PaperlayerView exact revision on arXiv

Source-linked AI summary

Sensitivity analysis methods in the biomedical sciences

George Qian, Adam Mahdi

arXiv:2001.03965v1q-bio.QMstat.AP

TL;DR

Biomedical models often contain uncertain data, stochastic processes, and many parameters, making it important to determine which inputs drive output uncertainty. This review compares sensitivity-analysis methods, software, and practical workflows, and applies Morris and Sobol methods to biomedical models. It shows that method choice and interpretation must account for model structure, nonlinearities, interactions, and the limitations of local and screening approaches.

  • Problem

    Biomedical modelling involves uncertain processes and data alongside many parameters that can affect model behaviour and interpretation, while good sensitivity-analysis practice remains limited.

  • Method

    The paper reviews sensitivity methods, software, and implementation guidance, illustrates methods on a simple model, and applies Morris and Sobol analyses to biomedical models.

  • Results

    Sensitivity analysis identifies influential inputs and interactions, while local and Morris methods have scope limitations for nonlinear models and detailed interaction interpretation.

  • Takeaways & Limitations

    Careful selection, implementation, and interpretation of sensitivity methods are needed to support model analysis and generate new hypotheses.

  • Takeaways & Limitations

    Local methods can produce different importance rankings at different parameter-space points for nonlinear models, and Morris cannot identify specific interacting factors.

Abstract

from arXiv · show

Sensitivity analysis is an important part of a mathematical modeller's toolbox for model analysis. In this review paper, we describe the most frequently used sensitivity techniques, discussing their advantages and limitations, before applying each method to a simple model. Also included is a summary of current software packages, as well as a modeller's guide for carrying out sensitivity analyses. Finally, we apply the popular Morris and Sobol methods to two models with biomedical applications, with the intention of providing a deeper understanding behind both the principles of these methods and the presentation of their results.

1 Introduction

Sensitivity analysis (SA) identifies how uncertainty in model outputs is apportioned among uncertain inputs, complementing uncertainty analysis, which characterizes output uncertainty. This review introduces SA methods for biomedical applications, explains their advantages and limitations, and provides examples, software guidance, and a practical workflow.

  • SA identifies the main input sources of uncertainty in model outputs, whereas uncertainty analysis characterizes the output uncertainty itself.
  • SA is especially important in biomedical modelling because biological processes and collected data are uncertain, while models often contain many parameters affecting behaviour and interpretation.
  • Existing SA reviews often target specific application areas or emphasize global methods, leaving room for an elementary biomedical-focused introduction.
  • The paper surveys methods, applications, advantages, limitations, software, and a practical workflow, then applies Morris and Sobol methods to algebraic and time-dependent biomedical models.

2 Basic definitions and concepts

Sensitivity analysis is part of model analysis and supports ranking, screening, and mapping of model inputs, with method choice depending on the goal and the model’s structure. Local, global, OAT, and MAT methods differ in explored input space, computational behavior, and suitability for nonlinear models.

  • Basic model concepts: A mathematical model maps inputs X1, ..., Xn to an output Y, which may be single-valued or vector-valued.
  • Aims of sensitivity analysis: SA can rank influential inputs, screen inputs with little effect, or map input values associated with stability, maxima, or optima.
  • Choosing an SA method: No single method suits all SA applications, and most techniques support ranking or screening rather than mapping.
  • Local and global methods: Local methods assess sensitivity at one point, whereas global methods average sensitivities across multiple points in input space.
  • Local and global methods: Local methods are simple and inexpensive but may mislead for nonlinear models, while global methods can indicate interactions and better explore nonlinear behavior.
  • OAT and MAT methods: OAT methods change one input per simulation and are relatively resilient to convergence problems, whereas MAT methods change multiple inputs and are preferred for nonlinear functions despite higher cost and instability risk.
  • Sampling: Sampling procedures generate input values for numerical SA, with random sampling risking clustered points and Latin hypercube sampling promoting more even factor-space coverage.

