .4519 .0371 1.961 6 UARM G .4781 .0262 1.408 7 USHABD .5114 .0333 1.840 8 UFORSUP .5373 .0258 1.452 9 UARMDIAG .5807 ' .0434 2.589 8 USHROT .5780 -.0027 9 . TRUNKFLE .6278 .0497 3.340 10* TRUNKROT .6613 .0335 2.377 11 WT .6787 .0174 1.247 12 LWRISTST .6924 .0136 0.976 13 F/E GHIP .7101 .0177 1.283 14 LFORROT .7200 .0099 0.708 15 STICK L .7342 .0142 1.012 16 LFORARMG .7463 .0122 0.863 15 UARM'G .7461 -.0002 16 UFORARMG .7654 .0192 1.476 15 F/E GHIP .7594 -.0059 16 LSHROT .7720 .0126 0.993 17 USHROT .7878 .0158 1.263 18 F/E GHIP .8010 .0132 1.060 19 D/G EXT .8095 .0085 0.673 20 LFORPRO .8250 .0155 1.242 21 LELBFLEX .8417 .0166 1.367 20 LFORROT .8411 -.0005 19 STICK L .8371 -.0041 20 LARMDIAG .8687 .0316 3.375 21 F/E DHIP .9040 .0352 4.770 22 UELBFLEX .9122 .0083 1.130 23 DRHIPFLE .9258 .0135 2.003 24 GLHIPEXT .9416 .0158 2.708 25 HT .9489 .0073 1.292 24 F/E GHIP .9487 -.0002 25 LFORROT .9521 .0034 0.639 26 AGE .9584 .0063 1.216 0.160 0.013 0.455 0.045 0.357 0.044 Cutoff of variables entering in selection of best equation 42: variables entering using the A l l Possible Subsets Regression program. The suggested best equation is identified by this program arid is noted in table 4.4 with an asterisk. One can see from these tables that i t is possible to account for virtually a l l the variance by letting a l l the variables enter. After the 38th step in the Stepwise solution the Multiple R has a value of .9790, indicating that 96% of the information required to predict puck velocity is known. With a small sample size i t M s possible to account for virtually a l l the variance when the subjects to number of variables ratio is small. With thirty-five subjects and twenty-six variables entered in . the equation at the 38th step, the subjects to number of variables ratio is relatively small. Thus i t is necessary to limit the number of variables entering the equation. The determination of a cutoff point for variables entering the equation is usually based on monitoring the standard error of estimate. As variables start to enter the equation the standard error should decrease. However, as the subject to variable ratio gets smaller the standard error should reach a minimum and eventually start to increase. With this data, however, the standard error of estimate never did achieve a minimum (see Figure 1). Thus alternate c r i t e r i a were required for the selection of the best equation. It was decided to plot the number of variables against each of (i) the standard error of estimate, ( i i ) the adjusted R squared, and ( i i i ) the multiple R. The adjusted R squared value considers the number of variables in the equation and the sample size. The BMD P Series Manual (1977,p.424) gives the formula used to calculate this value. Ideally, the best regression equation results when the value reaches a maximumj however, with this data, the adjusted R squared never 43 Table 4.4. Order of Entering of Variables Using A l l Possible Subsets Program No. of Variable R Squared Adjusted R Variables Entered Replaced Squared 1 LSHADD .2035 .1793 2 UFORROT .2909 .2466 3 USHROT .3656 .3042 4 UWR'ISTFX .4148 .3368 5 L F/E EL .4519 .3574 6 UARM G .4781 .3663 7 USHABD .5114 .3848 8 UFORSUP .5373 .3949 9 UARMDIAG .5807 .4298 10 TRUNKFLE .6278 .47.27 11 TRUNKROT .6621 .5004 12 WT .6787 .5035 13 STICK L WT .7057 .5235 LFORARMG 14 LSHROT UARM G .7290 .5393 LWRISTST 15 WT STICK L .7468 .5469 UARM G LFORARMG F/E GHIP 16 STICK L WT .7682 .5622 LFORARMG UARM G DRHIPFLE 17 LWRISTG .7773 .5546 18* UARM G UWRISTFX .8331 .6454 UWRISTST USHABD D/G EXT F/E GHIP U F/E EL 19 USHABD .8384 .6337 20 F/E GHIP .8451 .6238 * Selected as best equation by adjusted R Squared c r i t e r i a of program 44 reaches a maximum. Considering a l l three graphs, Figures 1,2 and 3, i t seems natural to stop variables from entering the equation when the absolute values of the slopes begin to reach a minimum. Analyzing Figures 1,2 and 3 for the stepwise program, the point at which a l l three graphs level off occurs at the 12th step. This yields an equation with ten variables, and is the best regression equation according to the cutoff criterion. The cutoff of variables is noted by a dashed line on the three graphs. The equation cut at this point accounts for 66% of the variance. By running both Stepwise and A l l Possible Subsets Regression programs i t was possible to see i f both programs entered the variables in the same order. Comparing the two programs for the f i r s t ten variables (Tables';4.3 and 4.4) one finds only one discrepancy. The Stepwise program enters the f l e x i b i l i t y measure trunk rotation while the A l l Possible Subsets program enters the f l e x i b i l i t y measure shoulder rotation of the upper arm. Since the Stepwise program had a higher multiple R value i t was selected. Thus the best regression equation for predicting puck velocity i s : Y(puck velocity) = X^(arm girth of upper arm)(.7:544) + (wrist f l e x i b i l i t y tipper arm) (.1374) + X^Oforearm rotation upper arm)(-.1779) + X.(trunk rotation)(-.0574) 4 + X 5(trunk flexion)(.1645) + X (arm abduction upper arm)(-1.1586) 6 + X^(arm adduction lower arm)(.8949) + X„(diagonal motion upper arm)(-.2990) o + Xg(forearm supination upper)(1.7023) + X^^flexion/ext. .elbow lower) (-7 .5737) + 73.0946. This equation has a multiple R value of .813 and a standard error of estimate of 4.52. Therefore the equation can account for 66% of the 45 Figure I. Standard Error of Estimate Versus Number of Variables S t a n d a r d E r r o r 0 i t 5 10 15 20 25 Figure 2i' Adjusted R Squared Versus Number of Variables A d j u s' t e d Figure M u 1 t i P 1 e R I .1 I i i> i i_ 0 5 10 15 20 . 3. Multiple R Versus Number of Variables i::.o V . 7 -. 6 -.5 .4 5 10 15 Number of Variables 20 25 25 46 var i a b i l i t y and gives a. prediction accuracy of + 9.0 K.P.H. 95% of the time. Discussion: Multiple Regression Using Raw Scores Anthropometric Variables Table 4.5 shows the order of entering of the Anthropometric variables. Table 4.5. Anthropometric Variables Order of Entering Step No. Variable F-to-Enter 6 Arm Girth Upper Arm 1.4076 13* Weight 1.2471 18* Forearm Girth Lower Arm 0.8632 35* Height 1.2919 38* Age 1.2162 No other anthropometric variables had a F-to-Enter of greater than .500 *- Variable did not enter best regression equation. In general, the six girth measures had very poor linear relationships with puck velocity. Arm girth, the anthropometric variable with the highest zero order correlation with puck velocity (.321), was the only anthropometric variable to enter the best regression equation. The relaxed arm girths primarily measure bone development while the forearm and wrist girths (both flexed) measure a combination of muscle and bone. One would expect the flexed forearm girths to be more highly correlated with the strength variables than the relaxed arm girths. This was the case. Since upper arm girth had a relatively 47 large correlation with puck velocity, and was not particularly well correlated with the'i.strength variables, i t seems reasonable for i t to play a somewhat independent role and hence enter the equation. The two arm length measures were essentially uncorrelated with puck velocity and thus did not enter the equation. Body weight had a correlation of .2000 with puck velocity; i t would have been the next variable ' entered had the equation not been< cutoff