Morris Sensitivity Analysis Overview

• Morris Sensitivity Analysis is a global, one-at-a-time technique that computes elementary effects to rank input factors in computational models.
• It uses finite differences aggregated over randomized trajectories to diagnose nonlinearity and interactions across high-dimensional spaces.
• This method offers a cost-effective screening tool for identifying key parameters in black-box and complex models, guiding further quantitative analysis.

Morris Sensitivity Analysis

Morris Sensitivity Analysis, also known as the elementary effects (EE) method, is a global, one-at-a-time (OAT) screening technique for identifying influential input factors in computational models. Developed by M.D. Morris (1991), it is particularly valued for its modest computational cost, straightforward implementation, and ability to efficiently screen high-dimensional and nonlinear systems for main effects and potential interactions, even when the model itself is a black-box function. The method has been established as a foundational tool in sensitivity analysis across applications including biomedical modeling, energy systems, environmental science, and algorithm configuration (Qian et al., 2020).

1. Theoretical Principles and Assumptions

Morris analysis assumes that all input factors can be transformed to be mutually independent and rescaled to the unit interval $[0,1]$ (non-uniform marginals are handled via inverse CDF transforms). The model is treated as a black-box mapping $f : [0,1]^k \to \mathbb{R}$, where $k$ is the number of input factors, and outputs are assumed to be scalar or have a chosen scalar summary. The method relies on the feasibility of evaluating $f$ at points that differ in only one coordinate.

The central idea is to extend the concept of a local finite-difference derivative to a global setting by performing multiple OAT perturbations at randomly sampled input locations. These finite differences, termed elementary effects, are aggregated to provide both an influence ranking and a qualitative assessment of nonlinearity and interaction effects (Qian et al., 2020, Kucherenko et al., 2014).

2. Formal Definition and Algorithmic Workflow

Given $k$ input factors $X = (X_1, ..., X_k)$, each scaled to $[0,1]$, Morris analysis proceeds as follows:

• Grid Construction: Select a grid resolution $p$ (number of levels per input), with typical choices $p \in \{4,6,10\}$ for a balance between resolution and cost.
• Step Size: The perturbation step is set to $\Delta = p/[2(p-1)]$, ensuring the perturbed points remain on the $p$-level grid.
• Trajectories: For each of $r$ randomly sampled trajectories:
  1. Randomly select a base point $x^{(j)}$ from the grid.
  2. Randomly permute the factor indices.
  3. Sequentially, for each factor $i$ in the permutation, move only $x_i$ by $\pm \Delta$ and compute:

  $$EE_i(x) = \frac{f(x_1, ..., x_i + \Delta, ..., x_k) - f(x_1, ..., x_i, ..., x_k)}{\Delta}$$

1. This results in precisely one elementary effect per factor per trajectory.

• Statistical Summaries:
  • Mean of signed effects: $\mu_i = \frac{1}{r}\sum_{j=1}^r EE_i^j$
  • Mean of absolute effects (Morris index): $\mu_i^* = \frac{1}{r}\sum_{j=1}^r |EE_i^j|$
  • Standard deviation: $$\sigma_i = \sqrt{(1/(r-1))\sum_{j=1}^r (EE_i^j - \mu_i)^2}$$

By design, $\mu_i^*$ is favored over $\mu_i$ as it prevents cancellation of positive and negative effects, yielding more robust importance scores. Total computational effort is $r(k+1)$ model evaluations (Qian et al., 2020, Sanchez et al., 2012, Ojha et al., 2022).

3. Interpretation of Morris Indices and Visual Analysis

The indices $\mu_i^*$ and $\sigma_i$ serve as qualitative diagnostics:

• $\mu_i^*$ (importance): Large values indicate a strong overall effect of factor $i$ on model output, regardless of direction.
• $\sigma_i$ (interaction/nonlinearity): Large $\sigma_i$ relative to $\mu_i^*$ suggest that the effect of $i$ varies across the input space, indicative of nonlinear behavior or interactions with other inputs.
• $\mu_i$ (monotonicity): If $|\mu_i| \approx \mu_i^*$, the input effect is predominantly monotonic; significant disparity indicates sign-changing (non-monotonic) behavior.

A standard practice is to present a $(\mu_i^*, \sigma_i)$ scatterplot (“Morris plot”):

Quadrant	Description
Lower left	Unimportant (small $\mu^*$, small $\sigma$)
Upper left	Nonlinear/interaction ($\mu^*$ small, $\sigma$ large)
Lower right	Linear/additive (large $\mu^*$, small $\sigma$)
Upper right	Both influential and heavily interacting ($\mu^*$, $\sigma$ large)

Thresholds or guide lines (e.g., for $\sigma/\mu^*\approx 0.1,\,0.5,\,1$) are sometimes superimposed for classification (Qian et al., 2020, Sanchez et al., 2012, Islam et al., 2022).

4. Extensions and Methodological Variants

Higher-Order Effects

Campolongo & Braddock’s extensions capture pairwise interactions by defining second-order incremental ratios and correcting for additive effects:

$$SEE_{ij}(u) = \frac{f(u + e_i\Delta + e_j\Delta) - f(u)}{\Delta^2}$$

$$EE_{ij}(u) = \left|SEE_{ij}(u) - \frac{EE_i(u)}{\Delta} - \frac{EE_j(u)}{\Delta}\right|$$

Aggregating these across trajectories yields $(\mu_{ij}^*, \sigma_{ij})$ for ranking interactions (Sanchez et al., 2012).