3 Review of SA methods

The review compares commonly used sensitivity methods through their assumptions, computational properties, and applications to a shared model. It illustrates one-way, two-way, derivative-based, and other methods while emphasizing that method suitability depends on model structure and analysis goals.

  • The review introduces commonly applied SA methods and illustrates each with a simple model whose inputs are uniformly distributed on [0, 1].
  • One-way sensitivities: One-way SA changes one factor while holding others constant and is most suitable for linear models; in the example, X4 produces the greatest output variation.
  • Multi-way sensitivities: Two-way SA changes two factors simultaneously, remains local, and is best suited to linear models; the example identifies [X1, X4] as producing the greatest output change magnitude.
  • Derivative-based local sensitivities: Derivative-based local indices quantify output change using absolute, relative, or variance-based scaling, with relative indices enabling comparisons across differently scaled factors.
  • Derivative-based local sensitivities: Local derivative sensitivities depend on the evaluation point, except for the model’s linear additive term X4.

3.4 Morris method

The Morris method extends local derivative ideas into a global, semi-quantitative screening technique. Its mean absolute elementary effects indicate factor influence, while their dispersion signals nonlinearities or interactions, but it cannot distinguish those mechanisms or identify interacting factor pairs.

  • The Morris method approximates derivatives with finite differences called elementary effects, computed after incrementing or decrementing one rescaled factor at a time.
  • The method summarizes repeated elementary effects using µ* and σ, where larger µ* indicates greater output influence and larger σ indicates nonlinearity or interactions.
  • Uses and limitations: Morris is typically used for factor screening because it provides semi-quantitative information rather than definitive importance boundaries.
  • Uses and limitations: Morris detects involvement in nonlinearities or interactions but cannot identify which mechanism or interacting factors produce the effect, motivating more expensive variance-based methods when detail is required.
  • Example: In the model example, X1 and X4 have the largest µ* values, while X2 and X3 have similar σ values because they interact.
  • Example: X4’s Morris index remains unchanged with step size because X4 appears only as a linear term, whereas the first three factors’ indices vary unless the step is sufficiently small.
  • Example: All four example factors influence output variance because each has a µ* value much greater than zero.

3.5 Sobol method

The Sobol method decomposes model variance into contributions from individual factors and their interactions, using first-order and total-order indices. It provides detailed variance-based sensitivity information but requires computationally expensive sampling and index calculations.

  • 3.5 Sobol method: Sobol indices partition model variance among individual input factors and their interactions.The method decomposes variance into contributions from each factor and all its interactions.
  • 3.5 Sobol method: First-order index Si measures the direct variance contribution of Xi, whereas total-order index STi includes every effect involving Xi.These indices are defined through conditional variances and distinguish main effects from total effects.
  • 3.5 Sobol method: Fixing Xi is interpreted through the expected variance reduction represented by Si; fixing all other factors leaves the variance represented by STi.This interpretation links the two indices to alternative variance-reduction interventions.
  • 3.5 Sobol method: STi supports factor screening because a zero total-order index identifies a non-influential factor, while Si supports ranking when interactions are small.Higher-order indices can additionally quantify pairwise and higher-order interactions.
  • 3.5 Sobol method: Sobol-index computation requires sampling input factors and then estimating Si and STi with dedicated algorithms.Sampling options include winding stairs and radial designs, while multiple algorithms estimate the indices.
  • 3.5 Sobol method: In model (2), X1 was most sensitive, while X2 and X3 were the only factors involved in interactions.Comparing Morris σi with Sobol STi − Si isolates nonlinear and interaction effects.

3.6 FAST and eFAST

FAST uses Fourier representations of model variance, while eFAST extends it to total-order indices. The methods provide quantitative sensitivity indices and can be more efficient than Sobol, but variance-based approaches remain computationally costly.