at ten variables. It appears that the increase in weight of hockey players in the past fifteen years (Ennos 1976) is not strongly linearly related to the increase in puck velocity in shooting. Height did not turn out to be an important variable in this analysis. Ennos (1976), Moyls and Nobbs (1979), and Selder (1964) found that the body types of collegiate hockey players were similar, and that playersr.at the college level have been increasingly heavier in recent years. However, one can only conclude from this study that height and weight are not as important in predicting puck velocity as one might be inclined to assume, looking at their studies. F l e x i b i l i t y Variables Table 4.6 shows the order of entering of the f l e x i b i l i t y variables Table 4.6. F l e x i b i l i t y Variables Order of Entering Step no. Variable F-to-Enter 2 Forearm Rotation Upper Arm 3.9482 3** Shoulder Rotation Upper Arm 3.6476 4 Wrist Flexion Extension Upper Arm 2.5239 12 Trunk Rotation 2.3773 16** Forearm Rotation Lower Arm 0.7076 22* Shoulder Rotation Lower Arm 0.9928 48 No other f l e x i b i l i t y variables had an F-to-Enter of greater than.5000. *- Variable not in best regression equation. **- Variable entered, was removed and entered again. Not in best regression equation. Chambers (1979) was the only authority to mention that a f l e x i b i l i t y measure is important in predicting puck velocity (he stated that trunk rotation played an important role in shooting) . In this study, three out of the ten variables entering the equation were f l e x i b i l i t y measures. Trunk rotation, forearm rotation and wrist f l e x i b i l i t y of •': the upper arm a l l proved to be important in predicting puck velocity. It should be noted that the upper and lower forearm rotation correlated f a i r l y highly with each other (r=.599). Thus forearm rotation of the lower arm is probably an important variable, but because of i t ' s correlation with the upper arm (which entered f i r s t ) , i t did not enter the best equation. It appears that the importance of f l e x i b i l i t y in shooting may be more important then has generally been thought. However, one?.observes that trunk rotation and upper forearm rotation have negative correlations (r's of -.0424 and -.2207 respectively). One can attempt to explain this by the fact that most of the strength variables are negatively correlated with the f l e x i b i l i t y variables.. Thus the higher the f l e x i b i l i t y score the lower the strength score. Shoulder f l e x i b i l i t y does not appear to follow this pattern and is correlated positively with most of the strength variables and puck velocity. 49 Strength Variables Table 4.7 shows the order of entering of the strength variables Table 4.7. Strength Variables Order of Entering Step no . Variable F-to-Enter 1 Arm Adduction at ShoulderT.Lower. Arm 8.4308 7 Arm-'Abduction at Shoulder Upper Arm 1.8402 8 Forearm Supination Upper Arm 1.4520 9 Diagonal Motion Upper Arm 2.5894 11 Trunk Flexion 3.3401 14* Wrist Strength Lower Arm 0.9760 26* Forearm Pronation Lower Arm 1.2419 2 7* Elbow Flexion Lower Arm 1.3748 30* Diagonal Motion Lower Arm 3.3748 32* Elbow Flexion Upper Arm 1.1296 33* Hip Flexion Drive Leg 2.0032 34* Hip Extension Glide Leg 2.7083 No other strength variables had a F-to-Enter value of greater than .5000. *- Variable not in best regression equation. In line with the predictions made by the authorities recorded in the census in chapter 2, the strength measures did have the greatest impact on the regression equation. The following strength measures: trunk flexion, arm<adduction at the shoulder of the lower arm, arm abduction at the shoulder of the upper arm, diagonal motion of the upper arm, and forearm supination .,of the upper arm a l l did enter the best regression equation. Alexander, Drake, Reichenback (1964)' investigated eight exercises, including arm adduction and wrist supination of upper arm, and found that an isometric exercise program consisting of these eight exercises significantly increased the velocity of the skating wrist and slap shots. It is interesting to note that two of their 50 exercises (arm adduction and wrist pronation of the upper arm) were variables which entered the best regression equation. Arm adduction at the shoulder of the lower arm was the best individual predictor of puck velocity, accounting for 20% of the uncertainty by i t s e l f . Wrist strength was not an important variable in terms of predicting puck velocity. Several of the authorities suggested that wrist strength was an important variable to consider when analyzing the causes of a fast shot. Moyls (1979) found l i t t l e or no relationship between puck velocity and grip strength. It appears that forearm supination and pronation may play a more important role : than wrist flexion, extension of grip strength. Muscle Ratio Variables Table 4.8 shows the order of entering of the muscle ratio variables. Table 4.8. Muscle Ratio Variables Order.of Entering Step no. Variable F-to-Enter 5 Flexion/Extension Elbow Lower Arm 1.9609 15* Flexion/Extension Hip Glide Leg 1.2832 25* Drive/Glide Leg Hip Extension 0.6732 31* Flexion/Extension Hip Drive Leg 4.7701 No other muscle ratio variables had a F-to-Enter value of greater than .5000. *-r .'Variable not in best regression equation. As indicated in Chapter 2, Jesse (1977) fel t that aa analysis of muscle ratios may be more significant than an analysis of individual 51 strengths. In this study, only^one muscle ratio entered the equation: the ratio of elbow flexion to extension of the lower arm proved to be important in predicting puck velocity. It was the f i f t h variable to enter the equation, and i t had a negative coefficient. The mean for this score was 1.34. The negative coefficient suggests that i t may be more efficient i f the ratio is closer to 1.0C". Thus by having the extensor and flexor muscles balanced the motion would be more efficient for the reasons suggested by Jensen and Schultz (1970) in Chapter 2. Therefore the extensor muscles in the arm should be strengthened to increase the efficiency of the motion. However one can only speculate as to the effect on shot velocity at this time. Further study in this area is needed. Several of these ratios enter the equation after the cutoff. Although flexion/extension of the hip for the glide leg had a f a i r l y large F-to-Enter value when i t entered the equation, this value did not become large enough to enter until after the 15th step. This was due to the high intercorrelations of some of the variables. However, i t appears that some of the uncertainty is not accounted for in the other variables and thus i t entered at the 15th step. Head Sequence Variable Head sequence was not entered into the analysis because every subject used the same sequence. A l l the subjects started with the head down looking at the puck. The head was l i f t e d after the execution of the shot was completed. Since none of the subjects varied from this sequence, one does not have to be be as concerned with the eye reflex effects on the shot as originally was thought. Since the pattern was consistent throughout the test i t is assumed that the neck and 52 labyrinthine reflexes outlined by Gardiner (1969), were consistent also. Results: Multiple Regression Using Nonlinear  Transformations From the i n i t i a