Using Physiological Big Data to Predict Cross Country Performance

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Acknowledgements

First, I would like to thank Dr. Daniel Lieberman for his support and advice throughout this study. His knowledge, patience, and guidance helped me immensely throughout the process. I also would like to thank my teammates, for this research would not have been possible without their consent for the use of their physiological data. I am also very grateful that WHOOP, Inc. provided me with the opportunity to do this research by allowing me to use their data. I would also like to thank Dr. Charles Czeisler, who helped me understand my results and helped me learn more about sleep, the circadian rhythm, and the body on such short notice. Lastly, I would like to thank Dr. Kevin Rader, Dr. Aaron Baggish, and Bo Waggoner for their help and advice along the way.

Table of Contents

Introduction ...... 1 Previous Relevant Research ...... 5 Summary ...... 9 Methods ...... 11 Subjects ...... 11 Study Duration and Treatments ...... 11 Measurements of Performance ...... 12 Physiological Data Measured ...... 14 Survey Data ...... 15 Statistical Methods ...... 16 Random Effects Models ...... 16 Varying Intercept Models ...... 16 Varying Slope and Intercept Models ...... 18 Difference from Classical Regression ...... 19 Likelihood Ratio Tests ...... 19 Data Aggregation ...... 21 Hypotheses Tested ...... 24 Set 1: Individual Physiological Variables and Performance ...... 24 Set 2: Investigating the REM Sleep and Total Sleep Relationship ...... 30 Set 3: Investigating Sleep One Night Before and Two Nights Before ...... 34 Set 4: Sleep and Heart Rate Variability ...... 37 Standardized Coefficients ...... 40 Results ...... 41 Summary Statistics: Sleep Stage Quantity ...... 41 Hypotheses Set 1 Important Results ...... 42 Hypotheses Set 2 Important Results ...... 50 Hypotheses Set 3 Important Results ...... 56 Hypotheses Set 4 Important Results ...... 62 Weekly Bike Efforts...... 67 Discussion ...... 69 Limitations ...... 76 Conclusions ...... 79

Appendix...... 80 A ...... 80 B ...... 89 C ...... 103 D ...... 109 E ...... 117 References ...... 132

Introduction

We currently live in the age of Big Data, where information is collected every second while the world attempts to derive meaning from it. Many industries are exploring how to use this endless river of information under the assumption that there is no such thing as too much data. One such example is the fitness industry. Over the last few years, wearable fitness tracking devices have become increasingly popular with exercise enthusiasts. The sensors from these devices record a variety of information, from exercise intensity to sleep quality, in an effort to inform individuals about their activity patterns and to influence them to live a healthier lifestyle. All of these data could potentially be useful for many individuals, especially endurance athletes such as cross country distance runners, whose performance is presumably affected by numerous factors. These factors include fitness, freshness on race day, running economy, ability to avoid injury, strategy, energy storage, weight, thermoregulation, and the ability to push oneself to exertion. While each of these factors likely plays a role in the performance of elite runners, it is reasonable to suggest that the most important of these to performance in endurance sports is one’s mental state, overall fitness, and how fresh one is on race day. There is no clear definition of one’s “mental state.” Mental state can be understood in many different ways, including but not limited to: how well one copes with stress, how one can deal with adversity, and how much one is willing to push through during a race despite fatigue and exhaustion on a given day. Though mental state cannot be definitively measured, it is very easy to understand how fitness levels can help determine the likely outcome of a race. For example, if Runner A is significantly fitter than Runner B, then Runner A has a much greater chance to perform better than Runner B, even if Runner B has perfect form, avoids injury, and can push himself to maximal ability. “Fitness,” however, is also a somewhat fuzzy concept, especially from an athletic standpoint. There is no universally accepted definition of fitness nor any universally accepted way to measure it because there are many components of fitness, including maximal oxygen uptake (VO2 max), resting heart rate, lactate threshold, muscular strength, and flexibility.

Another important but often overlooked component of performance in endurance sports is freshness on race day. For instance, if Runner A and Runner B are at comparable fitness levels, but Runner A’s body is more worn down than Runner B’s leading up to a race, then Runner B is more likely to perform better than Runner A. When recognizing the idea that fitness and freshness play large roles in athletic performance, companies producing these wearable devices hypothesize that continuously collected data on physiological function relevant to performance can enhance training for distance runners, as it will provide insight into a runner’s fitness and freshness. This study investigates this hypothesis using a recently released performance optimization system designed primarily for elite athletes, developed by WHOOP, Inc., (Boston, MA, USA). The WHOOP system wirelessly combines a wrist-worn physiological data monitoring device with a cloud-based analytics server. At a rate of 100 HZ, the WHOOP strap measures ambient temperature, skin conductivity, acceleration via a 3-axis accelerometer, and changes in green light absorption (photoplethsmography). From fluctuations in the photoplethsmography (PPG) signal, WHOOP can detect R-R wave peaks (R-R intervals) in volumetric changes caused by cardiac contractions or heart beats, from which heart rate and heart rate variability can be derived. These derived and measured metrics are used by WHOOP to assess the body’s strain, recovery, sleep duration, and sleep quality through proprietary algorithms, none of which were made available for this study. WHOOP asserts that the measures of strain and recovery calculated from these raw variables can be analyzed by the coach and athlete to determine whether the athlete’s body is physiologically ready for a high level of training on any given day, and perhaps even predict higher or worse levels of performance. When recognizing that so many different factors affect performance in endurance sports, and many of these factors are difficult to control for and measure, one might question whether raw physiological data can be used to predict better or worse performance in a race environment. With endurance sports such as distance running, variables such as resting heart rate, HRV, and quality of sleep are hypothesized to be important physical predictors of performance. HRV in particular has become a very popular measure in research, as it is fairly simple to record and calculate (TaskForce, 1996). HRV is simply defined as the variation in time between heart beats. One’s heart rate, even at rest, is never completely

2 constant. Depending on the state of one’s body and current oxygen need, the central nervous system (CNS) will communicate to the heart through the autonomic nervous system (ANS) whether to pump more or less blood to muscles. The variation in time intervals between heart beats during rest is explained by the effects of the sympathetic and parasympathetic nervous systems (Achten and Jeukendrop, 2003). The sympathetic nervous system acts to accelerate heart rate through the action of norepinephrine; it is used when the body needs its muscles to work hard, whether for survival or athletic purposes. The parasympathetic nervous system acts via the vagus nerve to slow heart rate, allowing the body to repair its muscles and conserve resources. A healthy nervous system balances both parts of the ANS, allowing the body to perform physical tasks when needed and rest and recover otherwise. Research has suggested that dominance of the sympathetic nervous system, and therefore a lower HRV, illustrates a decrease in activity in the vagus nerve, the main nerve that regulates the body’s resting state. The decrease in vagal activity leads to cardiac electrical instability, which will cause the heart to work too hard and increase cardiac mortality (Task Force, 1996). A balanced CNS is therefore seen as essential for good health. Research on elite endurance athletes has suggested that daily changes in HRV could possibly provide insight into performance (Plews et al., 2012; Kiviniemi et al., 2007). If an elite distance runner’s sympathetic nervous system is dominant relative to the parasympathetic nervous system, then heart rate will continue to accelerate, raising blood pressure and constricting blood vessels. With heart rate continuing to accelerate and no signal from the parasympathetic nervous to slow it down, HRV will decrease as the variation between heart beats will become minimal. According to this theory, decreased HRV indicates that a runner has an accelerated heart rate, meaning his heart is still straining to send additional blood to tired muscles. In other words, a decreased HRV suggests that the runner still needs time to recover and repair muscles, and his/her body is not quite ready to work hard and perform to maximal ability. On the other hand, increased HRV is explained by a more balanced ANS. When the ANS is balanced, the sympathetic nervous system and the parasympathetic nervous system will both influence a runner’s heart; the first will act to accelerate heart rate, while the second will act to slow heart rate. The heart will then have a greater variance in time between beats due to accelerating and

3 decelerating heart rate, leading to a higher HRV. This could imply that a runner’s body is ready to work hard and strain more, as the muscles need less blood for repair and recovery. However, it has been shown that changes in vagal activity, and therefore changes in HRV, are influenced by many factors besides exercise and physical strain. Epinephrine and norepinephrine are the two main hormones that are released as a result of sympathetic dominance, and both can be released due to physiological and psychological stressors, insufficient sleep, and panic (TaskForce, 1996; Kivienmi et al., 2007; Kivienmi et al., 2011). Therefore, it could be difficult to prove that athletic performance can be strictly associated with HRV. Very little research has been done on the links between sleep and athletic performance, mainly due to the limited availability of athletes able or willing to participate in such studies (Fullagar et al., 2014). It is commonly understood that more sleep is very important for athletic performance (Venter, 2012). Sleep is defined as “the natural state and regular state of inactivity in which consciousness ceases and the bodily functions slow down or cease” and is critical for restoring the body physically and mentally (Watson, 1976). A typical night’s sleep is broken into 90-minute cycles, with periods of Rapid Eye Movement (REM) sleep and Non Rapid Eye Movement (NREM) sleep (Shapiro, 1981). NREM sleep has four stages; during these stages blood pressure, heart rate and respiration decrease while growth hormone for tissue regeneration and nervous system recuperation is released (Weitzman, 1976). During REM sleep, the eyes move back and forth quickly and blood flow increases, leading to increased blood pressure, heart rate, and body temperature (Venter, 2012). Because athletes, and especially endurance runners, wear down their bodies every day while training, it is easy to understand how sleep, and especially NREM sleep, is important in the recovery process. However, less is known about the effects of REM sleep on athletes. Studies have suggested that REM sleep plays an important role in brain activation and emotional regulation (Seigel, 2005). Research has also indicated that a lack of REM sleep negatively affects memory required to perform procedural tasks, including both motor and cognitive skills such as reaction time in handball goalkeepers (Dotto, 1996; Jarraya S. et al., 2014). It is also understood that while the repetition of a task, such as running to train for an endurance competition, is important for teaching the body to learn how to perform a task,

4 there is additional learning that takes place during sleep; studies suggest that REM sleep is essential for this memory consolidation (Walker and Stickgold, 2005). Based on this research, it could be argued that REM sleep also has a role in optimizing athletic performance alongside NREM sleep; while NREM sleep helps an endurance athlete be physically ready to perform, REM sleep may potentially help an endurance athlete be mentally ready for the focus, strain, strategy and emotion that arise during competition. In the endurance running community, coaches will often tell their athletes that having a poor night’s sleep the night before a race will not matter too much. Instead, many coaches will claim that two nights before a competition is the most important night of sleep (Glover, 1996). This claim is presumably based on experience, but is untested and its basis is unclear; it could be that many athletes get less sleep due to anxiety before competition (Fullagar et al., 2014), and therefore should get enough sleep two nights before to minimize the damage of losing sleep the night prior to a competition. Despite this emphasis on two nights before a competition by the endurance running community, there has been no evidence or research that the two nights before a race is more important than the night before. This study explores whether continuously collected physiological data on the Harvard men’s cross country team using the WHOOP Strap during the 2015 cross country season can be used to predict performance in elite distance runners. A set of initial hypotheses investigates whether variables such as resting heart rate, heart rate variability, and sleep have a significant effect on performance. Then multivariable models are used to investigate the following questions: whether the effects of sleep on performance varies depending on the amount of REM sleep, whether the night before a competition or the second night before a competition indicate a better performance. These hypotheses are all tested using regression analysis with random effects controlling for each subject.

Previous Relevant Research There has been some research on the relationship between sleep, HRV, and other HR-related data and performance for elite endurance athletes. Research has shown that increased sleep is associated with significant improvements in physical measurements such as improved reaction time, sprint time,

5 shooting accuracy, and decreased sleepiness for eleven members of the Stanford University varsity men’s basketball team (Mah et al., 2010). Physical measurements were tracked at every practice during the study, as drills in a practice setting give a more controlled environment than a game. Sleep activity was measured daily through actigraphy devices, which monitor gross motor activity and can be used to evaluate sleep and activity patterns. These devices were constantly worn on the wrist except during practices or games, and the subjects also wrote sleep journals to help the researchers identify sleeping periods. The Epsworth Sleepiness Scale was used daily during the baseline period and the end of the extension period to evaluate daytime sleepiness. A baseline period of sleep-wake patterns was calculated for 2-4 weeks depending on the athlete’s academic schedule, and then the subjects attempted to sleep longer for 5-7 weeks. The study had a few notable limitations, especially with measurement errors when athletes recorded naps in a moving vehicle for transportation to and from competition. This study was unable to have a control group due to the small sample size of athletes interested in participating. Furthermore, the subjects were familiar with many of the physical tests, as they had experience with them in practice. The study therefore does not address sleep’s effect on in-game performance and the uncertainties that arise; the authors indicate that this would be a useful investigation. There has also been some research on the effects of sleep deprivation on endurance performance. One study examined eleven recreationally-active males through endurance treadmill tests. The subjects were assigned to do two tests, 7 days apart from each other; each test started with 30 minutes at 60% VO2, then a 30 minute self-paced distance test in which the subject tried to cover as much distance as possible. The subjects did one test with normal sleep and the other with 30 hours of sleep deprivation; the subjects were randomly assigned to one sleep treatment first before the first trial, then performed the second test with the other treatment. While there was no significant change in core temperature, thermoregulatory response, or cardio-respiratory response with sleep deprivation, it was observed that distance traveled on the treadmill during the tests decreased significantly (0.23 km) with sleep deprivation (Oliver et al., 2009). The study indicates that the perception of effort may be altered when a subject is lacking sleep. However, it should be noted that a treadmill endurance test does not have the environment or additional conditions of a race atmosphere, and thus it is not reasonable to extrapolate

6 the findings to race performance. The subjects also were recreationally-fit males and not elite athletes. Additionally, the study does not investigate the effects of sleep restriction on endurance performance, which is far more likely to occur for athletes leading up to a race rather than complete sleep deprivation. With regards to HRV, there has also been research observing higher HRV with endurance improvements, suggesting that higher intensity of training leads to significant changes in cardiovascular autonomic regulation (Kiviniemi et al., 2010). This study analyzed twenty-five male distance runners who had some prior experience in distance running and put them through a 28-week training program for a marathon. The first 14 weeks of training were a “baseline” in which the subjects trained at the same volume as before the study, and the last 14 weeks were a period in which training volume and intensity were increased. The second period contained 3-week cycles that alternated two weeks of high intensity with a recovery week of lower intensity. Throughout the training, the subjects measured their HR during exercise using heart rate monitors and kept a training log. R-R intervals were recorded and nocturnal HRV was calculated using a Suunto Memory Belt for three consecutive nights before and after both training periods, and the averages over two-day spans were used due to fairly frequent errors in R-R interval recordings. At the end of the first 14 weeks, HRV and resting heart rate did not significantly change. After the second 14-week period, HRV increased and resting heart rate decreased significantly. Additionally, mean marathon time improved by 8.2% compared to previous best performances, suggesting that a higher HRV is associated with better performance. This study did not have any control groups as all subjects went through the same training plan. A few studies have also examined if changes in HRV are a sign of fitness and potentially overtraining, but with mixed results. One study tracked daily HRV in two elite triathletes, one male and one female, and observed that HRV could be a useful metric to project mal-adaptation or overtraining (aka non-functional overreaching) (Plews et al., 2012). Each athlete’s morning resting HRV was monitored daily for a 10-week period, with a race in week 7 and high training loads leading up to the race. The athletes also filled out an electronic questionnaire every morning about their sleep quality, muscle soreness, stress, and fatigue. R-R intervals were recorded using Polar RS800cx heart rate monitors

7 and were used to calculate HRV, both single-day and rolling average. The female athlete went through a period where she struggled to achieve her desired training load, continued to train, and was unable to finish her race during the event due to fatigue. She then was diagnosed as having shingles and spent the next few weeks recovering. The male athlete trained very well and was able to achieve a personal best in his competition. The study noted that resting heart rate increased for the female athlete and decreased for the male athlete, and also reported that the female athlete’s HRV decreased significantly leading up to her competition while the male athlete’s HRV did not vary significantly. There have also been some studies on the connection between changes in HRV and fitness using elite rowers. One study observed an Italian junior rowing team by recording HRV for three hour sessions once a week, and noticed a decrease in HRV despite an increase in fitness (as measured by VO2 max) (Iellamo et al., 2002). Another study, however, noticed in three elite Olympic rowers that increases in HRV before a major competition was associated with a positive performance (Plews et al., 2013). While all of these studies could have benefitted from a much larger sample size, the results suggest that a decreasing trend in HRV can signal that an athlete is overtraining and failing to perform to his desired level. A few studies have investigated training healthy, recreationally-fit athletes using HRV to determine training load, and noticed that both training load and improvements in training increased significantly for an HRV-based training program. One study randomized the subjects into a predefined training group, a HRV-based training group, and a control group with no training. The predefined training group had a weekly training plan with two three-day cycles followed by a rest day, then a low-intensity exercise day followed by two high-intensity days. The HRV-based training group determined their intensity based on their daily HRV measurements. The subjects performed a graded maximal exercise test on the treadmill before and after the 4-week training period. Improvement in fitness was measured by VO2 max and training load was quantified by maximum running velocity. The subjects in the HRV-based group had their R-R intervals measured every morning upon awakening using a Polar S810 monitor for a 5-minute interval, and their HRV was calculated. Increased HRV led to higher intensity training that day, and decreased HRV led to lower intensity training. The HRV-based training group had a statistically significant improvement in VO2 max and in training load by 0.9 km/h, while the predefined training

8 group saw only a significant increase in training load by 0.6 km/h and the control group saw no significant increases. The increase in training load for the HRV-based training group was significantly higher than the predefined training group, while the increase in VO2 max was not significantly different (Kiviniemi et al., 2007). This study was then expanded by examining 24 healthy moderately active men and 36 healthy women over an 8-week training period. Subjects were randomized into three groups: predefined training, HRV- based training, and a control group that did no structured training. VO2 max and training load was measured for all six groups. As in the previous study, the HRV-based groups’ R-R intervals were measured using a Polar RS800 monitor for a 2-minute period upon awakening to calculate HRV. Increased HRV led to increased intensity, while decreased HRV led to decreased intensity. They observed that training load increased significantly for HRV-based training for the men’s HRV-based group, but not for the women’s group (Kiviniemi et al., 2010). For both studies, the measurements of R-R intervals were made at home by the athletes rather than done in a laboratory to allow for as normal conditions as possible when waking up.

Summary A few past studies have investigated how more sleep is linked with better athletic performance, how HRV can be used for guiding training, how HRV can be used to detect overtraining, and how higher HRV suggests greater endurance capability. Because sleep quality, greater VO2 max, and preventing overtraining have all been shown to be reasonably useful indicators of improved fitness and better athletic performance, it is reasonable to hypothesize that HRV and sleep data can help predict endurance performance. However, while studies have either focused on recreationally-fit runners or a very small sample of elite athletes for different sports, there has not been any research on HRV and sleep-related data for elite endurance runners, especially at the level of NCAA Division I cross country runners. All previous research has also utilized only heart rate and sleep data collected for small intervals of time; “big data” on heart rate and sleep collected from wearable fitness devices such as the WHOOP Strap have never been continuously collected over an extended period of time. Better understanding is also needed to learn about how changes in HRV in elite athletes relate to their performances in actual

9 competition. Most of the studies reviewed above mainly examine how HRV changes indicate improvements on athletic performance tests, VO2 measurements or the ability to handle an increased training load; there is very little research on how HRV fluctuations indicate how athletes will perform in the primary event for which they train. While it is commonly understood that sleep is beneficial for athletes, there is also very little research into the details of sleep’s effect on performance for elite athletes, especially in the primary event for which they train.

Methods Subjects Ten healthy male distance runners from the Harvard varsity men’s cross country team were recruited for the study after the Harvard track and field team purchased WHOOP Straps for some of the program’s student athletes. The ages of the subjects ranged from 18-22 years, with weights between approximately 117-167 pounds and heights between approximately 62 and 75 inches. The study was restricted by the coaching staff to only male subjects. The WHOOP Straps were distributed to athletes who volunteered to wear them, though the faster athletes on the team were given priority. All subjects then gave informed consent to participate in this study and for the use and analysis of all the raw physiological data collected from their WHOOP Straps. The study protocol was approved by the Committee on the Use of Human Subjects, which acts as the Institutional Review Board for Harvard College.

Study Duration and Treatments The subjects received WHOOP Straps for data collection on September 18, 2015, which was after the team’s first race of their season. Each subjects was instructed to wear the WHOOP Straps each day and night as often as possible until just after the end of the season. For both workouts and sleeps, the subjects recorded the activity’s start and end time using WHOOP’s mobile application which connected his mobile device to the WHOOP optimization system via Bluetooth. The subjects had the option of starting and ending an activity as it was happening, or several hours after the activity took place. Data collection from the WHOOP Strap lasted 66 days. In addition to competing in the cross country season, each subject participated in a weekly bike effort every Monday or Tuesday with the exception of illness or injury. The subjects were given flexibility to do the bike efforts on certain Tuesdays rather than Monday because some weeks of training required a hard running workout on Monday; some athletes preferred not to do the hard bike effort before the intensive running workout to conserve their energy. The subjects were also given an optional weekly survey each Saturday to evaluate their diet and stress levels during the previous week. While more

11 details are mentioned in the results section, the data from the bike efforts were eventually discarded.

Measurements of Performance A subject’s performance was measured in two ways: cross country race results and a weekly bike effort. The cross country race results for the Harvard men’s cross country team were collected from public websites at the end of the season. These results were chosen as a measurement of performance because they were from actual competition, which most previous studies do not utilize. Because the subjects raced on different cross country courses throughout the season, a slower time in tougher conditions, such as on a hillier course or in extreme weather, may suggest a “better” performance than would a faster time in easier conditions. Table 1 provides a short description of each race course the subjects race on during the season, as well as how many of the nine subjects race on the course during that race. Only one course, Van Cortlandt Park, was raced on twice: the Ivy League Heptagonal Championships at the end of October and the IC4A Championships three weeks later. However, the two races were defined in the model as “different” courses due to different circumstances in both races. When racing the Ivy League Championships, subjects had gone two weeks without a race, and the field of athletes began the race at a very fast pace. In the IC4A Championships, the subjects had only one week without racing prior to the competition, leading subjects to have less rest leading up to the race. Additionally, the field began the race at a very slow, tactical pace. Because of these factors affecting race times, it is proposed in this study that equivalent performances in both the 2015 IC4A championship and 2015 Ivy League Championship would have a different time in the IC4A Championship than the Ivy League Championship, and therefore the two competitions were treated as separate “courses.” Variables identifying subjects and each race were used to control for both subject and race course conditions in regression analysis with random effects. Since race courses were either 5 kilometers, 8 kilometers, or 10 kilometers, the average pace per kilometer in seconds was used as a dependent variable in all models. This was determined to be a reasonable approach because it helped diminish the effect of any outlier race results for a

12 subject; race times for longer distances will be much slower than race times for shorter distances, but the average pace per kilometer will not differ as much. In addition, two of the 8-kilometer races were on a very hilly course, which causes the average pace per kilometer for those two races to be slower than flatter courses and prevents the average pace per kilometer on the 10-kilometer course to be an outlier.

Table 1: Race courses during the Harvard cross country (XC) season.

Course and Distance Race Name and Description and Reputation Number of Date Subjects Entered

Franklin Park 8K Coast-to-Coast An early season meet with several high quality 8 Battle at Division I XC teams. The 8K course has a couple Beantown of hills but is overall pretty fair. In general it is Championship, known as a fast course when conditions are 9/25/2015 good. The top nine runners on the Harvard team were entered.

Franklin Park 5K Coast-to-Coast Athletes not entered in the championship race 1 Battle at were entered. Beantown Open Race, 9/25/2015

Wisconsin 8K Wisconsin Adidas A large invitation with many nationally-ranked 5 Invitational, teams. Harvard brought its top seven runners. 10/16/2015

Brown 8K Brown The Harvard team sent its remaining runners to 2 Invitational, an invitational at Brown. The course is fairly 10/16/2015 standard without too many hills.

Van Cortlandt Park 8K Ivy League The Ivy League Championship and IC4A 7 Heptagonal Championship were both held on this course. Championships, The IC4A Championship is a historic end-of- 10/30/15 season meet with teams that do not qualify for the NCAA Championship. The Harvard men sent IC4A all healthy runners to the IC4A meet and the top Championships, nine runners to the Ivy League Meet. The course 11/21/2015 has a reputation of being brutally tough and is by far the hilliest course the team raced on.

Franklin Park 10K NCAA Northeast The qualifier for the NCAA Championship. The 5 Regional, race is a 10K race, which is longest the team 11/13/2015 races during the season and the increase in distance can be challenging for many athletes.

The weekly bike effort was included to allow for a more standardized performance measure. Every Monday or Tuesday, before the team’s afternoon practice, the subjects used the same exercise bike at the same resistance level. The subjects biked hard for a 5-minute interval, and the total distance recorded at the end of the 5-minute interval was recorded.

