One-Way Sensitivity Analysis

One-Way Sensitivity Analysis: Tornado Diagrams, Parameter Ranges and a Worked Example

One-way sensitivity analysis is the simplest form of deterministic sensitivity analysis used in model-based economic evaluation, and its results are usually displayed as a tornado diagram. It identifies which single inputs, moved across a defensible range, shift the incremental cost-effectiveness ratio or net benefit far enough to matter for the decision. This page sets out the calculation, how parameter ranges should be chosen, how a tornado diagram is built and ordered, and how switching values are found through threshold analysis. A fully specified illustrative model for a fictional drug reports the ICER at every low and high input value and ranks the inputs by swing. The later sections cover the limits of the method, including correlation, interaction and non-linearity, and explain why the NICE methods manual (PMG36) prefers probabilistic results for the committee's preferred estimate.

What one-way sensitivity analysis does

One-way sensitivity analysis shows which individual inputs most influence a selected output and whether a plausible change in any one input could alter the decision. The method is commonly abbreviated as OWSA and is often displayed in a tornado diagram. It is an influence analysis, not a complete measure of joint parameter uncertainty.

The calculation behind the analysis

Let a model output be $Y=f(\theta_1,\ldots,\theta_k)$, where $\theta_j$ is the input being examined and $\theta_{-j}^{0}$ represents all other inputs fixed at their base-case values. The one-way low and high results are calculated separately for every selected input.

$$ Y_j^{L}=f!\left(\theta_j^{L},\theta_{-j}^{0}\right) $$

and

$$ Y_j^{U}=f!\left(\theta_j^{U},\theta_{-j}^{0}\right), $$

where $\theta_j^{L}$ and $\theta_j^{U}$ are the lower and upper values of input $j$, and $Y_j^{L}$ and $Y_j^{U}$ are the resulting outputs. The displayed impact width is usually

$$ R_j=\left|\max!\left(Y_j^{L},Y_j^{U}\right)-\min!\left(Y_j^{L},Y_j^{U}\right)\right|, $$

where $R_j$ is the width of the bar for input $j$, often called its swing.

The lower input value does not necessarily produce the lower output value. Recording both the input endpoint and its resulting output prevents the directions from being mislabelled.

Choosing the output that answers the decision question

The output should be selected before running the analysis and should match the decision being informed. Common outputs include incremental cost, incremental health effect, incremental net monetary benefit, budget impact, event counts, and sometimes the incremental cost-effectiveness ratio.

At a willingness-to-pay threshold $\lambda$, incremental net monetary benefit is

$$ INMB=\lambda\Delta E-\Delta C, $$

where $\Delta E$ is incremental effectiveness and $\Delta C$ is incremental cost. Net benefit often behaves more reliably than an ICER when incremental effects approach zero, when dominance changes, or when the cost-effectiveness plane crosses quadrants.

Choosing defensible parameter ranges

An OWSA result is only as meaningful as the range assigned to each input. Ranges should represent a defensible uncertainty interval or decision-relevant scenario rather than an automatic percentage applied uniformly to unrelated parameters. For a statistically estimated parameter, the ISPOR-SMDM good practice report on parameter estimation and uncertainty notes that the range can come directly from the estimation, for example a 95% confidence interval, on the scale on which the parameter was estimated.

A uniform $\pm20%$ range may be useful as an explicitly labelled exploratory assumption when better evidence is unavailable. The same ISPOR-SMDM report accepts arbitrary percentage variation as a measure of sensitivity but not as a representation of uncertainty, because a precisely estimated parameter and a poorly estimated one receive the same relative range. Such a range should not be presented as though it were an evidence-based confidence interval.

Worked example: a six-input tornado analysis for the fictional drug Quintavel

This example is illustrative. Quintavel is a fictional drug, and every number below is invented to show the method; none describes a real product, trial, price or disease estimate. The model compares Quintavel with standard care in patients at risk of a costly clinical event, such as a hospital admission for disease progression, over a fixed horizon. All costs and QALYs are per patient and are treated as already discounted, so the model has no time dimension.