l computer runs i t appeared that several of the variables were not related to puck velocity in a linear fashion. Transformations of the data were used in an attempt to produce a better prediction equation, possibly incorporating variables which were not linearly related. The following transformations were applied to the 2 3 variables: x , x , log x, and \/x. As a result of these procedures only the strength variables showed an increase in their correlation coefficient with puck velocity, and that occurred when the cubic function was used. For every strength measure, the correlation with puck velocity increased when the data was transformed by a cubic function. It i s of some interest that none of the anthropometric of f l e x i b i l i t y variables improved when transformed by any of these functions. Table 4.9 shows the correlation values of the transformed variables with puck velocity. With the i n i t i a l 48 variables, and the cubed strength variables now in the data storage, the Stepwise and A l l Possible Subsets Regression programs were rerun. Table 4.10 shows the order in which the variables entered in the Stepwise procedure. Table.4.11 shows the A l l Possible Subsets entering order. . After ten steps :the transformed run has (in both procedures) a multiple R of .889, 8% better than the untransformed regression of the better (Stepwise) procedure at the same point. The cutoff point was determined by the same procedure as was used'previously on the untransformed data. Figures 4, 5, and 6 show the number of 53 Table 4.9. Product-Moment Correlation Table of Cubed Strength Variables and Puck Velocity PUCKVEL TRUNKFLE3 .2334 TRUNKEXT3 .1934 GLHIPFLE3 .2165 GLHIPEXT3 .3929 DRHIPFLE3 .2840 DRHIPEXT3 .3888 USHABD3 .3922 LSHADD3 .4727 UARMDIAG3 .2492 LARMDIAG3 .4409 UFORSUP3 .4938 LFORPRO3 .1555 UWRISTST3 .4439 LWRISTST3 .3727 UELBFLEX3 .2815 UELBEXT3 .1401 LELBFLEX3 .3299 LELBEXT3 .3000 54 Ifiblec 4.10. Order of Entering of Variables Using Stepwise Program for Transformed Data No. of Variable • R : . Increase in F-To- F-To-Variables Entered Removed' Squared R Squared Enter Remove 1 3 UFORSUP .2438 .2438 .10.640 2 LSHADD3 .3302 .0864 4.126 3 UELBFLEX .4156 .0855 4.533 4 UELBFLEX3 .5690 .1534 • 10.676 5 UWRISTST3 .6036 .0345 2.526-6 STICK L .6463 .0427 3.383. -7 TRUNKROT .6807 .0344 2.907 8 GLHIPEXT .7353 .0546 5.364 A 9 LSHROT .7588 .0235 2.433 10 TRUNKFLE3 .7739 .0152 1.'612 11 USHABD .7838 .0098 1.047 12 AGE .7960 .0122 1.313 13 L F/E EL .8140 .0181 2.040 14 WT .8283" .0143 1.667 15 HT .8692 .0408 5.930 16 LSHADD .8899 .0208 3.393 17 UWRISTG .9042 .0143 2.533 18 LARMDIAG3 .9207 .0165 3.332 17 TRUNKFLE3 .9174 -.0033 .6708 18 LWRISTG .9309 .0135 3.7-117 19 GLHIPEXT3 .9420 .0111 2.876 2.0 USHROT .9490 .0071 1.937 21 UARM G .9530 .0039 1.171 22 LWRISTFX .9570 .0040 1.212 * Cutoff of variables entering in selection of best equation 55 Table 4.11. Order of Entering of Variables Using A l l Possible Subsets Program for Transformed Data No. of Variables Variable Entered Replaced R Squared Adjusted R Squared 1 UFORSUP3 .2438 .2209 2 LSHADD3 .3302 .2883 3 UELBFLEX .4156 .3591 4 UELBFLEX3 .5690 .5115 5 UWRISTST3j .6035 .5352 6 STICK L .6463 .5705 7 TRUNKROT .6807 .5979 8 GLHIPEXT .7353 .6538 9 LSHROT .7587 .6719 10 TRUNKFLE3 .7739 .6797 11 USHABD .7838 .6804 12 AGE .7959 .6846 13 L F/E EL .8140 .6989 14 WT .8283 .7081 15 HT .8691 .7658 16 LSHADD .8899 .7920 17 UWRISTG GLHIPEXT GLHIPEXT .9167 .8334 18 LARMDIAG3 .9370 .8661 19 USHROT .9434 .8717 20 GLHIPEXT .9490 .8762 21 UARM G .9529 .8769 * 22 LWRISTFX .9570 .8781 * Selected as best equation by adjusted R Squared c r i t e r i a of program 56 variables versus the standard error of estimate, adjusted R squared, and multiple R respectively. The cutoff point was nine variables. Moreover the f i r s t nine variables to'enter the Subsets procedure were exactly the same as the f i r s t nine for the Stepwise. The best regression equation for the transformed variables i s : Y(puck velocity) = X][ (stick length) (. 3405) + X£(shoulder rotation lower arm)(.0626) + X 3(trunk rotation)(-.0972) + X^(hip extension glide leg)(-.2334) + X,.(elbow flexion upper arm)(-1.9188) 3 + X (arm adduction lower arm) (.0001) 6 3 + X 7(forearm supination upper arm) (.0116) 3 + X Q(wrist extension upper arm) (.0001) 3 + Xg(elbow flexion upper arm) (.0003) + 103.1292. This equation has a multiple R of .871 and a standard error of estimate of 3': 74. Discussion: Nonlinear Versus Linear Equations On examining the transformed regression equation one sees that four transformed variables (cubed strength) entered the equation. One concludes that the cubic function fit t e d these variables better than the linear function since the cubed strength variables tended to have higher correlations with puck velocity than the linear variables. In other words, mathematically the cubic curve function fit t e d the relationship better than the linear function. In terms of improving one's shot this phenomenon remains unexplained. Two linear strength variables entered this equation. Neither of these entered the untransformed regression equation. One observes 57 Figure .4. Standard Error of Estimate Versus Number of Variables for Transformed Data S. t a n d a r d E r r o r 4 h 3 h 0 "10 15 20 25 Figure 5. Adjusted R Squared Versus Number of Variables for Transformed Data A d j u s t e d 0 10 15 20 25 Figure 6:. Multiple R Versus Number of Variables for Transformed Data 1.0 M u 1 t i .9 r P 1 R .7 .6 •5 r • 0 5 '.10 15 Number of Variables 20 •25 58 further that two f l e x i b i l i t y variables entered this equation. Stick length was the remaining variable to enter this equation. Trunk rotation was the only untransformed.variable to enter both equations. Using the regression equation without the transformations, each subject's puck velocity was predicted, and the residuals between the observed and predicted were calculated. Table 4.12 shows the observed, predicted and 95% confidence intervals. From this table one can see that a l l the subjects pred icted scores f e l l within the 95% confidence limits for each observation. The same procedure :.was carried out on the equation derived from the transformed data. Table'4.13 shows the results for this equation. Again, a l l the residuals were within the 95% confidence intervals for each observation. Since the confidence intervals are f a i r l y small one can conclude that .both regression equations are .fairly good predictors of puck velocity. A comparison of the two equations seems to show that the inclusion of the cubic strength function improves the equation. It has a higher multiple R value by .06 and employs one less variable in the equation. The standard error of estimate is also lower. Inspection of the confidence intervals forithe two equations, indicates that the confidence intervals are smaller for the transformed equation. By sorting the data into ascending order of puck velocity and plotting the residuals of the subjects in that order, one can check the bias of the equations. Figure 7 shows that the regression equation tends to estimate too high for the lower scores and too low for the higher scores. This 'is-.a kindr.of' -an'averagirig-jef f ect, tending to draw 59 Table 4.12. Observed and Predicted Scores with 95% Confidence Intervals Subject Observed Residual dEredicte'd^ :c9.5-.-l;„ "Confidence Intervals .