Physiological Data Measured Data on total sleep duration, the duration of each sleep stage, sleep latency, the number of REM cycles, resting heart rate, and heart rate variability was recorded in the nights leading up to a competition and a bike effort. All of the physiological predictor variables were measured using the WHOOP performance optimization system. WHOOP, Inc. agreed to provide the data required for this study. Ambient temperature, skin conductivity, acceleration via a 3-axis accelerometer, and changes in the PPG signal are measured at a rate of 100HZ. Using the fluctuations in the PPG signal, the system detects R-R intervals and derives calculations of heart rate and HRV. WHOOP uses a proprietary algorithm that determines each stage of sleep as well as sleep latency based on a function of motion, skin conductivity, and heart rate. For the calculation of heart rate and HRV, WHOOP measures the variance in frequency between R-R intervals during sleep. The system uses a proprietary algorithm to determine the last slow-wave sleep stage in a subject’s sleep. Then the system analyzes the last 5-minute interval of the stage with a padding time of 1 minute from the end to avoid analyzing the transition between the slow-wave sleep stage to the next stage, as the physiological data will vary if the beginning of the next sleep stage is accidentally captured in the interval. Five minutes is the standard amount of time recommended for measuring HRV in clinical and physiological studies (TaskForce, 1996). The square root of the mean sum of squared differences between R-R intervals (rMSSD) was calculated within this 5- minute interval at the end of sleep, as research shows that it reflects vagal activity (TaskForce, 1996) and is shown to be a more accurate calculation than other indices (Al Haddad et. al, 2011). This recording of data is different from previous studies on elite athletes, as data is usually collected once the subject has woken up. HRV is calculated during sleep rather than during the day due to there being less factors affecting the fluctuations of heart rate when a subject is asleep.

Survey Data In addition to the physiological data collected from the WHOOP Straps, an optional survey was given to the subjects weekly on each Saturday. The survey asked the subjects to rate their stress on a scale from 0 to 5; provide their weight if they measured it during the week; approximate how many hours of schoolwork they had during the week; and discuss their diet during the past week in as much detail as possible. Table 2 contains the questions asked to the subjects and overall response rate.

Table 2: Survey Questions

Number Question Response Rate

1 On the following numerical scale, how stressed were you last week? 86.1%

0) As if you were relaxing during summer vacation, without a care in the world.

1) You actually have things going on in life, but not really worried by them.

3) Typical week of stress for a Harvard student – some busy work, a little down time, etc.

5) You had multiple exams, lots of homework, an essay, your girlfriend broke up with you, etc.

2 If you weighted yourself in the weight room on Monday, how much did you weigh? 61.1%

3 About how many hours did you spend doing schoolwork this past week? 84.7%

4 Please provide a description of what you ate and drank for breakfast, lunch and dinner yesterday, 86.1% as well as any snacks. Is this diet typical of the rest of the week?

Statistical Methods Random Effects Models To analyze the effects of an individual physiological variable on expected race performance, a random effects model was used. Random effects models are extensions of linear regression models for grouped data in which coefficients are allowed to vary for different groups (Gelman and Hill, 2006). While other models were considered, this random effects setup was selected as the most appropriate model for predicting race performance. Individual subjects are all different in how their race performances are associated with physiological variable covariates. For example, one subject’s race performance may have a slightly different association with sleep duration the night before a race from another subject; in other words, getting more sleep might make less of a difference in how one subject races versus another subject. Therefore, it makes sense intuitively to allow for coefficients to vary for each subject. There are two possible designs for constructing a random effects model: the varying intercept model and the varying slope and intercept model (Gelman and Hill, 2006).

Varying Intercepts Model A random effects model with varying intercept for race performance in race [i] for subject [j] is generally defined as:

(1)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + 휖𝑖𝑗 2 휖𝑖𝑗 ~ 𝑁(0, 𝜎 )

휇0𝑗 ~ 𝑁(0, 훺휇)

In this model 𝑦𝑖 corresponds to the average pace per kilometer in seconds that a subject runs in race [i] and 𝑥𝑖 is some quantitative measurement of the physiological variable in a given time window before race [i]. For example, one possible model might have the value of 𝑥𝑖 as the sleep duration (in minutes) a subject gets the night before race

[i]. Another model might have the value of 𝑥𝑖 as a weighted average of HRV (in seconds) in

16 the three nights before a race, with greater weights assigned to the nights closer to the race and smaller weights assigned to the nights farther away from the race.

𝛽1 represents the coefficient for the physiological variable in question and is interpreted as the expected effect of a one-unit change in the physiological variable on the expected pace per kilometer of race [i] for subject [j]. For example, a value of -0.04 for 𝛽1 where 𝑥𝑖 represents the sleep duration in minutes would indicate that for every 1-minute increase in sleep duration, a subject’s expected pace per kilometer in race [i] would decrease by 0.04 minutes. In this model, (𝛽0 + 휇0𝑗) represents the intercept term, which can be understood as the expected race performance when the physiological variable in question has a value of zero. Because these physiological variables do not normally take a value of zero, it does not make sense to analyze the intercept in detail; however, it could be argued that the intercept gives a broad understanding of how “good” the subject is, as a smaller intercept suggests a faster subject. The value of 𝛽1, however, is of great importance, as it could indicate a physiological variable’s significant association with expected race performance. 휖𝑖𝑗 is the error in the model, and can essentially be understood as random noise that affects race pace per kilometer. In other words, 휖𝑖𝑗 captures the effects of all other factors that may influence the expected race performance in race [i] for subject [j] other than the predictor variable 𝑥. The error is normally distributed, with an expected value of zero and a variance of 𝜎2.

For multivariable models predicting pace per kilometer of race [i], or 𝑦𝑖, for subject [j], the varying intercept model can simply be expanded for k physiological variables:

(2)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥1𝑖 + 𝛽2𝑥2𝑖 + ⋯ + 𝛽𝑘𝑥𝑘𝑖 + 휖𝑖𝑗

In (2), the interpretation of the intercept is the same as in (1), and the interpretation of each coefficient 𝛽1, 𝛽2, … 𝛽𝑘 is the same as the coefficients in classic multivariable regression: holding all other predictor variables constant, an increase in 𝑥1𝑖 by one unit is expected to change the expected value of 𝑦𝑖 by 𝛽1. The error term 휖𝑖𝑗 also takes the same interpretation as in (1).

Varying Slope and Intercept Model (3)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥𝑖 + 휖𝑖𝑗 2 휖𝑖𝑗 ~ 𝑁(0, 𝜎 ) 휇0𝑗 0 [ ] ~ 𝑀𝑉𝑁 ([ ] , 훺휇) 휇1𝑗 0 2 𝜎휇0 𝜎휇01 훺휇 = [ 2 ] 𝜎휇10 𝜎휇1

In the varying slope and intercept model, the interpretations of 𝛽0, 휇0𝑗, 𝛽1, and 휖𝑖𝑗 are the same as in the varying intercept model. In the varying slope and intercept model, the term 휇1𝑗 is added to the slope of 𝑥. With 휇1𝑗 added to the slope 𝛽1 for the physiological predictor variable 𝑥, (𝛽1 + 휇1𝑗) is now interpreted as the effect of a one-unit change in variable 𝑥𝑖 on expected race performance.

For multivariable models predicting pace per kilometer of race [i], or 𝑦𝑖, for subject [j], the varying slope and intercept model in (3) can be expanded for k physiological variables, following the same idea as in (2):

(4)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥1𝑖 + (𝛽2 + 휇2𝑗)𝑥2𝑖 + ⋯ + (𝛽𝑘 + 휇𝑘𝑗)𝑥𝑘𝑖 + 휖𝑖𝑗

In (4), 휇1, 휇2, … 휇𝑘 are terms added to the slopes 𝛽1, 𝛽2, … 𝛽𝑘 respectively. In other words, the coefficient for each of the k predictor variables are allowed to vary for each subject. Interpretation of the coefficients in (4) is the same as in (2). When illustrating models in the remainder of this section, both varying intercept and varying slope and intercept models will be written out. Both types of models were analyzed and compared in this study; the effects of different physiological predictor variables can then be seen both when slopes are allowed to vary for each subject, and when slopes are not allowed to vary for each subject.

Difference from Classical Regression These random effects models (1) and (3) differ from the classic regression model because of the inclusion of the additional term 휇0𝑗 in (1) and 휇0𝑗 and 휇1𝑗 in (3). The term

휇0𝑗 in (1) can be interpreted as a random term added to the intercept 𝛽0, and this variation in the intercept can be explained at the individual subject level. In other words, the model allows for the intercept to vary for each individual subject. In (1), 휇0𝑗 is normally 2 distributed with mean zero and variance 𝜎휇0. In (3), the term 휇0𝑗 has the same general interpretation, while the term 휇1𝑗 is also added to the model. 휇1𝑗 can be interpreted as a random term added to the slope 𝛽1 for a physiological predictor variable 𝑥, which essentially allows the effect of a one-unit change in variable 𝑥 to change the expected race pace per kilometer by a different value for each subject. To put it simply, the slope is allowed to vary for each individual subject. In (3), Both 휇1𝑗 and 휇0𝑗 are multivariable normal with mean zero and covariance matrix 훺휇 respectively. In 훺휇, 𝜎휇01 = 𝜎휇10, and is essentially interpreted as the covariance between the intercept and the slope of the model. Another important difference between random effects models and classic regression models is the use of R2 to indicate goodness of fit of a model. In classic regression, R2 is:

∑ 2 2 𝑖 𝑒𝑖 푅 = 1 − 2 ∑𝑖(𝑥𝑖 − 𝑥̅)

2 Where 𝑒𝑖 is the squared residual, or the squared difference between the observed value and the predicted value from the model, and 𝑥̅ is the mean of the observed data. The quantity is not clearly defined for random effects models, as experts do not agree on whether to include the variance of the random effects for each subject, and if so how to include the variance (Gelman and Hill, 2006). Therefore, in this study, 푅2 was not utilized to investigate a model’s goodness of fit and was not reported in any results.

Likelihood Ratio Tests In a few situations this study utilized likelihood ratio tests to compare two different models to see which model fits the data better. In this case, one of the compared models is

19 an expansion of the other model. For a general multivariable model, the hypotheses of the likelihood ratio test is: (5)

퐻𝑜: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + ⋯ + 𝛽𝑚𝑥𝑚

퐻퐴: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + ⋯ + 𝛽𝑚𝑥𝑚+ 𝛽𝑚+1𝑥𝑚+1 + ⋯ + 𝛽𝑚+𝑛𝑥𝑚+𝑛

In this case, the null hypothesis is that the smaller model, with 𝑚 predictor variables, is the model that best fits the data. The alternative hypothesis is that the larger model, with 𝑛 additional predictor variables and 𝑚 + 𝑛 total predictor variables, is the model that best fits the data. This was determined to be a useful tool for this study; there are several situations in which random effects models predicting race performance include interactions between physiological predictors or additional control variables such as race course. Simply put, the likelihood ratio test is used here to help indicate whether more complex random effects models, with additional variables added alongside physiological predictor variables, better predict race performance. The null hypothesis for the likelihood ratio test requires the use of the likelihood function. The likelihood function is essentially the probability of seeing the observed data given a set of model parameter values 휃:

𝐿(휃|𝑥) = 𝑝(𝑥|휃)

To illustrate with the null hypothesis above, the likelihood function would be the probability of seeing the observed data (the 𝑥𝑖’s) given the parameters 휃𝑜 = 𝛽0,

𝛽1, 𝛽2, … 𝛽𝑚 that were fitted for the random effects model. The likelihood function of the model under the alternative hypothesis would be the probability of seeing the observed data given the fitted parameters 휃퐴 = 𝛽0, 𝛽1, 𝛽2, … 𝛽𝑚, 𝛽𝑚+1, 𝛽𝑚+𝑛. It is important to remember that in random effects models, as in classical regression, the 𝛽’s are the maximum likelihood estimators of the true parameter value. In other words, the 𝛽’s essentially maximize the likelihood function.

With the likelihood for the null model 𝐿(휃𝑜|𝑥) and the likelihood for the alternative model 𝐿(휃퐴|𝑥), the likelihood ratio test statistic 푇 can be calculated:

𝐿(휃 |𝑥) ᴧ = 𝑜 𝐿(휃퐴|𝑥)

푇 = −2 ln(ᴧ) = −2 ln(𝐿(휃𝑜|𝑥)) + 2 ln(𝐿(휃퐴|𝑥)) 2 푇 ~ 휒𝑑𝑓=𝑛

When the sample size is very large, 푇 is asymptotically chi-squared with degrees of freedom equal to 𝑛 (DeGroot, 1986), where 𝑛 is the difference in the number of parameters between the two compared models. In this study, 𝑛 is the number of additional parameters in the larger model, such as the number of race course coefficients or additional interaction terms. If the value of 푇 exceeds some critical value, then the null hypothesis is rejected and the larger model is determined to have a better fit than the smaller model. In other words, it can be argued that a model with extra parameters such as race course control variables or interaction terms is better able to predict race performance than a smaller model with only physiological variables. While in this study the sample size of race performances is rather small, the likelihood ratio test was still used to give a general idea of whether extra parameters added to a model would be more likely to successfully predict race performance. The likelihood ratio test can also be used to compare a varying intercept model to a varying slope and intercept model.

Data Aggregation When analyzing data for this study, data was aggregated into many different summary forms for analysis. In other words, there were several ways to aggregate physiological data when analyzing a variable such as a subject’s sleep duration for several nights leading up to a competition. Four types of data aggregation were used in this study: one night before a competition, two nights before a competition, a weighted average system with weights decaying linearly, and a weighted average system with weights

21 decaying exponentially. Table 3 contains a short description of each type of data aggregation. For obvious reasons, it makes sense to analyze a subject’s sleep and heart rate data the night before a competition; it is reasonable to argue that the quality and quantity of sleep, heart rate variability, and resting heart rate on the night before a competition may indicate better or worse performance. Additionally, it is of scientific interest to examine whether these physiological variable measurements two nights before a race indicate better or worse performance. This would be used to investigate the claims of coaches in the endurance running community that rest two nights before a competition is more important than one night (Glover, 1996).

Table 3: Data Aggregation Definitions

Variable Type Definition

One Night Before A measured variable for the night before a competition.

Two Nights Before A measured variable for two nights before a competition.

Weighted System A weighted average where weights decrease linearly for data on nights farther from a competition.

Exponential Weighted System A weighted average where weights decrease exponentially for data on nights farther from a competition.

It is also intuitive to examine how a subject’s physiological variables in the days leading up to a competition are associated with performance. One could use a simple average measurement for a physiological variable across multiple nights prior to competition, but it makes more sense to use a weighted average system. It is reasonable to assume that a subject’s volume of sleep and heart rate data on nights closer to competition should have greater weight than data several nights from competition; for example, a subject’s sleep four nights before a competition is not as likely to indicate better or worse performance as a subject’s sleep one or two nights before a race. The first weighted

22 average system used is a standard weighted average for subject [j] in the 𝑛 nights leading up to race [i] (called 𝑊퐴𝑗𝑖) of the following design, for the physiological variable 𝑥:

𝑛𝑥 + (𝑛 − 1)𝑥 + (𝑛 − 2)𝑥 + ⋯ + 2𝑥 + 𝑥 𝑊퐴 = 1 2 3 𝑛−1 𝑛 𝑗𝑖 𝑛 + (𝑛 − 1) + (𝑛 − 2) + ⋯ + 2 + 1

In this system, 𝑥1 represents the measurement of the physiological variable on the night before a race, 𝑥2 represents the measurement of the physiological variable on the night two nights before a race, 𝑥3 for three nights, and so on. The weights of the physiological variables in the weighted average decrease linearly for nights farther away from competition. To provide a concrete example, suppose Subject A has 480 minutes of sleep the night before race [i], 520 minutes two nights before race [i], and 460 minutes of sleep three nights before race [i]. His weighted average of sleep for three nights before race [i] under this system would be:

(3 ∗ 480) + (2 ∗ 520) + (1 ∗ 460) 𝑊퐴 = = 490 minutes 퐴𝑖 (3 + 2 + 1)

The second system is an exponential weighting system, with weights decaying at a 2 rate of . This rate was subjectively selected because it allows for weights to decay for 3 nights farther from competition, but also allows for nights closer to competition to still maintain a substantial effect on the average. For the 𝑚𝑡ℎ night before a competition, the 2 weight of the measurement takes a value of ( )𝑚−1. This exponential weighted average for 3 subject [j] in the 𝑛 nights leading up to race [i] (called 퐸𝑊퐴𝑗𝑖) has the following design:

2 2 2 2 𝑥 + 𝑥 + ( )2𝑥 + ⋯ + ( )𝑛−2𝑥 + ( )𝑛−1𝑥 1 3 2 3 3 3 𝑛−1 3 𝑛 퐸𝑊퐴 = 𝑗𝑖 2 2 2 2 1 + + ( )2 + ⋯ ( )𝑛−2 + ( )𝑛−1 3 3 3 3

As in the standard weighting average system, 𝑥1 represents the measurement of the physiological variable on the night before a race, 𝑥2 represents the measurement of the physiological variable on the night two nights before a race, 𝑥3 for three nights, and so on. In the exponential weighted average system, weights decay exponentially instead of linearly. Using the previous example of Subject A’s sleep three nights before race [i], it can be shown that Subject A’s 퐸𝑊퐴 for sleep three nights before is:

2 2 480 + ( ∗ 520) + (( )2 ∗ 460) 3 3 퐸𝑊퐴 = ≈ 488.42 minutes 퐴𝑖 2 2 1 + + ( )2 3 3

Note that in this system, more weight is put on the nights closer to a race than the nights farther from a race compared to the standard weighting system with weights decreasing linearly. This was named the exponential weighting system in this study because the decay of weights is a function with the number of nights from a race included in the exponent. Both weighting systems allow for nights closer to competition to have a greater impact on predicting performance than nights farther from competition. There is no empirical evidence or research suggesting that these weighting systems are ideal for modeling one’s sleep and heart rate data leading up to competition; it may also be the case that the ideal weighting systems may vary on the individual level. Simply put, physiological variables such as sleep may affect one individual across time differently than another individual. Further research is recommended to develop an appropriate weighting system.

Hypotheses Tested Set 1: Individual Physiological Variables and Performance There were four sets of hypotheses that were tested during this study. The first set of hypotheses involved investigating whether individual physiological variables in the nights leading up to a race have a significant association with race performance. These hypotheses were tested using both types of random effects models, with 𝑥 representing a different physiological variable aggregated in one of the four forms of data aggregation

24 mentioned in Table 3. The two types of weighted averages of physiological variables were examined for two nights, three nights, and four nights before a race; it was assumed that barring extreme circumstances, the quality of sleep or heart rate data from five nights or more from a race is very unlikely to indicate better or worse performance. Nine physiological variables collected from each subject’s WHOOP Strap were examined, as displayed in Table 4. For each of the nine physiological variables, for each type of data aggregation, the following hypothesis test was set up for the random effects model slope coefficient:

퐻𝑜: 𝛽퐻𝑅푉 = 0 퐻퐴: 𝛽퐻𝑅푉 ≠ 0

퐻𝑜: 𝛽𝑅퐻𝑅 = 0 퐻퐴: 𝛽𝑅퐻𝑅 ≠ 0 … …

퐻𝑜: 𝛽𝑅𝐸푀 𝑐𝑦𝑐𝑙𝑒 𝑐𝑜𝑢𝑛𝑡 = 0 퐻퐴: 𝛽𝑅𝐸푀 𝑐𝑦𝑐𝑙𝑒 𝑐𝑜𝑢𝑛𝑡 ≠ 0

Table 4: Physiological Variables

Variable Units

rMSSD Heart Rate Variability seconds

Resting Heart Rate beats/minute

Total Sleep Duration minutes

Light Sleep Duration minutes

REM Sleep Duration minutes

Slow Wave Sleep Duration minutes

Sleep Latency minutes

Time In Bed minutes

REM Cycle Count

Simply put, it was concluded whether the maximum likelihood estimator for each 𝛽 is significantly different from zero from the data, and thus whether there may be significant

25 association between the physiological variable of interest and race performance. The sign of the coefficient is also of importance, as it may indicate whether the variable is positively or negatively associated with race performance. To adjust for multiple comparisons in this set of hypotheses and decrease the probability of a Type I error by rejecting a null hypothesis when it should not be, the alpha level for these tests was set using the Bonferroni correction:

0.05 𝛼 = ∗ 𝑁

Here the value of 𝑁 is set to the number of physiological variables analyzed. When performing multiple hypothesis tests on many aggregations of these several physiological predictor variables, it is quite likely that a null hypothesis is rejected when it should not be due to random chance (Ramsey and Schaefer, 2012). Therefore, the alpha level of 0.05 is adjusted: 𝑁 is set to equal 9 here, and the adjusted alpha level is:

𝛼∗ = 0.00556

This allows for the overall probability of a Type I error throughout these tests to be 0.05. In addition to testing whether each individual predictor variable has a significant association with race performance alone, random effects models were examined that included indicator variables for each race course that the athletes raced on during the season. Because cross country races take place on different courses, the pace per kilometer each subject runs on each course will likely vary depending on course length, difficulty, and weather conditions, even given equivalent performances. This random effects model effectively controls for race course and can be defined as: (6)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + 𝛾1휃1𝑖 + 𝛾2휃2𝑖 + ⋯ + 𝛾푀−1휃(푀−1)𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥𝑖 + 𝛾1휃1𝑖 + 𝛾2휃2𝑖 + ⋯ + 𝛾푀−1휃(푀−1)𝑖 + 휖𝑖𝑗

In this model, the 휃’s are essentially indicator variables for the 𝑀 race courses that the subjects raced on during the cross country season. In this design, 𝑀 − 1 indicator variables are included in the model for 𝑀 − 1 of the race courses. One of the courses is not given a 휃 and is a “baseline” course (Gelman and Hill, 2006). This was done in this study to prevent complete linear dependence of each of the 휃’s. For example, if 𝑀 indicator variables were included in the model for each of the 𝑀 race courses, there would be an infinite amount of unique set of coefficients 𝛾1, 𝛾2, … 𝛾푀 that would best fit the model. The value of one coefficient could be increased or decreased, and the other remaining coefficients could then be adjusted to lead to a model that has the exact same fit (Gelman and Hill, 2006). Therefore, 𝑀 − 1 indicator variables are used. In this study there were six race courses, so 𝑀 = 6. With this setup, the coefficient for each race course 𝛾 is used to compare the effect of each race course compared to the “baseline” course, using the indicator variables

휃1, 휃2 … 휃푀−1 to declare which course a race takes place on. For example, if race [i] takes place on the “baseline” course, all of the 휃𝑖 ‘s equal zero. If race [i] does not take place on the “baseline” course, the 휃𝑖 corresponding to race [i]’s race course will equal 1, while the other 휃𝑖’s will equal 0. To illustrate, if race [i] takes place on course [3], then 휃3𝑖 = 1 and

휃1𝑖, 휃2𝑖, 휃4𝑖 … 휃(푀−1)𝑖 = 0. The model then becomes, for the varying intercept model and varying slope and intercept model respectively:

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + 𝛾3 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥𝑖 + 𝛾3 + 휖𝑖𝑗

Each 𝛾𝑖, or the coefficients associated with each race course variable 휃𝑖, essentially represents the change in 𝛽0 for each race course from the “baseline” course. In simpler words, each value of 𝛾 adjusts the expected race performance based on the race course that race [i] took place on. With this expanded model, the null hypothesis is that for each 𝑥 and for each 휃, its coefficient is equal to zero, with the alternative hypothesis that its coefficient is not equal to zero. Note that for this and other multivariable models in later analysis, the alternative hypothesis is not that all of the coefficients are significantly different from zero.

The alternative hypothesis is that, for a specific predictor variable, its coefficient is significantly different from zero when all other variables are held constant. In other words, the null hypothesis is that a predictor variable (or a race course variable, as in (6)) has no significant association with race performance, while the alternative hypothesis is that a predictor variable has a significant association with race performance. The alpha level for these tests was set to 0.05. For any physiological variable 𝑥 that had significant associations with race performance, additional multivariable random effects models with 𝑥 were examined. These models controlled for more factors by including race course indicator variables, and these models were analyzed to see if the significant association remained after these race controls were in place. Then a likelihood ratio test was used to compare the model based on 𝑥 alone with the model based on 𝑥 with race course controls included. This test can help indicate whether the addition of these race course indicator variables significantly improve the model’s ability to predict race performance based on 𝑥. Following the setup in (5), the hypotheses for the random effects models with varying intercepts is:

퐻푂: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + 𝛾2휃2𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

And for random effects models with varying slopes and intercepts:

퐻푂: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥𝑖𝑗 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇0𝑗)𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + 𝛾2휃2𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

For these likelihood ratio tests, the alpha level was set to 0.05. If the null hypothesis is rejected, it is an indication that models with race course indicator variables may be a better predictive model for race performance compared to models without race course indicator variables. In other words, it suggests that it is important take into account the race course when determining whether a physiological variable is significantly associated with performance.