  • 3.6 FAST and eFAST: FAST approximates total model variance with Fourier coefficients and attributes variance to harmonics associated with each factor.The original FAST formulation computes first-order indices; eFAST extends it to total-order indices.
  • 3.6 FAST and eFAST: eFAST calculates all first- and total-order indices in one set of model evaluations, making it more efficient than Sobol.Both methods nevertheless provide variance-based quantitative sensitivity information.
  • 3.6 FAST and eFAST: FAST/eFAST is more computationally expensive than derivative-based, correlation, and regression sensitivity methods.This is the principal computational trade-off identified for these variance-based methods.
  • 3.6 FAST and eFAST: FAST/eFAST has been applied to reliability and risk models, atopic dermatitis, and thermodynamic gene-transcription models.The cited applications span engineering, hydrology, and biomedical modelling.
  • 3.6 FAST and eFAST: For model (2), FAST/eFAST produced results similar to those obtained with Sobol.The comparison is reported for the results presented in Table 6.

3.9 Importance index

The importance index measures a factor’s fractional contribution to total model variability through the ratio of factor and output variances. In the example, all factors received the same importance index because they shared a uniform distribution.

  • 3.9 Importance index: The importance index Ii is the ratio between the variance of factor Xi and the variance of output Y.It is defined as a simple variance-based sensitivity measure.
  • 3.9 Importance index: Ii represents a factor’s fractional contribution to the model’s total variability and is considered global because it uses the entire input range.The global interpretation follows from accounting for the full range of Xi.
  • 3.9 Importance index: I1 = I2 = I3 = I4 = 1/14 when every factor follows a uniform distribution from 0 to 1.The equality of the indices reflects the identical distributions used for all factors in model (2).

3.10 Correlation and regression methods

Correlation and regression methods quantify factor importance through linear or monotonic relationships between inputs and outputs, supporting factor ranking. Their interpretation depends on verifying the relevant linearity or monotonicity assumptions.

  • Correlation and regression methods estimate factor sensitivity from associations between each input and the model output, enabling factor ranking.The reviewed methods include three linear-correlation measures and two monotonicity-based measures.
  • Pearson’s correlation coefficient measures the linear relationship between input Xi and output Y, with values from −1 to 1.A value of 0 implies no linear correlation, while −1 and 1 denote perfectly linear relationships.
  • Standardised regression coefficients normalise inputs and outputs before regression, removing the influence of measurement units.The resulting coefficients provide sensitivity measures for linear models.
  • Partial correlation coefficients account for correlations among independent variables by measuring the adjusted linear relationship between Xi and Y.PCCi can be interpreted as the new information gained by introducing the ith factor.
  • Spearman and partial rank correlation coefficients address nonlinear models by ranking variables and quantifying monotonic relationships.Their indices range from −1 to 1, with 1 and −1 indicating monotonically increasing and decreasing functions.
  • In the example model, X1 and X4 were the most influential parameters, while Pearson and Spearman results were similar and partial counterparts also agreed.The similarity arose because the model formed a monotonic function from three added terms.
  • Scaling the X1^2 term by 10 increased PCC1 to around 0.96 and SRCC1 to 1.00 because that term dominated the output.The example illustrates how coefficient values depend on the relative contribution of model terms.
  • Correlation methods can become misleading when model relationships violate their assumptions of linearity or monotonicity.Shifting the quadratic term made it non-monotonic and caused both correlation coefficients to decrease sharply.

3.11 Sensitivities in the ODE setting

Sensitivity analysis for ODE models describes how state variables depend on parameters over time. The review covers direct, Green function, automatic differentiation, and finite-difference approaches, as well as time-resolved applications.

  • Biomedical models often use ODEs, with state vector x evolving over time as a function of parameters θ.Sensitivity analysis quantifies how parameter changes influence model outputs x(t).
  • Sensitivity functions describe the time evolution of model-output dependence on model parameters, with n × p sensitivities for n states and p parameters.The indices i and j identify state variables and parameters, respectively.
  • ODE sensitivities can be evaluated at selected time points or across the entire interval on which the model is defined.The choice depends on whether sensitivity is needed at particular instances or throughout the trajectory.
  • The finite-difference method estimates sensitivities using one baseline model solution and p additional solutions with one perturbed parameter at a time.It is therefore an OAT method that keeps other parameters at nominal values.
  • The direct method differentiates the ODE system with respect to parameters and solves the model together with auxiliary sensitivity equations.It requires the state trajectory and can be computationally suboptimal or unstable for certain stiff kinetic problems.
  • The Green function method computes sensitivity coefficients from integrals over Green functions for the auxiliary equations.The review identifies Analytically Integrated Magnus as one implementation.
  • Automatic differentiation computes derivatives of arbitrary order through the chain rule and is implemented in packages including ACADO and CasADi.
  • Time-dependent sensitivities can guide measurement timing by identifying periods when states are most responsive to parameter changes.Sensitivity functions provide temporal information relevant to parameter estimation and data collection.

