Biostatistics D Final Project

A.Reviewing and adressing Missing Data and Outliers in the Dataset

After thoroughly reviewing the dataset, we observed significant patterns of missing data, particularly in key variables such as “avg_pain” and “avg_stress,” which both exhibited 45.5% missing data. These variables play a critical role in our analysis, especially in testing hypotheses related to the effects of HRV-B training and mediation models.

In addition to the missing data, we identified several problematic outliers. These include extreme values in “age_in_years” (values below 1 year) - 4 values, “height_in_cm” (values above 250 cm) - 6 Values, “total_duration_in_min” (values exceeding 6000 minutes) - 14 values, and “avg_stress” (values above 10) - 1 value. These extreme values were deemed implausible and could distort the results of the analysis if left unaddressed. Consequently, these outliers were removed from the dataset to ensure more accurate modeling and to avoid skewing the findings.

With all missing values and outliers removed, the dataset is now prepared for further analysis. From this point forward, no missing or extreme values will affect the results, allowing us to focus on the key research questions with a more consistent and reliable dataset.

A1.Descriptive Statistics of the Study Population

The descriptive statistics of the study population reveal key sociodemographic and anthropometric insights. As visualized in the charts, the age distribution of participants shows a predominance of individuals between 20 to 60 years, with a notable peak around the 30 to 40-year range. The range of ages spans from approximately 16 years to 93 years, with a fairly consistent distribution across both genders.

In terms of weight, the majority of the participants fall within the 60 to 90 kg range, with a few outliers indicating participants weighing either below 50 kg or over 120 kg. The weight distribution shows a higher density around the 70 kg mark, indicating a balanced distribution of weight in the study population.

The height distribution shows the majority of participants measuring between 160 cm and 180 cm, with a clear peak around 170 cm. A small number of participants were taller, with a few extreme cases above 200 cm, but these are rare. The spread in height appears to follow a relatively normal distribution.

The gender distribution, as shown in the bar plot, indicates a slightly higher number of female participants compared to males in the study. This pattern aligns with the overall demographic breakdown in the sample, where 56% of the participants were female and 44% male.

These charts provide a foundational overview of the population’s demographics, which will be crucial in interpreting the HRV-B training and its potential effects on pain and stress levels, as well as other variables.

A2.Training and Week Index Distribution: Overview of Participation Trends

The bar plots illustrate the distribution of the number of records across the week index and the number of days of training in a week. The chart on the left shows a higher concentration of records at the start of the intervention (Week 0), followed by a more gradual decrease across subsequent weeks. This suggests a higher initial response rate in the early stages of the HRV-B training intervention, tapering off as the intervention progresses.

The chart on the right highlights the number of days participants trained per week. Most participants reported training between 2 to 4 days per week, with the highest concentration at 3 days. Fewer participants reported training 1 or 7 days a week, indicating that most participants engaged in moderate training frequency rather than extreme values. These insights provide a clearer understanding of participation trends during the HRV-B intervention, which may be relevant in modeling the effectiveness of the training on pain and stress outcomes.

A3.Average Total Duration by Week Index and Days of Training

Average Total Duration by Week Index and Days of Training The chart shows the average total training duration (in minutes) across the weeks of HRV-B training, grouped by the number of training days per week. Participants who trained 7 days a week (pink line) consistently recorded the highest training duration, with peaks over 3000 minutes, showing significant variability across weeks. Those training fewer days (1-2 days, red and yellow lines) maintained lower, more stable durations, below 500 minutes.

Overall, as the number of training days increases, the total training duration rises, with more frequent training leading to higher cumulative minutes. Week-to-week fluctuations are most pronounced for those training 7 days a week.

A4.Correlation Between Stress, Pain, Week Index, and Days of Training

The table below presents the correlation matrix for the key variables in the dataset: avg_stress, avg_pain, week_index, and days_training_in_week. Notably, avg_stress and avg_pain are the only dependent variables in this analysis, reflecting the primary outcomes of interest in assessing intervention effects.