In addition, for any variable 𝑥 that had a significant association with race performance when controlling for race course, another model with interaction terms between race course and 𝑥 was examined. To illustrate, the random effects model with varying slope and intercept for 𝑥 on race performance with race course indicator variables and interaction terms would be: (7)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇0𝑗)𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + ⋯

… + 𝜌1(𝑥𝑖 ∗ 휃1𝑖) + ⋯ + 𝜌5(𝑥𝑖 ∗ 휃5𝑖) + 휖𝑖𝑗

In this equation, 𝜌1, 𝜌2, … 𝜌5 represent the interaction coefficients between 𝑥 and the race course indicator variables 휃1, 휃2, … 휃5 respectively. The value of 𝑥𝑖 ∗ 휃5𝑖 becomes 1 if race[i] takes place on race course [5], and 0 otherwise. If a race course takes place on race course [5], the equation for (7) becomes:

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇0𝑗 + 𝜌5)𝑥𝑖 + 𝛾5 + 휖𝑖𝑗

Similarly to the effect of the coefficient 𝛾 in (6), the coefficient 𝜌5 is the change in the slope of 𝑥 for race course [5] from the “baseline” race course. For example, if 𝜌5 = 0.16, it indicates that the effect of a one-unit increase in 𝑥 on race performance increases by 0.16 when a race takes place on course [5] compared to the “baseline” course. If the value of a 𝜌 is significantly different from zero statistically, it can be interpreted that the effect of 𝑥 on race performance changes significantly depending on the race course. To investigate whether the addition of the race course interaction terms significantly improves the model’s ability to predict race performance for a physiological variable 𝑥, a likelihood ratio test was then used to compare the random effects model with interactions to the model without interactions. The idea is the same as comparing models with race course variables without race course variables, as the ratio of likelihood functions is calculated. A test comparing models with varying intercepts would have the following null and alternative hypothesis:

퐻𝑜: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + ⋯

… + 𝜌1(𝑥𝑖 ∗ 휃1𝑖) + ⋯ + 𝜌5(𝑥𝑖 ∗ 휃5𝑖) + 휖𝑖𝑗

The null and alternative hypothesis for comparing models with varying slopes and intercepts would be:

퐻𝑜: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇0𝑗)𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇0𝑗)𝑥𝑖 + ⋯ + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + ⋯

… + 𝜌1(𝑥𝑖 ∗ 휃1𝑖) + ⋯ + 𝜌5(𝑥𝑖 ∗ 휃5𝑖) + 휖𝑖𝑗

If the null hypothesis is rejected, it is an indication that interaction terms between race course and the physiological variable 𝑥 are important to include in a model predicting race performance. In other words, it suggests that the estimated effect of 𝑥 on performance changes significantly depending on the race course. For these likelihood ratio tests, the alpha level was set at 0.05.

Set 2: Investigating the REM Sleep and Total Sleep Relationship After the hypotheses of Set 1 were investigated, it was concluded that two variables, total sleep duration and REM sleep duration, had significant associations with race performance. More details about these conclusions can be seen in the results section and the discussion section. This next set of hypotheses investigates the relationship between REM sleep duration, total sleep duration, and race performance using multivariable random effects models. This analysis was unplanned but due to the statistical significance of both variables and the unclear direct effects of REM sleep on athletic performance, it seemed interesting to explore these variables further.

The correlations and covariances between REM sleep duration and total sleep duration leading up to a race, using the data aggregation methods in Table 3, were first calculated to check for any possible signs of collinearity between the two. Afterwards, multivariable random effects models containing both REM sleep duration and total sleep duration, aggregated using the same method, were examined. In other words, REM sleep duration one night before a race was included with total sleep duration one night before a race, while REM sleep duration exponentially weighted over three nights was included with total sleep duration exponentially weighted over three nights. These models were analyzed to see whether the significant association these variables have with race performance remains when both are included in a model together. The models for race result 𝑦 follow the following design, for the varying intercept model and the varying slope and intercept model respectively: (8)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑠𝑙𝑒𝑒𝑝)𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀 𝑠𝑙𝑒𝑒𝑝)𝑖 + 휖𝑖𝑗

As in random effects models in Set 1, the coefficients are allowed to vary for each subject. For random effects models with varying intercept, the null and alternative hypotheses are:

퐻𝑜: 𝛽1 = 0, 𝛽2 = 0

퐻퐴: 𝛽1 ≠ 0, 𝛽2 ≠ 0

In other words, the null hypothesis is that REM sleep duration and total sleep duration have no significant association with performance when holding the other variable constant, while the alternative hypothesis is that either REM sleep duration or total sleep duration, or both, have a significant association with performance when holding the other variable constant. These hypotheses are investigated for REM sleep duration and total sleep duration for one night before a race, two nights before, weighted two nights before, weighted three nights, and exponentially weighted two nights before. As in Set 1, the sign of

31 the coefficients are also of interest, as they indicate whether a variable is positively or negatively associated with race performance. The alpha level here was set to 0.05. Multivariable models with REM sleep duration and total sleep duration that also included interactions were then analyzed. This model essentially investigates whether the estimated effect of REM sleep duration on race performance changes with more or less total sleep leading up to a race. Both the varying intercept and varying slope and intercept models are similar to (8) but with an interaction term 𝜌 added:

(9)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + 𝛽2𝑥(𝑅𝐸푀)𝑖 + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙)𝑖 ∗ 𝑥(𝑅𝐸푀)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀)𝑖 + ⋯

… + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙)𝑖 ∗ 𝑥(𝑅𝐸푀)𝑖) + 휖𝑖𝑗

While 𝜌 is not easy to interpret for two quantitative variables such as sleep duration, it follows a similar interpretation as in (7); it essentially represents the change in slope for

𝑥𝑡𝑜𝑡푎𝑙 when 𝑥𝑅𝐸푀 is held constant, and vice versa. Generally speaking, if 𝜌 is significantly different from zero, it indicates that the effect of one variable on race performance may change depending on the volume of the other. While the conclusions are detailed later in the results section and the discussion section, it was concluded that some interaction coefficients were significantly different from zero. As in Set 1, a likelihood ratio test is used to get a general idea of whether including these interactions significantly improve prediction for race performance. The null and alternative hypotheses for the varying intercept models are:

퐻𝑜: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + 𝛽2𝑥(𝑅𝐸푀)𝑖 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + 𝛽2𝑥(𝑅𝐸푀)𝑖 + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙)𝑖 ∗ 𝑥(𝑅𝐸푀)𝑖) + 휖𝑖𝑗

In the varying slope and intercept model, the null and alternative hypotheses are:

퐻𝑜: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀)𝑖 + 휖𝑖𝑗

퐻퐴: 𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀)𝑖 + ⋯

… + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙)𝑖 ∗ 𝑥(𝑅𝐸푀)𝑖) + 휖𝑖𝑗

If the null hypothesis is rejected, it is an indication that REM sleep and total sleep interactions may improve prediction for race performance. In other words, it is an indication that the expected effect of one variable on race performance may vary depending on the other variable. The alpha level for these tests was set to 0.05. Because stress and sleep have a complex relationship, and stress can affect sleep quality and quantity (Van Reeth et al., 2000; Âkerstedt 2006), an additional stress variable was then added to random effects models with REM sleep and total sleep duration to see if the significant associations between REM sleep, total sleep, and race performance is affected when essentially controlling for stress. The stress variable was taken from the survey data collected on the subjects weekly, as there was an 86.1% response rate as shown in Table 2. The stress responses for rating stress during the previous week, originally on a scale of 0-5, were modified slightly: responses of 0 and 1 were given a value of 0, responses of 2 and 3 were given a value of 2.5, and responses of 4 and 5 were given a value of 5. This was done to divide responses into three distinct levels: not very stressed, very stressed, and somewhere in the middle. The original survey contained six levels instead of three due to research indicating surveys with too few response categories do not allow for subjects to maximize their discriminatory power and express an accurate reply (Komorita and Graham, 1965). This multivariable model with stress included follows the same format as in (9), but with the stress variable added. (10)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + 𝛽2𝑥(𝑅𝐸푀)𝑖 + 𝛽3𝑥(𝑠𝑡𝑟𝑒𝑠𝑠)𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀)𝑖 + 𝛽3𝑥(𝑠𝑡𝑟𝑒𝑠𝑠)𝑖 + 휖𝑖𝑗

The null and alternative hypotheses in this question is, for each 𝑥, that its 𝛽 equals zero and that its 𝛽 does not equal to zero, respectively, with the alpha level set to 0.05.

Following the analysis of multivariable models for total sleep, REM sleep, stress, and race performance, additional multivariable models were analyzed with race course indicator variables added to control for race course: (11)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + 𝛽2𝑥(𝑅𝐸푀)𝑖 + 𝛽3𝑥(𝑠𝑡𝑟𝑒𝑠𝑠)𝑖 + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙)𝑖 + (𝛽2 + 휇2𝑗)𝑥(𝑅𝐸푀)𝑖 + 𝛽3𝑥(𝑠𝑡𝑟𝑒𝑠𝑠)𝑖 + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 휖𝑖𝑗

As in (9) the null hypothesis is that the coefficient for each 𝑥 and 휃 is equal to zero, while the alternative hypothesis is that the coefficient for each 𝑥 and 휃 is not equal to zero. As in (10), the alpha level was set to 0.05.

Set 3: Investigating Sleep One Night Before vs. Two Nights Before To address the assertion made in the distance running community that sleep two nights before a competition is more important than sleep one night before a competition (Glover, 1996), multivariable random effects models were examined with sleep variables both one night before a race and two nights before a race. The sleep variables examined were total sleep duration, light sleep duration, slow wave sleep duration, REM sleep duration, and sleep latency. Each model has both the variable for a sleep statistic one night before a race and the variable for two nights before a race. To illustrate clearly, an example model will have the REM sleep duration for a subject one night before a race and the REM sleep duration two nights before a race. This example for both the varying intercept model and the varying slope and intercept model is demonstrated in (12):

(12)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + 휖𝑖𝑗

The duration of each stage of sleep and total sleep duration are obvious variables to use to investigate the idea that sleep two nights before is more important than sleep one night before. Additionally, sleep latency was also included in analysis, as the amount of time it takes to fall asleep is affected by stress (Âkerstedt, 2006) and could provide an interesting insight into an athlete’s race performance. The correlation between each stage of sleep one night before a race and two nights before a race was first calculated, to address any potential collinearity concerns. It was then investigated whether the coefficients 𝛽1 and 𝛽2 in (12) are significantly different from zero, using the following hypothesis test for each of the five sleep variables:

퐻푂: 𝛽1 = 0, 𝛽2 = 0

퐻푂: 𝛽1 ≠ 0, 𝛽2 ≠ 0

If one coefficient is significantly different from zero while another is not, it indicates that the night of sleep linked with the significant coefficient may have a greater expected effect than the other night of sleep; generally speaking, it may suggest that one night of sleep is more important than the other. As in Sets 1 and 2, the sign of the coefficient is also of great interest, as it indicates whether a sleep variable is positively or negatively associated with race performance. The alpha level here was set to 0.05. For sleep variables in multivariable models that had at least one coefficient significantly different from zero, additional models were examined to investigate whether there are significant interactions between sleep one night before and two nights before. This was done by expanding the model in (12) to add an interaction term, for each of the five types of sleep variables: (13)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + ⋯

… + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + ⋯

… + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

As in (7) and (9), 𝜌 contains the same general interpretation in (13). It can be explained as the expected change in the effect of a one-unit increase of one 𝑥 when the other 𝑥 is increased. While this has an abstract interpretation, the significance of this interaction term can be generally thought of as sleep two nights before a race significantly changing the effect of sleep one night before a race on expected performance. In other words, a significant interaction term may suggest that more sleep two nights before a race may change how sleep one night before a race is associated with race performance. As in Set 2, multivariable models were examined that included the stress variable from survey data. As sleep can be affected by stress (Van Reeth et al., 2000; Âkerstedt 2006), adding this variable can essentially help control for certain patterns of sleep. The random effects model from (13), for each of the five sleep variables, was expanded to:

(14)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + ⋯

… + 𝛽3𝑥𝑠𝑡𝑟𝑒𝑠𝑠 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡) + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠) + ⋯

… + 𝛽3𝑥𝑠𝑡𝑟𝑒𝑠𝑠 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

As in random effects models in Set 2, the null hypothesis was that, for each 𝑥, its coefficient is equal to zero, while the alternative hypothesis is that its coefficient is not equal to zero. To control for race course, race course indicator variables were also added, in an expansion of the random effects model in (13): (15)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡) + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠) + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡) ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)) + 휖𝑖𝑗

The null and alternative hypotheses were the same as in (14), with the alpha level set to 0.05. These models were examined to see whether the conclusions on associations between sleep one night before a race and two nights before a race were consistent once race course controls were in place.

Set 4: Sleep and Heart Rate Variability As discussed in the introduction, previous research has suggested that sleep and heart rate variability are two major physiological variables that could be used to predict better or worse performance in athletes, especially endurance runners, as both play a role in the recovery of an athlete. Both HRV and sleep are analyzed in multivariable random effects models to investigate any potential associations with race performance with both variables included. While more is mentioned in the results and discussion section, total sleep duration, REM sleep duration, and sleep latency were included in these models due to their significance in Sets 1-3. In every model, the sleep variable and rMSSD HRV are aggregated using the same method. To provide some examples: total sleep two nights before a race is only included in a model with rMSSD HRV two nights before a race, and sleep latency weighted two nights before a race is only included in a model with rMSSD HRV one night before a race. An example of such a model would be as follows in (16):

(16)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 휖𝑖𝑗

The correlation between each stage of sleep one night before a race and two nights before a race was first calculated, to address any potential collinearity concerns. Similarly to the design in Set 3, it was then investigated whether the coefficients 𝛽1 and 𝛽2 in (16) are significantly different from zero, using the following hypothesis test for each of the five sleep variables:

퐻푂: 𝛽1 = 0, 𝛽2 = 0

퐻푂: 𝛽1 ≠ 0, 𝛽2 ≠ 0

A coefficient significantly different from zero indicates that on a specific night before a race, or when aggregated in some form in the nights leading up to a race, the variable’s measurement may have a significant association with race performance when the other variable is factored into the model and held constant. In simpler terms, significant coefficients can indicate whether sleep, HRV, or both may be used to predict better or worse race performance. The alpha level here was set to 0.05. An expansion of (16), with interaction terms, was then analyzed to investigate whether HRV and sleep leading up to a race have any significant interactions: (17)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + ⋯

… + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + ⋯

… + 𝜌(𝑥(𝑡𝑜𝑡푎𝑙 𝑠𝑙𝑒𝑒𝑝 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑟푀𝑆𝑆𝐷 퐻𝑅푉 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖) + 휖𝑖𝑗

Though the interaction term 𝜌 has an abstract interpretation, its general idea follows the same interpretation as the interaction term in (7), (9), and (13). Its sign and significance may indicate if one variable can change the other variable’s association with race performance. In general terms, a significant interaction term may suggest, for example, that a lower HRV, and more sympathetic dominance, may change the expected effect of more sleep in the nights leading up to a race on expected race performance. As in similar random effects models in Set 2 and 3, the null hypothesis was that, for each 𝑥 and the interaction, its coefficient is equal to zero, while the alternative hypothesis is that its coefficient is not equal to zero. The alpha level was set to 0.05. The stress variable from survey data was then added to help control for stress in these multivariable random effects models, as research has shown that stress can affect

38 both HRV and sleep (TaskForce, 1996; Van Reeth et al., 2000; Âkerstedt 2006). This can be modeled by expanding on (16): (18)

… + 𝛽3𝑥(𝑠𝑡𝑟𝑒𝑠𝑠)𝑖 + 휖𝑖𝑗

As explained later in the results and discussion section, no interaction terms were significant, and therefore they were left out of the models in (18). As in (17), the null hypothesis was that, for each 𝑥, its coefficient is equal to zero, while the alternative hypothesis is that its coefficient is not equal to zero. This essentially analyzes whether sleep, HRV, and stress are significantly associated with race performance when all are included in a model together. The alpha level here was set to 0.05. To control for race course, race course indicator variables were also added, in an expansion of the random effects model in (16): (19)

𝑦𝑖 = 𝛽0 + 휇0𝑗 + 𝛽1𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖 + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡)𝑖 ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)𝑖) + 휖𝑖𝑗

𝑦𝑖 = 𝛽0 + 휇0𝑗 + (𝛽1 + 휇1𝑗)𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡) + 𝛽2𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠) + ⋯

… + 𝛾1휃1𝑖 + ⋯ + 𝛾5휃5𝑖 + 𝜌(𝑥(𝑅𝐸푀 𝑜𝑛𝑒 𝑛𝑖𝑔ℎ𝑡) ∗ 𝑥(𝑅𝐸푀 𝑡𝑤𝑜 𝑛𝑖𝑔ℎ𝑡𝑠)) + 휖𝑖𝑗

These models were examined to see whether the conclusions on associations between rMSSD HRV and sleep before a race were consistent once race course controls were included. The alpha level here was set to 0.05.

Standardized Coefficients Because the units of predictors in these sets of hypotheses range from beats per minute to minutes to seconds, interpretation of coefficients may be difficult to understand for those without the scientific background and expertise of these physiological variables. To address this issue, each variable 𝑥 in the data was standardized:

𝑥 − 𝑥̅ 𝑥∗ = 𝑖 𝑖 𝑠

Here 𝑥̅ is the mean of the variable across all subjects, and 𝑠 is the standard deviation of the variable across all subjects. Note that within-subject means and standard deviations are not used. In doing this calculation, any coefficients in random effects models in this study will now be expressed in the unit of standard deviations, which allows for a general and clear understanding. For example, with standardized data, a 𝛽1 associated with the variable 𝑥 that has a value of 0.31 indicates that for every increase in 𝑥 by one standard deviation above the mean of 𝑥, the expected race performance increases by 0.31 standard deviations from the mean race performance. To convert the standardized coefficient into the original units, the standardized coefficient is multiplied by the standard deviation of race performance (the dependent variable) across all subjects divided by the standard deviation of 𝑥 across all subjects:

𝑠𝑦 𝛽1 𝑜𝑟𝑖𝑔𝑖𝑛푎𝑙 𝑢𝑛𝑖𝑡𝑠 = 𝛽1 𝑠𝑡푎𝑛𝑑푎𝑟𝑑𝑖𝑧𝑒𝑑 ∗ 𝑠𝑥

These standardized coefficients are used in all random effects results presented in this study, and the converted coefficient in the original units is also provided with some results.

Results

Summary Statistics: Sleep Stage Quantity Before investigating the sets of hypotheses, summary statistics for all nights of sleep during the study were calculated. Table 5 contains the mean duration of each sleep stage for all nights across all subjects. The table also compares the percent duration of each stage to the expected percent duration for the average individual. Based on these statistics, the average sleep efficiency across all subjects was 85.74%. Tables A1 and A2 in the appendix contain expanded summary statistics for a night’s sleep during the study. Each sleep variable, resting heart rate, rMSSD HRV, REM cycle counts, and total time in bed were included. Table A1 contains summary statistics across all subjects while Table A2 contains summary statistics within each subject.

Table 5: Average percentage of time in each stage of sleep across all subjects.

Sleep Stage Mean Duration Percent Duration of Expected Percent Duration Total Sleep

Light Sleep 316.64 min 67.9% 50%

Slow Wave Sleep 77.07 min 16.5% 15-23%

REM Sleep 72.28 min 15.8% 20-25%

Total Sleep Duration 465.69 min 100% 100%

Source for expected percent duration: (Sheldon et al, 2014).

Across all subjects, the average night of sleep during the study was approximately 7 hours and 45 minutes, with a standard deviation of 1 hour and 11 minutes. Within subjects, the average night of sleep ranged from approximately 7 hours to 8 hours and 26 minutes. Across all subjects the average total time spent in bed each night was approximately 9 hours and 3 minutes, with a standard deviation of 1 hour and 18 minutes. The within- subjects average for total time in bed each night ranged from approximately 8 hours and 37 minutes to 10 hours and 1 minute. Across all subjects, the average resting heart rate for all nights of sleep was 46.67 beats per minute with a standard deviation of 5.71 beats per minute. The average resting heart rate for individuals ranged from 42.07 beats per minute

41 to 57.42 beats per minute. Table A3 in the appendix contains between-subjects summary statistics for every variable in every method of data aggregation leading up to a race. For example, the first section of the Table A3 contains the mean and standard deviation of rMSSD heart rate variability for one night before a race, two nights before a race, weighted two nights before a race, etc. With regards to race performance, the average race pace across all subjects was 194.68 seconds per kilometer (s/km), or roughly 3 minutes and 14 seconds, with a standard deviation of 2.83 s/km. Table A4 in the appendix contains the mean race pace and standard deviation for each subject. Figure 1 below examines the distribution of race paces across all subjects.

Figure 1: Distribution of race performance across all subjects.

Hypotheses Set 1 Important Results Table 6 and Table 7 contain notable results from models for race performance with individual physiological variables. As explained in the methods section, the converted coefficient is calculated by multiplying the standardized coefficient by the ratio of the standard deviation of race pace to the standard deviation of the predictor variable.

Table 6: Results for significant individual predictor variables. See Appendix B1- B18 for complete results for all predictor variables.

Variable Standardized Converted Standard t-value p-value Coefficient Coefficient Error Varying Intercept Models Total Sleep Weighted Two Nights -0.5322 -0.0289 0.1694 -3.141 0.00369

Total Sleep Weighted Three Nights -0.5799 -0.0316 0.1798 -3.266 0.00292

REM Sleep Two Nights Before -0.5661 -0.0278 0.1426 -3.970 0.00048

REM Sleep Weighted Two Nights -0.5076 -0.0232 0.1540 -3.297 0.00242

REM Sleep Weighted Three Nights -0.5009 -0.0236 0.1577 -3.176 0.00323

REM Sleep Weighted Four Nights -0.4941 -0.0242 0.1528 -3.234 0.00272

REM Sleep Exponentially Weighted Two Nights -0.4583 -0.0211 0.1510 -3.036 0.00479

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.6892 -0.0338 0.1711 -4.029 0.00401

These coefficients for these predictors rejected the null hypothesis at the alpha level of p < 0.00556

In both varying intercept models and varying slope and intercept models, the standardized coefficient for REM sleep duration two nights before a race was negative and significantly different from zero at the alpha level of 0.00556. When converting the standardized coefficient back to original units, the converted coefficient for REM sleep duration two nights before was -0.0278 in the varying intercept model and -0.0338 in the varying slope and intercept model. This suggests that for these subjects, an increase in REM sleep duration two nights before a race is associated with a faster race pace by 0.0278 and 0.0338 seconds per kilometer with all other factors held constant. REM sleep duration weighted over two, three, and four nights and exponentially weighted over two nights were also significantly associated with better race performance in the varying intercept model, while total sleep duration weighted two and three nights before a race is significantly associated with better race performance in the varying intercept model. Table 7 notes the results of certain physiological variables that did not have a standardized coefficient significantly different from zero at the alpha level of 0.00556.

Table 7: Notable results for other individual predictor variables. See Appendix B1-B18 for complete results.

Variable Standardized Standard t-value p-value Coefficient Error Varying Intercept Models rMSSD Heart Rate Variability One Night Before 0.1827 0.1924 0.948 0.351

rMSSD Heart Rate Variability Two Nights Before 0.1443 0.2366 0.61 0.554

Resting Heart Rate One Night Before -0.0101 0.2322 -0.046 0.964

Resting Heart Rate Two Nights Before 0.0502 0.2317 0.217 0.831

Total Sleep Duration One Night Before -0.4560 0.1806 -2.525 0.0181

Total Sleep Duration Two Nights Before -0.4770 0.1739 -2.743 0.0107

Slow Wave Sleep Duration Weighted Two Nights 0.0417 0.1758 0.237 0.814

Slow Wave Sleep Duration Weighted Three Nights 0.0333 0.1753 0.190 0.851

Sleep Latency One Night Before -0.2319 0.1824 -1.272 0.216

Sleep Latency Two Nights Before 0.4327 0.1979 2.187 0.0375

Varying Slope and Intercept Models rMSSD Heart Rate Variability One Night Before 0.0978 0.2127 0.460 0.659

rMSSD Heart Rate Variability Two Nights Before -0.0288 0.3093 -0.093 0.929

Resting Heart Rate One Night Before -0.0149 0.2335 -0.064 0.95

Resting Heart Rate Two Nights Before 0.1686 0.2405 0.701 0.501

Total Sleep Duration One Night Before -0.4082 0.2140 -1.907 0.101

Total Sleep Duration Two Nights Before -0.4815 0.2204 -2.184 0.0628

Slow Wave Sleep Duration Weighted Two Nights 0.0843 0.2170 0.388 0.716

Slow Wave Sleep Duration Weighted Three Nights 0.1059 0.2537 0.418 0.695

Sleep Latency One Night Before -0.2285 0.1794 -1.273 0.215

Sleep Latency Two Nights Before 0.4599 0.1932 2.380 0.0256

Tables B1-B18 in the appendix contain complete results for all single-variable random effects models.

Table 8 displays results for multivariable random effects models containing the significant predictor variables from Table 6 and additional race course indicator variables. Table 8 only shows the results for the physiological coefficients; complete results with race course coefficients included can be found in Table B19 in the appendix. At the alpha level of 0.05, the standardized coefficient for REM sleep duration two nights before a race remained negative and was significantly different from zero in both the varying intercept and varying slope and intercept model, though its magnitude decreased slightly. The other physiological variables from Table 6 no longer had a statistically significant standardized coefficient when race course indicator variables were added.

Table 8: Results for physiological predictor variables in multivariable random effects models with race course controls. These physiological variables from Table 6 had significant coefficients, and race course indicator variables were added to the models with those variables.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models REM Sleep Two Nights Before -0.3117 0.1291 -2.415 0.0265

REM Sleep Weighted Two Nights -0.2619 0.1357 -1.930 0.0665

REM Sleep Weighted Three Nights -0.2174 0.1337 -1.626 0.1172

REM Sleep Weighted Four Nights -0.2025 0.1327 -1.526 0.1401

REM Sleep Exponentially Weighted Two Nights -0.2262 0.1108 2.041 0.0543

Total Sleep Weighted Two Nights -0.0102 0.1665 -0.062 0.9534

Total Sleep Weighted Three Nights -0.0088 0.1734 0.051 0.9598

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.3365 0.1336 -2.519 0.0301

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p<0.05. See Appendix B19 for coefficients for race course control indicator variables.