Model structure and inputs

The model has one pathway. Standard care carries a baseline event risk, Quintavel multiplies that risk by a relative risk, and each event avoided saves the event cost and a QALY loss while Quintavel adds its own acquisition and monitoring costs.

The six inputs, their base-case values and their illustrative ranges are shown below. The relative risk interval is asymmetric because ratio measures are estimated on the log scale; 0.60 ÷ 0.45 and 0.80 ÷ 0.60 both equal 1.33 to two decimal places.

Base-case result

In the base case, 0.50 × (1 − 0.60) = 0.20 events are avoided per patient. Incremental QALYs are 0.20 × 2.5 = 0.50, and incremental cost is £13,000 plus £1,000 less £2,000 of avoided event costs, which gives £12,000. The base-case ICER is 12,000 ÷ 0.50 = 24,000, or £24,000 per QALY, and INMB at the illustrative threshold of £30,000 per QALY is £15,000 less £12,000, or £3,000.

One-way results and tornado order

Each row below changes a single input to its low value, restores the base case, changes the same input to its high value, and records both results. The rows are sorted by ICER swing, which is the order of the bars from top to bottom in the tornado diagram.

The top bar can be checked by hand. At the low relative risk of 0.45, 0.50 × 0.55 = 0.275 events are avoided, incremental QALYs are 0.275 × 2.5 = 0.6875, incremental cost is £14,000 less £2,750, or £11,250, and the ICER is £11,250 ÷ 0.6875, or £16,364 to the nearest pound. At the high relative risk of 0.80, 0.50 × 0.20 = 0.10 events are avoided, incremental QALYs are 0.25, incremental cost is £13,000, and the ICER is £52,000.

Reading the Quintavel results

The relative risk dominates the diagram, and its bar is lopsided around the £24,000 base case because the ICER is a ratio: as incremental QALYs shrink, the ICER rises faster than it falls when they grow. The INMB bars give the same ranking here, but INMB is linear in the relative risk, so its bar is symmetric in the quantity of events avoided. Two endpoints cross the illustrative £30,000 threshold: the high relative risk and the low baseline risk. For the baseline risk, the low input produces the high ICER, which is why endpoints must be recorded rather than inferred from bar colour.

The ranking reflects the chosen ranges as much as the model. A wider price scenario for Quintavel would move acquisition cost up the diagram without any change in the evidence about effectiveness.

Finding decision thresholds

Threshold analysis extends OWSA by identifying the value at which the preferred decision changes. It is more informative than endpoint testing when a decision boundary lies inside the chosen range. The NICE methods manual (PMG36, section 4.7.22) describes this value as a switching value and asks that the analysis show how far it lies from the current best estimate of the parameter.

For a two-strategy comparison using net benefit, the break-even condition is

$$ INMB=\lambda\Delta E-\Delta C=0, $$

where the decision changes as INMB passes through zero. If incremental cost is fixed, the threshold incremental effect is

$$ \Delta E^*=\frac{\Delta C}{\lambda}, $$

where $\Delta E^*$ is the break-even incremental effect. In the worked example,

$$ \Delta E^*=\frac{12{,}000}{30{,}000}=0.40\text{ QALYs}, $$

where the base-case incremental cost and the illustrative threshold have been substituted. If incremental effectiveness is fixed, the break-even incremental cost is $\Delta C^*=\lambda\Delta E=£15{,}000$.

Switching values can also be solved for the model inputs themselves. In the Quintavel model, setting INMB to zero for the relative risk gives 1 − RR = 14,000 ÷ (0.50 × 85,000), because 30,000 × 2.5 + 10,000 = 85,000 and 0.50 × 85,000 = 42,500, so the switching relative risk is approximately 0.671. The same approach gives a switching baseline risk of approximately 0.412, a switching acquisition cost of £16,000 and a switching QALY loss of 2.0 per event. The first two lie inside their tested ranges and the last two lie outside, which matches the tornado results. A reported threshold should state whether the crossing was solved analytically, found with a numerical root finder, or approximated from a grid.

