Inhale, Exhale, Analyze: Using Linear Mixed Models to Predict Airway Resistance and Reactance Measures

UWF STA 6257 Capstone Project on Linear Mixed Models (LMMs)

Joshua J. Cook, M.S., ACRP-PM, CCRC

jcook0312@outlook.com

Syed Ahzaz H. Shah, B.S.

shs17@students.uwf.edu

Jacob Hernandez, B.S.

jacob.hernandez0830@gmail.com

Sara Basili, M.S.

saraelizabethbasili@gmail.com

April 16, 2024

Introduction to Linear Mixed Models (LMMs)

Introduction

Understanding Linear Mixed-Effects Models (LMMs)

Linear mixed-effects models are advanced statistical tools designed to handle complex data structures.
These models are essential when dealing with hierarchical organization, repeated measures, and random effects in datasets.
LMMs are particularly useful when traditional ANOVA or regression assumptions—like independence of observations, homoscedasticity, and normality of residuals—are not met.

Software Tools and Resources for LMMs

Tools for Implementing LMMs

The development and use of LMMs are supported by several software packages and programming languages.
Key resources include the lme4 package in R, detailed by Bates et al. (2015), which simplifies the fitting of mixed models, especially those with crossed random effects.
For Python users, Pymer4 developed by Jolly (2018) integrates Python with R’s lme4 package, broadening accessibility to these advanced methods.

Applications of LMMs Across Disciplines

Broad Applications of LMMs

LMMs find diverse applications across various scientific domains, addressing unique analytical challenges.
- In healthcare, LMMs model pandemic-related mortality changes (Verbeeck et al., 2023) and analyze longitudinal data in clinical trials (Touraine et al., 2023).
- In ecology, studies by Harrison et al. (2018) and Bolker et al. (2009) discuss their use in analyzing complex ecological data.
- In psychology and neuroscience, LMMs tackle the complexities of repeated measures and nested data structures (Magezi, 2015; Aarts et al., 2015).

Methods - Mathematical Foundations

Linear Algebra

Foundations

LMMs leverage linear algebra and in our case, we are explaining the mathematical concepts for a two-level longitudinal random intercepts model. Index i is used to denote the participant and index t is used to denote the different time points of the observation

\[ Y=X\beta + Zu+ \epsilon \]

Equation 1: the base linear mixed model.

Y is the response vector. Shape N x 1 where N is the number of the number of repeated measures
X is the design matrix for fixed effects. Shape N x p where p is the number of regression coefficients
β is the vector of regression coefficients. Shape P x 1
Z is the design matrix for random effects. Shape N x J where J number of subjects
u is the vector of random effects. Shape J x 1 vector
ϵ is the vector of residual errors. Shape N x 1 vector

Assumptions

The relationship between the predictors and response variable is assumed to be linear, within each level of random effects.
Random effects (u) are assumed to follow a normal distribution with mean zero and variance-covariance matrix G.

\(\gamma \sim N(0,G)\)
Residual errors (ϵ ) are assumed to follow a normal distribution with mean zero and variance-covariance matrix R.

\(\epsilon \sim N(0,R)\)
Random effects (u) and residual errors (ϵ ) are assumed to be independent.
Homoscedasticity is assumed for the residuals across all levels of the independent variables.

Implementation in R

Data is loaded from a CSV file using the read.csv function
Fitting Data to LMMs
- The lme() function from the nlme package has parameters to specify random effects structure and estimation method.
- lmer() function from the lme4 package has similar syntax to the lme() function but differs in how it handles random effects specifications
Hypothesis Testing
- Evaluated using F-tests, Likelihood ratio test, and Shapiro-Wilks tests

The Capstone Project Data

Dataset Overview

Key attributes and measurements in the dataset.
Categorical and numerical variables.
Presence of missing values, espsecially in the Fres_PP variable.

Why Linear Mixed Models (LMMs)?

Suitability of LMMs for the dataset.
Multiple observations over time for the same participants.
Handling unbalanced groups, as observed in participant dropout over time.

EDA - Categorical Variables

EDA - Numerical Variables

Outlier Detection and Summary Statistics

Presence of outliers in variables and their implications.

Participant Dropout Analysis

Significance of participant dropout over time.
Ability of LMMs to handle unbalanced groups

Analysis & Results

The Initial Model

One Random Effect

In this dataset:

Measures of airway resistance and reactance are the variables of interest: R5Hz_PP, R20Hz_PP, X5Hz_PP, Fres_PP.
Controlled variables are present such as Group, Age, Weight, Height, and other Co-morbidities. These are the fixed effects.
Random variability may exist between individual observations which are nested in each subject. These represent the random effects. In the initial model, Subject_ID was treated as the sole random effect.