4 Visual methods

Visual methods communicate input-output relationships and sensitivity indices, offering intuitive views of influential factors, interactions, nonlinearities, and time-dependent changes. Different plots emphasize different aspects of sensitivity results.

  • Visual sensitivity tools provide intuitive views of important factors, complementing quantitative sensitivity indices.They are classified as qualitative because they communicate results graphically rather than through numerical indices alone.
  • Scatterplots show model output against a selected input and can reveal sensitivity magnitude, nonlinearities, and functional relationships subjectively.They are commonly used as an initial visualisation of input-output behaviour.
  • Cobweb plots display each input factor on a separate axis and connect successive input samples to show multivariable sample structure.Conditional cobwebs can retain only lines satisfying a specified threshold for one variable.
  • Tornado plots show percentage output changes when each factor is increased or decreased while other factors remain at their nominal values.They are typically used for one-way sensitivity results.
  • Morris plots place the mean absolute elementary effect against its standard deviation to show factor ranking and interactions.Small values indicate little output effect or few interactions, whereas large values indicate greater effects or interactions.
  • Bar plots can compare first-order and total-order Sobol indices for multiple input variables, while pie charts show index magnitudes as proportional slices.
  • Radar graphs summarise many variables, and circular diagrams encode first- and total-order sensitivities with circle sizes and interactions with connecting lines.
  • Matrix plots show pairwise relationships in a variable grid, while phase portraits track sensitivity and interaction measures across time.Time-dependent sensitivity plots are useful when parameter importance changes over a model trajectory.

5 Software for performing SA

The review surveys software packages for sensitivity analysis across MATLAB, Python, R, C++, Fortran, and standalone or GUI-based environments. Packages differ in supported methods, model types, visualisation, and related modelling tasks.

  • Sensitivity-analysis software is available in several programming languages and supports a range of local, global, regression, and ODE-focused techniques.
  • Dakota performs global sensitivity analysis including Morris and Sobol methods and has been applied to immunology models.
  • Data2Dynamics supports parameter estimation, uncertainty analysis, and local sensitivity analysis for biochemical reaction-network models.
  • DyGloSA performs global sensitivity analysis for differential-equation models, focusing on critical transitions in systems with bifurcations.
  • GUI-HDMR constructs metamodels and calculates variance-based sensitivity analysis through both graphical and scripting interfaces.
  • PeTTSy implements perturbation-theory and sensitivity-analysis techniques for large and complex ODE models.
  • PSUADE provides global sensitivity analysis including Morris and FAST methods in a C++ package originally developed for hydrology.
  • R packages include global methods such as Morris, FAST, and Sobol, with numerical schemes including Jansen and Saltelli implementations.The review also identifies ODEsensitivity for ODE models.

6 Framework for applying SA

Applying sensitivity analysis requires defining its purpose, inputs, method, and interpretation, while avoiding choices that make results misleading or computationally infeasible. The framework emphasizes matching methods to model structure, computational resources, and the uncertainty or nonlinearities being investigated.