- Mean Observation Plus-Minus Plus-Minus 7 75.7 -2.519 78.219 5.621 10.89 19' 76.7 -5.485 82.185 4.705 10.45 5 78.0 -6.012 84.012 5.370 10.76 15 78.0 -0.772 78.772 4.891 10.53 12 79.3 -6.023 85.323 3.900 10.11 25 80.0 -1.206 81.206 4.906 10.54 18 80.0 0.630 79.370 5.332 10.75 35 81.0 -9.644 90.644 4.862 10.52 31 81.3 -2.258 83.558 4.374 10.30 10 83.0 -0.103 83.103 5.360 10.76 34 83.0 -2.259 85.259 6.407 11.32 2^3 83.3 -1.325 84.625 4.107 10.19 22 83.7 3.353 80.347 4.860 10.52 14 84.3 1.864 82.436 3.708 10.04 11 84.7 -1.030 85.730 5.158 10.66 8 85.0 5.362 79.638 4.607 10.40 32 85.3 -0.599 85.899 4.111 10.19 27 " 86.0 -3.130 89.130 4.961 10.57 2 86.3 5.797 80.503 5.807 10.99 3 86.3 -5.086 91.386 4.518 10.37 16 87,7 5.111 82.589 5.043 10.60 4 88.3 1.817 86.483 3.475 '9.55 33 : " 88.7 -1.386 90.086 6.356 11.29 6 89.0 -2.461 91.461 6.163 11.18 . 21 89.7 2.229 87.471 5.909 11.04 20 90.0 2.989 87.011 4.190 10.23 17 90.3 ".4.517 85.783 7.442 11.93 26 92.0 -1.912 93.912 4.985 10.58 30 92.3 ,2.823 89.477 5.727 10.95 1 93.7 4.006 89.694 5.394 10.78 9 93.7 -1.812 95.512 5.178 10.67 13 94.7 2.985 91.715 6.200 11.20 29 „ ' ,. 96:3; •'2.947 93:353 5.465 10.81 24 97.3 .3.133 94.167 5.537 10.85 28 105.0 "4.562 99,538 6.050 11.12 60 Table 4.13. Observed and Predicted Scores with 95% Confidence Intervals for Transformed Data Subject Observed Residual Predicted 95"'% Confidence Intervals Mean Observation Plus-Minus Plus-Minus 7 75.7 : -4.240 79.940 3.175 8.327 19 76.7 -4.768 81.468 3.793 8.582 5 78.0 3.964 74.054 4.433 8.883 15 78.0 0.158 77.842 4.438 8.886 12 79.3 -3.789 83.089 3.801 8.585 25 80.0 -2.754 82.754 2.763 8.179 18 80.0 -3.644 83.644 2.014 7.957 35 81.0 -5.842 86.842 2.910 8.230 31 81.3 -0.594 81.894 4.030 8.689 10 83.0 -1.476 84.476 2.561 8.113 34 83.0 -3.227 86.227 5.400 9.403 23 83.3 0.258 83.042 3.026 8.271 22 83.7 -3.470 87.170 4.151 8.746 14 84.3 3.525 80.775 4.513 8.923 11 84.7 1.841 82.859 3.230 8.348 8 85.0 0.548 84.452 2.384 8.059 32 85.3 -1.852 87.152 3.744 8.560 27 86.0 -0.690 86.690 5.538 9.483 2 86.3 -0.653 86.953 4.115 8.729 3 86.3 -1.568 87.868 4.186 8.763 16 87.7 -0.297 87.997 2.863 8.213 4 88.3 -0.448 88.748 4.041 8.694 33 88.7 2.065 86.635 3.545 8.475 6 89.0 -0.579 89.579 4.191 8.765 21 89.7 5.401 84.299 4.553 8.944 20 90.0 0.866 89.134 3.963 8.658 17 90.3 6.407 83.893 4.585 8.960 26 92.0 .2v669 89.331 3.121 8.306 30 92.3 -0.316 92.616 4.800 9.072 1 93.7 5.152 88.548 4.336 8.835 9 93.7 -3.777 97.477 5.083 9.225 13 94.7 5.893 88.807 4.748 9.044 29 96.3 -3.940 92.360 4.338 8.836 24 97.3 .1.237 96.063 5.243 9.314 28 105.0 0.078 104.920 6.468 10.050 61 the scores towards the mean. The corresponding Figure 8, for the equation using strength transformations, displays the same bias although the residuals are on the average smaller. Thus one can conclude that both equations are slightly biased and tend to have an averaging effect on the scores. ' Both the transformed data and untransformed data equations are useful but for different reasons. From the practical'.standpoint the untransformed data equation i s more useful to the coach or player who i s trying to find the causes of a fast shot. From a s c i e n t i f i c approach, one may want to predict puck velocity more accurately and with fewer variables- thus the transformed data equation would be more useful. Finally, one must conclude that the two regression equations calculated, are1 f a i r l y good predictors of puck velocity. One could obtain better prediction equations by letting more variables enter these equations. The aim of this investigation, however, is to present prediction equations which are not too cumbersome and awkward to use, and yet are s t i l l adequate predictors. Some accuracy of prediction i s lost but usefullness is enhanced. Also, for further.investigation of the causes of a faster shot, the reduced equations present the variables most li k e l y to be important. The best possible prediction equation, selected by a maximum adjusted R squared criterion for the A l l Possible Subsets run resulted when twenty-two variables had entered the equation using the transformed' data. This was too large for practical use and thus was discarded for the smaller equations. 62 Figure 7. Residuals Plotted in Order of Shot Velocity from Lowest to Highest -12.0 -6.0 -12.0 -6.0 12.0 Scale of Residuals 63 Figure 8 . Residuals Plotted in Order of Shot Velocity from Lowest to Highest for Transformed Data -12.0 -6.0 '(T 6.0 12.0 i * -12.0 -6.0 0 Scale of Residuals 6.0 12.0 64 Chapter 5 SUMMARY AND CONCLUSIONS Summary The main purpose of this study was to investigate the relationships between the v e l o c i t y o f the ice hockey wrist shot and selected human factors in order that one can predict the velocity of the wrist shot. A second objective was to identify the modifiable human factors most highly related (linearly) with puck velocity in order to guide further research investigating the causal factors of a high velocity shot. Thirty-five volunteer hockey players of varying a b i l i t y were measured on forty-eight variables. With these measures, multiple regression techniques were used to analyze and develop a best regression equation. It was found that as a group, the strength variables: arm adduction at the shoulder of the lower arm, trunk flexion, arm abduction at the shoulder of the upper arm, diagonal motion of the upper arm, and forearm rotation, had the most impact on the equation. Three f l e x i b i l i t y variables, one anthropometric variable and one muscle ratio variable were the other variables to enter the equation. The equation accounted for 66.1% of the variance. The single most important variable in terms of predicting puck velocity turned out to be arm adduction at the shoulder of the lower arm; i t accounted for 20% of the variablity. It was found that by cubing the strength variables one could increase the multiple R by 65 .06, using one less variable. Thus with the transformed data, puck velocity can be-.predicted more accurately with fewer variables. Both equations have f a i r l y small ninety-five percent confidence intervals. The equation for the transformed data produced confidence intervals which were slightly smaller than those for the untransformed data. Both equations tended to be slightly biased, having an averaging effect on the scores: the high scores tended to be underestimated and the low scores tended to be overestimated. The best regression equation using the untransformed data was Y(puck velocity)= X^UARM G)(.7544) + X 2 (UWRISTFX) (.1374)+'X (UF0RR0T) (-.1779) + X,(TRUNKROT)(-.0574) + X (TRUNKFLE)(.1645) 4 5 + X, (USHABD) (-1.1586) + X-,(LSHADD) ( .8949) + X 0 (UARMDIAG) o / o (-.2990) + X(UFORSUP)(1.7023) + X 1 Q(L F/E EL)(-7.5737) + 73.0946. This equation has a multiple R value of .813 and a standard error of estimate of 4.52. The best equation using transformed data was Y(puck velocity) = X^(STICK L)(.3405) + X2(LSHROT)(.0626) + X3(TRUNKROT) (-.0972) + X,(UELBFLEX)(-1.9188) + Xc(GLHIPEXT) 4 5 (-.2334) + X^(LSHADD)3(.0001) + X_(UF0RSUP)3(.0116) 6 / + Xo(UWRISTST)3(.0001) + X_(UELBFLEX)3(.0003) + o 9 103.1292. This equation has a multiple R of .871 and a standard error of estimate of 3.74. By using either of these equations the puck velocity in the wrist shot can be predicted f a i r l y well for any subject having s k i l l s consistent with those used in this study. 