Deterministic Sensitivity Analysis

Deterministic Sensitivity Analysis: Concept Architecture

Orientation and learning roadmap

Deterministic sensitivity analysis links deliberate changes in model inputs to changes in results. The architecture below develops that logic from question formulation and parameter selection through one-way and multi-way analysis, scenario design, presentation, interpretation, and decision-focused validation.

What deterministic sensitivity analysis tests

Deterministic sensitivity analysis recalculates a model under deliberately chosen alternative values or assumptions. Each recalculation creates a traceable link between a specified change and the resulting change in costs, health outcomes, net benefit, or the preferred strategy. Because the tested values are fixed rather than sampled from probability distributions, the analysis is transparent and reproducible but does not estimate the probability that a decision is correct.

• Deterministic sensitivity analysis can identify inputs and assumptions that materially influence model results.
• Deterministic sensitivity analysis can show whether a preferred strategy changes within a justified range.
• Deterministic sensitivity analysis can locate thresholds at which the decision changes.
• Deterministic sensitivity analysis cannot by itself quantify joint parameter uncertainty or produce a probability of cost-effectiveness.

Start from a controlled base case

A deterministic analysis is interpretable only when it is anchored to a clearly specified base case. The base case defines the population, comparators, perspective, time horizon, discounting, model structure, evidence sources, parameter values, and decision rule against which each alternative result is compared. Analysts should preserve this configuration and alter only the values or assumptions named in each test.

Let the model output be (Y=g(\boldsymbol{\theta},\mathbf{s})), where (\boldsymbol{\theta}) contains numerical parameters and (\mathbf{s}) contains structural or methodological choices. The base-case result is:

[ Y_0=g(\boldsymbol{\theta}_0,\mathbf{s}_0) ]

A deterministic test (k) substitutes a specified alternative value or assumption and produces:

[ Y_k=g(\boldsymbol{\theta}_k,\mathbf{s}_k) ]

The effect of the test may be summarized as an absolute change, (\Delta Y_k=Y_k-Y_0), a percentage change, or a change in the preferred strategy. The chosen summary must match the decision question; a small numerical change can still be important if it crosses a decision threshold.

Choose the form of analysis that matches the uncertainty

Deterministic sensitivity analysis is an umbrella term for several related methods. The correct form depends on whether the uncertainty concerns one parameter, a combination of parameters, a coherent alternative scenario, or the value at which a conclusion reverses. Labelling each test precisely prevents a one-way parameter check from being mistaken for a broader assessment of model uncertainty.

One-way analysis is easy to audit but cannot reveal every interaction. Multi-way analysis can reveal combinations that matter, but the combinations should be selected for a reason rather than through arbitrary searching. Scenario analysis is often the most appropriate deterministic method for structural, methodological, implementation, or evidence-choice uncertainty because those alternatives cannot always be represented by changing one number.

Select defensible values and ranges

The credibility of a deterministic result depends on the credibility of the values tested. Ranges should represent plausible uncertainty, meaningful policy alternatives, observed variation, or clearly labelled stress tests. A convenient percentage around the base case is not automatically appropriate, especially for probabilities, utilities, relative effects, correlated quantities, or parameters constrained by clinical logic.

Suitable sources for deterministic values may include:

• Confidence or credible intervals from the evidence used to estimate a parameter.
• Alternative estimates from relevant studies, meta-analyses, registries, or jurisdictions.
• Minimum and maximum values supported by clinical, epidemiological, or operational evidence.
• Alternative prices, uptake levels, time horizons, discount rates, or implementation assumptions relevant to the decision maker.
• Expert-elicited bounds when empirical evidence is unavailable, with the elicitation method and uncertainty recorded.
• Explicit stress-test values used to expose model behaviour, clearly distinguished from evidence-based plausible ranges.

Parameter bounds must respect logical and mathematical constraints. Probabilities and proportions must remain between 0 and 1, mutually exclusive probabilities may need to sum to 1, and linked costs or event rates may need to move together. When changing one input would create an internally inconsistent model, the related quantities should be varied jointly in a scenario rather than presented as an independent one-way test.

Calculate decision-relevant outcomes

The analysis should report outputs that remain interpretable across all tested values. Incremental cost, incremental health outcome, and incremental net monetary benefit are usually more stable than an incremental cost-effectiveness ratio when results move between cost-effectiveness quadrants or the denominator approaches zero. Reporting only a changed ICER can obscure dominance, sign changes, and switches in the preferred strategy.

For a new intervention (A) compared with comparator (B):

[ \Delta C=C_A-C_B ]

[ \Delta E=E_A-E_B ]

[ ICER=\frac{\Delta C}{\Delta E} ]

At willingness-to-pay threshold (\lambda), incremental net monetary benefit is:

[ INMB=\lambda\Delta E-\Delta C ]

A positive INMB favours intervention (A) at the stated threshold, while a negative INMB favours comparator (B). For more than two mutually exclusive strategies, the analysis should evaluate the preferred option using a consistent net-benefit framework or repeat the full incremental analysis; it should not compare every strategy only with an arbitrary common comparator and assume the ranking is valid.