  • Step 1: Define the purpose of the SA: Define whether the analysis will rank factors or screen them, and specify which output measures will be examined.
  • Step 2: Choose input factors, their ranges and distributions: Choose biomedical input ranges and distributions carefully because the selected parameter space can change factor rankings.Nominal values may come from expert measurement or preliminary parameter estimation; uniform distributions are commonly assumed when distributions are unclear.
  • Step 3: Select SA method(s): Select methods according to model structure and computational cost: global approaches are needed for nonlinear or interacting models, whereas local methods are cheaper but less informative away from the operating point.Variance-based methods such as Sobol can provide quantitative detail but may be computationally prohibitive for complex biomedical models.
  • Step 4: Visualisation and interpretation: Visualize and interpret sensitivity results in the biomedical context, considering implications for model selection, reduction, and practical identifiability.
  • Common pitfalls: One-at-a-time methods explore less than one-thousandth of a 12-parameter space and make interactions difficult to assess.Their simplicity therefore limits coverage of the parameter space and can leave many model evaluations uninformative.
  • Common pitfalls: Sensitivity estimates depend on sampling choices, and negative Sobol indices indicate large approximation errors that require increasing the sample size.Robustness can be assessed with boxplots, confidence intervals, and convergence plots across resamples or sampling sequences.
  • Common pitfalls: Local sensitivity rankings for nonlinear models are meaningful only at the calculation point, with factor importance changing between parameter settings.When all factors changed from 0.1 to 0.9, the ordering of X1 and X4 swapped.

7 Application of SA in biomedical sciences

The paper applies Morris and Sobol sensitivity analysis to algebraic and ODE models of cancer and cell differentiation, identifying influential parameters, interactions, nonlinearities, and measurement implications.

  • Algebraic model: cancer prediction: In the cancer model, cell divisions d and mutation rate u most affect output variance and strongly interact with other factors, while k has moderate influence and interactions.Factors m and n have low output influence and small interactive effects.
  • Algebraic model: cancer prediction: The cancer-model result is mathematically explained by u and d occurring deep in the equation, where variations are compounded by later exponents involving k and n × m.
  • ODE model: cell differentiation in the colon: In the cell-differentiation model, Morris analysis identifies five of eleven parameters as influential and interactive, with α3 producing the greatest effect on N0.α2, k0, α1, and m0 also have non-negligible influence, while sensitivity peaks around 10-20 days.
  • ODE model: cell differentiation in the colon: Sobol total-order indices reproduce the Morris ranking, with α3 most influential, α2 and k0 contributing less, and α1 and m0 contributing only slightly.Unlike Morris µ∗ rankings, total-order Sobol indices quantify each parameter’s contribution to output variance.
  • ODE model: cell differentiation in the colon: Comparing Morris σ with Sobol first- and total-order indices indicates that most observed σ values arise from nonlinear effects rather than parameter interactions.The small differences between first- and total-order Sobol indices support this interpretation.
  • Physiological interpretation: Physiologically, stem-cell proliferation α3 affects the stem-cell population more than combined changes in death and differentiation rates, while transit-cell proliferation β3 most affects differentiated populations.The latter result contrasts with an initial expectation that removal or transformation rates would dominate fully differentiated-cell variance.
  • Physiological interpretation: Sensitivity results can guide parameter estimation by identifying which outputs to measure and when, while parameters unidentifiable from N0 require measurements of N1 and N2.For N0, sensitivities are highest near steady state, making that period suitable for estimating influential parameters.

8 Conclusions

The paper frames sensitivity analysis as essential for biomedical modelling, where stochastic biology, uncertain data, and many influential parameters complicate model interpretation. It reviews methods and software, proposes an analysis framework, and applies global methods to algebraic and ODE cancer-biology models.

  • The paper reviews sensitivity-analysis methods and software and provides a framework focused on biomedical sciences and biology.
  • The paper demonstrates global sensitivity analysis with two models in cancer biology: an algebraic model and an ODE model.
  • Sensitivity analysis is particularly important in biology and medicine because biological processes are stochastic, acquired data are uncertain, and models often contain many influential free parameters.These factors affect model behaviour and interpretation, motivating systematic sensitivity analysis.
  • Global sensitivity analysis is usually more appropriate than local analysis for prevalent nonlinear systems-biology models.Local methods examine sensitivities in only a small parameter-space region, making their conclusions meaningful only for linear models.
Loading 2001.03965v1…