Morris Method

Morris Method: Concept Architecture

The Morris method screens model inputs by measuring the effect of changing one input at a time at several locations in the input space. It offers a relatively inexpensive first look at which uncertain assumptions may matter to a model's output. The page explains the elementary-effect calculation, the trajectory design, the three common summaries, and the limits of using screening results in health economic decisions.

Measure a change at more than one starting point

Let $Y=f(x_1,\ldots,x_k)$ be a deterministic model output, with $k$ uncertain inputs mapped to a specified grid in $[0,1]^k$. For an allowed positive step $\Delta$ in input $i$, hold other inputs fixed and compute its elementary effect at a grid point $x$ where the shifted point is feasible. A physical input value may be obtained from its normalized coordinate using a declared range or transformation.

$$\mathrm{EE}_i(x)=\frac{f(x+\Delta e_i)-f(x)}{\Delta},$$

where $e_i$ changes only coordinate $i$. The numerator is a change in model output and the denominator is a change in the normalized input here, so the resulting units are output units per unit of normalized input. If physical-unit increments are used instead, the effect's numerical scale and units change. Choose ranges that represent the uncertainty or scenarios of interest; a ranking is conditional on those ranges, the transformation, and the output being analyzed.

A single one-way sensitivity analysis around one baseline provides just one local change per input. Morris evaluates elementary effects across multiple points, giving evidence about whether an input has small effects throughout the sampled region or a varying effect that the baseline alone misses. The method is a screening design, not a proof that an input is causally unimportant in every setting.

Build trajectories efficiently

In a trajectory design, each path starts at a valid grid point and changes one input at a time. A complete trajectory over $k$ inputs has $k+1$ model evaluations and yields one elementary effect for each input. With $r$ trajectories, the standard ungrouped design uses $r(k+1)$ model evaluations before any optional reuse of identical points. A trajectory can change inputs in different orders, and the starting points, directions, grid levels, and step must all be recorded.

The randomized paths sample effects over multiple combinations of inputs. Some implementations select a subset of well-spread trajectories from a larger candidate pool; this changes the design but not the basic meaning of an elementary effect. Correlated inputs, impossible combinations, and parameter constraints need an explicit design: independently moving one input may otherwise create infeasible patient profiles or model settings. Merely running software with default ranges does not make a sensitivity analysis appropriate to the clinical or economic question.

Summarize effects without losing their direction

For input $i$, let $\mathrm{EE}_{ij}$ be its effect from trajectory $j$, with $j=1,\ldots,r$ and $r>1$ for a sample standard deviation. Three widely used summaries answer different questions. None is a variance-based Sobol index or a percentage of output variance explained.

