Global Sensitivity Screening With Morris Trajectories

Why Use Morris Trajectories?

A decision model may depend on many uncertain inputs, while each model run takes enough time that an exhaustive sensitivity analysis is impractical. The Morris method offers a computationally economical first screen: it perturbs one input at a time at multiple locations in the input domain, then summarizes the resulting finite-difference effects. Its purpose is to identify inputs worth closer investigation, not to assign a definitive fraction of output variance to each input.

For a MIDD decision, the output might be a predicted probability of meeting a development target, an exposure measure, or a decision score. The method is applied to a specified scalar output and a specified set of uncertain inputs. If the decision depends on several outputs, screen each one separately; an input can be influential for one output and relatively unimportant for another.

Definition: For a model $y=f(x_1,\ldots,x_k)$, the elementary effect of input $i$ at a point $x$, using a signed step $\Delta_i$, is $$ EE_i(x)=\frac{f(x+\Delta_i e_i)-f(x)}{\Delta_i}, $$ where $e_i$ is the vector that changes coordinate $i$ by one unit and leaves the other coordinates unchanged. It is a finite change in model output per unit change in the chosen input coordinate.

Scale Inputs Before Constructing Trajectories

The inputs may have different physical units and plausible ranges. First define a decision-relevant range for each input and map it to a common scaled coordinate, usually $x_i\in[0,1]$. For a physical input $z_i$ with range $[L_i,U_i]$, a simple linear scaling is $x_i=(z_i-L_i)/(U_i-L_i)$. Thus an elementary effect calculated on this scale is output change per unit of scaled range, not per milligram, day, or other physical unit.

The range should represent values that are plausible and relevant to the decision, rather than arbitrary numerical limits. The earlier material on selecting distributions for uncertain model inputs is relevant when defining those ranges and the way uncertainty is represented. A standard Morris design samples a rectangular input domain and changes coordinates independently. If inputs must remain dependent or satisfy constraints, independently changing one input can create implausible combinations; a constrained or dependence-aware design is then needed, and its effects require careful interpretation.

A conventional design places each scaled input on a grid of $p$ levels, from 0 to 1, where $p$ is an even integer. A commonly used step size is

\delta=\frac{p}{2(p-1)}.

For example, with $p=4$, the grid is $\{0,1/3,2/3,1\}$ and $\delta=2/3$. Trajectory starting points are selected so that every planned positive or negative step stays on the grid and within $[0,1]$. A trajectory visits $k+1$ points: one starting point and one step for each of the $k$ inputs, in a selected order. Each step changes only the input being examined. Repeating this design at $r$ starting points produces $r$ elementary effects for each input, usually at a cost of $r(k+1)$ model evaluations if evaluations are not reused.

Define the question and domain.
Choose the scalar decision output, the uncertain inputs, and plausible ranges or transformations for those inputs.

Set the grid and step.
Scale inputs, choose an even number $p$ of levels, and select a valid step such as $\delta=p/[2(p-1)]$.

Generate trajectories and evaluate the model.
At each trajectory point, change one coordinate by a signed step and record the output difference.

Summarize and check stability.
Calculate the effects’ signed mean, mean absolute effect, and spread for each input; examine whether rankings are stable across trajectories.

Elementary Effects and Their Summaries

Because a trajectory step can be positive or negative, the signed denominator matters. If the input moves downward, both the output change and $\Delta_i$ are signed; the effect remains output change per unit change in the scaled coordinate. For input $i$, let the $r$ trajectory effects be $EE_{i1},\ldots,EE_{ir}$. The standard screening summaries are:

\mu_i=\frac{1}{r}\sum_{j=1}^{r}EE_{ij},\qquad \mu_i^*=\frac{1}{r}\sum_{j=1}^{r}|EE_{ij}|,\qquad \sigma_i=\sqrt{\frac{1}{r-1}\sum_{j=1}^{r}(EE_{ij}-\mu_i)^2}.

The signed mean $\mu_i$ indicates the average direction of change across the sampled steps, but effects of opposite signs can cancel. The mean absolute effect $\mu_i^*$ avoids that cancellation and is commonly used to rank overall influence. The sample standard deviation $\sigma_i$ describes how much the effects vary across the sampled trajectories. A large spread can signal nonlinearity, interactions with other inputs, or both; it does not by itself identify which explanation applies.