66 At this point nothing can be said about the causes of a faster s shot, but the groundwork has been laid for further investigation of potential causal variables. Conclusions 1) Using either transformed of untransformed data equations i t is possible to predict puck velocity f a i r l y accurately. 2) The strength variables, as a group, play the most important role in predicting puck velocity. 3) Arm adduction at the shoulder of the lower arm is the best single variable for predicting puck velocity. 4) The f l e x i b i l i t y of a shooter may play a more important role in terms of puck velocity than considered by most authorities. 5) The variables entering into the untransformed data equation may be suspected of playing a v i t a l role in the causes of a hard shot. 6) Using the transformed data, the a b i l i t y to predict puck velocity is improved and fewer variables are required. Recommendations for Further Study 1) Further research should be conducted to investigate the causes of a faster shot by implementing the variables in the untransformed data equation into a training program and monitoring the effects that this program has on shot velocity. 67 REFERENCES Alexander, J.F., Haddow, J.B., & Schultz, G.A. Comparison of the Ice Hockey Wrist and Slap Shot for Speed and Accuracy. Research  Quarterly, 1963,34,259-266. Alexander, J.F., Drake, C.J., Reichenbach, P.J., & Haddow, J.B. Effect of Strength Development on Speed of Shooting of Varsity Ice Hockey Players. Research Quarterly, 1964,35,101-106. BMDP Biomedical Computer Programs P-Series. Berkely:University California Press, 1977. Brown, S. Personal Communication. 1980. Chambers, D. Complete Hockey Instruction. Markam:LR, 1979. Corbett, R. Radar Systems: An Introductory Brief on their Theory, Use and Practical Applications. Unpublished paper for graduate course, University of British Columbia, 1979. Daish, CB. Ball Games. London:English Universities Press, 1972. Ennos, W. A Comparison of Somatotype Information on the University of B.C. Varsity Ice Hockey Teams. Unpublished paper for graduate course, University of British Columbia. Esposito, P., & Esposito, T. We Can Teach You to Play Better Hockey. New York:Hawthorn, 1975. Five Exercises to Help You Develop Shooting Power. Sports Talk, 1978,7. Gardiner, E.B. Proprioceptive Reflexes and their Participation in Motor S k i l l s . Quest, 1969,12,1-25. Gilbert, R., & Park, B. Playing Hockey the Professional Way. Montreal: Holt-Rinehart-Winston,"1973. Halliwell, W.R., Groppel, J.L., & Ward, T. A Kinematic Analysis of the Snap Shot in Ice Hockey as Executed by Professional Hockey Players. Unpublished paper, Movement Science Program, Florida State Univ. 1978. Hayward, B. Shooting. Sports Talk, 1978,7,2-3. Hindmarch, R.G. Personal Communication. Hodge, K., & Awrey, D. Power Hockey. New York:Atheneum, 1967. Howe, G. Here's Howe. Toronto:Copp Clark, 1963 68 Hull, R.M. Hockey is My Game. Don Mills:Longman1s, 1967. Jensen, C., & Schultz, G. Applied Kinesiology. "New York:McGraw-Hill, 1970. Jesse, J. Hidden Causes of Injury, Prevention, and Correction for  Running Athletes and Joggers. Pasadena:Athletic Press, 1977. Johnston, M. He Shoots, He Scores. Young Athlete, 1976,3,46-47 Jones, R.E. Shoot. Hockey, 1979,5,40-41. King, D. Personal Communication. 1980. Kingston, G. Personal Communication. 1980. Leighton, J.R. The Leighton Flexometer and F l e x i b i l i t y Test. Journal  of the Association for Physical and Mental Rehabilitation, 1966,20, 86-93. MacDonald, D. The Art of Shooting. Hockey Today, 1978-79,13-15. MacGillivary, W., & Watson, R. A Kinesiological Analysis of the Skating Wrist Shot in Ice Hockey. Unpublished paper for graduate course, University of Alberta, 1964. Mahovalich, F. Ice Hockey: How to Play Hockey and Enjoy It. London: Pelham,-1965. Montreal Olympic Games Anthropometrical Project: Anthropometrics Instruments and Measurements. Unpublished paper, San Diego State University, 1976. Moyls, P.W. The Effect of Dominant Hand Placement on Accuracy and Velocity of the Standing Snap Shot. Unpublished paper for graduate course, University of British Columbia, 1979. Moyls, P.W., & Nobbs, L.A. A Comparison Study: Somatotypes of the University of B.C. Varsity Ice Hockey Teams (1964,1976,1979), Physical Education Graduate Students (1979) and Other Groups. Unpublished paper for graduate course, University of British Columbia, 1980. Patrick, L., & Monahan, L. Lets Play Hockey. Toronto:MacMillan, 1957. Reed, A.T., Cotton, C , Hansen, H., & Gauthier, R. Upper Body Strength and Handedness Shooting Characteristics of Junior and Professional Hockey Players. Paper presented at International Congress of Sports Sciences; Edmonton, Alberta, July, 1979. Robb, M.D. The Dynamics of S k i l l Aquisition. Englewood C l i f f s P r e n t i c e -Hall, 1972. 69 Selder, D. Anthropometric, Cardiovascular and Motor Performance of University Ice Hockey Players. Unpublished paper, Masters thesis, University of British Columbia, 1964. Tarasov, A., & Persson, V. Tarasov's Hockey Technique. Montreal:Holt-Rinehart-Winston, 1973. Tremble, R. Off-Ice Hockey D r i l l s . Athletic Journal, 1978,59,42. Watt, T. Personal Communication. 1980. Wiren,~G. Human Factors Influencing the Golf Drive for Distance. Ph.D." Dissertation, University of Oregon, 1968. 70 Appendix A VARIABLE TESTING PROCEDURES 71 Appendix A VARIABLE TESTING PROCEDURES Anthropometric Procedures Height Instrument- Stadiometer Position- Standing Position, barefoot, heels together and arms hanging naturally by the sides. Subject looks straight ahead. Horizontal line of sight from orbitale to traglion. (Frankfurt plane) Measurement- Subject instructed to take deep breath and stand as t a l l as possible. Stadiometer lowered to vertex of head. Reading taken, subject relaxes. Body Weight Instrument- Balance beam scale Measurement- Standing position, barefoot, minimal clothing. Reading taken, subject relaxes. Arm Length Instrument- Stadiometer Position- Standing position, heels together, arms at side, palms facing back. Measurement- Stadiometer lowered to acromiale. Reading taken, subject relaxes. Subject leans away but holds foot position as stadiometer lowered. Standing position resumed, stadiometer raised" to index finger height, reading taken subject relaxes. Instrument- tape (for a l l girths) Arm Girth Measurement- Arm extended.andrrelaxed hanging loosely at the side, tape placed midway between the acromion process and the elbow joint. Maximum girth is recorded. Forearm Girth Measurement- Elbow extended, forearm supinated and fingers extended. Maximum girth of the forearm is recorded. Wrist Girth Measurement- Elbow flexed to 90 degrees, forearm supinated and fingers extended. Minimum girth distal to styloid process is recorded. Stick Length Measurement- Measured from heel of blade to top of stick. 