Average Stress (avg_stress) exhibits a slight positive correlation with week_index (0.28), suggesting that stress levels have a tendency to increase slightly over time during the intervention period. Average Pain (avg_pain) shows a moderate negative correlation with week_index (-0.34), indicating a general decrease in pain over time, aligning with the observed trends in the data. The correlation between days_training_in_week and avg_stress is weak (0.07), indicating minimal association between training frequency and stress levels. Similarly, the correlation between days_training_in_week and avg_pain is slightly negative (-0.08), suggesting a minor association between increased training frequency and reduced pain levels.

These observations underscore that while avg_stress and avg_pain are the primary dependent outcomes in this dataset, their relationships with other variables, such as time and training frequency, highlight nuanced patterns in the intervention’s effects.

A1.Modeling The Trajectory Of Pain, Stress And HRV-B Training Duration Fluctuations

The charts below illustrate the trajectories of average pain, average stress, and HRV-B training duration over both week index and days of training per week.

Pain Trajectory The “Trajectory of Average Pain over Week Index” shows a notable decline in pain levels, especially in the initial weeks, with pain stabilizing around week 4. This suggests that continuous HRV-B training may contribute to pain reduction over time. The “Trajectory of Average Pain over Days Training in Week” displays a decrease in pain levels with more training days per week. This trend reinforces the potential impact of frequent HRV-B training on reducing pain levels.

Stress Trajectory In the “Trajectory of Average Stress over Week Index”, there is an initial increase in stress levels, which then plateaus. This pattern might reflect participants’ adaptation to the HRV-B intervention, with stress stabilizing as they acclimate to the training.

The “Trajectory of Average Stress over Days Training in Week” chart shows a slight increase in stress as the number of training days rises. This could imply that while frequent training benefits pain management, it may contribute to mild stress elevation.

HRV-B Training Duration Trajectory The “Trajectory of HRV-B Duration over Week Index” reveals a gradual decrease in training duration over time, potentially indicating adaptation or increased efficiency in training. Conversely, the “Trajectory of HRV-B Duration over Days Training in Week” shows a steady increase with more training days, highlighting the importance of training frequency in sustaining HRV-B engagement.

A2.Linear Mixed Effects Model For Repeated Measures On Avg_Pain, Avg_Stress, Total_Duration_In_Min

The table below summarizes linear mixed effects models used to analyze the trajectory of pain, stress, and HRV-B training duration over time, represented by the week_index variable. Each model evaluates the effect of time on these outcomes separately.

For pain and stress, the week_index coefficient is statistically significant (p < 0.001), suggesting that week index is associated with changes in these outcomes over time. Specifically, the negative estimate for week_index in the pain model (-0.219) indicates a gradual decrease in pain over time, whereas the positive estimate for stress (0.178) suggests a slight increase in stress over the weeks.

The HRV-B training duration model also shows a significant negative association with week_index (-44.010), indicating a decrease in HRV-B duration over time. This may suggest that the frequency or intensity of HRV-B training diminishes as the weeks progress.

These results provide insights into the temporal trends for pain, stress, and HRV-B duration, with time-related changes for each variable that could inform further analyses or interventions aimed at stabilizing or improving these measures over time.

B.Testing The Effect Of The HRV-B Training On Pain, Stress And HRV-B Training Duration Fluctuations

B1.Linear Mixed Effects Model for Repeated Measures of Average Pain, Average Stress, and Total Training Duration with Fixed Effects of Training Days and Random Intercepts by ID

The table below presents linear mixed effects models evaluating the impact of the number of HRV-B training days per week on pain, stress, and HRV-B training duration. Each model includes the variable days_training_in_week to assess whether training frequency influences these outcomes.

