Mathematical modeling in biology requires careful evaluation of model performance and reliability. Sensitivity analysis examines how changes in input parameters affect model outputs, helping identify key drivers and understand model behavior under various conditions.

Model validation assesses how well a model represents real-world systems by comparing predictions with observed data. Techniques like goodness-of-fit measures, residual analysis, cross-validation, and bootstrapping help ensure model accuracy and applicability in biological research.

Sensitivity Analysis Methods

Local vs Global Sensitivity Analysis

• Local sensitivity analysis examines how small changes in input parameters affect model outputs
  • Focuses on a specific point in the parameter space
  • Calculates partial derivatives of the output with respect to each input parameter
  • Provides information about model behavior around a particular set of parameter values
• Global sensitivity analysis investigates the effects of input variations across the entire parameter space
  • Considers the full range of possible input values
  • Accounts for interactions between different parameters
  • Offers a more comprehensive understanding of model behavior under various conditions

Advanced Sensitivity Analysis Techniques

• Morris method serves as a screening tool for identifying influential parameters in complex models
  • Uses a one-at-a-time approach to vary input factors
  • Calculates elementary effects to assess parameter importance
  • Efficiently handles models with many input parameters
• Sobol indices quantify the contribution of each input parameter to the overall model output variance
  • Decomposes the total variance of the model output into individual and interaction effects
  • Provides a measure of the relative importance of each parameter
  • Allows for the identification of key drivers in the model
• Latin hypercube sampling improves the efficiency of sensitivity analysis by ensuring better coverage of the parameter space
  • Divides the range of each input parameter into equally probable intervals
  • Selects sample points to represent the full range of each variable
  • Reduces the number of model evaluations required compared to random sampling

Model Validation Techniques

Assessing Model Performance

• Model validation evaluates how well a mathematical model represents the real-world system it aims to describe
  • Compares model predictions with observed data or experimental results
  • Helps determine the reliability and applicability of the model
  • Involves multiple steps and techniques to ensure model accuracy
• Goodness-of-fit measures quantify the agreement between model predictions and observed data
  • Includes metrics such as R-squared, root mean square error (RMSE), and mean absolute error (MAE)
  • R-squared indicates the proportion of variance in the dependent variable explained by the model
  • RMSE and MAE provide information about the average magnitude of prediction errors
• Residual analysis examines the differences between observed values and model predictions
  • Helps identify patterns or trends in model errors
  • Includes plotting residuals against predicted values or independent variables
  • Can reveal issues such as heteroscedasticity or non-linearity in the model

Advanced Validation Methods

• Cross-validation assesses how well a model generalizes to independent datasets
  • Involves partitioning the data into training and testing sets
  • K-fold cross-validation divides the data into k subsets, using each subset as a test set
  • Leave-one-out cross-validation uses a single observation as the test set and repeats for all observations
  • Helps detect overfitting and estimate model performance on new data
• Bootstrapping generates multiple datasets by resampling the original data with replacement
  • Allows for estimation of model parameter uncertainty
  • Provides confidence intervals for model predictions
  • Can be used to assess the stability of model results