Table 9 contains results for likelihood ratio tests comparing the models from Table 6 to the models from Table 8. In other words, Table 9 examines whether the addition of race course indicator variables seems to significantly improve prediction for race performance for the physiological variables with significant standardized coefficients.

Table 9: Likelihood ratio test results for race course controls. Each variable listed below was a significant physiological predictor variable from Table 6. For each predictor variable, an expanded model was examined with race course indicator variables and the coefficients are presented in Table 8. This table contains the results of a likelihood ratio test comparing, for each variable, the smaller model without race course indicator variables to the larger model with race course variables to see which model is a better fit.

Variable Chi-Squared value Pr(>Chisq) Varying Intercept Models REM Sleep Two Nights Before 36.09 0.000002648

REM Sleep Weighted Two Nights 35.62 0.000003266

REM Sleep Weighted Three Nights 37.66 0.000001312

REM Sleep Weighted Four Nights 36.64 0.000002071

REM Sleep Exponentially Weighted Two Nights 37.37 0.000001489

Total Sleep Weighted Two Nights 31.94 0.00001678

Total Sleep Weighted Three Nights 34.65 0.000005043

Varying Slope and Intercept Models REM Sleep Two Nights Before 34.92 0.000004461

Bolded ANOVA tests are tests that reject the null hypothesis at Pr(>Chisq) < 0.05 See Appendix B20 for complete likelihood ratio test results with additional statistics.

For all models in Tables 6 and 8, the larger model that had additional race course indicator variables had a significantly better fit than the smaller model. Additional random effects models were examined that included interaction terms between REM sleep duration two nights before a race and each race course indicator variable. Both the varying intercept model and the varying slope and intercept model were analyzed. The summary of results for these models can be found in Table 10. Note that Table 10 only shows the results for the physiological coefficients; complete results with race course coefficients and the interaction terms can be found in Table B21 in the appendix. The coefficient for REM sleep duration two nights before a race remained negative and significant zero at the alpha level of 0.05 in the varying intercept model, but not in the varying slope and intercept model. The coefficients also increased in magnitude compared to the coefficients in models with no race course interaction terms.

A likelihood ratio test was then utilized to address whether the addition of the interaction term provides a significantly better fit to the model predicting race performance with REM sleep duration two nights before a race. The results of the likelihood ratio test are shown in Table 11, and indicate that the model with interaction terms does not have a significantly better fit than the model without interaction terms.

Table 10: Results for random effects models including REM sleep duration two nights before a race with race course interactions included.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Model REM Sleep Two Nights Before -0.4043 0.1642 2.463 0.0298

Varying Slope and Intercept Model REM Sleep Two Nights Before -0.4914 0.2326 -2.113 0.0897

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p<0.05 See Appendix B21 for complete results including race course coefficients and interaction coefficients.

Table 11: Likelihood ratio test results for the interaction between race course and REM sleep duration two nights before. This table contains the results of a likelihood ratio test comparing, for both the varying intercept and varying slope and intercept models, the smaller model for REM sleep two nights before without race course interactions to the larger model with race course interactions to see which model is a better fit.

Variable Chi-Squared value Pr(>Chisq) Varying Intercept Models REM Sleep Two Nights Before 6.86 0.2311

Varying Slope and Intercept Models REM Sleep Two Nights Before 9.09 0.1055

See Appendix B22 for complete likelihood ratio test results with additional statistics.

Figure 2, Figure 3, and Figure 4 address regression diagnostics for the two models from Table 8 that have coefficients for REM sleep duration two nights before as well as race course indicator variables. Figure 2 displays the distribution of residuals for the varying intercept model and also utilizes a residual plot comparing the fitted values and residuals.

Figure 3 displays the distribution of residuals and the residual plot for the varying slope and intercept model. Figure 4 is a plot illustrating the association between REM sleep two nights before a race and performance across all subjects.

Figure 2: Diagnostics for varying intercept models with REM sleep duration two nights before and race course control variables. Residual Histogram Fitted Vs. Residuals

0.4

0.0

count

Residuals 2

-0.4

-0.5 0.0 0.5 -1 0 1 2 Residuals Fitted Figure 3: Diagnostics for varying slope and intercept models with REM sleep duration two nights before and race course control variables. Residual Histogram Fitted Vs. Residuals

7.5 0.4

5.0 0.0

count

Residuals

2.5

-0.4

0.0

-1.0 -0.5 0.0 0.5 1.0 -1 0 1 2 Residuals Fitted

Figure 4: The association between REM sleep two nights before and performance across all subjects.

To address the question of whether the varying intercept model or the varying slope or intercept model is a better fit for modeling performance based on REM sleep two nights before, a likelihood ratio test is analyzed. The complete results for the test is shown in Table 11. Figure 5 then illustrates the association between REM sleep two nights before a race and performance for three individual subjects, with a line of best plotted for each subject.

Table 11: Likelihood ratio test results comparing the REM sleep duration two nights before varying intercept model and the varying slope and intercept model.

Variable Df Deviance AIC BIC Chi-Squared Chi Df Pr(>Chisq) value Varying Intercept Model REM Sleep Two Nights Before 10 46.825 66.825 81.482

Varying Slope and Intercept Model REM Sleep Two Nights Before 12 46.801 70.801 88.390 0.0231 2 0.989

Figure 5: Example of different slopes and intercepts between subjects.

Hypotheses Set 2 Important Results Before examining multivariable random effects models including both REM sleep and total sleep duration leading up to a race, the correlation between REM sleep and total sleep was examined in Table 12, while their general association across all nights for all subjects is shown in Figure 6. From the data, it appears that REM sleep and total sleep duration in the same period leading up to a race do not have a very high correlation.

Table 12: The correlation between REM and total sleep duration in the nights leading up to a race across all subjects. See Appendix C1 for correlations within subjects.

Variables Correlation Covariance

REM and Total Sleep One Night Before 0.310 0.318

REM and Total Sleep Two Nights Before 0.193 0.198

REM and Total Sleep Weighted Two Nights 0.198 0.202

REM and Total Sleep Weighted Three Nights 0.137 0.140

REM and Total Sleep Expo. Weighted Two Nights 0.229 0.234

Figure 6: The general association between REM sleep and total sleep across all nights for all subjects.

Table 13 displays notable coefficients from both varying intercept and varying slope and intercept containing total sleep duration and REM sleep duration leading up to a race. All models utilized predictors with data aggregated in the same method; for example, REM sleep duration exponentially weighted two nights was paired with total sleep duration exponentially weighted two nights. Total and REM sleep duration one night before a race, two nights before a race, weighted two nights, weighted exponentially two nights, and weighted three nights were examined. Ten models were analyzed in total. Table 13 only displays the coefficients that were significantly different from zero at the alpha level of 0.05, with those coefficients also converted to the original units. The coefficients for REM sleep duration two nights before, weighted two nights before, weighted exponentially two nights before and weighted three nights before a race were significantly different from zero for both varying intercept and varying slope and intercept models, while the coefficients for total sleep duration weighted two and three nights before a race were significantly different from zero. The REM sleep duration coefficient values had similar magnitudes to the REM sleep coefficients in Set 1. Table C2 in the appendix contains complete results and coefficients for each of these models. All coefficients for a sleep variable the night before a race were not significantly different from zero. Interaction terms were added to the ten random effects models in Table C2. The notable results for those models is displayed in Table 14, while the remaining results can

51 be found in Table C3 in the appendix. All REM sleep duration coefficients from Table 13 remained significantly different from zero at the alpha level of 0.05 when interaction terms were added to their models. The interaction term between REM sleep and total sleep duration exponentially weighted two nights was significantly different from zero in both the varying intercept model and the varying slope and intercept model. The coefficients for total sleep weighted two nights and three nights were no longer significantly different from zero once interaction terms were added. For both varying intercept and varying slope and intercept models, a likelihood ratio test was utilized to address whether the addition of interaction terms significantly improved the fit of models for REM and total sleep duration two nights before and exponentially weighted two nights before a race. The results of these tests can be seen in Table 15.

Table 13: Significant results for significant physiological variables in multivariable models with both REM and total sleep duration. See Appendix C2 for complete results.

Variable Standardized Converted Standard t-value p-value Coefficient Coefficient Error Varying Slope Models REM Sleep Two Nights Before -0.4645 -0.0228 0.1461 -3.180 0.00378

REM Sleep Weighted Two Nights -0.4028 -0.0184 0.1515 -2.660 0.0124

Total Sleep Weighted Two Nights -0.4082 -0.0222 0.1638 -2.492 0.0186

REM Sleep Weighted Three Nights -0.4004 -0.0188 0.1540 -2.599 0.014

Total Sleep Weighted Three Nights -0.4542 -0.0247 0.1743 -2.606 0.014

REM Sleep Exponentially Weighted Two Nights -0.3566 -0.0164 0.1647 -2.166 0.0382

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.6070 -0.0298 0.1650 -3.678 0.00817

REM Sleep Weighted Two Nights -0.4753 -0.0218 0.1764 -2.694 0.0474

REM Sleep Weighted Three Nights -0.4836 -0.0227 0.1805 -2.678 0.0497

REM Sleep Exponentially Weighted Two Nights -0.4413 -0.0203 0.1700 -2.595 0.0403

All coefficients are results from models with physiological variables rejecting the null hypothesis at the alpha level of p<0.05

Table 14: Important results for REM sleep and total sleep interactions. See Appendix C3 for complete results including all coefficients and interaction terms.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Model REM Sleep Two Nights Before -0.6026 0.1671 -3.606 0.00143

REM Sleep Weighted Two Nights -0.6100 0.1993 -3.063 0.00461

REM Sleep Weighted Three Nights -0.5821 0.1972 -2.952 0.00605

REM Sleep Expo. Weighted Two Nights -0.5501 0.1791 -3.072 0.0045

REM Sleep*Total Sleep Expo. Weighted Two Nights 0.4059 0.1876 2.164 0.0394

Varying Slope and Intercept Model REM Sleep Two Nights Before -0.5706 0.1547 -3.689 0.00768

REM Sleep Weighted Two Nights -0.6502 0.2212 -2.939 0.00674

REM Sleep Expo. Weighted Two Nights -0.4780 0.1646 -2.917 0.00984

REM Sleep*Total Sleep Expo. Weighted Two Nights 0.3687 0.1782 2.069 0.04903

All coefficients are results from models with physiological variables rejecting the null hypothesis at the alpha level of p<0.05

Table 15: Results for likelihood ratio tests for interactions between REM and total sleep duration two nights before and exponentially weighted two nights.

Variable Chi Degrees of Chi-Squared value Pr(>Chisq) Freedom Varying Intercept Models REM*Total Sleep Two Nights Before 1 3.490 0.0619

REM*Total Sleep Expo. Weighted Two Nights 1 4.891 0.027

Varying Slope and Intercept Models REM*Total Sleep Two Nights Before 1 4.754 0.02924

REM*Total Sleep Expo. Weighted Two Nights 1 4.061 0.04389

Bolded ANOVA tests are tests that reject the null hypothesis at Pr(>Chisq) < 0.05

In the varying intercept model for REM sleep and total sleep duration two nights before a race, the model with REM and total sleep interaction did not significantly improve model fit, while the addition of the interaction term for the varying slope and intercept

53 model significantly improved model fit. For both varying intercept and varying slope and intercept models containing REM and total sleep duration exponentially weighted over two nights, the addition of the interaction term significantly improved model fit. Multivariable random effects models with REM sleep and total sleep duration including the stress variable from survey data, without interactions, were then analyzed. The significant coefficients after the addition of the stress variable are highlighted in Table 16, with complete results in Table C4 in the appendix. Due to incomplete survey data for some subjects leading up to certain races, some models did not converge and could not be analyzed. Most coefficients were not significantly different from zero at the alpha level of 0.05, but the coefficient for REM sleep duration two nights before a race was significantly different from zero in both the varying intercept and varying slope and intercept models.

Table 16: Significant results for multivariable random effects models for REM sleep and total sleep duration. See Appendix C4 for complete results including stress coefficients.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Model REM Sleep Two Nights Before -0.5165 0.2088 -2.474 0.032

Varying Slope and Intercept Model REM Sleep Two Nights Before -1.0933 0.3013 -3.628 0.0255

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p<0.05

The varying intercept model for REM sleep and total sleep two nights before a race with stress included was then expanded to include race course variables. The complete results of this model are shown in Table 17. The coefficient for total sleep two nights before a race was significantly different from zero in this model at the alpha level of 0.05. It was also noted that some race course coefficients could not be calculated due to not enough data leading up to those races for some subjects on those race courses. Figure 7 helps evaluate regression diagnostics for this model by displaying the distribution of residuals and the residual plot against the fitted values for race performance. The varying slope and

54 intercept model could not be analyzed, as too much missing survey data before certain races for a few subjects caused the model to fail to converge.

Table 17: Results for REM and total sleep duration with stress and race course controls included in the varying intercept model.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Model REM Sleep Two Nights Before 0.1301 0.1801 0.723 0.4849

Total Sleep Two Nights Before -0.3717 0.1657 -2.243 0.0464

Stress During Week 0.0367 0.1001 0.367 0.7205

Franklin Park 10K 0.0175 0.6143 0.029 0.9778

Van Cortlandt Park HEPS 8K 1.1954 0.7508 1.592 0.1396

Van Cortlandt Park ICAAAA 8K 1.3882 0.6617 2.098 0.0598

Wisconsin 8K -0.636 0.6284 -1.013 0.3330

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p<0.05

Figure 7: Diagnostics for varying slope and intercept models with REM sleep two nights before and race course control variables.

Residual Histogram Fitted Vs. Residuals

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Hypotheses Set 3 Important Results Before examining models containing sleep variables both one night and two nights before a race, the correlations between each sleep variable one night and two nights before were examined both across subjects and within subjects. The correlation for each sleep variable across all subjects is displayed in Table 18, while the correlation for each sleep variable within subjects can be found in Table D1 in the appendix. REM sleep duration one night before a race seemed to have a fairly strong positive correlation with REM sleep duration two nights before a race, while the rest of the sleep variables did not have a very strong correlation. Sleep latency one night before and two nights before had almost no correlation across all subjects. However, when examining each individual subject, it was noted that the correlation between total sleep duration the night before and two nights before was very strong and positive (>0.9) for a few subjects, while total sleep for other subjects showed almost no correlation. Additionally, slow wave sleep duration and sleep latency one night and two nights before also had very different correlations within subjects; some subjects had a very strong and positive correlation for both variables, while others had a very strong and negative correlation.

Table 18: Correlation between sleep one night before and two nights before across all subjects.

Variables Correlation Covariance

Total Sleep One Night and Two Nights Before 0.291 0.303

Light Sleep One Night and Two Nights Before 0.512 0.544

REM Sleep One Night and Two Nights Before 0.728 0.786

Slow Wave Sleep One Night and Two Nights Before 0.473 0.427

Sleep Latency One Night and Two Nights Before -0.098 -0.104

See Appendix D1 for correlations within subjects.

Important results for both varying intercept and varying slope and intercept models are displayed in Table 19, with complete results displayed in Table D2 in the appendix. Table 19 notes models containing REM sleep, total sleep, and sleep latency both one and

56 two nights before a race. For both the varying intercept and varying slope and intercept model, the coefficient for both REM sleep duration and sleep latency two nights before a race was significantly different from zero at the alpha level of 0.05. For REM sleep duration two nights before a race, the coefficient converted to original units was -0.0256 and - 0.0287 for the varying intercept and varying slope and intercept model respectively. Both were very similar to the converted coefficients for REM sleep two nights before a race in Table 6. The coefficient for sleep latency two nights before a race was positive and significant; the converted coefficients were 0.0908 and 0.1011 in the varying intercept model and the varying slope and intercept model respectively. This suggests that an additional minute of sleep latency two nights before a race is significantly associated with an increase in one’s expected race pace by roughly 0.1 s/km when all other variables are held constant. In addition, the coefficient for total sleep two nights before a race was negative and significant at the alpha level of 0.05 for the varying intercept model, but not for the varying slope and intercept model. In all models, the coefficients for one night before the race was not significantly different from zero.

Table 19: Important results for multivariable random effects models examining sleep one night before versus two nights before. See Appendix D2 for complete results for all sleep variables.

Variable Standardized Converted Standard Error t-value p-value Coefficient Coefficient Varying Intercept Models REM Sleep One Night Before 0.0424 0.0016 0.2632 0.161 0.8736

REM Sleep Two Nights Before -0.5169 -0.0256 0.2314 -2.234 0.0372

Total Sleep One Night Before -0.2510 -0.0125 0.2022 -1.241 0.2299

Total Sleep Two Nights Before -0.4406 -0.0221 0.1925 -2.289 0.0335

Sleep Latency One Night Before -0.1864 -0.0076 0.1720 -1.084 0.2910

Sleep Latency Two Nights Before 0.4445 0.0908 0.2041 2.178 0.0412

Varying Slope and Intercept Models REM Sleep One Night Before 0.0505 0.0019 0.2669 0.189 0.8521

REM Sleep Two Nights Before -0.5794 0.0287 0.2405 -2.409 0.0314

Sleep Latency One Night Before -0.1687 -0.0684 0.1724 -0.978 0.3437

Sleep Latency Two Nights Before 0.4946 0.1011 0.1984 2.493 0.0214

Bolded coefficients displayed here are results for physiological variables rejecting the null hypothesis at the alpha level of p<0.05

Table 20: Important results for multivariable random effects models for REM sleep duration, total sleep duration, and sleep latency interactions for one night before and two nights before a race.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Model Total Sleep Two Nights Before -0.4673 0.1877 -2.489 0.0227

Sleep Latency Two Nights Before 0.5342 0.2489 2.146 0.0439

Varying Slope and Intercept Model Total Sleep Two Nights Before -0.5909 0.2458 -2.403 0.0447

Sleep Latency Two Nights Before 0.5548 0.2284 2.429 0.0261

See Appendix D3 for complete results including all coefficients and interaction coefficients

Table 20 displays notable results for expanded sleep latency, REM sleep duration, and total sleep duration models by adding interaction terms for sleep one and two nights before a race. For both the varying intercept model and varying slope and intercept model, the coefficients for total sleep duration two nights before and sleep latency two nights before were significantly different from zero at the alpha level of 0.05, with coefficient magnitude not changing too much. The coefficients for REM sleep duration two nights before, in both the varying intercept and varying slope and intercept models, were not significantly different from zero. In all models, the coefficient for the sleep variable one night before the race was not significantly different from zero. In addition, no interaction terms were significantly different from zero. Table D3 in the appendix contains complete results for all models and coefficients. Figure 8 illustrates the association between total sleep two nights before a race and race performance. The left image is a plot across all subjects, while the right image is a plot for five of the subjects with lines of best fit. Four subjects were not included in the right side of the plot, as they only raced three times or less. Table 21 displays results for multivariable varying intercept models for REM sleep, total sleep, and sleep latency one and two nights before a race, but with the addition of the stress variable from survey data. Only the coefficient for sleep latency two nights before a

58 race was significantly different from zero at the alpha level of 0.05; the coefficient was positive and also had a slightly higher magnitude than coefficients from previous models.

Figure 8: The association between total sleep duration two nights before and race performance.

Table 21: Results for varying intercept models for race performance with sleep latency, total sleep, and REM sleep duration one and two nights before a race, with the inclusion of a stress variable.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.3285 0.4460 -0.737 0.492

REM Sleep Two Nights Before -0.2451 0.3794 -0.646 0.542

Stress During Week 0.1490 0.1223 1.218 0.276

Model 2

Total Sleep One Night Before -0.0381 0.3249 -0.117 0.909

Total Sleep Two Nights Before -0.4780 0.3670 -1.307 0.221

Stress During Week 0.1809 0.1656 1.092 0.306

Model 3

Sleep Latency One Night Before -0.2745 0.1821 -1.507 0.2089

Sleep Latency Two Nights Before 0.5820 0.1772 3.284 0.0257

Stress During Week 0.1659 0.1191 1.393 0.2263

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05 Note: varying slope and intercept models did not converge due to not enough stress data for certain races.

All other coefficients were not significantly different from zero. Because there was not enough stress data, the varying slope and intercept model did not converge and could not be analyzed. Table 22 and Table 23 display notable results for extended models from Table 21 that added race course indicator variables instead of the stress variable. Table 22 shows the notable coefficients for these models, while Table 23 displays notable race course coefficients in both the varying intercept and varying slope and intercept models for race performance based on sleep latency (see Models 3 and 6 in Table D4). The race course coefficients listed were significantly different from zero at the alpha level of 0.05, and were the coefficients for the Franklin Park 5K, the Franklin Park 10K, and the two Van Cortlandt Park races. Complete results with coefficients can be found in Table D4 in the appendix. Of note, the coefficients for REM sleep duration two nights before had a negative and significant coefficient at the alpha level of 0.05 in both the varying intercept and varying slope and intercept models, while the coefficient for sleep latency one night before was positive and significant.

Table 22: Significant results for REM sleep, total sleep and sleep latency one and two nights before a race with race course indicator variables added.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models REM Sleep Two Nights Before -0.4518 0.1179 -3.831 0.00282

Sleep Latency One Night Before 0.3409 0.1059 3.218 0.008412

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.4478 0.1190 -3.764 0.00498

Sleep Latency One Night Before 0.3765 0.1048 3.592 0.005589

All coefficients represent results that reject the null hypothesis at the alpha level of p<0.05 See Appendix D4 for all results including race course coefficients.

Figure 9 illustrates the association between sleep latency one night before a race and race performance, and Figure 10 illustrates the association between sleep latency two nights before a race and race performance. Like Figure 8, the left image is a plot across all

60 subjects, while the right image is a plot for five of the subjects with lines of best fit. Four of the subjects only race three times or less, so they were not included in the plot.

Table 23: Significant results for REM sleep, total sleep and sleep latency one and two nights before a race with race course indicator variables added.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Franklin Park 10K 0.6976 0.2972 2.347 0.040467

Franklin Park 5K -1.4650 0.5454 -2.686 0.0197

Van Cortlandt Park HEPS 8K 1.7107 0.3117 5.488 0.000243

Van Cortlandt Park ICAAAA 8K 1.5695 0.3719 4.220 0.001381

Varying Slope and Intercept Models Franklin Park 10K 0.6690 0.2861 2.338 0.0444

Franklin Park 5K -1.4465 0.5274 -2.743 0.0190

Van Cortlandt Park HEPS 8K 1.7053 0.3029 5.629 0.000284

Van Cortlandt Park ICAAAA 8K 1.6524 0.3593 4.599 0.001005

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05 See Appendix D4 for all results including race course coefficients

Figure 9: The association between sleep latency one night before a race and race performance.

The regression diagnostics for the four models displayed in Table 22 can be found in Figures D1, D2, D3, and D4 in the appendix.

Figure 10: The association between sleep latency two nights before a race and performance.

Hypotheses Set 4 Important Results Before examining multivariable random effects models including both rMSSD HRV and sleep, the correlations between the three notable sleep variables from Sets 1-3 and HRV were examined both across subjects and within subjects. Table 24 contains the correlations across subjects while Table E1 in the appendix contains the correlations within subjects.

Table 24: Correlations between rMSSD HRV and sleep across all subjects.

Variables Correlation Covariance

rMSSD HRV and REM Sleep One Night Before 0.257 0.263

rMSSD HRV and REM Sleep Two Nights Before 0.374 0.383

rMSSD HRV and REM Sleep Weighted Two Nights 0.408 0.418

rMSSD HRV and Total Sleep One Night Before -0.009 -0.009

rMSSD HRV and Total Sleep Two Nights Before -0.205 -0.210

rMSSD HRV and Total Sleep Weighted Two Nights -0.201 -0.207

rMSSD HRV and Sleep Latency One Night Before -0.103 -0.106

rMSSD HRV and Sleep Latency One Night Before e 0.426 0.437

rMSSD HRV and Sleep Latency One Night Before 0.167 0.164

See Appendix E1 for correlations within subjects.

Across subjects, there were no strong correlations between any of the three major sleep variables and rMSSD HRV. Within subjects, there were a few individual who had very strong negative correlations with REM sleep duration and rMSSD HRV. For both total sleep duration and sleep latency, there were individuals with strong positive correlations (>0.75) with rMSSD HRV and with strong negative correlations (<-0.75).

Table 25: Results of note for sleep and rMSSD HRV random effects models.