\mu_i^*=\frac{1}{r}\sum_{j=1}^{r}|\mathrm{EE}_{ij}|,\qquad \sigma_i=\sqrt{\frac{1}{r-1}\sum_{j=1}^{r}(\mathrm{EE}_{ij}-\mu_i)^2}.$$ | Summary | What a large value suggests | What it does not establish | | --- | --- | --- | | Signed mean $\mu_i$ | A consistent directional change, if effects share a sign. | A small value does not prove the input is unimportant; signs can cancel. | | Mean absolute effect $\mu_i^*$ | Large typical change magnitude over sampled points. | It is not an uncertainty-weighted causal effect or a variance share. | | Spread $\sigma_i$ | Effects vary across points, possibly from nonlinearity or interactions. | It cannot separate those causes on its own. | The revised $\mu_i^*$ is the **mean of absolute effects**, not the absolute value $|\mu_i|$ of the signed mean. A high $\sigma_i$ signals that the local effect differs across sampled locations; nonlinear response to input $i$, interactions with other inputs, or both may be responsible. With few trajectories these summaries can be unstable. Examine individual effects, convergence across more trajectories, and Monte Carlo uncertainty in the screening estimate before making strong exclusions. ## Worked example: an effect hidden by sign cancellation Consider the illustrative normalized-input function $f(x_1,x_2)=(x_1-0.5)^2+2x_2$ with both inputs between zero and one and step $\Delta=0.5$. The output is an arbitrary model score, not an empirical health outcome. The two paths below are a didactic grid example, not a claim that two trajectories are sufficient for real screening. | Path | Consecutive points | Consecutive model outputs | Effect of $x_1$ | Effect of $x_2$ | | --- | --- | --- | ---: | ---: | | 1 | $(0,0)\to(0.5,0)\to(0.5,0.5)$ | $0.25\to0\to1$ | $-0.5$ | $2$ | | 2 | $(0.5,0)\to(0.5,0.5)\to(1,0.5)$ | $0\to1\to1.25$ | $0.5$ | $2$ | For $x_1$, the two effects are $(0-0.25)/0.5=-0.5$ and $(1.25-1)/0.5=0.5$. Consequently $\mu_1=0$, $\mu_1^*=0.5$, and the two-effect sample standard deviation is $\sigma_1=\sqrt{0.5}\approx0.707$. The signed mean cancels, but the input plainly changes the output; in this function the variation arises from *curvature*, with no interaction between $x_1$ and $x_2$. For $x_2$, both effects are 2, so $\mu_2=\mu_2^*=2$ and $\sigma_2=0$. This example demonstrates the summaries, not a reliable empirical ranking from only two paths. ## Use the screen in health economic models A health economic model can have many uncertain parameters, including event risks, treatment effects, utilities, costs, discount-related assumptions, and treatment duration. Apply Morris separately to a clearly defined output such as total cost, QALYs, or incremental net monetary benefit at a stated willingness-to-pay value. An input can rank highly for one output and much lower for another; report the decision outcome used for screening. For incremental net monetary benefit, a possible output is $f(x)=\lambda\Delta Q(x)-\Delta C(x)$, where $\lambda$ is the stated value per QALY, $\Delta Q$ incremental QALYs, and $\Delta C$ incremental cost. The effect of an input on this output may differ from its effects on QALYs and costs separately. An input with a near-zero mean signed effect could still cause decisions to switch at different parameter values; inspect the distribution and direction of effects and use a decision-focused follow-up analysis when appropriate. Morris can prioritize model checking, data collection, and more computationally demanding analyses, but it is not a substitute for probabilistic sensitivity analysis (PSA). PSA propagates joint parameter uncertainty under specified distributions and correlations; Morris probes response to designed input changes over selected ranges. One-way sensitivity analysis typically uses one baseline and an input range; Morris repeats local one-at-a-time changes across multiple paths. Variance-based global methods, such as Sobol indices, quantify a different notion of influence when their sampling assumptions hold. ## What may distort the ranking Screening results depend on the analyst's choice of input ranges and mappings. A wide arbitrary range can inflate one input's apparent importance relative to a tightly constrained input; a log-scale versus linear-scale mapping changes what a fixed normalized step represents. Specify whether inputs are sampled independently, whether ranges represent evidence-based uncertainty or exploratory stress tests, and whether structural alternatives belong in separate model runs. The standard trajectory design varies one input while others remain fixed, so constraints and correlations require deliberate treatment. Grouped factors can help keep related changes together, but group effects are not the individual effects of each member. Stochastic simulators add run noise to elementary effects: control random numbers or replicate runs and quantify Monte Carlo error so it is not mistaken for response nonlinearity. Discontinuities and threshold decisions can also produce highly variable effects; investigate where they occur rather than assuming a smooth derivative. High $\mu_i^*$ is a screening signal, not a policy priority by itself. Check the plausibility of the input range, the magnitude in decision-relevant units, and the impact on the preferred strategy. A low rank is conditional on the sampled domain and output; it is not permission to delete a parameter that is structurally essential or supported by strong clinical evidence. ## A reproducible workflow The workflow makes clear which choices shape the results. It also separates screening from the follow-up analyses needed for a decision. Document the design before interpreting a ranking. 1. **Define the output.** Name the population, perspective, comparator, time horizon, discounting, and exact cost, QALY, or decision measure. 2. **Specify inputs and ranges.** Record distributions or evidence for bounds, normalization, transformations, constraints, and assumed dependence. 3. **Choose the design.** State grid levels, step, trajectories, selection method, random seed, and expected model-run count. 4. **Run and check paths.** Confirm each model call uses a feasible input combination and that output differences match the intended input move. 5. **Calculate summaries.** Report $\mu$, $\mu^*$, $\sigma$, units, effects or uncertainty intervals, and stability across additional trajectories. 6. **Investigate signals.** Explore high-magnitude or high-spread inputs, sign changes, interactions, and decision thresholds with suitable further analysis. 7. **Report limits.** State which output and domain were screened, unresolved dependence or simulation noise, and how the findings affect subsequent PSA or validation. ## Sources and further study The original paper establishes the trajectory-based elementary-effects screen. The later methodological paper and software documentation clarify the revised absolute-effect summary and implementation choices. The worked example above is newly constructed and independently calculated. 1. Morris MD. [Factorial Sampling Plans for Preliminary Computational Experiments](https://doi.org/10.1080/00401706.1991.10484804). Original 1991 method introducing randomized one-factor-at-a-time trajectories and elementary effects. 2. Campolongo F, Cariboni J, Saltelli A. [An Effective Screening Design for Sensitivity Analysis of Large Models](https://publications.jrc.ec.europa.eu/repository/handle/JRC37360). European Commission Joint Research Centre record for the revised screening design. 3. SALib developers. [Morris analysis documentation](https://salib.readthedocs.io/en/main/api/SALib.analyze.html#salib-analyze-morris-module) and [Morris sampling documentation](https://salib.readthedocs.io/en/latest/api/SALib.sample.morris.html). Primary implementation references for trajectory sampling and the reported $\mu$, $\mu^*$, and $\sigma$ measures.