Dependent Inputs

The classical method mandates input independence. For dependent parameters, copula-based sampling preserves the native joint dependency structure. Latin Hypercube Sampling with Dependence (LHSD) draws from a copula-fitted structure and maps samples to the grid, ensuring OAT perturbations remain within the feasible region (Tene et al., 2018).

Heterogeneous and Spatio-Temporal Data

Morris analysis adapts to settings with static and dynamic (e.g., time-varying or spatially distributed) features by accumulating output changes over all indices and time steps before computing normalized indices. For example, in time-series deep learning applied to COVID-19 forecasting, the normalized Morris index aggregates over the county-day grid and scales by feature variance (Islam et al., 2022).

Latin Hypercube Sampling and Space-Filling Designs

While standard Morris uses random grids, Morris-LHS enhances coverage and reduces sample clumping by constructing base points using Latin Hypercube designs in the $k$-dimensional grid, then applying OAT steps. This improves the stability of $\sigma$ estimates in large $k$ (Ojha et al., 2022).

5. Computational and Practical Aspects

Parameter	Typical Recommendation	Notes
$p$	4–10 (even)	Finer grid ↔ increased cost
$\Delta$	$p/[2(p-1)]$	Ensures grid-aligned steps
$r$	10–20 (screening), 50–100+ (stable rankings)	Larger $r$ stabilizes summary statistics

Normalization of input domains (to $[0,1]$), space-filling trajectory design, and reporting of grid/trajectory parameters (including random seeds) are critical for reproducibility (Qian et al., 2020, Ojha et al., 2022). For models with different input units, elementary effects should be normalized for comparison.

Morris analysis is parallelizable, with each trajectory or OAT perturbation evaluated independently (Tene et al., 2018). Compared to Sobol or variance-based methods, Morris requires significantly fewer model runs for main-effect screening in high-dimensional systems, providing at least a four-fold reduction in evaluation cost for comparable convergence (Feistner et al., 15 Aug 2025).

6. Applications and Case Studies

Morris analysis is widely adopted across scientific and engineering domains as a primary global screening tool. Notable examples include:

• Biomedical Sciences: Identifying influential parameters in mechanistic models of cancer risk, with $p=100$, $r=100$ yielding clear identification of additive versus nonlinear/interactive inputs (Qian et al., 2020).
• Energy Systems: Screening nine input parameters in a molten carbonate fuel cell model ($p=20$, $r=100$), the method aligned with variance-based Sobol and regression-based analyses but at lower cost (Radaideh et al., 2019).
• Battery Modelling: Uncertainty quantification for equivalent circuit model parameters, with experimental validation confirming that parameters flagged as negligible by Morris can be held constant without loss of predictive fidelity (Zhao et al., 2016).
• Groundwater Flow: Parameter screening in a steady-state aquifer model, efficiently reducing seven candidate parameters to three for calibration, combining multiple output metrics (RMSE, fluxes) for comprehensive assessment (Baker et al., 2022).
• Algorithm Hyperparameter Tuning: Ranking and diagnosing main effects and interactions for the configuration of evolutionary algorithms, with Morris and Morris-LHS providing effective triage for prioritizing parameter adjustment (Ojha et al., 2022).
• Spatio-Temporal Deep Learning: Adapted Morris analysis for interpretability of county-level COVID-19 forecasts, differentiating critical influences among dynamic and static features (Islam et al., 2022).

7. Advantages, Limitations, and Best-Practice Recommendations

Advantages

• Scalable to high-dimensional models ($O(rk)$ model evaluations)
• Non-intrusive and highly parallelizable
• Reasonably robust to model failures or missing data
• Provides both importance rankings and qualitative diagnostics for nonlinearity and interaction

Limitations

• Only semi-quantitative: $\mu^*$ and $\sigma$ lack direct variance-based interpretation
• Cannot localize which pairs of factors interact—captures only that some interaction/nonlinearity occurs
• Sensitivity to choice of $p$, $\Delta$, and $r$; subjective thresholds for interpreting $\mu^*$ and $\sigma$
• Assumes (in the classical form) input independence, though extensions for dependencies exist (Tene et al., 2018)
• For precise variance apportionment, must be supplemented by variance-based (e.g., Sobol) or derivative-based methods (Kucherenko et al., 2014)

Recommendations

1. Normalize all input spaces; use inverse-CDF for non-uniform marginals.
2. Start with $p = 4$–10, $r = 10$–20 for initial screening; increase $r$ if rankings shift.
3. Always analyze convergence of $\mu^*$ and $\sigma$ with increasing $r$.
4. For time-dependent or multivariate outputs, compute $\mu^*$ and $\sigma$ at each required output time or for each error metric (Islam et al., 2022, Feistner et al., 15 Aug 2025).
5. Apply Morris screening prior to resource-intensive quantitative sensitivity analysis on the subset of important factors.
6. Use copula-based or stratified sampling designs in the presence of input dependencies or for improved coverage (Tene et al., 2018, Ojha et al., 2022).

In summary, Morris Sensitivity Analysis offers a computationally efficient and methodologically flexible approach for initial sensitivity screening, especially suited to complex, high-dimensional, or costly computational models. It remains foundational for model reduction, interpretation, experimental design, and as a precursor to deeper quantitative sensitivity analysis techniques (Qian et al., 2020, Kucherenko et al., 2014).