72 Fl e x i b i l i t y Procedures Instrument- Leighton flexometer Wrist Flexion and Extension Starting position- Sitting position in standard armchair, back straight, forearm resting on chair arm, f i s t doubled and extended beyond end of chair arm, palm of hand to be measured turned up. Instrument fastened to thumb-side of f i s t , (common chair and table of suitable height may be substituted for armchair) Movement- Count (1) f i s t moved upward and backward in an arc as far as possible, dial locked (2), f i s t moved forward, downward and backward in an arc as far as possible, pointer locked (3), subject relaxes, reading taken. Caution- Forearm maytnot be raised from chair during movement. Radial-Ulnar Supination and Pronation Starting position- Sitting position in standard armchair, back straight, forearms resting on chair arms, f i s t doubled and extended beyond end of chair arm, wrist of arm to be measured held straight. Strap is grasped in hand, fastening instrument to front of f i s t , (common chair and table of suitable height may be substituted for armchair) Movement- Count (1) thumbside of the f i s t turned outward and downward as far as possible, dial locked, (2) thumbeside of f i s t turned upward, downward and inward as far as possible, pointer locked,'(3) subject relaxes,::reading taken. Caution- Body and forearm must remain stationary, except for specified movement. No leaning of the body may be permitted. Shoulder Rotation Starting position- Standing position at projecting corner of wall, arm to be measured extended sideward and bent to right angle at elbow, shoulder extended just beyond projecting corner, opposite-arm at side of body, back to wall, shoulder blades, buttocks and heels touching wall. Instrument fastened to side of forearm. Movement- Count (1) forearm moved downward and backward in an arc as far as possible, d i a l locked, (2) forearm moved forward, upward and backward in an arc as far as possible, pointer locked (3) subject relaxes, reading taken. Caution- Upper arm being measured must be held directly sideward arid parallel with the floor during movement. Heels, buttocks, and shoulders must touch wall at a l l times. Trunk Rotation Starting position- Supine position on bench, legs"together, knees raised above hips, lower legs parallel to bench and body. Assistant holds subjects shoulders. Instrument fastened to middle rear of upper legs, strap going around both legs. Movement- Count (1) knees lowered to the l e f t as: far as possible, dial locked, (2) knees brought back to starting position and lowered to the right as far as possible, pointer locked, (3) subject relaxes, 73 reading taken. Caution- Subjects shoulders must not be permitted to rise from bench during movement. Knees must be moved directly sidewards at the height of the hips, not above or below. Strength Procedures Instrument- Cybex II Trunk Flexion Attachments-: Small half T with pad Setting- 30 degrees per second, Scale- 180 foot pounds Starting position- Supine position parallel to arm of cybex. Arms at side, legs secured at ankles and superior to knees. Joint of cybex arm lines up with hip joint. Pad placed on • chest at lowest point on sternum. Movement- Subject flexes at waist and drives chest upward and forward as far as possible. Caution- Legs must not l i f t off table. Trunk Extension Attachments- Small half T with pad Setting- 0 degrees per second Scale- 180 or 360 foot pounds Starting position- Lying supine on stomach parallel to arm of cybex, arms at side, head facing forward. Pad placed on 5th, 6th and 7th cervical vertibrae. Joint of cybex lines up with hip joint. Movement- Subject tries to flex backward and upward as hard as possible. Hip Flexion Attachments- Large half T with pad. Setting- 30 degrees per second Scale- 180 or 360 foot pounds Starting positon- Supine position with pad secured 5 inches above the maleolus, parallel to arm of cybex, arms at side. Joint of cybex lines up with hip joint. Movement- Subject flexes leg up towards ceiling. Caution- Leg must remain straight. Hip Extension Attachments- Large half T with pad Setting- 30 degrees per second Scale- 180 or 360 foot pounds Starting position- Lying position on stomach, parallel to::arm of cybex, legs hanging over table at hips, free leg touching floor. Joint of cybex lines up with hip joint. Pad strapped to calf 5 inches above maleolus. Movement- Subject extends leg towards ceiling. Caution- Leg must remain straight. 74 Shoulder Abduction and Adduction Attachments- Large or small half T,UTattachment at joint. Setting- 30 degrees per second Scale- 180 foot pounds Starting position- Lying on side facing cybex. Joint of cybex lines up with shoulder joint. Subject grasps handle at side with palms facing body with upper arm for arm abduction. Lower arm grasps handle above the head with palm facing up for arm adduction. Movement- For arm abduction, upper arm moves from side to above head. For arm adduction, lower arm moves from above the head down to the side. Caution- Arm must not bend during the movement. Shoulder Diagonal Motion Attachments- Large half T, U attachment at joint. Setting- 30 degrees per second, cybex t i l t e d 30 degrees. Scale- 180 foot pounds Starting position- Table angled 60 degrees to cybex. Supine position, joint of cybex directly above shoulder joint. Upper arm grasps handle at side. Lower arm grasps handle above head. Movement- Upper arm moves from side across body up above head. Lower arm moves from above head down across body to side. Same grips as used in shoulder adduction and abduction. Caution- Arm must remain straight throughout motion. Forearm Supination and Pronation Attachments- Large U at joint Setting- 30 degrees per second. Scale- 30 foot pounds Starting position- Sitting position facing the arm of cybex, arm held to bench by other arm. Movement- Lower arm rotated from palms up\v posit ion to palms down position. Upper arm rotated from palms down position to palms up position. Caution- Arm must stay in contact with bench. Wrist Flexion and Extension Attachments- Small half T with hand grip, U attachment at joint. Setting- 30 degrees per second Scale- 30 foot pounds Starting position- Sitting position parallel to arm of cybex, arm on bench held down by other arm. Movement- Lower arm grips palms up and flexes towards ceiling. Upper arm, 'grips'palms down; and extends to ceiling. Caution- Arm must stay in contact with bench. Elbow Flexion and Extension Attachments- Small half T with hand grip Setting- 30 degrees per second Scale- 180 foot pounds Starting position- Supine position parallel to arm of cybex, arms at sides, joint of cybex lines up with elbow joint. Movement- Flexion movement starts at side and flexes upward to 75 shoulder. Extension movement starts in f u l l flexion at shoulder and extends down towards bench. Caution- Forearm and wrist must be straight throughout motion. 76 Appendix B FLEXIBILITY WARMUP 77 Appendix B FLEXIBILITY WARMUP The warmup consisted of four simple exercises designed to simulate the motions required in the f l e x i b i l i t y tests. Wrist Flexion and Extension Starting position- Standing position, arms at sides, elbows flexed to 90 degrees in front of the body, hands clenched. Movement- Wrists extended up towards ceiling and then down towards floor. Repeated ten times. Wrist Supination and Pronation Starting position- Same as wrist flexion and extension. Movement- Palms rotated from palms down to palms up and back. Repeated ten times. Shoulder Rotation Starting position- Standing position, arms out to side, elbows bent to 90 degrees, hands pointed to ceiling. Movement- Keeping upper arms parallel to floor, shoulder rotates so hands are now pointing to floor. Repeated ten times. Trunk Rotation Starting position- Standing position, hands on hips. Movement- Subject rotates at waist to l e f t and then to right. Repeated ten times. 