A worked one-way example

Suppose a new intervention has base-case incremental cost of £4,800 and incremental effectiveness of 0.30 quality-adjusted life-years compared with usual care. At a threshold of £20,000 per QALY, the base-case INMB is £1,200, so the intervention is preferred. The analyst tests the treatment acquisition cost at a justified lower and upper value while holding all other base-case inputs fixed.

[ INMB=(£20{,}000\times0.30)-£4{,}800=£1{,}200 ]

The analysis shows a decision reversal within the tested cost range. It does not show how likely the upper value is, nor does it account for simultaneous uncertainty in effectiveness, utilities, event rates, or other costs. Those questions require additional analysis, including probabilistic sensitivity analysis when joint parameter uncertainty is relevant.

Find a threshold value

Threshold analysis solves for the input value at which the decision metric reaches a specified boundary. In a two-strategy cost-effectiveness analysis using INMB, the decision switches when (INMB=0). A threshold is useful only when the model relationship, feasible parameter range, and direction of preference are reported.

If acquisition cost (P) is the uncertain component of incremental cost and all other incremental cost is (\Delta C_{other}), then:

[ 0=\lambda\Delta E-(P+\Delta C_{other}) ]

and the break-even acquisition cost is:

[ P^*=\lambda\Delta E-\Delta C_{other} ]

For nonlinear or discontinuous models, the threshold may require numerical search rather than algebra. Analysts should check for multiple crossings, flat regions, or changes caused by model logic, because a single reported switching value can be misleading when the outcome is not monotonic.

Present one-way results with a tornado diagram

A tornado diagram orders one-way results by their effect on a selected output. Each horizontal bar normally extends from the result at the lower tested value to the result at the upper tested value, with the most influential parameters shown first. The diagram is a prioritisation display, not a probability distribution and not a ranking of evidence quality.

Every tornado diagram should identify the outcome, base-case reference, parameter labels, tested values, units, and direction of the low and high cases. Bars based on incomparable or unjustified ranges can produce a misleading ranking because their lengths reflect both model responsiveness and the width chosen for each range. Parameters with structural or non-monotonic effects may require separate plots or scenario tables rather than a conventional two-ended bar.

Implement the analysis in a spreadsheet or model

A reproducible implementation separates base-case inputs, alternative values, calculations, and results. The model should restore the base case before every run so that one test cannot accidentally carry into the next. Automated data tables, scripts, or macros may be used, but the resulting values should be traceable to named inputs and tested against manual calculations.

A practical results table includes:

Quality assurance should confirm that low and high inputs are passed to the intended cells or variables, formulas recalculate fully, named ranges are unique, and outputs return to their original base-case values after testing. Analysts should also test a small set of cases manually and confirm that parameter changes move the model in a clinically and mathematically plausible direction.

Interpret deterministic and probabilistic analysis together

Deterministic and probabilistic sensitivity analyses answer different questions. Deterministic analysis explains what changes the result, tests defined alternatives, and locates decision thresholds. Probabilistic analysis propagates joint parameter uncertainty through repeated draws to estimate the distribution of outcomes and the probability that each strategy is cost-effective at different thresholds.

Agreement between the two approaches can strengthen understanding, while disagreement can reveal interactions, correlations, nonlinearities, or structural choices that need further investigation. Neither approach repairs weak evidence, an invalid model structure, or implausible assumptions.

Common errors and safeguards

Deterministic analyses can look persuasive even when their ranges or outputs are poorly chosen. The main safeguards are transparent justification, internal consistency, and reporting that separates plausible uncertainty from hypothetical stress testing. Reviewers should be able to reproduce every test from the base case.

• Do not describe arbitrary ±10% or ±20% ranges as evidence-based unless they genuinely reflect the relevant uncertainty.
• Do not vary correlated or logically linked inputs independently when doing so creates impossible combinations.
• Do not infer probabilities from tornado-bar lengths or from the number of scenarios favouring each strategy.
• Do not rank parameters solely by absolute output change when different parameters were tested over incomparable ranges.
• Do not rely on ICERs alone when incremental effects change sign, approach zero, or move between quadrants.
• Do not treat one-way analysis as a complete assessment of joint uncertainty, interaction, or structural uncertainty.
• Do not hide a decision reversal by reporting only the magnitude of the output change.
• Do not call a deliberately extreme scenario plausible without evidential support.

Reporting a decision-ready analysis

A decision-ready report links every deterministic test to a specific uncertainty and explains why its values matter. It presents the base case, tested alternatives, sources, outputs, decision threshold, and any change in the preferred strategy in a form that can be audited. The conclusion should distinguish robust results from conditional results and identify uncertainties that require probabilistic analysis, further evidence, or expert judgment.

At minimum, reporting should include the model outcome tested, all alternative values and units, the rationale for each range or scenario, incremental costs and outcomes, the decision metric, the preferred strategy, and a clear account of thresholds or reversals. When structural or methodological choices dominate the result, the report should say so directly rather than implying that parameter precision alone will resolve the uncertainty.