Pain Model the coefficient for days_training_in_week is negative (-0.049) and marginally significant (p = 0.051), suggesting a slight trend toward reduced pain with increased training days, though this effect is not strong enough to reach conventional significance levels.

Stress Model days_training_in_week has a positive but non-significant coefficient (0.022, p = 0.343), indicating no significant association between the number of training days and stress levels.

HRV-B Training Duration Model shows a significant positive association with days_training_in_week (estimate = 363.635, p < 0.001), indicating that as the number of training days per week increases, the duration of HRV-B training also significantly increases. This suggests that participants are spending more time in HRV-B training as they engage in it more frequently.

In summary, while increased training days show a slight trend toward reducing pain and significantly increase training duration, they do not significantly impact stress levels. These findings offer partial support for the efficacy of HRV-B training in influencing pain and training duration but not stress.

A.Testing The Mediating Role Of Stress In The Effect Of HRV-B Training Duration On Pain

To evaluate the mediating role of stress in the effect of HRV-B training duration on pain, three linear mixed models were analyzed

Mediated Effect (table one) This model examines the effect of average stress and week index on average pain. The estimate for the week index effect is -0.216, with a significant p-value (< 0.001), indicating that, on average, pain decreases by 0.216 units per week. The effect of average stress on pain is -0.013, with a non-significant p-value (0.482). This suggests that, although there is a small negative relationship, stress does not significantly impact pain in this model.

Direct Effect (table two) This model assesses the effect of HRV-B training duration and week index on average stress. The estimate for the effect of HRV-B training duration on stress is 0.000 (with a significant p-value < 0.001), implying that HRV-B training duration has a statistically significant but very minimal positive impact on stress. Practically, this suggests that increases in HRV-B training duration are associated with almost negligible increases in stress levels. The week index effect on stress is 0.187, indicating a small but significant weekly increase in stress.

Total Effect (table three) This model includes HRV-B training duration, week index, and average stress as predictors of pain, representing the total effect of HRV-B training on pain. The estimate for week index is -0.227, with a significant p-value (< 0.001), showing that pain decreases by approximately 0.227 units per week. The HRV-B training duration effect on pain is 0.000 (significant at p < 0.001), suggesting that HRV-B training has a statistically significant but practically minimal effect on pain reduction. The effect of average stress on pain remains non-significant, with an estimate of -0.006 and a p-value of 0.767, further indicating that stress does not meaningfully contribute to pain in this model.

Based on the Mediation Analysis Summary, the mediation effects of stress on the relationship between HRV-B training duration and pain were evaluated. The Average Causal Mediation Effect (ACME), Average Direct Effect (ADE), and Total Effect all have estimated values of 0.000, with confidence intervals that suggest no effect in either direction (all confidence intervals are [0.00, 0.00]). The p-values associated with ACME and Proportion Mediated indicate no statistical significance (p = 0.860 for ACME and Prop. Mediated), reinforcing the finding that stress does not play a meaningful mediating role in this relationship.

In conclusion, these results suggest that HRV-B training duration does not impact pain through stress levels, as evidenced by the lack of significant mediation effect in the observed model. This further supports the notion that any pain reduction associated with HRV-B training duration likely operates through direct mechanisms rather than indirectly through changes in stress.

The diagram below illustrates a conceptual model that tests the hypothesis of whether stress mediates the effect of HRV-B training duration on pain. The arrows depict different types of effects between HRV-B training duration, stress, and pain, with the key pathways and their observed statistical significance detailed below:

Direct Effect: This pathway represents the impact of HRV-B training duration on stress, with a near-zero but statistically significant positive effect. The small estimate (0.000 with p < 0.001) suggests that increasing the duration of HRV-B training has a minimal yet statistically measurable impact on stress levels.

Mediated Effect: This pathway examines whether stress influences pain outcomes as a mediator. However, the analysis reveals that the effect of stress on pain is statistically non-significant (estimate = -0.013, p = 0.482), indicating that stress does not meaningfully alter pain levels in this model. Thus, stress does not mediate the relationship between HRV-B training duration and pain reduction.