Variable Standardized Converted Standard Error t-value p-value Coefficient Coefficient Varying Intercept Models REM Sleep One Night Before -0.4302 -0.0161 0.1811 -2.375 0.0256

rMSSD HRV One Night Before 0.2403 14.470 0.1791 1.341 0.1920

REM Sleep Two Nights Before -0.5548 -0.0272 0.1440 -3.853 0.000657

rMSSD HRV Two Nights Before 0.2317 19.286 0.2424 0.956 0.3501

REM Sleep Weighed Two Nights -0.5385 -0.0246 0.1549 -3.476 0.00159

rMSSD HRV Weighted Two Nights 0.3083 23.517 0.1970 1.565 0.1293

Total Sleep One Night Before -0.5376 -0.0267 0.1756 -3.062 0.00535

rMSSD HRV One Night Before 0.3010 18.124 0.1703 1.819 0.0813

Total Sleep Two Nights Before -0.4897 -0.0243 0.1733 -2.826 0.00916

rMSSD HRV Two Nights Before 0.1812 15.082 0.2568 0.706 0.4894

Sleep Latency Two Nights Before 0.4256 0.0870 0.2017 2.110 0.0451

rMSSD HRV Two Nights Before 0.0791 6.583 0.2387 0.331 0.7450

Varying Slope and Intercept Models REM Sleep One Night Before -0.4807 -0.0180 0.1763 -2.726 0.0119

rMSSD HRV One Night Before 0.1908 11.489 0.2262 0.843 0.4377

REM Sleep Two Nights Before -0.5766 -0.0283 0.1324 -4.354 0.00023

rMSSD HRV Two Nights Before 0.0038 0.3163 0.3496 0.011 0.9916

REM Sleep Weighted Two Nights -0.5933 -0.0272 0.1519 -3.905 0.000561

rMSSD HRV Weighted Two Nights 0.1438 10.969 0.2468 0.583 0.5767

Total Sleep Weighted Two Nights -0.5139 -0.0543 0.1808 -2.842 0.0244

rMSSD HRV Weighted Two Nights 0.2622 20.001 0.2023 1.296 0.2076

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05 Note: Some varying slope and intercept models failed to converge. See Appendix E2 for all results

Table 25 contains notable results containing rMSSD HRV and the notable sleep variables. Both varying intercept models and varying slope and intercept models were analyzed. All models contained predictors with data aggregated in the same method. For example, REM sleep weighted two nights was paired with rMSSD HRV exponentially weighted two nights. None of the rMSSD HRV coefficients were significantly different from zero at the alpha level of 0.05. However, all three REM sleep coefficients (one night, two nights, and weighted two nights before a race) were significantly negative at the alpha level of 0.05 in both the varying intercept and varying slope and intercept models. The coefficients for REM sleep duration two nights before a race converted to original units for both the varying intercept and varying slope and intercept models were -0.0272 and -0.0283 respectively. These were very close in magnitude to coefficients calculated in previous models. In addition, the coefficient for sleep latency two nights before a race was significantly positive in the varying intercept model, while the coefficients for total sleep duration one night before a race and two nights before a race were significantly negative in the varying intercept model. The coefficient for total sleep duration weighted two nights before a race was significantly negative in the varying slope and intercept models. Complete results for all models examined can be found in Table E2 in the appendix. Some of the varying slope and intercept models failed to converge, so those models could not be analyzed. In addition, all models from Table E2 in the appendix were also expanded to include interaction terms. Table 26 contains the notable results. No interaction terms were significantly different from zero at the alpha level of 0.05. Additionally the overall results remained the same from Table 25, with very minor changes in the magnitude of coefficients. No rMSSD HRV coefficients were significantly different from zero. Complete results, including the interaction coefficients, can be found in Table E3 in the appendix. As in the models without interactions, some varying slope and intercept models failed to converge. The models from Table E2 in the appendix were then expanded to include the stress variable from survey data, and the significant coefficients are displayed in Table 27. All three REM sleep coefficients in the varying intercept model remained significantly negative at the alpha level of 0.05. Additionally, the coefficients for REM sleep duration two nights

64 before and the coefficient for REM sleep duration weighted two nights were also significantly negative. The magnitude of these coefficients increased slightly.

Table 26: Results of note for sleep and rMSSD HRV random effects models with interactions included.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models REM Sleep One Night Before -0.4650 0.1929 -2.410 0.0241

rMSSD HRV One Night Before 0.2723 0.1871 1.455 0.1585

REM Sleep Two Nights Before -0.6844 0.1964 -3.484 0.0021

rMSSD HRV Two Nights Before 0.3185 0.23313 1.366 0.1912

REM Sleep Weighed Two Nights -0.6495 0.1753 -3.704 0.000928

rMSSD HRV Weighted Two Nights 0.3815 0.1863 2.048 0.0538

Total Sleep One Night Before -0.5794 0.1887 -3.071 0.00517

rMSSD HRV One Night Before 0.3851 0.2111 1.824 0.0801

Total Sleep Two Nights Before -0.4747 0.1776 -2.673 0.0132

rMSSD HRV Two Nights Before 0.1833 0.2598 0.706 0.4894

Total Sleep Weighted Two Nights -0.5882 0.1713 -3.434 0.00182

rMSSD HRV Two Nights Before 0.2874 0.2148 1.338 0.1926

Varying Slope and Intercept Models REM Sleep One Night Before -0.5064 0.1839 -2.754 0.0116

rMSSD HRV One Night Before 0.2238 0.2399 0.933 0.3965

REM Sleep Two Nights Before -0.7021 0.1764 -3.980 0.000554

rMSSD HRV Two Nights Before 0.0876 0.3218 0.272 0.7941

REM Sleep Weighed Two Nights -0.6778 0.1680 -4.034 0.000441

rMSSD HRV Weighted Two Nights 0.2337 0.2392 0.977 0.3614

Total Sleep One Night Before -0.4989 0.2123 -2.351 0.0484

rMSSD HRV One Night Before 0.3581 0.1906 1.879 0.0755

Total Sleep Weighted Two Nights -0.4988 0.1856 -2.687 0.0298

rMSSD HRV Two Nights Before 0.2336 0.2125 1.099 0.2824

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05 Note: Some varying slope and intercept models failed to converge. See Appendix E3 for all results including coefficients for interaction terms

Table 27: Notable results for sleep and rMSSD HRV models with stress variable included.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models REM Sleep One Night Before -0.5861 0.1992 -2.942 0.0408

REM Sleep Two Nights Before -0.6959 0.1790 -3.888 0.00317

REM Sleep Weighted Two Nights -0.5669 0.2239 -2.532 0.0302

Total Sleep Two Nights Before -0.6085 0.2530 -2.406 0.0316

Sleep Latency Two Nights Before 0.4836 0.2692 1.796 0.0100

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.4897 0.1090 -4.494 0.00424

REM Sleep Weighted Two Nights -0.5796 0.2269 -2.554 0.0305

All coefficients represent results that reject the null hypothesis at the alpha level p<0.05 Note: Some varying slope and intercept models failed to converge. See Appendix E4 for all results including coefficients for stress.

The coefficient for total sleep duration two nights before was also negative and significant with an increase in magnitude from the original coefficient in Table 25, while the coefficient for sleep latency two nights before a race was positive and significant without changing much in magnitude. No rMSSD HRV coefficients were significantly different from zero. As in previous models, some varying slope and intercept models did not converge and could not be analyzed. Complete results can be found in Table E4 in the appendix. The models from Table E2 were then expanded by adding the race course indicator variables, with Table 28 containing the notable results. Similar to the results from Set 3, the coefficient for REM sleep duration two nights before a race and sleep latency one night before a race were significantly different from zero at the alpha level of 0.05 in the varying intercept model. In addition, sleep latency weighted two nights before a race was significantly different from zero as well in the varying intercept model. In the varying slope and intercept model, both sleep latency coefficients were significant, but the coefficient for REM sleep duration two nights before a race was not significant. As in previous models examined in Set 4, no rMSSD HRV coefficients were significantly different from zero.

Table 28: Results of note for sleep and rMSSD HRV random effects models with race course controls included.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models REM Sleep Two Nights Before -0.3010 0.1335 -2.255 0.0381

rMSSD HRV Two Nights Before -0.1137 0.1947 -0.584 0.5650

Sleep Latency One Night Before 0.3778 0.1064 3.551 0.003903

rMSSD HRV One Night Before -0.0700 0.1007 -0.691 0.5021

Sleep Latency Weighted Two Nights 0.3072 0.0932 3.298 0.00377

rMSSD HRV Weighted Two Nights 0.1474 0.1500 0.983 0.3353

Varying Slope and Intercept Models REM Sleep Two Nights Before -0.2686 0.1290 -2.082 0.2382

rMSSD HRV Two Nights Before -0.1115 0.3103 -0.370 0.7232

Sleep Latency One Night Before 0.3525 0.0962 3.664 0.00355

rMSSD HRV One Night Before -0.0609 0.1122 -0.542 0.6110

Sleep Latency Weighted Two Nights 0.3087 0.0954 3.235 0.00654

rMSSD HRV Weighted Two Nights 0.1341 0.1470 0.912 0.3938

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05 Note: Some varying slope and intercept models failed to converge. See Appendix E5 for all results including race course coefficients.

In general, when considering the coefficients for REM sleep duration two nights before a race and sleep latency one and two nights before a race, results remained almost the same as in Set 3.

Weekly Bike Efforts The weekly bike efforts by the subjects were originally going to be used as a second metric of performance. However, after analyzing initial models using the bike effort data, the results were discarded. There were no significant associations between any physiological variables and distance traveled on the bike, potentially due to flawed study design. It is suggested here to improve on this experimental design by having a longer period for the hard bike efforts, such as 30 minutes or more. A short period, such as the 5-

67 minute effort for this study, might have trouble producing significant differences in distance traveled, while a longer period would allow more time for an individual who is more recovered to increase the gap between an effort when less recovered. In other words, a longer time interval could potentially cause variables such as worse sleep quality or lower HRV, and therefore fatigue, to have a greater impact on distance traveled during the effort. Additionally, the bike efforts were not treated the same as a race; subjects typically were training through the bike efforts and often the efforts took place a couple of days after a race. It is suggested here that a future study design should avoid having conflicting performance metrics.

Discussion

The accuracy of the data collected from the WHOOP Strap must first be addressed. With any wearable device, there is the possibility of a signal processing or measurement error. This is especially true when tracking complex actions, such as sleep, that are a function of many variables. As an example, one subject’s data was removed from the study due to a signal processing error that miscalculated his heart rate. Additionally, when examining the summary statistics in Table 5, it is important to recognize that for these subjects, the percentage of REM sleep duration on average was less than what is expected, while the percentage of light sleep duration was more than what is expected. Of course, this is presuming that the subjects are “normal.” This may not be the case; as student-athletes with potentially many stressors, it is quite possible that the subjects have different sleep patterns compared to “normal” people. The estimated sleep efficiency of the subjects, calculated at 85.74%, is considered normal; a 90% sleep efficiency is considered very good, while lower than 85% is considered poor (Sheldon et al., 2014). In any case, when considering the hypotheses in Set 1, increased REM sleep and total sleep duration in various aggregations leading up to a race were significantly associated with better race performance when analyzed without additional control variables included. Other physiological variables did not have significant associations with performance. rMSSD HRV, which has been suggested in research to be a potential indicator of better or worse performance, did not have a significant association with performance. When race course variables were included in models with significant physiological variables, it effectively controlled for races on different race courses. With those controls in place, the significant association between increased REM sleep duration two nights before a race and better race performance remained in both varying intercept and varying slope and intercept models. The likelihood ratio test in Table 9, not surprisingly, indicated that the race course coefficients significantly improved the model’s ability to predict race performance. While interaction terms did not significantly improve prediction, the coefficient for REM sleep duration two nights before a race remained significant in the varying intercept model.

When considering the hypotheses in Set 2, one may think that that REM sleep duration might just be a function of total sleep duration; in other words, more sleep overall would cause more REM sleep. Figure 6 illustrates that for these subjects, there was really no direct relationship between the amount of REM sleep and total sleep across all nights for all subjects. Additionally, the models in Tables 13 and 14 indicate that when both are factored into the equation, increased REM sleep duration and increased total sleep duration both had significant associations with better race performance. For the most part, the interactions between the total sleep and REM sleep, especially in the two nights before a race, seemed to significantly improve model fit in the likelihood ratio tests displayed in Table 15. When both the stress variable from survey data and race course variances were included, increased total sleep two nights before a race had a significant association with better race performance. The comparison between one night before a race and two nights before a race was addressed in Set 3. Table 18 indicates that there was a rather strong correlation between REM sleep duration two nights before a race and one night before a race across all subjects, with most subjects also having strong correlations individually. For the remaining sleep variables, the correlations between one night before a race and two nights before a race varied for different subjects; some subjects had strong positive correlations, some had strong negative correlations, and some had very weak to essentially no correlations. Original models without additional controls in place suggested that only sleep variables two nights before a race were significantly associated with race performance, with one night before a race not significant. When the models were expanded to included interaction terms, the results remained consistent. Once race course control variables were added to the models, increased REM sleep duration two nights before a race continued to have a significant association with better race performance, while increased sleep latency one night before a race continued to have a significant association with worse race performance. Set 4 addresses multivariable models and rMSSD HRV paired with the three notable sleep variables. Table E1 in the appendix illustrates that on the individual subject level, the correlations between rMSSD HRV and REM sleep duration, rMSSD HRV and total sleep duration, rMSSD HRV and sleep latency varied. Some individuals had strong positive

70 correlations, some individuals had strong negative correlations, and some individuals had very weak correlations. In all models analyzed in Tables 24-28 and Appendix E, rMSSD HRV had no significant association with race performance. However, the sign of the rMSSD HRV coefficient still proves interesting. In the initial models displayed in Table E2 in the appendix, all coefficients for rMSSD were either positive or essentially zero. When interaction terms were added to those models in Table E3, the coefficients for rMSSD HRV generally remained positive. However, when the stress variable from survey data was added to the models, Table E4 illustrates that many of the rMSSD HRV coefficients became negative. When race course indicator variables were added to the models shown in Table E5, the coefficients for rMSSD HRV mainly hovered around zero, with a few exceptions that were either positive or negative. In Set 4, both varying intercept and varying slope and intercept models in Table E2, with no additional variables added, indicate that increased REM sleep duration both one and two nights before a race had a significant association with better race performance. In addition, increased total sleep duration one and two nights before the race had a significant association with better race performance in the varying intercept model; increased total sleep weighted over two nights in the varying slope and intercept model had a significant association with better race performance. Increased sleep latency two nights before the race had a significant association with worse race performance. With interaction terms between sleep and rMSSD HRV added in Table E3, the same associations were discovered, with the exception of sleep latency no longer having a significant association with race performance. When the stress variable from survey data was added to the models in Table E4, the significant association between REM sleep duration and race performance remained in both the varying intercept and varying slope and intercept models, while increased sleep latency two nights before a race had a significant association with worse race performance in the varying intercept model. Finally, when race course indicator variables were added to the models shown in Table E5, increased REM sleep duration two nights before a race had a significant association with better race performance in the varying intercept model, while increased sleep latency one night before a race had a significant association with worse race performance in both the varying intercept and varying slope and intercept models.

As a whole, the results from the four sets of hypotheses indicate that, even with factors controlled for such as race course difficulty and stress, increased total sleep and REM sleep duration have significant associations with better performance, especially two nights before a race. On the other hand, increased sleep latency leading up to a race seemed to have a significant association with worse race performance. The assumptions of random effects models appear to be satisfied with this analysis. Allowing varying coefficients for each subject helps address the fact that race results for each subject may not be completely independent. The race course control variables help take into account the specific race that an athlete is racing in, and it could be argued that they somewhat reflect the fact that a subject may be getting “better” as the season progresses. When determining whether modeling race performance with varying slopes and intercepts is more ideal than just varying intercepts, Table 11 indicated that there was not a significant difference between the two models. However, visualizations such as Figures 5, 8, 9, and 10 suggest that models allowing for varying slopes may be more applicable, which is not surprising; different physiological variables likely have different effects for each subject. In any case, the results did not vary too much between models. The diagnostic plots for notable models displayed in Figures, 2, 3, 4, and 7, as well as D1-D4 in the appendix, illustrate that the residuals for the models of note appear to be normally distributed with a mean around zero across all subjects. The residuals plotted against fitted values also indicate that the race performance has constant variance across all subjects, though variance within-subjects will be discussed later in the limitations section. The significant associations between sleep and race performance are not necessarily surprising. Increased sleep latency the night before a race or two nights before a race could be an indication of extreme stress or nerves, and could foreshadow a worse performance. For example, the varying intercept and varying slope and intercept models summarized in Table 22 and Table 23 suggest that increased sleep latency one night before a race had a significant association with worse race performance when controlling for race course. Examining the significant race course coefficients in Table 23 provides some additional insight; the two races at Van Cortlandt Park (a very challenging course, as mentioned in Table 1) and the Franklin Park 10K race (the longest distance the athletes raced) both had significant associations with worse race performance compared to other courses. While the

72 difficulty of these courses likely caused slower race times compared to other courses, it is also reasonable to speculate that the subjects were very nervous the night before these challenging races; therefore, their sleep latency increased. However, one could also argue that being nervous is not always a bad thing. Previous research suggested that lower HRV indicates that the body may be more ready to perform on race day (Plews et al., 2013). In this study, the results in Tables E2-E5 suggest that one night before a race, having a lower HRV has an association with better performance (though the association is not significant). The positive rMSSD HRV coefficients in those results could also reflect the idea that sympathetic dominance one night before a race, and therefore lower HRV, is better for race performance. This makes sense, as the rising blood pressure and accelerated heart rate occurring during race conditions can be beneficial; it could be argued that the body’s response to race conditions is parallel to the body’s “survival instincts” that is associated with lower HRV. Past research has also indicated that HRV can be used to detect overtraining (Plews et al., 2012), signal greater endurance capability (Kiviniemi et al., 2010), and used daily for guided training (Kiviniemi et al., 2007; Kiviniemi et al, 2010). It may be that HRV fluctuations vary greatly on the individual level; therefore, it could be very difficult to use HRV to successfully model performance. While there were no significant associations between HRV and performance in this study, it is suggested here that future research be done to investigate the idea of sympathetic dominance leading to better race performance. The significant association between increased REM sleep duration and race performance is more complicated, and there are many possible reasons for why this association was observed in these subjects. As mentioned previously, it could be that REM sleep duration is simply correlated with total sleep duration. Even though Figure 6 and Table 12 indicated that this is not necessarily true across all subjects, a few subjects had strong correlations between REM and total sleep as shown in Table C1 in the appendix. Further post hoc research into links between REM sleep and the body reveal some additional interesting insights into why this association was observed. Most notably, research has shown that the REM sleep propensity during the night is correlated with the body temperature’s circadian rhythm; when the body’s core temperature is at its lowest, the percentage of REM sleep during sleep is at its highest (Czeisler et al., 1980; Czeisler et

73 al., 1979). Core body temperature endogenously oscillates throughout the day. When the body is under normal conditions, its core temperature has its peak in the late afternoon to early evening and its nadir in the early hours of the morning (Winget et al., 1985). As a result, the majority of REM sleep during a normal night of sleep usually takes place in the early morning hours during the nadir in core body temperature. REM sleep propensity can therefore be interpreted as a marker or reflection of circadian timing. Research has shown that day-active species such as humans tend to have trouble deviating from their normal circadian rhythms (Dijk and Czeisler, 1994), and therefore can find it difficult to sleep outside traditional hours. If sleep is disrupted in the middle of the night and one struggles to fall back asleep, REM sleep duration will likely decrease the most as REM sleep takes place in the early morning. In other words, if one’s sleep pattern is thrown off on a given night for any particular reason, REM sleep is the first stage of sleep to be affected, as the majority of it takes place in the later part of sleep. Many factors affect one’s sleep patterns; as an example, artificial lights entering the eyes from devices such as iPads and eReaders essentially disrupt the circadian rhythm, and can therefore effect sleep quality (Chang et al., 2015; Czeisler, 2013). Athletes typically have less sleep or interrupted sleep for reasons such as nerves the night before a race (Fullagar et al., 2014) or jet lag from travel (Manfredini et al., 1998), so a decrease in REM sleep is fairly understandable one night before competition. However, less REM sleep two nights before could potentially be an indication of chronic stressors or other factors chronically affecting sleep patterns. In addition to its link with quality of sleep, the circadian rhythm also has a connection to athletic performance. In the early evening, there is a very powerful surge in core body temperature that has been associated with quicker reaction times (Dijk and Czeisler, 1994; Cajochen et al., 1999). In general, this surge helps explain why many athletic records are broken in the evening. The timing of an athlete’s peak core body temperature is considered essential for optimizing athletic performance. A person’s regular circadian clock and sleep-wake pattern take time to adjust, especially when traveling across multiple time zones: A study of National Football League games from 1978-1987 showed that teams from the west coast lost approximately 14-16% more games when traveling east for afternoon games (Jehue et al., 1993). The study suggested that circadian rhythms of the players from the west coast teams were likely desynchronized due to having not fully adapted to the

74 time change. In other words, their body clocks were behind schedule when playing the game on the east coast in the afternoon, and their lower body temperature was associated with slower reaction time and worse athletic performance. In general, research has indicated that performing at the ideal time in the circadian clock is associated with an increase in athletic performance by up to 10% (Manfredini et al., 1998). When recognizing this research, the association observed between REM sleep and athletic performance is more intuitive. It is quite possible that the less REM sleep duration observed two nights before a race is a proxy for a poorly synchronized or disrupted circadian rhythm, and therefore more REM sleep two nights before a race can be seen as a marker for more consistent sleep patterns. This could essentially be evidence supporting the idea of “banking sleep” leading up to a race mentioned previously in the introduction; since it is likely that athletes lose sleep because of nerves the night before a race (Fullagar et al., 2014), getting great sleep two nights before a race may provide extra insurance to keep one’s body in prime position to perform well despite inconsistent sleep the night before. However, it is difficult for the results of this study to conclude precisely that a desynchronized circadian rhythm leads to worse race performance; some of the races took place as early as 10:00 in the morning, while other races took place as late as 3:30 in the afternoon. Regularity of sleep, including the time athletes went to bed, was not analyzed closely for this study, nor was the irregularity of race times considered. It is suggested here that future studies investigating sleep and performance take into account regularity and timing of sleep as well as sleep duration, as it could provide more insights into the relationship between sleep, the circadian rhythm, and performance. The expected change in performance associated with REM sleep duration and sleep latency does not seem to be very large at first glance. For example, the coefficient for REM sleep duration two nights before a race in the varying slope and intercept model in Table 19, when converted to original units, was -0.0287. This suggests that for every additional minute of REM sleep two nights before a race, the expected race pace is expected to be faster by about 0.0287 seconds/km when all other factors are held constant. However, an extra 30 minutes of REM sleep two nights before equates to roughly 0.861 seconds/km, which is therefore associated with an expected improvement in race performance by about 6.9 to 8.6 seconds for races ranging from 8-10 kilometers. This drop in race time can make

75 a huge difference in how well an athlete places in large competitions such as NCAA cross country races. In Table 19, the varying slope and intercept model’s coefficient for sleep latency two nights before was 0.1011 when converted to original units. This corresponds to a 0.1 seconds/km increase in expected race pace for each additional minute of sleep latency when all other factors are held constant; an additional 10 minutes of sleep latency two nights before a race is therefore associated with an expected 1-second improvement in race pace per kilometer, and therefore a slower race time by 8-10 seconds over an 8 or 10 kilometer race. The significant association generally observed in this study between sleep and race performance, especially two nights before a race, adds to previous research on sleep and athletic performance. While significant associations have been observed between both more sleep and athletic performance tests (Mah et al., 2011), and sleep deprivation and endurance treadmill distance tests (Oliver et al., 2009), this study adds research containing results for significant associations between sleep and the results of actual competitions for which the subjects train. This study also adds research with elite endurance runners as subjects, which has not yet been done. The unfounded claim that sleep two nights before a race is more important than the night before (Glover, 1996), made by many coaches in the endurance running community, is also supported in this study. With the associations between more sleep and better race times for the Harvard cross country team in this study, it seems that physiological “big data” can be utilized to predict better or worse athletic performance, especially with the continuing development of the tools to track these variables.

Limitations It must first be recognized that these subjects were collegiate student-athletes with multiple priorities. A high level of stress and pressure comes with performing at a high level, especially in NCAA cross country, when performance means a great deal to the subjects. It is emphasized here that one cannot necessarily use these results to make inference on a different population. In other words, the significant associations seen here between sleep and performance may not carry over for the professional athlete, the recreational enthusiast, or the everyday person.