The Initial Model

One Random Effect

The Initial Model

One Random Effect

Equation 2. The initial LMM.

Implementation

#lme()

# Fit models using a tidy and clear approach
model_lme <- lme(
  fixed = cbind(R5Hz_PP, R20Hz_PP, X5Hz_PP, Fres_PP) ~ BMI + Asthma + ICS + LABA + Gender + Age_months + Height_cm + Weight_Kg,
  random = list(Subject_ID = pdIdent(~1)),
  data = x_clean,
  method = "REML"
)

#lmer() 

model_lmer <- lmer(
  formula = R5Hz_PP + R20Hz_PP + X5Hz_PP + Fres_PP ~ BMI + Asthma + ICS + LABA + Gender + Age_months + Height_cm + Weight_Kg + (1 | Subject_ID),
  data = x_clean
)

Evaluation

Akaike Information Criterion (AIC) - indicator of model fit without unnecessary complexity.
- AIC for lme = 1898.95 (selected as initial model)
- AIC for lmer = 2517.37
Assumptions Check - normality.

Finding: in this case, P = 0.00001163. P < 0.05, so the null hypothesis was rejected, suggesting that the residuals were not normally distributed. This model did not satisfy the assumptions of LMMs.

The Imputed Model

Satisfying Assumptions

Upon further inspection, outliers were present in most variables.
To improve model performance, these outliers were imputed using the threshold values (i.e., winsorization).
Confirmation of outlier removal was completed using boxplots.
All metrics were then reevaluated.

Evaluation

AIC for lme = 1790.91 (better!)

Finding: in this case, P = 0.05066. P > 0.05, so we failed to reject the null hypothesis, suggesting that the residuals were normally distributed after threshold imputation. This model now satisfies the assumptions of LMMs.

The Final Model

Two Random Effects and Final Fixed Effect

This was a longitudinal study involving multiple observations for each subject over time, and subjects are grouped into two categories (children with sickle cell disease and African-American children with asthma).

Thus, in this final model:

we modeled Group as a fixed effect since we were interested in the effect of the group itself on the outcome.
Subject_ID should be a random effect to account for the repeated measures within subjects.
Observation_number was included as a random slope within Subject_ID (i.e., nested within Subject_ID).
The same visualizations and tests were completed to assess the LMM assumptions.

The Final Model

Equation 3. The final LMM.

Implementation

model_lme_imputed_final <- lme(fixed = cbind(R5Hz_PP, R20Hz_PP, X5Hz_PP, Fres_PP) ~ BMI + Asthma + ICS + LABA + Gender + Age_months + Height_cm + Weight_Kg + Group,
                         data = x_clean_imputed,
                         random = list(Subject_ID = pdIdent(~1 + Observation_number)),
                         method = "REML")

Evaluation

AIC for lme = 1801.60 (better than initial, but worse than imputed?)

Findings:

In this case, P = 0.0529. P > 0.05, so we failed to reject the null hypothesis, suggesting that the residuals were normally distributed after threshold imputation. This final model also satisfies the assumptions of LMMs.
The AIC penalizes model complexity to avoid overfitting, suggesting that the added effects of Group and Observation_number may not be sufficiently increasing model accuracy compared to complexity.
However, these effects may still be relevant given the research goal of the project despite the slight increase in AIC, and thus were left in the final model.

Conclusion

Overview of Model Evaluations

In our analysis, we compared three Linear Mixed Models: the base model, the model with imputed values, and the final adjusted model, to predict airway resistance and reactance effectively.
We focused on Mean Squared Error (MSE) and Mean Absolute Error (MAE) to assess model performance.

Findings: The final imputed model achieved the lowest MSE and MAE, indicating superior performance over the other models.

Sample Predictions vs. Actual Data

Figure 24 illustrates a side-by-side comparison of the predicted versus actual values for R5Hz_PP, a measure of airway resistance and reactance, for 10 random subjects.
The close alignment between predicted and actual values represents a low residual error, confirming the model’s high accuracy in predicting R5Hz_PP.

Conclusion

Our analysis demonstrates that linear mixed models are exceptionally versatile and can effectively handle complex datasets with multiple layers of correlation and missing data, incorporating both fixed and random effects seamlessly.
Our final model accurately predicts airway resistance and reactance given demographic and co-morbidity data, which could aid in better understanding and managing respiratory functions in children with conditions such as Sickle Cell Disease and asthma.

Acknowledgements

The authors thank Dr. Achraf Cohen, for his ongoing mentorship and support.

Questions are welcome and encouraged!