Why an Elementary Effect Is a Useful Local Summary

For a differentiable model, change only coordinate $i$ by a signed step $h$. Holding the other coordinates fixed, the fundamental theorem of calculus gives

\frac{f(x+h e_i)-f(x)}{h} =\int_0^1 \frac{\partial f(x+t h e_i)}{\partial x_i}\,dt.

Thus an elementary effect is the average of the partial derivative along the step, rather than a derivative at just one point. For a linear model in that coordinate, it equals the same slope wherever the step is taken. If the slope changes with location, or if the effect of changing $x_i$ depends on other coordinates, the elementary effects can vary between trajectories. That variation motivates the use of $\sigma_i$ as a screening diagnostic.

Worked Example: Calculating One Three-Input Trajectory

Consider a hypothetical deterministic model with output in decision-score units:

$$ f(A,B,C)=6+10A+6B+3C. $$

Use four grid levels, so $\delta=4/[2(4-1)]=2/3$. One valid trajectory starts at $(A,B,C)=(0,1,1/3)$, then changes $A$ upward, $B$ downward, and $C$ upward. The values stay on the four-level grid. At the starting point, $f(0,1,1/3)=6+0+6+1=13$.

After changing $A$, the point is $(2/3,1,1/3)$, with output $f(2/3,1,1/3)=6+20/3+6+1=59/3$, or about $19.67$. Therefore, $EE_A=[(59/3)-13]/(2/3)=(20/3)/(2/3)=10$ score units per unit of scaled $A$.

Next, $B$ changes from 1 to $1/3$, so its signed step is $-2/3$. The new output is $f(2/3,1/3,1/3)=6+20/3+2+1=47/3$. Hence, $EE_B=[(47/3)-(59/3)]/(-2/3)=(-12/3)/(-2/3)=6$ score units per unit of scaled $B$.

Finally, $C$ changes from $1/3$ to 1. The output becomes $f(2/3,1/3,1)=6+20/3+2+3=53/3$. Thus, $EE_C=[(53/3)-(47/3)]/(2/3)=(6/3)/(2/3)=3$ score units per unit of scaled $C$. These are effects at one trajectory only; additional trajectories are needed to assess how they vary over the domain.

Worked Example: Interpreting the Spread Across Trajectories

Suppose three hypothetical trajectories for input $B$ give elementary effects $-1$, $3$, and $2$, each in score units per unit of scaled $B$. The signed mean is $\mu_B=(-1+3+2)/3=4/3$, or about $1.333$. The absolute mean is $\mu_B^*=(1+3+2)/3=2$.

For the sample spread, the deviations from the mean are $-7/3$, $5/3$, and $2/3$. Their squared deviations sum to $[49+25+4]/9=78/9=26/3$. Since $r=3$, divide by $r-1=2$ before taking the square root: $\sigma_B=\sqrt{(26/3)/2}=\sqrt{13/3}\approx2.082$ score units per unit of scaled $B$.

The positive signed mean does not mean that every trajectory has a positive effect: one of the three effects is negative. Here the mean absolute effect is the more useful summary for judging magnitude, while the sizeable spread warns that the effect depends on where the trajectory is evaluated. The spread alone cannot establish whether nonlinearity, interactions, or both are responsible.

Ranking Inputs Under a Limited Run Budget

Morris screening is most useful when model evaluations are costly and the immediate goal is to reduce a long input list to a manageable set. The number of trajectories $r$ is a practical design choice: more trajectories generally provide more information about the range of effects, but require more model runs. A ranking is a screening result conditional on the chosen output, ranges, scaling, grid, and trajectory sample.

Worked Example: Ranking Four-Trajectory Summaries

Suppose a hypothetical analysis uses $r=4$ trajectories and yields the following effects, all in score units per unit of scaled input: $A=(-1,1,2,3)$, $B=(-1.6,-1.4,-1.5,-1.5)$, and $C=(0.2,0.4,0.8,1.4)$. For $A$, $\mu_A=(-1+1+2+3)/4=1.25$ and $\mu_A^*=(1+1+2+3)/4=1.75$. Its sample standard deviation is $\sqrt{8.75/3}\approx1.708$, since the squared deviations from 1.25 sum to 8.75.