Indirect Effect: This path theoretically represents the combined effect of HRV-B training duration on pain through stress. However, given the lack of significant mediation by stress, this indirect effect is minimal and not statistically meaningful in contributing to pain reduction.

Total Effect: This pathway encompasses both direct and indirect influences of HRV-B training duration on pain. The overall effect of HRV-B training duration on pain is statistically significant (estimate = 0.000 with p < 0.001), but its practical impact remains minimal. The week index in the model shows a larger, significant negative impact on pain (estimate = -0.227, p < 0.001), suggesting that pain reduction over time is primarily attributed to other factors rather than mediated by stress.

In summary, this mediation model confirms that while HRV-B training duration has a statistically significant yet minimal effect on pain, this reduction does not occur through changes in stress levels. Instead, the observed pain reduction likely operates through other mechanisms not captured by stress as a mediating factor. This supports the conclusion that stress does not play a meaningful role as a mediator in this relationship.

B.Testing The Potential Moderating Role Of Weight_In_Kg, Height_In_Cm, Age_In_Years, And Gender In This Model

Weight The table below examines the potential moderating role of weight in the relationship between HRV-B training duration, stress, and pain. The results show that weight_in_kg itself does not have a significant direct effect on pain (estimate = -0.006, p = 0.395). The interaction terms HRV-B training duration and weight (total_duration_in_min, estimate = 0.000, p = 0.595) and stress and weight (avg_stress, estimate = 0.000, p = 0.977) are also non-significant. This suggests that weight does not moderate the effects of HRV-B training duration or stress on pain levels. Therefore, in this model, weight does not play a significant role as a moderator in the relationship between HRV-B training, stress, and pain.

Height The table below evaluates the potential moderating role of height in the effects of HRV-B training duration and stress on pain. The results indicate that height_in_cm has a small but significant direct effect on pain (estimate = -0.032, p = 0.006), suggesting that taller individuals may experience slightly lower pain levels. However, the interaction terms—HRV-B training duration and height (total_duration_in_min, estimate = 0.000, p = 0.847) and stress and height (avg_stress, estimate = 0.003, p = 0.110)—are non-significant. This implies that height does not moderate the effects of HRV-B training duration or stress on pain levels. Therefore, while height has a direct association with pain, it does not play a moderating role in this model.

Age The table below explores the potential moderating role of age on the effects of HRV-B training duration and stress on pain. The direct effect of age_in_years on pain is not significant (estimate = -0.010, p = 0.232), indicating that age alone does not have a direct impact on pain levels in this model. Additionally, the interaction terms—HRV-B training duration and age (total_duration_in_min , estimate = 0.000, p = 0.598) and stress and age (avg_stress , estimate = 0.002, p = 0.071) are also not significant. This suggests that age does not significantly moderate the effects of HRV-B training duration or stress on pain. In summary, age does not play a moderating or direct role in influencing pain levels in this context.

Gender The table below examines the potential moderating role of gender on the effects of HRV-B training duration and stress on pain. The direct effect of gender (Female) on pain is not significant (estimate = 0.376, p=0.114), indicating that gender alone does not have a direct impact on pain levels in this model. Additionally, the interaction terms HRV-B training duration and gender (estimate = 0.000, p=0.357) and stress and gender (estimate = -0.063, p=0.078) are also not significant. This suggests that gender does not significantly moderate the effects of HRV-B training duration or stress on pain. In summary, gender does not play a moderating or direct role in influencing pain levels in this context.

Conclusion The analysis shows that weight, height, age, and gender do not significantly moderate the relationship between HRV-B training duration, stress, and pain. While height has a small direct association with pain, none of these demographic variables play a moderating role in influencing pain levels in this model.