Additionally, it takes an endurance athlete’s body several months to adjust to increased levels of training to improve race performance. For an NCAA cross country athlete, formal training begins in the early summer and continues throughout the fall. Most NCAA cross country athletes also compete in track and field distance events during the winter and spring seasons, meaning that an athlete trains year-round and always continues to build fitness. This study, which lasted 66 days, captures only a very small portion of time that would be useful to observe; future studies would benefit from having a much longer period to analyze the effects of HRV and sleep quality on performance in endurance competitions. A longer period would allow for more races to be included, allowing for greater statistical power to capture any possible significant associations between HRV, sleep-related data and race performance. In regard to study design, the faster runners on the team were asked to wear the WHOOP Straps before the other runners on the team, meaning the WHOOP Straps were not randomly assigned. A future study would benefit from having a larger sample size of athletes of different abilities. This may provide greater statistical power to capture any significant associations and would also allow better insight into whether changes in HRV and sleep-related data may affect performance differently for runners of greater or worse ability. When using the WHOOP Strap to record physiological data, missing data during the study must be taken into account. Subjects occasionally removed the WHOOP Straps temporarily, and as a result R-R intervals are not recorded for that interval of time. This missing data may be trivial for small intervals, but could prove problematic if an athlete removed his device for an extended period. For example, one subject removed his WHOOP Strap, forgot to put it back on, and lost it for several days. The impact on the study was minimal as this specific subject sustained a season-ending injury shortly afterwards, so he did not race again after his device went missing. A second example involved several of the subjects removing their WHOOP Straps on an evening out, as they observed that the devices could easily be knocked off the wrist when in a large crowd. This night involved the consumption of alcohol, which will have affected the physiological variables such as resting heart rate, sleep quality, and HRV. Because these subjects did not wear WHOOP Straps on

77 this night, the physiological data could not be collected and any changes in these variables could not be factored into the analysis. As mentioned earlier, one of the main assumptions when using a random effects regression model in this setting is that there is constant variance around the predicted race result. In other words, it is assumed that each subject has the same expected variation in his actual race performance from his expected performance. The diagnostics plots for the notable models indicate that across all subjects, the assumption of constant variance seems to be satisfied. It can be argued that this is not the case in general, however, as one can assert that every runner’s variance in performance is different for many possible reasons: some runners are better on courses with more hills; some runners are more consistent in their racing abilities; different racing strategies can lead to more tactical races in slower paces; and the psychological component and pressure of racing can affect runners in different ways given different conditions. Figure 5, for example, shows that one subject appears to have better race performances on nights with less sleep duration two nights before a race. Controlling for all of these variables that come into play during a race environment is very difficult and is a distinct limitation in study design that must be addressed. Finally, it must be recognized that there are some confounding variables affecting a runner’s race performance throughout the season that can potentially mask the effect of better sleep or HRV data, such as the subject getting fitter throughout the course of the season or becoming more acclimated to racing by the end of the season; for example, a good race can lead to increased confidence in the next race. While a lot of these factors would be hard to control for under any study design, future studies involving this type of physiological data could benefit from using competitions that are easier to compare with each other, such as track and field events or endurance treadmill tests, and could also benefit by using standard measurements such as VO2 max to gauge an athlete’s fitness over time alongside race performance.

Conclusion This study has shown that wearable devices appear to have the ability to recognize trends and associations between physiological variables and performance. This seems especially true to me after observing the relationship between sleep two nights before a

78 race and better performance, which is always inferred but has never been specifically backed up by research. I believe that this study indicates the utility of physiological big data; the ability to track the body’s patterns, signals, and physiology on a day-to-day basis is intriguing in itself, but the fact that this data can signal improved or impaired athletic performance is truly fascinating. From the perspective of an athlete or a coach, physiological big data can be especially useful. The ability to quantify how worn down, tired, or energized the body is by tracking sleep and autonomic response can be an invaluable advantage, especially in the grueling sport of endurance running. I would argue that using this technology to learn about the factors affecting one’s individual sleep patterns and autonomic nervous system response can help put yourself in the best possible position to optimize performance. However, it must be noted that like any other technology that utilizes big data, it cannot perfectly predict how well one performs. Too many variables come into play during a race, and the most important variable cannot be tracked: how well one competes. When toeing the line for a race, one must be ready to grind until the end and put one foot in front of the other as fast as possible, no matter how fatigued the body is on race day. Having said that, while the improvements in performance made by taking advantage of this physiological data are minimal compared to the big picture, it can give one the slightest edge. And at the elite level, where every second counts, that slightest edge can make all the difference.

Appendix Appendix A: Summary Statistics

Table A1: Overall summary statistics for a night’s sleep.

Variable Units Mean Standard Deviation

Time In Bed minutes 543.15 78.19

Total Sleep Duration minutes 465.69 71.20

Light Sleep Duration minutes 316.34 88.05

REM Sleep Duration minutes 72.28 63.82

Slow Wave Sleep Duration minutes 77.07 41.31

Sleep Latency minutes 19.39 10.93

REM Cycle Counts 5.20 2.88

Resting Heart Rate beats/minute 46.67 5.71

rMSSD HRV seconds 0.0968 0.0425

Table A2: Within-subjects summary statistics for a night’s sleep.

Variable Subject Units Mean Standard Deviation

Time In Bed Subject A minutes 600.60 102.63

Subject B 516.66 50.43

Subject C 562.78 60.91

Subject D 531.17 79.16

Subject E 537.80 52.50

Subject F 575.56 66.36

Subject G 560.20 27.88

Subject H 533.61 50.12

Subject I 523.26 75.21

Total Sleep Duration Subject A minutes 503.67 51.31

Subject B 445.14 54.55

Subject C 474.73 68.28

Subject D 461.87 70.53

Subject E 420.69 64.52

Subject F 494.07 59.73

Subject G 506.29 33.57

Subject H 451.05 44.77

Subject I 460.84 75.48

Variable Subject Units Mean Standard Deviation

REM Sleep Duration Subject A minutes 56.12 41.26

Subject B 51.18 41.31

Subject C 125.13 108.70

Subject D 63.80 22.80

Subject E 111.33 65.99

Subject F 68.53 58.98

Subject G 40.52 24.74

Subject H 136.06 70.10

Subject I 35.50 18.89

Slow Wave Sleep Duration Subject A minutes 98.63 49.05

Subject B 72.85 31.94

Subject C 86.70 31.92

Subject D 113.11 45.37

Subject E 44.77 27.65

Subject F 95.28 42.36

Subject G 69.23 34.93

Subject H 75.83 31.27

Subject I 56.55 29.35

Variable Subject Units Mean Standard Deviation

Sleep Latency Subject A minutes 23.44 16.79

Subject B 16.15 6.51

Subject C 22.37 15.32

Subject D 18.01 7.65

Subject E 18.56 6.90

Subject F 24.67 13.09

Subject G 15.79 6.85

Subject H 18.90 11.40

Subject I 15.84 7.21

REM Cycle Counts Subject A 5.45 2.95

Subject B 4.67 2.45

Subject C 5.68 3.58

Subject D 5.30 2.14

Subject E 7.05 2.98

Subject F 5.02 2.37

Subject G 4.38 2.79

Subject H 7.41 2.41

Subject I 3.40 2.13

Variable Subject Units Mean Standard Deviation

Resting Heart Rate Subject A beats/minute 46.18 2.84

Subject B 45.40 2.99

Subject C 41.56 2.43

Subject D 48.57 5.16

Subject E 44.53 4.38

Subject F 42.07 2.39

Subject G 57.42 3.70

Subject H 44.43 3.33

Subject I 50.03 4.43

rMSSD HRV Subject A seconds 0.0731 0.0377

Subject B 0.0803 0.0203

Subject C 0.1327 0.0226

Subject D 0.1018 0.0209

Subject E 0.1440 0.0360

Subject F 0.0637 0.0199

Subject G 0.0495 0.0129

Subject H 0.1193 0.0271

Subject I 0.0675 0.0275

Table A3: Summary statistics for aggregated data leading up to a race across all subjects. rMSSD (Heart Rate Variability)

Variable Units Mean Standard Deviation

One Night Before seconds 0.0953 0.0470

Two Nights Before 0.0928 0.0340

Weighted Two Nights 0.0963 0.0371

Weighted Three Nights 0.0961 0.0356

Weighted Four Nights 0.0950 0.0356

Exponentially Weighted Two Nights 0.0955 0.0365

Exponentially Weighted Three Nights 0.0961 0.0348

Exponentially Weighted Four Nights 0.0952 0.0371

Resting Heart Rate

Variable Units Mean Standard Deviation

One Night Before beats/minute 46.30 5.85

Two Nights Before 46.63 5.53

Weighted Two Nights 46.22 5.14

Weighted Three Nights 46.17 4.99

Weighted Four Nights 46.07 4.86

Exponentially Weighted Two Nights 46.18 5.09

Exponentially Weighted Three Nights 46.11 5.04

Exponentially Weighted Four Nights 45.92 4.85

Total Sleep Duration

Variable Units Mean Standard Deviation

One Night Before minutes 460.32 57.05

Two Nights Before 475.23 56.38

Weighted Two Nights 466.18 52.03

Weighted Three Nights 468.48 52.01

Weighted Four Nights 466.30 52.78

Exponentially Weighted Two Nights 475.48 71.75

Exponentially Weighted Three Nights 477.54 60.54

Exponentially Weighted Four Nights 466.28 66.37

Light Sleep Duration

Variable Units Mean Standard Deviation

One Night Before minutes 290.66 76.76

Two Nights Before 317.24 77.64

Weighted Two Nights 300.51 69.70

Weighted Three Nights 305.41 71.75

Weighted Four Nights 303.91 73.46

Exponentially Weighted Two Nights 314.85 86.35

Exponentially Weighted Three Nights 319.47 84.84

Exponentially Weighted Four Nights 308.81 85.16

REM Sleep Duration

Variable Units Mean Standard Deviation

One Night Before minutes 90.22 75.62

Two Nights Before 71.13 57.67

Weighted Two Nights 81.35 61.84

Weighted Three Nights 79.34 60.19

Weighted Four Nights 77.47 57.82

Exponentially Weighted Two Nights 75.35 61.45

Exponentially Weighted Three Nights 74.46 57.41

Exponentially Weighted Four Nights 73.15 52.37

Slow Wave Sleep Duration

Variable Units Mean Standard Deviation

One Night Before minutes 79.43 37.14

Two Nights Before 86.85 41.77

Weighted Two Nights 84.32 35.12

Weighted Three Nights 83.73 32.23

Weighted Four Nights 84.91 31.98

Exponentially Weighted Two Nights 85.28 39.74

Exponentially Weighted Three Nights 83.61 34.01

Exponentially Weighted Four Nights 84.31 31.58

Sleep Latency

Variable Units Mean Standard Deviation

One Night Before minutes 18.38 6.98

Two Nights Before 22.55 13.85

Weighted Two Nights 20.06 5.91

Weighted Three Nights 20.05 5.91

Weighted Four Nights 21.29 10.95

Exponentially Weighted Two Nights 21.30 9.90

Exponentially Weighted Three Nights 19.87 6.94

Exponentially Weighted Four Nights 20.26 11.25

Time In Bed

Variable Units Mean Standard Deviation

One Night Before minutes 542.20 61.07

Two Nights Before 566.25 60.18

Weighted Two Nights 552.98 61.07

Weighted Three Nights 554.60 62.20

Weighted Four Nights 551.94 62.98

Exponentially Weighted Two Nights 566.35 103.07

Exponentially Weighted Three Nights 560.87 75.53

Exponentially Weighted Four Nights 549.32 76.28

REM Cycle Counts

Variable Units Mean Standard Deviation

One Night Before 5.89 2.83

Two Nights Before 6.12 3.13

Weighted Two Nights 6.02 2.37

Weighted Three Nights 5.88 2.24

Weighted Four Nights 5.83 2.06

Exponentially Weighted Two Nights 5.95 2.79

Exponentially Weighted Three Nights 5.60 2.22

Exponentially Weighted Four Nights 5.66 1.91

Table A4: Summary statistics for race performance.

Variable Units Subject Mean Standard Deviation

Average Race Pace seconds/km Overall (Across Subjects) 194.68 2.83

Subject A 195.69 4.36

Subject B 190.69 5.44

Subject C 200.34 8.90

Subject D 191.51 4.38

Subject E 199.92 6.61

Subject F 192.16 7.87

Subject G 199.53 4.15

Subject H 186.08 3.81

Subject I 200.37 14.55

Appendix B: Extended Hypotheses Set 1 Results

Table B1: rMSSD heart rate variability results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before 0.1827 0.1924 0.948 0.351

Two Nights Before 0.1443 0.2366 0.61 0.554

Weighted Two Nights 0.1490 0.2064 0.722 0.480

Weighted Three Nights 0.1511 0.2127 0.710 0.488

Weighted Four Nights 0.1373 0.2134 0.643 0.531

Exponentially Weighted Two Nights 0.0939 0.2148 0.437 0.670

Exponentially Weighted Three Nights 0.1310 0.2309 0.567 0.580

Exponentially Weighted Four Nights 0.1447 0.2270 0.638 0.537

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B2. rMSSD heart rate variability results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before 0.0978 0.2127 0.460 0.659

Two Nights Before -0.0288 0.3093 -0.093 0.929

Weighted Two Nights -0.0481 0.2373 -0.203 0.847

Weighted Three Nights -0.0344 0.2387 -0.144 0.889

Weighted Four Nights -0.0564 0.2440 -0.231 0.824

Exponentially Weighted Two Nights -0.2104 0.2994 -0.703 0.518

Exponentially Weighted Three Nights 0.0891 0.2329 0.383 0.709

Exponentially Weighted Four Nights 0.0334 0.2477 0.135 0.896

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B3. Resting heart rate results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.0101 0.2322 -0.046 0.964

Two Nights Before 0.0502 0.2317 0.217 0.831

Weighted Two Nights 0.0160 0.2181 0.073 0.943

Weighted Three Nights 0.0263 0.2212 0.119 0.907

Weighted Four Nights 0.0283 0.2190 0.129 0.899

Exponentially Weighted Two Nights 0.1044 0.2130 0.490 0.629

Exponentially Weighted Three Nights 0.1139 0.2222 0.513 0.616

Exponentially Weighted Four Nights 0.0389 0.2126 0.183 0.857

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B4. Resting heart rate results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.0149 0.2335 -0.064 0.95

Two Nights Before 0.1686 0.2405 0.701 0.501

Weighted Two Nights 0.1076 0.1979 0.544 0.594

Weighted Three Nights 0.1187 0.1954 0.608 0.549

Weighted Four Nights 0.1262 0.1948 0.648 0.536

Exponentially Weighted Two Nights 0.1592 0.2007 0.794 0.438

Exponentially Weighted Three Nights 0.1610 0.3989 0.744 0.499

Exponentially Weighted Four Nights 0.0926 0.2569 0.361 0.733

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B5. Total sleep duration results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.4560 0.1806 -2.525 0.0181

Two Nights Before -0.4770 0.1739 -2.743 0.0107

Weighted Two Nights -0.5322 0.1694 -3.141 0.00369

Weighted Three Nights -0.5799 0.1798 -3.226 0.00292

Weighted Four Nights -0.4342 0.1676 -2.590 0.0144

Exponentially Weighted Two Nights -0.4742 0.1979 -2.397 0.0229

Exponentially Weighted Three Nights -0.6049 0.2402 -2.518 0.0169

Exponentially Weighted Four Nights -0.0940 0.1644 -0.571 0.572

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B6. Total Sleep Duration results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.4082 0.2140 -1.907 0.101

Two Nights Before -0.4815 0.2204 -2.184 0.0628

Weighted Two Nights -0.4484 0.1959 -2.289 0.0619

Weighted Three Nights -0.4723 0.2014 -2.345 0.0528

Weighted Four Nights -0.4020 0.1836 -2.190 0.0525

Exponentially Weighted Two Nights -0.4090 0.2636 -1.555 0.202

Exponentially Weighted Three Nights -0.4768 0.2707 -1.761 0.111

Exponentially Weighted Four Nights -0.2723 0.2174 -1.253 0.252

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B7. Light sleep duration results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before 0.0660 0.1923 0.343 0.734

Two Nights Before 0.0481 0.2193 0.220 0.828

Weighted Two Nights 0.1056 0.1904 0.555 0.583

Weighted Three Nights 0.1161 0.1948 0.596 0.556

Weighted Four Nights 0.1002 0.1840 0.544 0.590

Exponentially Weighted Two Nights 0.0695 0.2045 0.340 0.737

Exponentially Weighted Three Nights 0.0149 0.2130 0.070 0.945

Exponentially Weighted Four Nights 0.1159 0.1833 0.632 0.532

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B8. Light sleep duration results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before 0.0521 0.2018 0.258 0.801

Two Nights Before 0.1551 0.2445 0.634 0.541

Weighted Two Nights 0.1467 0.2165 0.678 0.514

Weighted Three Nights 0.1628 0.2272 0.716 0.491

Weighted Four Nights 0.1043 0.2005 0.520 0.613

Exponentially Weighted Two Nights 0.0740 0.2330 0.318 0.757

Exponentially Weighted Three Nights 0.0716 0.2446 0.293 0.776

Exponentially Weighted Four Nights 0.1053 0.2039 0.516 0.615

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B9. REM sleep duration results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.4006 0.1814 -2.208 0.0362

Two Nights Before -0.5661 0.1426 -3.970 0.00048

Weighted Two Nights -0.5076 0.1540 -3.297 0.00242

Weighted Three Nights -0.5009 0.1577 -3.176 0.00323

Weighted Four Nights -0.4941 0.1528 -3.234 0.00272

Exponentially Weighted Two Nights 0.4583 0.1510 -3.036 0.00479

Exponentially Weighted Three Nights -0.3925 0.1728 -2.271 0.0299

Exponentially Weighted Four Nights -0.4052 0.1505 -2.692 0.0111

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B10. REM sleep duration results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.3893 0.1768 -2.101 0.0413

Two Nights Before -0.6892 0.1711 -4.029 0.00401

Weighted Two Nights -0.5704 0.1720 -3.316 0.00943

Weighted Three Nights -0.5823 0.1819 -3.201 0.014

Weighted Four Nights -0.5718 0.1764 -3.242 0.0124

Exponentially Weighted Two Nights -0.5044 0.1619 -3.115 0.0125

Exponentially Weighted Three Nights -0.4655 0.1964 -2.370 0.0604

Exponentially Weighted Four Nights -0.4525 0.1662 -2.723 0.0217

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B11. Slow wave sleep duration results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before 0.0050 0.1799 0.028 0.978

Two Nights Before 0.1682 0.1948 0.863 0.395

Weighted Two Nights 0.0417 0.1758 0.237 0.814

Weighted Three Nights 0.0333 0.1753 0.190 0.851

Weighted Four Nights 0.1024 0.1698 0.603 0.550

Exponentially Weighted Two Nights 0.0232 0.1771 0.131 0.897

Exponentially Weighted Three Nights 0.1410 0.1812 0.778 0.442

Exponentially Weighted Four Nights 0.2663 0.1650 1.615 0.116

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B12. Slow wave sleep duration results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.0670 0.2825 0.044 0.965

Two Nights Before 0.3604 0.3108 1.160 0.350

Weighted Two Nights 0.0843 0.2170 0.388 0.716

Weighted Three Nights 0.1059 0.2537 0.418 0.695

Weighted Four Nights 0.2499 0.2415 1.035 0.349

Exponentially Weighted Two Nights 0.3462 0.3527 0.981 0.363

Exponentially Weighted Three Nights 0.1334 0.2739 0.487 0.641

Exponentially Weighted Four Nights 0.4297 0.2204 1.950 0.0874

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B13. Time in bed results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.5055 0.1740 -2.904 0.00736

Two Nights Before -0.2943 0.1911 -1.540 0.135

Weighted Two Nights -0.3889 0.1761 -2.208 0.0345

Weighted Three Nights -0.3712 0.1850 -2.006 0.0531

Weighted Four Nights -0.2706 0.1773 -1.527 0.136

Exponentially Weighted Two Nights -0.1826 0.1898 -0.962 0.343

Exponentially Weighted Three Nights -0.1015 0.2255 -0.450 0.655

Exponentially Weighted Four Nights 0.0672 0.1670 0.402 0.690

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B14. Time in bed results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.4359 0.2085 -2.091 0.107

Two Nights Before -0.3462 0.2546 -1.360 0.238

Weighted Two Nights -0.3871 0.2601 -1.488 0.202

Weighted Three Nights -0.3482 0.2907 -1.198 0.297

Weighted Four Nights -0.2815 0.2298 -1.225 0.310

Exponentially Weighted Two Nights -0.3944 0.3683 -1.071 0.344

Exponentially Weighted Three Nights -0.0468 0.5471 -0.086 0.939

Exponentially Weighted Four Nights -0.1582 0.2317 -0.683 0.532

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B15. Sleep latency results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.2319 0.1824 -1.272 0.216

Two Nights Before 0.4327 0.1979 2.187 0.0375

Weighted Two Nights 0.1882 0.1857 1.014 0.320

Weighted Three Nights 0.1686 0.1904 0.886 0.383

Weighted Four Nights 0.0945 0.1547 0.611 0.546

Exponentially Weighted Two Nights 0.2377 0.1801 1.313 0.200

Exponentially Weighted Three Nights 0.0776 0.1876 0.414 0.682

Exponentially Weighted Four Nights 0.0980 0.1567 0.626 0.536

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B16. Sleep latency results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.2285 0.1794 -1.273 0.215

Two Nights Before 0.4599 0.1932 2.380 0.0256

Weighted Two Nights 0.07145 0.2865 0.249 0.813

Weighted Three Nights 0.0403 0.2803 0.144 0.891

Weighted Four Nights 0.1669 0.2105 0.793 0.556

Exponentially Weighted Two Nights 0.2358 0.2015 1.170 0.270

Exponentially Weighted Three Nights 0.1212 0.2016 0.601 0.559

Exponentially Weighted Four Nights 0.2827 0.2575 1.098 0.416

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B17. REM cycle count results with varying intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.2278 0.1945 -1.171 0.252

Two Nights Before -0.4456 0.1591 -2.801 0.00886

Weighted Two Nights -0.4194 0.1589 -2.639 0.0127

Weighted Three Nights -0.4305 0.1585 -2.717 0.0104

Weighted Four Nights -0.4033 0.1562 -2.582 0.0143

Exponentially Weighted Two Nights -0.3143 0.1538 -2.043 0.0495

Exponentially Weighted Three Nights -0.2683 0.1735 -1.546 0.132

Exponentially Weighted Four Nights -0.2638 0.1635 -1.614 0.116

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B18. REM cycle count results with varying slope and intercept.

Variable Standardized Standard Error t-value p-value Coefficient

One Night Before -0.1026 0.2023 -0.507 0.622

Two Nights Before -0.4587 0.1663 -2.758 0.0135

Weighted Two Nights -0.4567 0.1694 -2.696 0.0173

Weighted Three Nights -0.4570 0.1678 -2.723 0.0161

Weighted Four Nights -0.4277 0.1656 -2.582 0.0209

Exponentially Weighted Two Nights -0.3515 0.1695 -2.074 0.0724

Exponentially Weighted Three Nights -0.2833 0.1858 -1.525 0.149

Exponentially Weighted Four Nights -0.3262 0.1784 -1.829 0.0929

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.00556

Table B19: Individual predictor random effects models with race course controls for significant REM sleep and total sleep variables.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1 REM Sleep Two Nights Before -0.3117 0.1291 -2.415 0.0265

Franklin Park 10K 0.2741 0.3969 0.690 0.5006

Franklin Park 5K -1.7944 0.7114 -2.522 0.0214

Franklin Park 8K -0.0671 0.3431 -0.196 0.8476

Van Cortlandt Park HEPS 8K 1.0198 0.4148 2.459 0.0267

Van Cortlandt Park ICAAAA 8K 0.9737 0.4522 2.153 0.0469

Wisconsin 8K -0.2100 0.4042 -0.520 0.6110

Model 2

REM Sleep Weighted Two Nights -0.2619 0.1357 -1.930 0.0665

Franklin Park 10K 0.4057 0.4138 0.980 0.3399

Franklin Park 5K -1.5432 0.7241 -2.131 0.0451

Franklin Park 8K 0.0044 0.3671 0.012 0.9905

Van Cortlandt Park HEPS 8K 1.1695 0.4104 2.850 0.0106

Van Cortlandt Park ICAAAA 8K 1.0582 0.4553 2.324 0.0314

Wisconsin 8K -0.1088 0.4308 -0.252 0.8035

Model 3

REM Sleep Weighted Three Nights -0.2174 0.1337 -1.626 0.1172

Franklin Park 10K 0.4591 0.414 1.108 0.2817

Franklin Park 5K -1.3649 0.7105 -1.921 0.0678

Franklin Park 8K 0.0270 0.3691 0.073 0.9424

Van Cortlandt Park HEPS 8K 1.2423 0.4073 3.050 0.00654

Van Cortlandt Park ICAAAA 8K 1.2235 0.4230 2.893 0.00922

Wisconsin 8K -0.0373 0.4274 -0.087 0.9314

Variable Standardized Standard Error t-value p-value Coefficient Model 4

REM Sleep Weighted Four Nights -0.2025 0.1327 -1.526 0.1401

Franklin Park 10K 0.4987 0.4211 1.184 0.2502

Franklin Park 5K -1.3882 0.7167 -1.937 0.0653

Franklin Park 8K 0.0598 0.3730 0.160 0.8742

Van Cortlandt Park HEPS 8K 1.2821 0.4135 3.100 0.0056

Van Cortlandt Park ICAAAA 8K 1.1894 0.4211 2.825 0.0104

Wisconsin 8K 0.0299 0.4301 0.070 0.9453

Model 5

REM Sleep Exponentially Weighted Two Nights -0.2262 0.1108 2.041 0.0543

Franklin Park 10K 0.5418 0.3937 1.376 0.1858

Franklin Park 5K -1.3804 0.6886 2.005 0.0583

Franklin Park 8K 0.0217 0.3629 0.060 0.9523

Van Cortlandt Park HEPS 8K 1.2549 0.3916 3.205 0.0050

Van Cortlandt Park ICAAAA 8K 1.1834 0.4172 2.837 0.0109

Wisconsin 8K -0.0025 0.4124 -0.006 0.9952

Model 6

Total Sleep Weighted Two Nights -0.0102 0.1665 -0.062 0.9534

Franklin Park 10K 0.6130 0.4491 1.365 0.1878

Franklin Park 5K -0.9052 0.8424 -1.075 0.2926

Franklin Park 8K -0.0124 0.4144 -0.030 0.9764

Van Cortlandt Park HEPS 8K 1.4077 0.4335 3.247 0.00436

Van Cortlandt Park ICAAAA 8K 1.5131 0.4527 3.342 0.00363

Wisconsin 8K 0.0974 0.4605 0.212 0.8347

Model 7

Total Sleep Weighted Three Nights -0.0088 0.1734 0.051 0.9598

Franklin Park 10K 0.6235 0.4381 1.423 0.1697

Variable Standardized Standard Error t-value p-value Coefficient Franklin Park 5K -0.9012 0.8373 1.076 0.2914

Franklin Park 8K -0.0004 0.4033 0.001 0.9993

Van Cortlandt Park HEPS 8K 1.4213 0.4214 3.373 0.00310

Van Cortlandt Park ICAAAA 8K 1.5407 0.4159 3.704 0.0015

Wisconsin 8K 0.1135 0.4464 0.254 0.8019

Varying Slope and Intercept Models Model 8 REM Sleep Two Nights Before -0.3365 0.1336 -2.519 0.0301

Franklin Park 10K 0.3056 0.3976 0.769 0.4544

Franklin Park 5K -1.7381 0.7059 -2.462 0.0244

Franklin Park 8K -0.0150 0.3461 -0.043 0.9661

Van Cortlandt Park HEPS 8K 1.0321 0.4153 2.485 0.0254

Van Cortlandt Park ICAAAA 8K 1.0133 0.4500 2.252 0.0387

Wisconsin 8K -0.18081 0.4053 -0.446 0.6619

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p < 0.05

Table B20: Likelihood ratio test results for race course controls.