For $B$, $\mu_B=(-1.6-1.4-1.5-1.5)/4=-1.5$ and $\mu_B^*=1.5$. Its squared deviations from $-1.5$ sum to 0.02, so $\sigma_B=\sqrt{0.02/3}\approx0.08165$. For $C$, $\mu_C=(0.2+0.4+0.8+1.4)/4=0.7$ and $\mu_C^*=(0.2+0.4+0.8+1.4)/4=0.7$. Its squared deviations from 0.7 sum to 0.84, giving $\sigma_C=\sqrt{0.84/3}\approx0.5292$.

Ranking by $\mu_i^*$ gives $A$ first at 1.75, $B$ second at 1.5, and $C$ third at 0.7. The negative signed mean for $B$, together with its small spread, is consistent with a fairly steady decreasing effect over the sampled steps. The larger spread for $A$ suggests that its effect varies across the sampled domain and warrants follow-up. These summaries support prioritization; they do not establish causal effects in clinical data or prove that a model input can be changed independently in practice.

Diagnostics, Alternatives, and Limitations

Treat the summaries as a screen, and check whether the design and calculations behave as intended:

Check the trajectory construction. Each step should change exactly one input, use the planned signed step, remain in the domain, and lie on the selected grid.
Check ranking stability. Compare summaries across additional or independently generated trajectory sets. If the ranking changes substantially, the run budget may be too small for a stable screen.
Check model evaluation quality. Failed runs, discontinuities, numerical instability, or stochastic simulation noise can distort finite differences. If the model is stochastic, control random-number variation or replicate evaluations sufficiently to distinguish input effects from simulation noise.
Check the scale and domain. A large effect over a broad assumed range may partly reflect that range choice. If inputs are mapped through non-linear transformations, the effect is per unit of the transformed coordinate, not necessarily per unit of the physical input.
Check dependence and feasibility. A standard trajectory changes one coordinate while holding the others fixed. If this produces impossible combinations for dependent inputs, do not interpret the resulting effects as if they described feasible joint uncertainty.

One-way sensitivity analysis changes an input around a reference case; it is useful for a focused local comparison but may miss effects that change elsewhere in the domain. Morris trajectories also use one-factor-at-a-time steps, but repeat them at multiple locations to screen globally. Variance-based methods, including Sobol indices covered in the later material, answer a different question by decomposing output variance under specified input distributions; they typically require many model evaluations. Morris is therefore a practical screening alternative when the input list is long or model runs are costly, not a substitute for every subsequent sensitivity analysis.

A high $\mu_i^*$ indicates large sampled elementary effects on the chosen scale, but not a unique mechanism, a probability of changing a decision, or a percentage contribution to output variance. A small value can also be conditional on the ranges and trajectories explored. Likewise, a large $\sigma_i$ is a warning of varying effects, not a standalone test for interaction. Follow-up analyses should target the decision question and the inputs or regions flagged by the screen.

Common pitfall: Reporting $\mu_i$ alone can make an influential input appear unimportant when positive and negative effects cancel. Calculate $\mu_i^*$ as well. Also use the signed step in the elementary-effect denominator: dividing by the absolute step instead can reverse the interpretation of effects from downward moves.

Key takeaway: Morris screening summarizes repeated, grid-based finite differences. Use $\mu_i^*$ to rank sampled influence, $\mu_i$ to describe average direction, and $\sigma_i$ to flag variation that may reflect nonlinearity or interactions. Interpret all three in light of the input domain, scaling, dependence, and trajectory stability.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

On a scaled input, the model output increases by 4 score units when the input changes by $+0.5$. What is the elementary effect, and what are its units?
An input has effects $-2$, $2$, and $2$. Calculate $\mu_i$ and $\mu_i^*$. Which better reflects the average magnitude of the sampled effects?
For four effects $(1,1,3,3)$, calculate the sample standard deviation $\sigma_i$.
Why can a large $\sigma_i$ not, by itself, be taken as proof of an interaction?

Solutions. 1. $EE=4/0.5=8$ score units per unit of scaled input. 2. $\mu_i=(-2+2+2)/3=2/3$, while $\mu_i^*=(2+2+2)/3=2$; $\mu_i^*$ reflects magnitude without cancellation. 3. The mean is 2, and the squared deviations sum to $1+1+1+1=4$; therefore $\sigma_i=\sqrt{4/(4-1)}=\sqrt{4/3}\approx1.155$. 4. Effects can vary because of nonlinearity as well as interactions, and their spread also depends on the sampled domain and trajectories. A large $\sigma_i$ flags variation but does not identify its cause.