A.Predictive Model of Pain Levels Using All Available Data Except Stress Levels

Model Summary The table below provides the fixed effects of the model, which predicts average pain based on week_index, total_duration_in_min (HRV-B training duration), weight_in_kg, height_in_cm, age_in_years, and gender. Significant predictors include:

week_index (estimate = -0.228, p < 0.001) Shows a significant negative effect on pain, suggesting that pain decreases over time.

total_duration_in_min (estimate = 0.000, p < 0.001) A very small but statistically significant effect, indicating that HRV-B training duration slightly impacts pain.

height_in_cm (estimate = -0.028, p = 0.001) Height has a significant negative effect, suggesting taller individuals report slightly less pain on average. Other predictors, such as weight, age, and gender, did not show statistically significant effects on pain in this model.

Model Validation The diagnostic plots provide insights into the model’s assumptions:

Histogram of Residuals with Normal Curve The residuals appear approximately normally distributed, aligning well with the overlaid normal curve, suggesting that the model errors are reasonably normal.

Q-Q Plot of Residuals Most points follow the theoretical quantile line closely, with slight deviations at the tails, further supporting the assumption of normality for residuals.

Residuals vs. Fitted Values Plot This plot shows no clear patterns, indicating homoscedasticity (constant variance) across the range of fitted values, which is a key assumption for linear models.

Model Performance The R-squared value of 0.596 suggests that approximately 59.6% of the variance in pain levels is explained by the model, indicating a moderate level of predictive accuracy.

Conclusion The model effectively predicts pain levels using demographic and training-related variables, with week index and height being particularly influential predictors. The R-squared value and diagnostic plots support the model’s validity, indicating that it meets key assumptions and captures a substantial portion of the variability in pain outcomes.

B.Predictive Model of Stress Levels Using All Available Data Except Pain Levels

Model Summary The table below provides a summary of the fixed effects in the model for predicting average stress levels, incorporating week_index, total_duration_in_min (HRV-B training duration), stress_duration_ratio, weight_in_kg, height_in_cm, age_in_years, and gender as predictors. Significant predictors include:

week_index (estimate = 0.185, p < 0.000) Indicates that stress levels slightly increase over time.

total_duration_in_min (estimate = 0.000, p < 0.000) Though the effect size is very small, HRV-B training duration has a statistically significant positive impact on stress levels.

stress_duration_ratio (estimate = -1.306, p < 0.001) Has a significant negative effect, suggesting that a higher ratio of stress duration is associated with reduced average stress levels.

age_in_years (estimate = 0.014, p = 0.001) and genderFemale (estimate = -0.498, p = 0.002) Both age and gender (with females reporting slightly lower stress) show statistically significant effects on stress.

Other predictors, such as weight and height, do not significantly impact stress in this model.

Model Validation The diagnostic plots provide an evaluation of the model’s assumptions:

Histogram of Residuals with Normal Curve The residuals are approximately normally distributed, aligning well with the normal curve, suggesting that the error terms follow a normal distribution.

Q-Q Plot of Residuals The residuals mostly follow the theoretical quantile line, with minor deviations at the tails, indicating approximate normality.

Residuals vs. Fitted Values Plot This plot shows a random scatter around the zero line, suggesting no clear patterns, which supports the assumption of homoscedasticity (constant variance).

Model Performance The R-squared value of 0.655 suggests that approximately 65.5% of the variance in stress levels is explained by the model, indicating a moderate-to-strong level of predictive accuracy.

Conclusions The model demonstrates that week index, HRV-B training duration, stress duration ratio, age, and gender significantly influence stress levels, with these predictors explaining approximately 65.5% of the variance. Notably, stress slightly increases over time during the intervention, while a higher stress duration ratio correlates with lower average stress levels. Age and gender also play significant roles, with females reporting slightly lower stress. Overall, this model provides valuable insights into the factors that contribute to stress fluctuations, emphasizing the combined impact of time, training, and demographic characteristics.