78 Appendix C PRODUCT-MOMENT CORRELATION MATRIX 19 20 21 22 23 Product-Moment Correlation Matrix 1 2 3 A 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1.000 2 0.192 1.000 3 0.113 0.435 1.000 4 0.097 0.493 0.217 1.000 5 0.134 0.788 0.306 0.568 1.000 6 0.119 0.787 0.270 0.475 0.961 1.000 7 0.042 0.273 0.036 -0.520 0.386 0.486 1.000 8 -0.276 0.060 0.424 -0.049 0.127 0.107 0.142 1.000 9 -0.131 0.219 0.473 0.048 0.281 0.284 0.215 0.921 1.000 10 0.104 -0.021 0.620 -0.086 -0.122 -0.109 -0.037 0.451 0.516 1.000 11 0.115 -0.034 0.575 -0.184 -0.151 -0.120 0.048 0.437 0.512 0.925 1.000 12 0.159 -0.103 0.564 -0.132 -0.162 -0.195 -0.061 0.400 0.381 0.799 0.741 1.000 13 0.170 -0.120 0.528 -0.135 -0.213 -0.222 -0.110 0.408 0.391 0.830 0.793 0.958 1.000 14 0.029 0.019 0.209 0.032 -0.043 0.039 -0.038 0.251 0.305 0.260 0.275 0.321 0.313 1.000 15 0.123 0.134 -0.099 0.093 0.086 0.126 0.080 0.177 0.269 -0.055 0.002 -0.258 -0.258 0.019 1.000 16 0.211 0.180 -0.347 0.139 0.231 0.273 0.146 -0.171 -0.021 -0.179 -0.232 -0.341 -0.286 -0.198 0.463 1.000 17 0.075 0.355 0.027 0.123 0.323 0.362 0.236 0.200 0.299 0.085 -0.062 -0.124 -0.120 0.145 0.269 0.380 1.000 18 0.075 0.324 -0.051 0.175 0.331 0.330 0.155 0.209 0.308 -0.046 -0.130 -0.231 -0.233 0.060 0.233 0.382 0.820 1.000 19 0.154 0.276 -0.019 0.070 0.345 0.341 0.274 0.105 0.213 -0.021 0.055 -0.205 -0.188 -0.221 0.414 0.435 0.408 0.587 l 000 20 -0.042 0.218 0.109 0.313 0.266 0.246 -0.066 0.036 0.143 -0.062 -0.065 -0.258 -0.275 -0.259 0.415 0.369 0.272 0.318 o!599 1.000 21 0.065 0.266 -0.137 -0.034 0.167 0.191 0.245 0.060 0.172 -0.069 -0.023 -0.204 -0.251 -0.042 0.084 0.256 0.249 0.335 0.292 o!l93 1.000 22 -0.227 -0.030 0.118 -0.176 -0.090 -0.098 0.105 0.269 0.346 0.232 0.324 0.155 0.111 0.192 0.002 -0.100 0.128 0.157 0.277 0.253 0.400 1.000 23 0.077 0.023 0.197 0.201 0.116 0.112 -0.114 0.097 0.152 0.240 0.167 0.114 0.133 0.143 0.070 0.082 0.335 0.371 0.054 o!277 -0.156 o!o30 1.000 24 0.073 -0.055 0.330 0.045 -0.117 -0.108 -0.173 0.291 0.303 0.393 0.378 0.219 0.295 0.153 0.074 0.029 0.099 0.159 0.116 0.192 -0.011 0.385 0.554 25 -0.142 0.014 0.445 0.041 0.054 0.062 -0.026 0.339 0.323 0.472 0.433 0.371 0.367 0.287 -0.150 -0.312 0.201 0.249 0.149 0.126 -0.124 0.402 0.601 26-0.098-0.182 0.268 0.010-0.184-0.178-0.197 0.277 0.260 0.363 0.346 0.193 0.234 0.195-0.001-0.144 0.090 0.167 0.053 0.159 0.006 0.465 0.548 27 -0.096 0.059 0.509 0.075 0.103 0.109 -0.011 0.351 0.331 0.484 0.393 0.469 0.441 0.311 -0.075 -0.311 0.173 0.107 0.003 0.067 -0.211 0.219 0.512 28 -0.218 -0.149 0.445 -0.272 -0.190 -0.154 0.081 0.478 0.468 0.569 0.611 0.469 0.468 0.304 -0.170 -0.368 -0.069 -0.024 0.010 -0.110 0.064 0.538 0.271 29 -0.041 -0.009 0.261 -0.144 -0.044 0.046 0.108 0.324 0.389 0.327 ,0.424 0.248 0.270 0.451 -0.159 -0.311 0.037 0.154 0.158 -0.118 0.319 0.461 0.173 30 0.119 0.058 0.595 -0.062 0.095 0.084 0.109 0.327 0.331 0.509 0.542 0.508 0.498 0.200 -0.336 -0.458 -0.173 -0.146 -0.062 -0.083 0.051 0.322 0.148 31 -0.099 -0.003. 0.378 -0.019 0.057 0.069 0.059 0.458 0.456 0.457 0.555 0.424 0.457 0.378 -0.133 -0.433 -0.078 -0.015 0.085 -0.170 0.097 0.480 0.204 32 0.009 0.071 0.524 -0.058 0.168 0.174 0.206 0.428 0.424 0.558 0.482 0.485 0.460 0.431 -0.165 -0.300 0.302 0.140 0.006 -0.129 -0.003 0.213 0.264 33 0.109 -0.153 0.363 -0.165 -0.070 -0.039 0.114 0.242 0.292 0.344 0.428 0.375 0.364 0.121 -0.013 -0.281 -0.142 -0.030 0.041 -0.096 -0.113 0.266 0.114 34 0.074 -0.094 0.236 -0.311 -0.186 -0.109 0.154 0.559 0.542 0.338 0.417 0.355 0.407 0.418 0.077 -0.211 -0.043 -0.020 0.002 -0.029 0.170 0.491 0.130 35 0.022 0.151 0.442 0.073 0.050 0.105 -0.012 0.353 0.447 0.487 0.581 0.278 0.351 0.344 0.020 -0.316 -0.035 0.103 0.154 -0.018 0.120 0.372 0.255 36 -0.123 -0.027 0.524 -0.070 -0.022 -0.017 0.055 0.408 0.430 0.611 0.668 0.492 0.495 0.181 -0.033 -0.383 -0.045 0.034 0.150 -0.057 -0.008 0.450 0.304 . 37 -0.204 0.037 0.232 0.078 0.118 0.167 0.025 0.344 0.422 0.475 0.396 0.263 0.292 0.106 0.038 0.021 0.160 0.214 0.313 0.273 0.173 0.441 0.316 38 -0.046 0.073 0.416 -0.008 0.033 0.061 0.065 0.360 0.385 0.524 0.601 0.432 0.459 0.197 -0.055 -0.389 -0.004 -0.021 0.048 -0.142 0.105 0.498 0.195 39 0.057 0.087 0.365 -0.078 0.100 0.153 0.198 0.387 0.415 0.529 0.537 0.437 0.486 0.236 -0.047 -0.159 0.093 0.110 0.224 0.046 0.016 0.425 0.304 40 0.284 -0.056 -0.184 0.082 -0.170 -0.150 -0.212 -0.084 -0.029 -0.110 -0.052 -0.267 -0.139 -0.138 0.303 0.404 -0.136 -0.112 -0.040 0.068 0.173 -0.057 -0.019 41 0.064 -0.246 -0.272 0.010 -0.271 -0.257 -0.234 -0.108 -0.079 -0.125 -0.021 -0.371 -0.254 -0.139 0.145 0.266 -0.104 0.019 0.058 0.136 0.257 0.226 0.082 42 -0.182 0.024 -0.094 -0.286 -0.102 -0.106 0.208 0.095 0.102 -0.065 0.043 -0.014 -0.061 0.005 -0.103 -0.133 -0.136 -0.134 0.142 -0.070 0.412 0.629 -0.731 43 -0.516 -0.492 -0.204 -0.151 -0.315 -0.325 -0.139 -0.055 -0.147 -0.063 -0.075 -0.051 -0.148 0.092 -0.225 -0.322 -0.054 0.007 -0.227 -0.078 0.050 0.265 0.042 44 0.091 0.067 0.069 0.054 0.068 0.068 0.019 -0.026 -0.034 -0.024 -0.133 0.195 0.134 0.093 0.156 0.021 -0.096 -0.318 -0.351 -0.210 -0.167 -0.457 -0.230 45 0.073 -0.082 0.333 -0.191 -0.169 -0.222 0.040 0.114 0.023 0.080 0.216 0.203 0.173 0.052 -0.046 -0.445 -0.254 -0.224 -0.213 -0.368 -0.276 -0.097 -0.103 46 -0.204 -0.063 -0.002 0.060 -0.146 -0.194 -0.217 -0.037 -0.067 -0.093 -0.037 -0.083 -0.102 -0.099 0.016 -0.200 -0.097 -0.165 -0.259 -0.208 0.084 0.041 -0.146 47 -0.052 -0.150 0.199 -0.050 -0.080 -0.108 -0.052 0.065 0.074 0.169 0.124 0.151 0.116 0.030 0.012 0.034 -0.057 0.092 0.156 0.113 -0.163 -0.095 0.177 48 -0.152 -0.004 -0.012 0.207 0.119 0.085 -0.104 0.048 0.071 0.216 0.052 0.069 0.065 -0.287 0.188 0.271 0.150 0.097 0.177 0.381 -0.107 0.073 0.239 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 45 46 47 48 1. AGE 11. LFORARMG 21. TRUNKROT 31. LARMDIAG 41. F/E DHIP 2. HT 12. UARM G 22. TRUNKEXT 32. UFORSUP 42. T F/E 3. WT 13. LARM G 23. TRUNKEXT 33. LFORPRO 43. D/G FLEX 4. STICK L 14. PUCKVEL 24. GLHIPFLE 34. UWRISTST 44. D/G EXT 5. UARM L 15. UWRISTFX 25. GLHIPEXT 35. LWRISTST 45. U F/E EL 6. LARM L 16. LWRISTFX 26. DRHIPFLE 36. UELBFLEX 46. L F/E EL 7. ARM/STI 17. USHROT 27. DRHIPEXT 37. UELBEXT 47. U/L F EL 8. UWR1STG 18. LSHROT 28. USHABD 38. LELBFLEX 48. U/L E EL 9. LWRISTG 19. UFORROT 29. LSHADD 39. LELBEXT 10. UFORARMG 20. LFORROT 30. UARMDIAG 40. F/E GHIP 1.000 0.691 0.94 7 0.633 0.654 0.439 0.423 0.513 0.333 0.411 0.571 0.594 0.508 0.536 0.489 0.682 0.325 • 0.364 -0.167 • -0.061 -0.219 • -0.111 -0.261 • 0.059 0.198 1.000 0.782 0.897 0.693 0.573 0.551 0.628 0.657 0.499 0.446 0.631 0.660 0.641 0.577 0.713 •0.427 •0.202 •0.198 0.300 0.275 0.061 0.252 • 0.151 0.196 1.000 0.696 0.721 0.488 0.408 0.538 0.409 0.485 0.537 0.618 0.564 0.568 0.517 0.655 0.153 0.327 •0.129 0.258 •0.261 •0.093 •0.210 0.099 0.209 1.000 0.608 0.425 0.539 0.566 0.717 0.553 0.400 0.532 0.530 0.595 0.499 0.643 -0.415 -0.419 -0.256 0.215 0.169 -0.122 -0.248 0.082 0.274 00 O 1.000 0.747 0.569 0.684 0.560 0.512 0.607 0.701 0.771 .581 .626 .625 .089 .078 .121 0.235 -0.227 0.116 -0.081 0.257 -0.001 1.000 0.563 0.761 0.433 0.356 0.576 0.747 0.594 0.455 0.571 0.539 -0.168 0.020 0.158 0.150 •0.308 0.055 •0.052 0.082 •0.387 1.000 0.716 0.539 0.444 0.415 0.504 0.578 0.375 0.586 0.648 -0.232 -0.207 0.118 -0.038 -0.081 0.160 -0.199 0.036 -0.112 1.000 0.539 0.430 0.543 0.754 0.731 0.514 0.789 0.712 -0.167 -0.062 0.146 0.072 -0.157 ' 0.120 -0.016 -0.068 -0.122 1.000 0.453 0.319 0.479 0.564 0.466 0.424 0.445 -0.432 -0.391 -0.079 0.181 0.105 0.046 •0.127 0.286 