Reading a tornado diagram correctly

A tornado diagram ranks horizontal bars by the difference between the outputs obtained at each input's two endpoints. The widest bar appears at the top, creating the characteristic tornado shape and showing which tested ranges have the greatest individual influence. The bars are drawn around a vertical line at the base-case output, and the decision threshold can be marked as a second line so that any bar crossing it shows an endpoint that changes the decision.

Each bar should identify the parameter, low and high input values, resulting outputs, base-case reference line, outcome units, and direction of change. Colour alone should not encode low and high endpoints because colour may be inaccessible and because the low input can produce the high output. The ISPOR-SMDM good practice report recommends a legend or table giving each parameter's bounds and their justification from the evidence base.

The ranking combines model sensitivity with range width. A parameter may appear influential because its evidence-based interval is wide, while another may have a steep local effect but a narrow uncertainty interval.

Local sensitivity and non-linearity

Endpoint-only OWSA can conceal curvature, discontinuities, and decision changes within the range. Evaluating several intermediate values reveals whether the relationship is approximately linear or whether the endpoints provide a misleading summary.

A local sensitivity measure at the base case is

$$ S_j=\left.\frac{\partial Y}{\partial\theta_j}\right|_{\theta=\theta^0}, $$

where $S_j$ is the slope of the output with respect to input $j$ at the base case $\theta^0$. A dimensionless elasticity can compare parameters measured on different scales:

$$ E_j=\left.\frac{\partial Y}{\partial\theta_j}\frac{\theta_j}{Y}\right|_{\theta=\theta^0}, $$

where $E_j$ is the proportional change in the output for a small proportional change in input $j$.

Derivatives describe local behaviour, whereas a tornado bar describes the effect across the selected endpoints. Neither should be interpreted without its scale and range.

What OWSA can and cannot show

OWSA is valuable because it is transparent, easy to audit, and effective at identifying important drivers. It can also expose unexpected model behaviour and indicate where better evidence or validation effort may have the greatest value.

OWSA cannot simultaneously propagate uncertainty in all inputs, preserve correlation among parameters, or estimate the probability that an intervention is cost-effective. A bar that crosses the threshold carries no information about how likely that endpoint is, so a one-way analysis describes influence and cannot measure decision uncertainty. Probabilistic sensitivity analysis is generally used for joint parameter uncertainty, while scenario analysis is better suited to coherent packages of structural or policy assumptions.

The NICE methods manual (PMG36) reflects this division. Section 4.7.21 treats univariate analysis as a way of identifying the key drivers of a model, but notes that such analyses become less helpful for representing combined uncertainty as the number of parameters grows. Section 4.7.12 states that the committee's preferred cost-effectiveness estimate should come from a probabilistic analysis when possible, unless the model is linear, and section 4.7.17 lists tornado diagrams as a useful presentation for deterministic results.

These methods are complementary rather than substitutes. A complete uncertainty assessment may use several of them for different questions.

Correlation and logical constraints

Changing one input while fixing all others can create combinations that are statistically unlikely or logically impossible. This occurs when probabilities must sum to one, regression coefficients are correlated, survival parameters jointly define a curve, or costs share common components.

Analysts should vary an independent parameterisation and recalculate dependent quantities automatically. If a one-at-a-time change cannot preserve the model's internal constraints, the result should be replaced by a coherent scenario or a joint analysis rather than reported as a valid OWSA.

Interaction is a separate limitation. In the Quintavel model the baseline risk and the relative risk multiply, so setting both to their unfavourable endpoints together gives 0.40 × 0.20 = 0.08 events avoided, 0.20 incremental QALYs, an incremental cost of £13,200 and an ICER of £66,000. No single bar in the tornado diagram shows this combined effect, which is why two-way and probabilistic analyses are needed alongside it.

A reproducible workflow

The analysis should be driven by an input register and a controlled base case. This prevents accidental changes to multiple parameters and makes every tornado bar traceable to two model runs.