Variable Race Degrees of Deviance Chi-Squared Pr(>Chisq) Course Freedom value Controls Varying Intercept Models REM Sleep Two Nights Before No 4 82.914

Yes 10 46.825 36.09 0.000002648

REM Sleep Weighted Two Nights No 4 87.912

Yes 10 87.912 35.62 0.000003266

REM Sleep Weighted Three Nights No 4 91.416

Yes 10 53.760 37.66 0.000001312

REM Sleep Weighted Four Nights No 4 93.019

Yes 10 56.380 36.64 0.000002071

100

Variable Race Degrees of Deviance Chi-Squared Pr(>Chisq) Course Freedom value Controls REM Sleep Exponentially Weighted Two Nights No 4 89.208

Yes 10 51.835 37.37 0.000001489

Total Sleep Weighted Two Nights No 4 88.706

Yes 10 56.769 31.94 0.00001678

Total Sleep Weighted Three Nights No 4 91.704

Yes 10 57.057 34.65 0.000005043

Varying Slope and Intercept Models REM Sleep Two Nights Before No 4 81.724

Yes 10 46.801 34.92 0.000004461

Bolded ANOVA tests are tests that reject the null hypothesis at Pr(>Chisq) < 0.05

101

Table B21: Interactions for race course controls and REM sleep two nights before.

Variable Standardized Standard t-value p-value Coefficient Error Varying Intercept Model REM Sleep Two Nights Before -0.4043 0.1642 2.463 0.0298

Franklin Park 10K 0.2786 0.4869 0.572 0.5808

Franklin Park 5K -1.8991 0.7877 -2.411 0.0340

Franklin Park 8K -0.2027 0.4592 -0.442 0.6687

Van Cortlandt Park HEPS 8K 1.0329 0.6562 1.574 0.1472

Van Cortlandt Park ICAAAA 8K 0.7701 0.6930 1.111 0.2896

Wisconsin 8K -0.4070 0.4639 -0.877 0.4023

REM Sleep Two Nights Before*Franklin Park 10K 0.4781 0.4781 0.639 0.5385

REM Sleep Two Nights Before*Franklin Park 8K 0.1122 0.2005 0.560 0.5897

REM Sleep Two Nights Before*Van Cortlandt Park HEPS 8K 0.2341 0.6324 0.370 0.7193

REM Sleep Two Nights Before* Van Cortlandt Park ICAAAA 8K -0.2148 1.0056 -0.214 0.8345

REM Sleep Two Nights Before*Wisconsin 8K 0.9708 0.5365 1.810 0.1016

Varying Slope and Intercept Model REM Sleep Two Nights Before -0.4914 0.2326 -2.113 0.0897

Franklin Park 10K 0.1959 0.4722 0.415 0.6890

Franklin Park 5K -1.9693 0.7552 -2.608 0.0298

Franklin Park 8K -0.1782 0.4876 -0.365 0.7242

Van Cortlandt Park HEPS 8K 1.1290 0.6547 1.725 0.1193

Van Cortlandt Park ICAAAA 8K 1.8295 0.7949 2.302 0.0402

Wisconsin 8K -0.4366 0.4448 -0.982 0.3555

REM Sleep Two Nights Before*Franklin Park 10K 0.4434 0.6975 0.636 0.5426

REM Sleep Two Nights Before*Franklin Park 8K 0.2838 0.1972 1.439 0.1841

REM Sleep Two Nights Before*Van Cortlandt Park HEPS 8K 0.4959 0.6324 0.784 0.4530

REM Sleep Two Nights Before* Van Cortlandt Park ICAAAA 8K 1.4837 1.1795 1.258 0.2334

REM Sleep Two Nights Before*Wisconsin 8K 1.2044 0.4772 2.524 0.0365

Bolded coefficients are results for physiological variables rejecting the null hypothesis at the alpha level of p < 0.05

102

Table B22: Likelihood ratio test results for REM sleep two nights before and race course control interactions.

Variable Interactions Degrees of Deviance Chi-Squared Pr(>Chisq) Freedom value Varying Intercept Models REM Sleep Two Nights Before No 10 46.825

Yes 15 39.962 6.86 0.2311

Varying Slope and Intercept Models REM Sleep Two Nights Before No 12 46.801

Yes 17 37.711 9.09 0.1055

Bolded ANOVA tests are tests that reject the null hypothesis at Pr(>Chisq) < 0.05

Appendix C: Extended Hypotheses Set 2 Results

Table C1: Correlations for REM and total sleep within subjects.

Variable Subject Correlation Covariance

REM and Total Sleep One Night Before Subject A 0.411 0.038

Subject B 0.491 0.152

Subject C 0.712 0.632

Subject D 0.962 0.018

Subject E 0.529 0.159

Subject F -0.221 -0.324

Subject G 0.565 -0.114

Subject H 0.925 -0.333

Subject I -0.282 0.132

REM and Total Sleep Two Nights Before Subject A 0.380 0.165

Subject B 0.605 0.450

Subject C 0.479 0.770

Subject D 0.176 0.0627

Subject E 0.870 0.831

Subject F 0.018 0.013

Subject G 0.155 0.007

103

Variable Subject Correlation Covariance Subject H 0.468 0.198

Subject I -0.109 -0.042

REM and Total Sleep Weighted Two Nights Subject A 0.757 0.387

Subject B 0.682 0.348

Subject C 0.944 1.323

Subject D 0.154 0.025

Subject E 0.790 0.351

Subject F 0.158 0.070

Subject G 0.304 0.043

Subject H 0.792 1.181

Subject I -0.081 -0.022

REM and Total Sleep Weighted Three Nights Subject A 0.822 0.350

Subject B 0.671 0.269

Subject C 0.984 1.062

Subject D 0.116 0.015

Subject E 0.850 0.295

Subject F 0.095 0.036

Subject G 0.136 0.015

Subject H 0.801 1.446

Subject I -0.133 -0.031

REM and Total Sleep Expo. Weighted Two Nights Subject A 0.897 1.233

Subject B 0.773 0.174

Subject C 0.477 0.470

Subject D 0.579 0.091

Subject E 0.885 0.552

Subject F -0.166 -0.067

Subject G -0.313 -0.012

Subject H 0.814 1.414

Subject I 0.204 0.041

104

Table C2: Complete results for multivariable random effects models with both REM sleep duration and total sleep duration leading up to a race.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.3027 0.1777 -1.704 0.101

Total Sleep One Night Before -0.3719 0.1813 -2.052 0.051

Model 2

REM Sleep Two Nights Before -0.4645 0.1461 -3.180 0.00378

Total Sleep Two Nights Before -0.3250 0.1597 -2.034 0.05284

Model 3

REM Sleep Weighted Two Nights -0.4028 0.1515 -2.660 0.0124

Total Sleep Weighted Two Nights -0.4082 0.1638 -2.492 0.0186

Model 4

REM Sleep Weighted Three Nights -0.4004 0.1540 -2.599 0.014

Total Sleep Weighted Three Nights -0.4542 0.1743 -2.606 0.014

Model 5

REM Sleep Exponentially Weighted Two Nights -0.3566 0.1647 -2.166 0.0382

Total Sleep Exponentially Weighted Two Nights -0.3046 0.2055 -1.482 0.1486

Varying Slope and Intercept Models

Model 6

REM Sleep One Night Before -0.2731 0.1543 -1.770 0.0961

Total Sleep One Night Before -0.3577 0.2083 -1.718 0.1297

Model 7

REM Sleep Two Nights Before -0.6070 0.1650 -3.678 0.00817

Total Sleep Two Nights Before -0.3252 0.2066 -1.574 0.15895

Model 8

REM Sleep Weighted Two Nights -0.4753 0.1764 -2.694 0.0474

Total Sleep Weighted Two Nights -0.3527 0.1934 -1.823 0.1165

105

Variable Standardized Standard Error t-value P-Value Coefficient Model 9

REM Sleep Weighted Three Nights -0.4836 0.1805 -2.678 0.0497

Total Sleep Weighted Three Nights -0.3715 0.2027 -1.833 0.1172

Model 10

REM Sleep Exponentially Weighted Two Nights -0.4413 0.1700 -2.595 0.0403

Total Sleep Exponentially Weighted Two Nights -0.2851 0.2592 -1.100 0.3121

Bolded coefficients represent coefficients that reject the null hypothesis at the alpha level p<0.05

Table C3: Complete results for multivariable random effects models with both REM sleep duration and total sleep duration leading up to a race, as well as their interactions.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.3740 0.2174 -1.721 0.0981

Total Sleep One Night Before -0.2889 0.2296 -1.258 0.2200

REM Sleep*Total Sleep One Night Before 0.1424 0.2401 0.593 0.5585

Model 2

REM Sleep Two Nights Before -0.6026 0.1671 -3.606 0.00143

Total Sleep Two Nights Before -0.1515 0.1821 -0.832 0.41332

REM Sleep*Total Sleep Two Nights Before 0.2886 0.1579 1.827 0.08008

Model 3

REM Sleep Weighted Two Nights -0.6100 0.1993 -3.063 0.00461

Total Sleep Weighted Two Nights -0.2063 0.1994 -1.035 0.3095

REM Sleep*Total Sleep Weighted Two Nights 0.3346 0.2049 -1.633 0.1130

Model 4

REM Sleep Weighted Three Nights -0.5821 0.1972 -2.952 0.00605

Total Sleep Weighted Three Nights -0.2691 0.2076 -1.296 0.2051

REM Sleep*Total Sleep Weighted Three Nights 0.3331 0.2221 1.500 0.1439

106

Variable Standardized Standard Error t-value P-Value Coefficient Model 5

REM Sleep Expo. Weighted Two Nights -0.5501 0.1791 -3.072 0.0045

Total Sleep Expo. Weighted Two Nights -0.2194 0.1981 -1.108 0.2768

REM Sleep*Total Sleep Expo. Weighted Two Nights 0.4059 0.1876 2.164 0.0394

Varying Slope and Intercept Models Model 6

REM Sleep One Night Before -0.3525 0.2037 -1.730 0.0992

Total Sleep One Night Before -0.2771 0.2437 -1.137 0.2743

REM Sleep*Total Sleep One Night Before 0.1424 0.2266 0.628 0.5369

Model 7

REM Sleep Two Nights Before -0.5706 0.1547 -3.689 0.00768

Total Sleep Two Nights Before -0.2019 0.2416 -0.836 0.4302

REM Sleep*Total Sleep Two Nights Before 0.4007 0.1791 2.237 0.0949

Model 8

REM Sleep Weighted Two Nights -0.6502 0.2212 -2.939 0.00674

Total Sleep Weighted Two Nights -0.1623 0.2023 -0.802 0.4712

REM Sleep*Total Sleep Weighted Two Nights 0.4466 0.2471 1.807 0.8173

Model 9

REM Sleep Weighted Three Nights -0.5907 0.1982 -2.980 0.0983

Total Sleep Weighted Three Nights -0.2373 0.2111 -1.124 0.3124

REM Sleep*Total Sleep Weighted Three Nights 0.3674 0.2386 1.540 0.3594

Model 10

REM Sleep Expo. Weighted Two Nights -0.4780 0.1646 -2.917 0.00984

Total Sleep Expo. Weighted Two Nights -0.2709 0.1949 -1.390 0.1868

REM Sleep*Total Sleep Expo. Weighted Two Nights 0.3687 0.1782 2.069 0.04903

Bolded coefficients represent coefficients that reject the null hypothesis at the alpha level p<0.05

107

Table C4: Complete results for multivariable random effects models for total and REM sleep duration with stress variable included.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.5702 0.2131 -2.676 0.0557

Total Sleep One Night Before 0.1724 0.1250 1.379 0.2293

Stress During Week -0.0551 0.2010 -0.263 0.8026

Model 2

REM Sleep Two Nights Before -0.5165 0.2088 -2.474 0.032

Total Sleep Two Nights Before -0.3345 0.2502 -1.337 0.208

Stress During Week -0.0554 0.1111 -0.498 0.631

Model 3

REM Sleep Expo. Weighted Two Nights -0.3817 0.2440 -1.564 0.137

Total Sleep Expo. Weighted Two Nights -0.0849 0.2817 -0.302 0.767

Stress During Week -0.0025 0.1414 -0.018 0.986

Varying Slope and Intercept Models Model 4

REM Sleep Two Nights Before -1.0933 0.3013 -3.628 0.0255

Total Sleep Two Nights Before -0.1382 0.2205 -0.627 0.5447

Stress During Week -0.0496 0.0999 -0.496 0.6344

Model 5

REM Sleep Expo. Weighted Two Nights -0.3245 0.2134 -1.522 0.160

Total Sleep Expo. Weighted Two Nights -0.1341 0.2652 -0.506 0.627

Stress During Week -0.0248 0.1311 -0.189 0.853

Bolded coefficients represent coefficients that reject the null hypothesis at the alpha level of p<0.05 Note: the model for REM sleep, total sleep, and stress the night before a race would not converge due to too much missing data.

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Appendix D: Extended Hypotheses Set 3 Results

Table D1: Correlation between one night before and two nights before a race for each subject.

Variable Subject Correlation Covariance Total Sleep One Night and Two Nights Before Subject A 0.989 0.596

Subject B -0.571 -0.419

Subject C -0.291 -0.305

Subject D 0.927 0.200

Subject E 0.004 0.002

Subject F 0.131 0.054

Subject G -0.005 -0.002

Subject H -0.641 -0.323

Subject I 0.681 0.929

Light Sleep One Night and Two Nights Before Subject A 0.096 0.043

Subject B -0.361 -0.145

Subject C 0.049 0.064

Subject D 0.625 0.165

Subject E 0.321 0.100

Subject F 0.340 0.206

Subject G 0.222 0.120

Subject H 0.318 0.091

Subject I 0.493 0.604

REM Sleep One Night and Two Nights Before Subject A 0.919 0.098

Subject B 0.858 0.647

Subject C 0.965 3.397

Subject D 0.570 0.011

Subject E 0.788 0.417

Subject F 0.708 0.247

Subject G 0.145 0.005

Subject H 0.589 0.517

Subject I 0.470 0.034

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Variable Subject Correlation Covariance Slow Wave Sleep One Night and Two Nights Before Subject A 0.206 0.234

Subject B 0.874 0.810

Subject C 0.492 0.349

Subject D 0.668 1.198

Subject E -0.849 -0.468

Subject F -0.945 -0.126

Subject G 0.531 0.368

Subject H 0.504 0.103

Subject I 0.583 0.383

Sleep Latency One Night and Two Nights Before Subject A 0.776 0.094

Subject B -0.152 -0.082

Subject C -0.018 -0.025

Subject D -0.866 -0.566

Subject E 0.733 -0.034

Subject F -0.323 -0.034

Subject G -0.732 -0.112

Subject H -0.613 -1.131

Subject I -0.093 -0.032

Table D2: Results for multivariable models for all sleep variables one night and two nights before a race.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before 0.0424 0.2632 0.161 0.8736

REM Sleep Two Nights Before -0.5169 0.2314 -2.234 0.0372

Model 2

Total Sleep One Night Before -0.2510 0.2022 -1.241 0.2299

Total Sleep Two Nights Before -0.4406 0.1925 -2.289 0.0335

110

Variable Standardized Standard Error t-value P-Value Coefficient Model 3

Light Sleep One Night Before 0.1493 0.2473 0.604 0.553

Light Sleep Two Nights Before -0.1421 0.2866 -0.496 0.624

Model 4

Slow Wave Sleep One Night Before -0.0792 0.1937 -0.409 0.688

Slow Wave Sleep Two Nights Before 0.3045 0.2401 1.268 0.219

Model 5

Sleep Latency One Night Before -0.1864 0.1720 -1.084 0.2910

Sleep Latency Two Nights Before 0.4445 0.2041 2.178 0.0412

Varying Slope and Intercept Models Model 6

REM Sleep One Night Before 0.0505 0.2669 0.189 0.8521

REM Sleep Two Nights Before -0.5794 0.2405 -2.409 0.0314

Model 7

Total Sleep One Night Before 0.0322 0.1737 0.186 0.855

Total Sleep Two Nights Before -0.4728 0.2558 -1.848 0.106

Model 8

Light Sleep One Night Before 0.2186 0.2219 0.985 0.338

Light Sleep Two Nights Before -0.0743 0.2985 -0.249 0.807

Model 9

Slow Wave Sleep One Night Before 0.2504 0.1863 1.344 0.243

Slow Wave Sleep Two Nights Before 1.136 0.6615 1.718 0.171

Model 10

Sleep Latency One Night Before -0.1687 0.1724 -0.978 0.3437

Sleep Latency Two Nights Before 0.4946 0.1984 2.493 0.0214

Bolded coefficients represent results that reject the null hypothesis at the alpha level p<0.05

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Table D3: Complete results for REM sleep, total sleep, and sleep latency one night before and two nights before with interaction terms included.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.0376 0.2908 -0.129 0.8983

REM Sleep Two Nights Before -0.5726 0.2873 -1.993 0.0633

REM Sleep One and Two Nights Interaction 0.0929 0.1658 0.560 0.5812

Model 2

Total Sleep One Night Before -0.2759 0.1966 -1.403 0.1779

Total Sleep Two Nights Before -0.4673 0.1877 -2.489 0.0227

Total Sleep One and Two Nights Interaction 0.2708 0.1517 1.785 0.0907

Model 3

Sleep Latency One Night Before -0.0333 0.2901 -0.115 0.9097

Sleep Latency Two Nights Before 0.5342 0.2489 2.146 0.0439

Sleep Latency One and Two Nights Interaction 0.2899 0.4368 0.664 0.5139

Varying Slope and Intercept Models Model 4

REM Sleep One Night Before 0.0338 0.2936 0.115 0.9095

REM Sleep Two Nights Before -0.5982 0.2923 -2.047 0.0615

REM Sleep One and Two Nights Interaction 0.0192 0.1775 0.108 0.9172

Model 5

Total Sleep One Night Before -0.0652 0.1674 -0.389 0.7018

Total Sleep Two Nights Before -0.5909 0.2458 -2.403 0.0447

Total Sleep One and Two Nights Interaction 0.2957 0.1438 2.057 0.0544

Model 6

Sleep Latency One Night Before -0.0506 0.2614 -0.194 0.8484

Sleep Latency Two Nights Before 0.5548 0.2284 2.429 0.0261

Sleep Latency One and Two Nights Interaction 0.2400 0.3910 0.614 0.5477

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Table D4: Complete results for REM sleep, total sleep, and sleep latency one night before and two nights before with race course indicator variables added.

Variable Standardized Standard Error t-value P-Value Coefficient Varying Intercept Models Model 1 REM Sleep One Night Before 0.1288 0.1246 1.034 0.3240

REM Sleep Two Nights Before -0.4518 0.1179 -3.831 0.00282

Franklin Park 10K 0.3668 0.2950 1.243 0.2413

Franklin Park 5K -2.1172 0.5524 -3.833 0.00263

Franklin Park 8K -0.0096 0.2513 -0.038 0.9707

Van Cortlandt Park HEPS 8K 0.8631 0.3041 2.839 0.0172

Van Cortlandt Park ICAAAA 8K 0.8144 0.3669 2.220 0.04855

Wisconsin 8K -0.3964 0.3445 -1.151 0.2754

Model 2

Total Sleep One Night Before 0.0600 0.1825 0.329 0.7467

Total Sleep Two Nights Before -0.0754 0.1650 -0.457 0.6537

Franklin Park 10K 0.5860 0.4404 1.320 0.2146

Franklin Park 5K -0.7955 0.8286 -0.960 0.3496

Franklin Park 8K -0.0832 0.4088 -0.204 0.8429

Van Cortlandt Park HEPS 8K 1.2298 0.4644 2.648 0.02624

Van Cortlandt Park ICAAAA 8K 1.6657 0.5100 3.269 0.00807

Wisconsin 8K -0.2663 0.5050 -0.527 0.6092

Model 3

Sleep Latency One Night Before 0.3409 0.1059 3.218 0.008412

Sleep Latency Two Nights Before 0.1204 0.1132 1.064 0.3084

Franklin Park 10K 0.6976 0.2972 2.347 0.040467

Franklin Park 5K -1.4650 0.5454 -2.686 0.0197

Franklin Park 8K -0.1267 0.2721 -0.466 0.6514

Van Cortlandt Park HEPS 8K 1.7107 0.3117 5.488 0.000243

Van Cortlandt Park ICAAAA 8K 1.5695 0.3719 4.220 0.001381

Wisconsin 8K 0.3920 0.3620 1.083 0.3034

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Variable Standardized Coefficient Standard Error t-value P-Value Varying Slope and Intercept Models

Model 4

REM Sleep One Night Before 0.1319 0.1270 1.038 0.3297

REM Sleep Two Nights Before -0.4478 0.1190 -3.764 0.00498

Franklin Park 10K 0.3475 0.2943 1.181 0.2665

Franklin Park 5K -2.1244 0.5523 -3.847 0.00269

Franklin Park 8K -0.0248 0.2518 -0.098 0.9237

Van Cortlandt Park HEPS 8K 0.8554 0.3033 2.820 0.0193

Van Cortlandt Park ICAAAA 8K 0.8140 0.3672 2.217 0.0493

Wisconsin 8K -0.4171 0.3437 -1.214 0.2528

Model 5

Total Sleep One Night Before 0.1467 0.1888 0.777 0.4878

Total Sleep Two Nights Before 0.0020 0.1765 0.011 0.9913

Franklin Park 10K 0.3800 0.4176 0.910 0.3861

Franklin Park 5K -0.1907 0.8134 -0.234 0.8249

Franklin Park 8K -0.1324 0.3666 -0.361 0.7259

Van Cortlandt Park HEPS 8K 1.2985 0.4193 3.097 0.0122

Van Cortlandt Park ICAAAA 8K 1.773 0.4916 3.607 0.00282

Wisconsin 8K -0.3994 0.4548 -0.878 0.3979

Model 6

Sleep Latency One Night Before 0.3765 0.1048 3.592 0.005589

Sleep Latency Two Nights Before 0.1618 0.1107 1.462 0.1994

Franklin Park 10K 0.6690 0.2861 2.338 0.0444

Franklin Park 5K -1.4465 0.5274 -2.743 0.0190

Franklin Park 8K -0.1428 0.2655 -0.538 0.6028

Van Cortlandt Park HEPS 8K 1.7053 0.3029 5.629 0.000284

Van Cortlandt Park ICAAAA 8K 1.6524 0.3593 4.599 0.001005

Wisconsin 8K 0.4250 0.3528 1.205 0.2579

Bolded coefficients are physiological predictors that reject the null hypothesis at the alpha level of p < 0.05

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Figure D1: Regression diagnostics for the varying intercept model examining latency one night before and two nights before a race. Residual Histogram Fitted Vs. Residuals 0.4 8

0.2

0.0

count

Residuals -0.2

-0.4

-0.75 -0.50 -0.25 0.00 0.25 0.50 -1 0 1 2 Residuals Fitted

Figure D2: Regression diagnostics for varying intercept model examining REM sleep one night before and two nights before a race. Residual Histogram Fitted Vs. Residuals

6 0.4

0.2

0.0

count

Residuals 2

-0.2

0 -0.4 -0.3 0.0 0.3 0.6 -1 0 1 2 Residuals Fitted

115

Figure D3: Regression diagnostics for varying slope and intercept model examining sleep latency one night before and two nights before a race. Residual Histogram

count

-0.75 -0.50 -0.25 0.00 0.25 0.50 Residuals

Figure D4: Regression diagnostics for varying slope and intercept model examining REM sleep one night before and two nights before a race. Residual Histogram Fitted Vs. Residuals

6 0.4

0.2

0.0

count

Residuals

-0.2

0 -0.4 -0.3 0.0 0.3 0.6 -1 0 1 2 Residuals Fitted

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Appendix E: Extended Hypotheses Set 4 Results

Table E1: Correlations for REM, total sleep, and sleep latency and HRV within subjects.