0.115 1.000 0.395 0.488 0.587 0.391 0.578 0.424 -0.135 -0.088 0.042 0.264 0.076 0.191 0.036 0.058 0.135 1.000 0.570 0.391 0.437 0.474 0.619 0.131 .0.154 0.214 •0.051 •0.159 •0.067 •0.216 •0.114 •0.071 1.000 0.696 0.531 0.615 0.640 -0.014 0.102 0.003 0.068 -0.235 0.089 -0.159 0.173 -0.097 1.000 0.602 0.824 0.635 -0.200 0.020 0.011 0.175 •0.319 0.311 0.087 0.313 0.141 1.000 0.530 0.669 -0.134 -0.025 0.037 0.129 -0.162 -0.553 -0.247 0.145 0.626 1.000 0.671 •0.110 0.040 0.154 0.096 •0.232 0.189 0.250 •0.269 0.094 1.000 •0.101 •0.019 0.073 •0.051 •0.220 •0.137 •0.527 •0.017 0.224 1.000 0.810 -0.022 -0.438 -0.015 -0.080 0.047 -0.130 0.005 1.000 0.068 1.000 -0.022 0.077 -0.508 -0.140 -0.001 -0.031 0.099 0.077 -0.041 -0.252 -0.023 -0.139 1.000 -0.137 1.000 0.025 -0.101 0.153 0.023 0.112 -0.102 0.069 0.096 1.000 0.358 0.186 -0.586 1.000 -0.319 1.000 -0.180 0.072 1.000 81 Correlation Matrix with Transformed Variables «9 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 1 -0.212 0.087 0.057 -0.150 -0.110 -0.101 -0.236 -0.037 0.105 -0.085 -0.017 0.144 0.103 -0.029 -0.124 -0.242 0.008 0.005 2 0.005 0.056 -0.040 0.015 -0.122 0.083 -0.130 -0.020 -0.031 -0.036 0.012 -0.134 -0.033 0.075 -0.025 0.033 0.092 0.060 3 0.151 0.159 0.333 0.383 0.293 0.509 0.385 0.208 0.501 0.333 0.488 0.323 0.291 0.384 0.481 0.225 0.414 0.355 4 -0.190 0.173 0.009 0.044 0.002 0.099 -0.265 -0.148 -0.049 -0.021 -0.075 -0.149 -0.312 0.006 -0.091 0.008 -0.004 -0.102 5 -0.064 0.127 -0.133 0.011 -0.155 0.098 -0.190 -0.084 0.030 -0.008 0.109 -0.027 -0.186 -0.041 -0.053 0.087 0.015 0.008 6 -0.072 0.125 -0.114 0.036 -0.140 0.117 -0.138 0.001 0.016 -0.002 0.135 0.001 -0.100 -0.003 -0.042 0.143 0.052 0.061 7 0.146 -0.085 -0.148 -0.066 -0.159 -0.038 0.081 0.079 0.049 -0.009 0.174 0.129 0.162 -0.054 0.050 0.062 0.046 0.126 8 0.310 0.092 0.314 0.341 0.318 0.392 0.451 0.270 0.317 0.315 0.430 0.277 0.469 0.333 0.376 0.402 0.332 0.395 9 0.386 0.144 0.324 0.337 0.305 0.389 0.436 0.327 0.310 0.321 0.438 0.317 0.457 0.396 0.401 0.455 0.372 0.407 10 0.240 0.203 0.410 0.467 0.388 0.520 0.511 0.301 0.490 0.436 0.647 0.251 0.341 0.485 0.593 0.462 0.549 0.533 11 0.341 0.132 0.374 0.448 0.359 0.438 0.573 0.402 0.519 0.541 0.578 0.342 0.395 0.587 0.675 0.376 0.632 0.523 12 0.191 0.097 0.241 0.339 0.218 0.458 0.422 0.221 0.453 0.420 0.540 0.334 0.386 0.339 0.484 0.256 0.487 0.444 13 0.136 0.113 0.308 0.352 0.250 0.450 0.425 0.233 0.436 0.429 0.540 0.314 0.428 0.400 0.482 0.288 0.504 0.498 14 0.233 0.193 0.216 0.393 0.284 0.389 0.392 0.473 0.249 0.441 0.494 0.155 0.444 0.373 0.281 0.140 0.330 0.300 15 -0.254 0.204 -0.019 -0.039 -0.042 -0.034 -0.210 -0.215 -0.002 -0.118 0.050 -0.466 -0.461 -0.186 -0.167 0.038 -0.154 -0.046 16 -0.004 0.010 0.108 -0.120 0.030 -0.044 -0.153 -0.117 -0.291 -0.198 -0.195 0.035 0.027 -0.061 -0.020 0.067 -0.056 -0.016 17 -0.150 0.050 -0.055 -0.307 -0.183 -0.314 -0.382 -0.297 -0.441 -0.472 -0.310 -0.257 -0.216 -0.392 -0.390 0.023 -0.377 -0.154 18 0.166 0.322 0.145 0.188 0.147 0.197 -0.061 0.070 -0.173 -0.150 0.265 -0.152 -0.019 -0.072 -0.060 0.155 -0.006 0.106 19 0.188 0.380 0.149 0.242 0.171 0.114 -0.026 0.161 -0.179 -0.095 0.064 -0.025 -0.036 0.060 0.032 0.196 -0.019 0.079 20 0.264 0.016 0.129 0.121 0.084 0.006 -0.021 0.127 -0.158 -0.018 -0.082 0.058 -0.038 0.114 0.107 0.281 0.002 0.158 21 0.227 0.201 0.183 0.057 0.142 0.061 -0.160 -0.131 -0.095 -0.223 -0.200 -0.075 -0.103 -0.058 -0.107 0.236 -0.198 0.024 22 0.431 -0.144 0.014 -0.077 0.030 -0.180 0.066 0.289 0.092 0.064 -0.020 -0.143 0.161 0.084 0.008 0.152 0.119 0.021 23 0.985 0.018 0.398 0.422 0.488 0.268 0.518 0.467 0.346 0.508 0.217 0.259 0.424 0.443 0.478 0.406 0.503 0.451 24 0.047 0.975 0.503 0.598 0.506 0.521 0.282 0.270 0.137 0.279 0.234 0.087 0.080 0.325 0.323 0.254 0.228 0.295 25 0.376 0.533 0.967 0.675 0.921 0.673 0.633 0.474 0.387 0.503 0.289 0.370 0.563 0.600 0.504 0.507 0.499 0.723 26 0.417 0.588 0.675 0.969 0.790 0.902 0.690 0.607 0.490 0.659 0.619 0.441 0.457 0.686 0.660 0.618 0.602 0.719 27 0.465 0.52) 0.907 0.768 0.965 0.728 0.708 0.536 0.394 0.557 0.372 0.420 0.530 0.640 0.569 0.531 0.531 0.693 28 0.249 0.499 0.635 0.836 0.716 0.980 0.602 0.455 0.472 0.574 0.664 0.502 0.434 0.557 0.517 0.586 0.531 0.660 11 nitl n"?o? n " f ^ ° - 7 5 2 ° - 6 5 7 ° - 9 8 5 ° - 7 7 1 °'* 7 7 ° - 7 2 8 ° - 5 5 0 ° ' A 5 7 ° - 6 0 6 °-' 7 6 6 °- 7 8° 0-559 0.665 0.649 3? ^ ° - 7 7 4 ° - 9 6 8 ° - 4 8 4 ° - 7 5 7 ° - 4 3 7 ° - 3 2 3 ° - 5 5 9 ° - 7 7 1 ° - 6 2 0 °-432 0.621 0 557 n: n- n°- °- ? °" °-"i " i f ?•?» ?- 6 8? ° - 5 2 0 °-* 2 3 0.378 0.492 0.553 0.338 0.602 0.601 32 0.492 0.209 0.538 0.655 0.592 0.602 0.686 0.721 0.689 0.948 0.546 0.386 0.472 0.766 0.735 0.464 0.791 0.694 33 0.266 0.240 0.319 0.612 0.428 0.703 0.545 0.415 0.511 0.526 0.967 0.408 0.364 0.461 0.551 0.451 0.476 0.436 34 0.317 0.075 0.333 0.449 0.414 0.530 0.482 0.340 0.382 0.387 0.442 0.973 0.422 0.443 0.564 0.359 0.584 0.367 35 0.515 0.139 0.585 0.480 0.552 0.459 0.609 0.541 0.407 0.498 0.336 0.430 0.954 0.597 0.409 0.488 0.498 0.667 36 0.393 0.228 0.580 0.682 0.639 0.591 0.703 0.710 0.448 0.739 0.478 0.413 0.542 0.962. 0.705 0.493 0.648 0.630 37 0.482 0.245 0.489 0.657 0.570 0.553 0.736 0.586 0.490 0.705 0.568 0.515 0.398 0.715 0.981 0.531 0.828 0.603 38 0.443 0.282 0.524 0.636 0.564 0.635 0.545 0.413 0.304 0.477 0.478 0.362 0.450 0.535 0.568 0.978 0.523 0.677 39 0.532 0.171 0.494 0.589 0.545 0.545 0.620 0.581 0.560 0.734 0.482 0.520 0.459 0.628 0.815 0.462 0.975 0.643 40 0.440 0.320 0.719 0.709 0.707 0.689 0.605 0.526 0.578 0.660 0.444 0.383 0.614 0.646 0.611 0.646 0.674 0.972 49 1.000 0.038 0.392 0.441 0.492 0.302 0.553 0.497 0.348 0.520 0.265 0.309 0.466 0.470 0.513 0.407 0.544 0.463 50 1.000 0.498 0.608 0.502 0.522 0.295 0.305 0.141 0.302 0.213 0.062 0.088 0.313 0.282 0.232 0.217 0.310 51 1.000 0.683 0.945 0.696 0.652 0.538 0.388 0.522 0.282 0.296 0.583 0.599 0.486 0.510 0.505 0.782 52 1.000 0.806 0.881 0.749 0.708 0.454 0.720 0.599 0.400 0.487 0.760 0.683 0.619 0.639 0.732 53 1.000 0.774 0.756 0.632 0.403 0.611 0.396 0.359 0.548 0.676 0.582 0.538 0.568 0.760 54 1.000 0.667 0.530 0.467 0.623 0.674 0.488 0.490 0.628 0.553 0.633 0.586 0.719 i.000 0.820 0.429 0.752 0.546 0.437 0.615 0.784 0.769 0.529 0.679 0.640 " x 0 0 0 0 - 4 3 3 0 - 7 7 1 0 - 4 1 6 0.310 0.525 0.771 0.632 0.380 0.651 0.549 " 1.000 0.671 0.518 0.360 0.340 0.427 0.475 0.271 0.573 0.539 „ 1.000 0.536 0.343 0.440 0.807 0.743 0.419 0.779 0.655 5 8 1.000 0.391 0.374 0.467 0.569 0.471 0.537 0.444 5 9 1.000 0.439 0.377 0.503 0.346 0.537 0.327 6 0 1.000 0.567 0.416 0.516 0.496 0.691 6 1 1.000 0.745 0.497 0.680 0.654 6 2 1.000 0.506 0.849 0.594 6 3 1.000 0.457 0.683 64 1.000 0.658 65 " 1.000 61. AGE 11. LFORARMG 21. 2. HT 12. UARM G 22. 3. WT 13. LARM G 23. 4. STICK L 14. PUCKVEL 24. 5. UARM L 15. HAND 25. 6. LARM L 16. UWRISTFX 26. 7. ARM/STI 17. LWRISTFX 27. 8. UWRISTG 18. USHROT 28. 9. LWRISTG 19. LSHR0T 29. 10. LFORARMG 20. UF0RROT 30. LFORROT 31. UARMDIAG 49. TRUNKROT 32. LARMDIAG 50. TRUNKFLE 33. UFORSUP 51. TRUNKEXT 34. LFORPRO 52. GLHIPFLE 35. UWRISTST 53. GLHIPEXT 36. LWRISTST 54. DRHIPFLE 37. UELBFLEX 55. DRHIPEXT 38. UELBEXT 56. USHABD 39. LELBFLEX 57. LSHADD 40. LELBEXT 58. TRUNK FLE, 59. UFORSUP, TRUNKEXT, 60. LFORPRO , GLHIPFLE, 61. UWRISTST, GLHIPEXT, 62. LWRISTST, DRHIPFLE, 63. UELBFLEX DRHIPEXT 64. UELBEXT , USHABD, 65. LELBFLEX LSHADD , 66. LELBEXT UARMDIAG, LARMDIAG 

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