1. Define the decision question and choose the output and its units.
2. Lock and reproduce the base-case input set and output.
3. Select parameters whose uncertainty or influence is relevant to the decision.
4. Assign evidence-based low and high values with a source and rationale.
5. Change one independent parameter to its low value and run the model.
6. Restore the base case, change the same parameter to its high value, and rerun the model.
7. Record endpoint inputs, endpoint outputs, model status, and any constraint violations.
8. Repeat the paired runs for every selected parameter.
9. Rank parameters by output width and identify decision-boundary crossings.
10. Validate influential or unexpected results with intermediate values and independent calculations.
11. Report excluded parameters, range limitations, and complementary uncertainty analyses.

For stochastic models, each endpoint should use adequate simulation precision and preferably common random numbers. Otherwise Monte Carlo noise may be mistaken for sensitivity to the parameter.

Spreadsheet implementation

A spreadsheet implementation should separate the base-case inputs, sensitivity ranges, model calculations, and OWSA results. Each result row should retain the parameter identifier so that sorting the tornado table does not detach labels from values.

Automated data tables, scripts, or macros can speed repeated runs, but the final results should be stored with the model version and recalculation status. Spot checks should reproduce selected rows by direct substitution into the model.

Validation checks

OWSA should be validated as both a model analysis and a data-processing workflow. The checks should confirm that only the intended input changes, the base case is restored between runs, and the displayed bars match the stored endpoint results.

• A base-to-base run should reproduce the locked base-case output exactly within the model's numerical tolerance.
• Each endpoint run should differ from the base case in only the intended independent input and its formula-driven dependants.
• Analytical special cases should reproduce the expected direction and magnitude of change.
• Parameters expected to be monotonic should be checked at intermediate values for reversals or discontinuities.
• The tornado ranking should use the absolute endpoint-output width rather than the parameter's numeric range.
• Decision changes should be checked against the stated decision rule and threshold.
• Stochastic endpoint results should be checked for adequate Monte Carlo precision.

Unexpected findings should trigger investigation rather than automatic deletion from the chart. They may reveal a model error, an interaction, a discontinuity, or a genuine feature of the decision problem.

Common interpretation errors

Several common shortcuts turn a useful diagnostic into a misleading claim. Clear reporting separates influence over a chosen range from statistical uncertainty and decision risk.

• A wide tornado bar does not prove that the parameter is poorly estimated.
• A narrow tornado bar does not prove that the parameter is known precisely.
• Holding correlated inputs fixed can create implausible combinations.
• Using arbitrary percentage ranges can distort the ranking across parameters.
• Reporting only endpoint outputs can hide non-linear behaviour inside the range.
• Ranking ICER bars can become misleading when incremental effects change sign or approach zero.
• Varying the cost-effectiveness threshold as if it were a model parameter mixes the decision rule with parameter uncertainty; the threshold is better handled by marking it on the diagram or by an acceptability curve.
• OWSA does not provide the probability that a strategy is cost-effective.
• OWSA does not capture structural uncertainty unless explicit structural scenarios are analysed.

The analyst should link each material limitation to its likely decision consequence. A limitation matters most when it could change the preferred option or the confidence placed in the result.

How results should be reported

Transparent reporting states the base-case output, outcome measure, parameter ranges, range sources, endpoint outputs, decision rule, and model version. It should identify any thresholds crossed and explain whether the relationship was checked between endpoints.

A tornado diagram should be accompanied by a results table or accessible text equivalent containing the underlying values. The report should also distinguish OWSA from probabilistic and scenario analyses and avoid claiming that a deterministic ranking fully characterises decision uncertainty.

Sources

• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 31 March 2026. Section 4.7, Exploring uncertainty.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press. 2006.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes, 4th edition. Oxford University Press. 2015.
• Stinnett AA, Mullahy J. Net health benefits: a new framework for the analysis of uncertainty in cost-effectiveness analysis. Medical Decision Making. 1998;18(2 Suppl):S68-S80.