Variable Subject Correlation Covariance

rMSSD HRV and REM Sleep One Night Before Subject A -0.304 -0.087

Subject B -0.414 0.053

Subject C -0.507 -0.450

Subject D -0.982 -0.014

Subject E 0.011 0.006

Subject F -0.938 -0.315

Subject G 0.701 0.092

Subject H 0.991 0.243

Subject I 0.004 0.001

rMSSD HRV and REM Sleep Two Nights Before Subject A -0.175 -0.076

Subject B -0.766 -0.160

Subject C -0.031 -0.017

Subject D -0.663 -0.158

Subject E -0.849 -0.152

Subject F 0.176 0.033

Subject G -0.786 -0.017

Subject H -0.984 -0.286

Subject I 0.756 0.119

rMSSD HRV and REM Sleep Weighted Two Nights Subject A 0.302 0.129

Subject B -0.574 -0.051

Subject C -0.412 -0.184

Subject D -0.882 -0.132

Subject E -0.230 -0.154

Subject F -0.065 -0.022

Subject G 0.896 0.070

Subject H -0.410 -0.106

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Variable Subject Correlation Covariance Subject I 0.522 0.047 rMSSD HRV and Total Sleep One Night Before Subject A 0.738 1.000

Subject B -0.081 -0.008

Subject C -0.201 -0.130

Subject D -0.893 -0.121

Subject E 0.852 0.829

Subject F -0.130 -0.055

Subject G 0.165 0.050

Subject H 0.968 0.109

Subject I -0.248 -0.179 rMSSD HRV and Total Sleep Two Nights Before Subject A -0.480 -0.246

Subject B 0.012 0.004

Subject C 0.333 0.076

Subject D 0.604 0.313

Subject E -0.865 -0.103

Subject F 0.812 0.129

Subject G -0.538 -0.075

Subject H -0.398 -0.166

Subject I 0.565 0.319 rMSSD HRV and Total Sleep Weighted Two Nights Subject A 0.772 0.816

Subject B -0.029 -0.002

Subject C -0.396 -0.064

Subject D -0.0976 -0.019

Subject E 0.329 0.196

Subject F 0.566 0.160

Subject G -0.030 -0.004

Subject H -0.750 -0.220

Subject I -0.116 -0.058

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Variable Subject Correlation Covariance rMSSD HRV and Sleep Latency One Night Before Subject A -0.293 -0.602

Subject B -0.403 -0.053

Subject C -0.18 -0.074

Subject D 0.827 0.195

Subject E 0.071 0.083

Subject F 0.448 0.110

Subject G -0.785 -0.269

Subject H 0.337 0.800

Subject I 0.746 0.622 rMSSD HRV and Sleep Latency Two Nights Before Subject A -0.490 -0.033

Subject B 0.463 0.067

Subject C 0.192 0.098

Subject D -0.351 -0.186

Subject E 0.460 0.064

Subject F -0.969 -0.098

Subject G -0.495 -0.020

Subject H 0.478 0.303

Subject I 0.953 0.120 rMSSD HRV and Sleep Latency Weighted Two Nights Subject A -0.687 0.143

Subject B -0.189 0.346

Subject C 0.164 1.990

Subject D -0.641 0.994

Subject E -0.189 1.794

Subject F -0.265 0.752

Subject G -0.872 0.323

Subject H -0.092 0.868

Subject I 0.799 0.425

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Table E2: Results for random effects models for sleep and rMSSD HRV.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.4302 0.1811 -2.375 0.0256

rMSSD HRV One Night Before 0.2403 0.1791 1.341 0.1920

Model 2

REM Sleep Two Nights Before -0.5548 0.1440 -3.853 0.000657

rMSSD HRV Two Nights Before 0.2317 0.2424 0.956 0.3501

Model 3

REM Sleep Weighed Two Nights -0.5385 0.1549 -3.476 0.00159

rMSSD HRV Weighted Two Nights 0.3083 0.1970 1.565 0.1293

Model 4

Total Sleep One Night Before -0.5376 0.1756 -3.062 0.00535

rMSSD HRV One Night Before 0.3010 0.1703 1.819 0.0813

Model 5

Total Sleep Two Nights Before -0.4897 0.1733 -2.826 0.00916

rMSSD HRV Two Nights Before 0.1812 0.2568 0.706 0.4894

Model 6

Total Sleep Weighed Two Nights -0.6059 0.1661 -3.647 0.001

rMSSD HRV Weighted Two Nights 0.3116 0.2087 1.493 0.1466

Model 7

Sleep Latency One Night Before -0.2119 0.1848 -1.147 0.264

rMSSD HRV One Night Before 0.1528 0.1931 0.791 0.436

Model 8

Sleep Latency Two Nights Before 0.4256 0.2017 2.110 0.0451

rMSSD HRV Two Nights Before 0.0791 0.2387 0.331 0.7450

Model 9

Sleep Latency Weighed Two Nights 0.2124 0.1862 1.141 0.264

rMSSD HRV Weighted Two Nights 0.1808 0.2133 0.848 0.408

120

Variable Standardized Standard Error t-value p-value Coefficient Varying Slope and Intercept Models Model 10

REM Sleep One Night Before -0.4807 0.1763 -2.726 0.0119

rMSSD HRV One Night Before 0.1908 0.2262 0.843 0.4377

Model 11

REM Sleep Two Nights Before -0.5766 0.1324 -4.354 0.00023

rMSSD HRV Two Nights Before 0.0038 0.3496 0.011 0.9916

Model 12

REM Sleep Weighted Two Nights -0.5933 0.1519 -3.905 0.000561

rMSSD HRV Weighted Two Nights 0.1438 0.2468 0.583 0.5767

Model 13

Total Sleep One Night Before -0.4716 0.2094 -2.252 0.0631

rMSSD HRV One Night Before 0.3021 0.1575 1.918 0.0685

Model 14

Total Sleep Weighted Two Nights -0.5139 0.1808 -2.842 0.0244

rMSSD HRV Weighted Two Nights 0.2622 0.2023 1.296 0.2076

Model 15

Sleep Latency One Night Before -0.1893 0.1812 -1.045 0.308

rMSSD HRV One Night Before 0.0907 0.2084 0.435 0.687

Model 16

Sleep Latency Weighted Two Nights 0.0259 0.2949 0.088 0.933

rMSSD HRV Weighted Two Nights -0.0050 0.2412 -0.021 0.984

Bolded coefficients are coefficients that reject the null hypothesis at the alpha level of p < 0.05 Note: Some varying slope and intercept models failed to converge.

121

Table E3: Sleep and rMSSD HRV Interactions

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.4650 0.1929 -2.410 0.0241

rMSSD HRV One Night Before 0.2723 0.1871 1.455 0.1585

REM Sleep*rMSSD HRV One Night Before 0.1632 0.2652 0.615 0.5447

Model 2

REM Sleep Two Nights Before -0.6844 0.1964 -3.484 0.0021

rMSSD HRV One Night Before 0.3185 0.23313 1.366 0.1912

REM Sleep*rMSSD HRV Two Nights Before 0.2274 0.1847 1.231 0.2292

Model 3

REM Sleep Weighted Two Nights -0.6495 0.1753 -3.704 0.000928

rMSSD HRV Weighted Two Nights 0.3815 0.1863 2.048 0.0538

REM Sleep*rMSSD HRV Weighted Two Nights 0.3237 0.1962 1.650 0.1096

Model 4

Total Sleep One Night Before -0.5794 0.1887 -3.071 0.00517

rMSSD HRV One Night Before 0.3851 0.2111 1.824 0.0801

Total Sleep*rMSSD HRV One Nights Before -0.1097 0.1891 -0.580 0.5677

Model 5

Total Sleep Two Nights Before -0.4747 0.1776 -2.673 0.0132

rMSSD HRV Two Nights Before 0.1833 0.2598 0.706 0.4894

Total Sleep*rMSSD HRV Two Nights Before 0.0961 0.1795 0.536 0.5969

Model 6

Total Sleep Weighted Two Nights -0.5882 0.1713 -3.434 0.00182

rMSSD HRV Weighted Two Nights 0.2874 0.2148 1.338 0.1926

Total Sleep*rMSSD Weighted Two Nights 0.0754 0.1782 0.423 0.6755

Model 7

Sleep Latency One Night Before -0.2478 0.1943 -1.275 0.216

rMSSD HRV One Night Before 0.1889 0.2026 0.933 0.360

Sleep Latency*rMSSD HRV One Nights Before -0.1049 0.1401 -0.733 0.472

122

Variable Standardized Standard Error t-value p-value Coefficient Model 8

Sleep Latency Two Nights Before 0.2524 0.2881 0.876 0.389 rMSSD HRV Two Nights Before 0.1204 0.2384 0.505 0.623

Sleep Latency *rMSSD HRV Two Nights Before 0.1909 0.2306 0.828 0.415

Model 9

Sleep Latency Weighted Two Nights 0.2319 0.2103 1.103 0.279 rMSSD HRV Weighted Two Nights 0.1777 0.2180 0.815 0.426

Sleep Latency *rMSSD Weighted Two Nights -0.0328 0.1631 -0.201 0.842

Varying Slope and Intercept Models Model 10

REM Sleep One Night Before -0.5064 0.1839 -2.754 0.0116 rMSSD HRV One Night Before 0.2238 0.2399 0.933 0.3965

REM Sleep*rMSSD HRV One Night Before 0.1585 0.2542 0.624 0.5394

Model 11

REM Sleep Two Nights Before -0.7021 0.1764 -3.980 0.000554 rMSSD HRV One Night Before 0.0876 0.3218 0.272 0.7941

REM Sleep*rMSSD HRV Two Nights Before 0.1947 0.1700 1.145 0.2625

Model 12

REM Sleep Weighted Two Nights -0.6778 0.1680 -4.034 0.000441 rMSSD HRV Weighted Two Nights 0.2337 0.2392 0.977 0.3614

REM Sleep*rMSSD HRV Weighted Two Nights 0.2743 0.1844 1.488 0.1481

Model 13

Total Sleep One Night Before -0.4989 0.2123 -2.351 0.0484 rMSSD HRV One Night Before 0.3581 0.1906 1.879 0.0755

Total Sleep*rMSSD HRV One Night Before -0.0887 0.1705 -0.521 0.6092

Model 14

Total Sleep Weighted Two Nights -0.4988 0.1856 -2.687 0.0298 rMSSD HRV Weighted Two Nights 0.2336 0.2125 1.099 0.2824

Total Sleep*rMSSD HRV Weighted Two Nights 0.0626 0.1862 0.336 0.7402

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Variable Standardized Standard Error t-value p-value Coefficient Model 15

Sleep Latency One Night Before -0.2475 0.1943 -1.274 0.219

rMSSD HRV One Night Before 0.1757 0.2082 0.843 0.457

Sleep Latency*rMSSD HRV One Night Before -0.1182 0.1454 -0.813 0.431

Model 16

Sleep Latency Weighted Two Nights -0.1409 0.4349 -0.324 0.7394

rMSSD HRV Weighted Two Nights -0.0038 0.2061 -0.018 0.9870

Sleep Latency*rMSSD HRV Weighted Two Nights 0.4513 0.2378 -1.898 0.0816

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.05 Note: Some varying slope and intercept models did not converge.

Table E4: Sleep and rMSSD random effects models with stress variable included.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.5861 0.1992 -2.942 0.0408

rMSSD HRV One Night Before 0.0561 0.1621 0.346 0.7455

Stress During Week 0.1959 0.1309 1.497 0.1899

Model 2

REM Sleep Two Nights Before -0.6959 0.1790 -3.888 0.00317

rMSSD HRV One Night Before -0.0058 0.2598 -0.022 0.9826

Stress During Week -0.0621 0.1121 -0.554 0.5930

Model 3

REM Sleep Weighted Two Nights -0.5669 0.2239 -2.532 0.0302

rMSSD HRV Weighted Two Nights 0.0283 0.2722 0.104 0.9186

Stress During Week 0.0364 0.1227 0.296 0.7720

Model 4

Total Sleep One Night Before -0.5115 0.3325 -1.538 0.152

rMSSD HRV One Night Before 0.3257 0.2862 1.138 0.283

Stress During Week 0.2710 0.1617 1.676 0.146

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Model 5

Total Sleep Two Nights Before -0.6085 0.2530 -2.406 0.0316 rMSSD HRV Two Nights Before -0.1969 0.2413 -0.816 0.4371

Stress During Week 0.0357 0.1388 -0.257 0.8005

Model 6

Total Sleep Weighted Two Nights -0.4065 0.2477 -1.641 0.126 rMSSD HRV Weighted Two Nights -0.0613 0.2696 -0.227 0.825

Stress During Week 0.0251 0.1419 0.177 0.862

Model 7

Sleep Latency One Night Before -0.3210 0.3288 -0.976 0.368 rMSSD HRV One Night Before 0.0858 0.2472 0.347 0.741

Stress During Week 0.3330 0.1852 1.799 0.108

Model 8

Sleep Latency Two Nights Before 0.4836 0.2692 1.796 0.0100 rMSSD HRV Two Nights Before -0.2379 0.2709 -0.878 0.3958

Stress During Week -0.0231 0.1597 -0.145 0.8872

Model 9

Sleep Latency Weighted Two Nights 0.1828 0.2302 0.794 0.444 rMSSD HRV Weighted Two Nights -0.1661 0.2844 -0.584 0.569

Stress During Week 0.0450 0.1520 0.296 0.772

Varying Slope and Intercept Models Model 10

REM Sleep Two Nights Before -0.4897 0.1090 -4.494 0.00424 rMSSD HRV One Night Before -0.7587 0.5653 -1.342 0.2353

Stress During Week 0.0605 0.0778 0.778 0.4614

Model 11

REM Sleep Weighted Two Nights -0.5796 0.2269 -2.554 0.0305 rMSSD HRV Weighted Two Nights -0.0001 0.2804 0.000 0.997

Stress During Week 0.0392 0.1209 0.324 0.7513

125

Variable Standardized Standard Error t-value p-value Coefficient Model 12

Total Sleep Weighted Two Nights -0.3699 0.2598 -1.424 0.197

rMSSD HRV Weighted Two Nights -0.1644 0.2934 -0.560 0.615

Stress During Week 0.0220 0.1357 0.162 0.874

Model 13

Sleep Latency Weighted Two Nights -0.3522 0.3267 -1.078 0.3244

rMSSD HRV Weighted Two Nights -0.6418 0.2980 -2.153 0.0896

Stress During Week 0.2439 0.0631 3.866 0.0202

Bolded coefficients are results that reject the null hypothesis at the alpha level of p < 0.05 Note: Several varying slope and intercept models did not converge.

Table E5: Sleep and rMSSD HRV random effects models with race course control variables.

Variable Standardized Standard Error t-value p-value Coefficient Varying Intercept Models Model 1

REM Sleep One Night Before -0.1301 0.1685 -0.772 0.4535

rMSSD HRV One Night Before 0.0677 0.1473 0.460 0.6529

Franklin Park 10K 0.4409 0.5374 0.820 0.4285

Franklin Park 5K -1.1531 0.7786 -1.481 0.1603

Franklin Park 8K -0.0440 0.4047 -0.109 0.9154

Van Cortlandt Park HEPS 8K 1.1819 0.4898 2.413 0.0339

Van Cortlandt Park ICAAAA 8K 1.2787 0.5199 2.459 0.0306

Wisconsin 8K -0.3402 0.557 -0.611 0.5523

Model 2

REM Sleep Two Nights Before -0.3010 0.1335 -2.255 0.0381

rMSSD HRV Two Nights Before -0.1137 0.1947 -0.584 0.5650

Franklin Park 10K 0.3629 0.4254 0.853 0.4064

Franklin Park 5K -1.7042 0.7476 -2.279 0.0344

Franklin Park 8K -0.0145 0.3521 -0.041 0.9678

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Variable Standardized Standard Error t-value p-value Coefficient Van Cortlandt Park HEPS 8K 1.1283 0.4582 2.463 0.0253

Van Cortlandt Park ICAAAA 8K 1.0897 0.5083 2.144 0.0463

Wisconsin 8K -0.1041 0.4411 -0.236 0.8164

Model 3

REM Sleep Weighted Two Nights -0.2810 0.1464 -1.919 0.0703 rMSSD HRV Weighted Two Nights 0.0860 0.1643 0.524 0.6054

Franklin Park 10K 0.2972 0.4698 0.633 0.5350

Franklin Park 5K -1.5674 0.7500 -2.090 0.0495

Franklin Park 8K -0.0230 0.3795 -0.061 0.9524

Van Cortlandt Park HEPS 8K 1.0819 0.4512 2.398 0.0279

Van Cortlandt Park ICAAAA 8K 0.9899 0.4906 2.018 0.0591

Wisconsin 8K -0.1999 0.4718 -0.424 0.6770

Model 4

Total Sleep One Night Before 0.0623 0.1935 0.322 0.7509 rMSSD HRV One Night Before 0.0118 0.1494 0.079 0.9377

Franklin Park 10K 0.6328 0.4879 1.297 0.2173

Franklin Park 5K -1.0741 0.8187 -1.312 0.2060

Franklin Park 8K -0.0288 0.4059 -0.071 0.9447

Van Cortlandt Park HEPS 8K 1.3510 0.4618 2.926 0.0124

Van Cortlandt Park ICAAAA 8K 1.5699 0.4879 3.217 0.0076

Wisconsin 8K -0.1293 0.5122 -0.253 0.8048

Model 5

Total Sleep Two Nights Before 0.0041 0.1451 0.028 0.9777 rMSSD HRV Two Nights Before -0.1410 0.1900 -0.742 0.4693

Franklin Park 10K 0.6342 0.4682 1.384 0.1835

Franklin Park 5K -0.7970 0.7856 -1.014 0.3215

Franklin Park 8K -0.0275 0.4214 -0.065 0.9488

Van Cortlandt Park HEPS 8K 1.5457 0.4874 3.171 0.00500

Van Cortlandt Park ICAAAA 8K 1.7327 0.4809 3.603 0.00183

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Variable Standardized Standard Error t-value p-value Coefficient Wisconsin 8K 0.0946 0.4835 0.196 0.8470

Model 6

Total Sleep Weighted Two Nights -0.0118 0.1800 -0.065 0.9484 rMSSD HRV Weighted Two Nights 0.00388 0.1761 0.022 0.9826

Franklin Park 10K 0.6133 0.4762 1.288 0.2125

Franklin Park 5K -0.9102 0.8684 -1.048 0.3047

Franklin Park 8K -0.0101 0.4216 -0.024 0.9812

Van Cortlandt Park HEPS 8K 1.4082 0.4600 3.061 0.00617

Van Cortlandt Park ICAAAA 8K 1.5105 0.4734 3.191 0.00478

Wisconsin 8K 0.0991 0.4812 0.206 0.8389

Model 7

Sleep Latency One Night Before 0.3778 0.1064 3.551 0.003903 rMSSD HRV One Night Before -0.0700 0.1007 -0.691 0.5021

Franklin Park 10K 0.9330 0.3355 2.781 0.0167

Franklin Park 5K -1.3682 0.5089 -2.688 0.0192

Franklin Park 8K -0.0815 0.2755 -0.296 0.7727

Van Cortlandt Park HEPS 8K 1.9032 0.3356 5.672 0.000104

Van Cortlandt Park ICAAAA 8K 1.7306 0.3014 5.742 0.000110

Wisconsin 8K 0.5751 0.3748 1.534 0.1551

Model 8

Sleep Latency Two Nights Before 0.1489 0.1266 1.176 0.2544 rMSSD HRV Two Nights Before -0.1294 0.1849 -0.700 0.4942

Franklin Park 10K 0.5353 0.4519 1.185 0.2511

Franklin Park 5K -0.9622 0.7390 -1.302 0.2066

Franklin Park 8K -0.0365 0.4058 -0.090 0.9293

Van Cortlandt Park HEPS 8K 1.4661 0.4456 3.290 0.00374

Van Cortlandt Park ICAAAA 8K 1.5412 0.4812 3.203 0.00421

Wisconsin 8K 0.0461 0.4697 0.098 0.9229

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Variable Standardized Standard Error t-value p-value Coefficient Model 9

Sleep Latency Weighted Two Nights 0.3072 0.0932 3.298 0.00377 rMSSD HRV Weighted Two Nights 0.1474 0.1500 0.983 0.3353

Franklin Park 10K 0.2054 0.4023 0.511 0.6152

Franklin Park 5K -1.4174 0.6108 -2.320 0.0311

Franklin Park 8K -0.1444 0.3303 -0.437 0.6673

Van Cortlandt Park HEPS 8K 1.3159 0.3560 3.697 0.00159

Van Cortlandt Park ICAAAA 8K 1.2458 0.3580 3.480 0.00260

Wisconsin 8K 0.0301 0.3795 0.082 0.9358

Varying Slope and Intercept Models Model 10

REM Sleep One Night Before -0.1676 0.1598 -1.049 0.3140 rMSSD HRV One Night Before 0.0408 0.1787 0.228 0.8305

Franklin Park 10K 0.4761 0.5179 0.919 0.3758

Franklin Park 5K -1.0189 0.6966 -1.463 0.1687

Franklin Park 8K -0.0176 0.3820 -0.046 0.9641

Van Cortlandt Park HEPS 8K 1.1477 0.4687 2.449 0.0307

Van Cortlandt Park ICAAAA 8K 1.2618 0.4810 2.623 0.0228

Wisconsin 8K -0.3491 0.5309 -0.658 0.5227

Model 11

REM Sleep Two Nights Before -0.2686 0.1290 -2.082 0.2382 rMSSD HRV Two Nights Before -0.1115 0.3103 -0.370 0.7232

Franklin Park 10K 0.3406 0.4309 0.791 0.4476

Franklin Park 5K -1.5290 0.7159 -2.136 0.0475

Franklin Park 8K 0.0052 0.3826 0.014 0.9899

Van Cortlandt Park HEPS 8K 1.2010 0.5043 2.399 0.0310

Van Cortlandt Park ICAAAA 8K 1.0461 0.5242 1.996 0.0788

Wisconsin 8K -0.0180 0.4444 -0.041 0.9691

Model 12

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Variable Standardized Standard Error t-value p-value Coefficient REM Sleep Weighted Two Nights -0.3292 0.1565 -2.103 0.0893 rMSSD HRV Weighted Two Nights -0.0128 0.2189 -0.058 0.9555

Franklin Park 10K 0.4938 0.4395 1.124 0.2809

Franklin Park 5K -1.2649 0.6396 -1.978 0.0704

Franklin Park 8K 0.1203 0.3533 0.340 0.7384

Van Cortlandt Park HEPS 8K 1.1591 0.4027 2.879 0.0153

Van Cortlandt Park ICAAAA 8K 1.1766 0.4347 2.707 0.0190

Wisconsin 8K -0.0005 0.4264 -0.001 0.9992

Model 13

Total Sleep One Night Before 0.0205 0.2180 0.094 0.9271 rMSSD HRV One Night Before -0.0431 0.1721 -0.251 0.8103

Franklin Park 10K 0.8348 0.4120 2.027 0.0675

Franklin Park 5K 0.5375 0.8436 0.637 0.5429

Franklin Park 8K -0.0226 0.3179 -0.071 0.9448

Van Cortlandt Park HEPS 8K 1.3223 0.3750 3.526 0.00503

Van Cortlandt Park ICAAAA 8K 1.7928 0.3984 4.500 0.00097

Wisconsin 8K -0.3070 0.4251 -0.722 0.4845

Model 14

Total Sleep Weighted Two Nights 0.1414 0.2082 0.679 0.5146 rMSSD HRV Weighted Two Nights -0.2060 0.2033 -1.013 0.3517

Franklin Park 10K 0.7847 0.4334 1.811 0.0890

Franklin Park 5K 0.6264 0.6074 1.031 0.3273

Franklin Park 8K 0.0198 0.3529 0.056 0.9560

Van Cortlandt Park HEPS 8K 1.6083 0.3966 4.056 0.000962

Van Cortlandt Park ICAAAA 8K 1.8391 0.4355 4.223 0.000950

Wisconsin 8K 0.1147 0.4178 0.274 0.7868

Model 15

Sleep Latency One Night Before 0.3525 0.0962 3.664 0.00355 rMSSD HRV One Night Before -0.0609 0.1122 -0.542 0.6110

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Variable Standardized Standard Error t-value p-value Coefficient Franklin Park 10K 0.9653 0.3275 2.947 0.0124

Franklin Park 5K -1.1803 0.4743 -2.488 0.0280

Franklin Park 8K -0.0349 0.2629 -0.133 0.8968

Van Cortlandt Park HEPS 8K 1.8528 0.3220 5.754 0.0000923

Van Cortlandt Park ICAAAA 8K 1.7701 0.2911 6.081 0.0000643

Wisconsin 8K 0.5731 0.3633 1.578 0.1410

Model 16

Sleep Latency Weighted Two Nights 0.3087 0.0954 3.235 0.00654

rMSSD HRV Weighted Two Nights 0.1341 0.1470 0.912 0.3938

Franklin Park 10K 0.2571 0.4107 0.626 0.5395

Franklin Park 5K -1.2999 0.6056 -2.147 0.0445

Franklin Park 8K -0.1183 0.3352 -0.353 0.7286

Van Cortlandt Park HEPS 8K 1.3522 0.3643 3.712 0.00163

Van Cortlandt Park ICAAAA 8K 1.3340 0.3726 3.581 0.00253

Wisconsin 8K 0.0486 0.3907 0.124 0.9025

Bolded coefficients are significant physiological predictors at the alpha level of p < 0.05. Note: several varying slope and intercept models did not converge.

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