Sensitivity analysis

Sensitivity analysis examines how calculation results change when inputs, parameters or assumptions are varied. In fire safety engineering, it helps identify which choices materially influence the predicted outcome or the resulting design decision. Notarianni distinguishes this investigation from selecting the correct value of an input: sensitivity reveals the consequences of alternatives, rather than deciding which alternative is true. Its scope includes both measured quantities and assumptions embedded in the analysis.

Sensitivity analysis supports uncertainty analysis by showing where imperfect information matters. It can also identify whether a comparative conclusion remains unchanged across plausible variations. In fire modelling, sensitivity may concern fire inputs, occupant behaviour, model parameters or numerical choices. The selected variations and outputs must match the question being investigated, because sensitivity at one condition does not describe every possible scenario.

What the analysis is intended to establish

A sensitivity study begins with an output or decision of interest. That might be a predicted temperature, a movement time or the relative performance of two designs. Changing an input is useful only when its effect is interpreted against that purpose. A large numerical change in an irrelevant output need not be important to the actual decision.

Sensitivity connects selected input variations to outputs and their effect on a decision. Based on Notarianni (2002), pp. 5-49–5-51.

Notarianni describes parametric analysis as examining a sequence of values for one input while holding others constant. Plotting the resulting output shows how that parameter influences the selected criterion. The approach can reveal a trend, a threshold or a region of weak response. It does not automatically describe the consequences of several uncertain inputs changing together.

Sensitivity and uncertainty are related but distinct. An output can respond strongly to a parameter that is known accurately. Another parameter may have a weaker response but a much wider uncertain range. Assessing importance requires attention to both the response and the uncertainty assigned to the input, rather than the slope alone.

The analysis can direct further investigation towards influential information. Notarianni describes importance analysis as identifying uncertain inputs that contribute most to output uncertainty. This can help prioritise better data or a more detailed model. The result depends on the model, outcome variable and uncertainties included in the study.

A sensitivity study also makes hidden assumptions easier to discuss. Assumptions about route use, departure, fire growth or boundary conditions can change a design result. Making their effects explicit helps separate consequences of evidence from consequences of analyst choices. The study still cannot replace evidence about which assumptions are appropriate.

Selecting inputs, ranges and comparisons

Input ranges need a stated basis. They may represent measurement uncertainty, plausible scenario variation or alternatives in an analysis assumption. These meanings should remain distinct because they support different conclusions. A range chosen for exploration is not automatically a probability distribution describing real-world occurrence.

Notarianni distinguishes empirical quantities from parameters expressing decisions or preferences. The latter do not necessarily have an unknown physical value that more measurement will reveal. Varying them can still show whether the design decision depends on the choice. Their treatment must make the underlying judgement visible rather than disguise it as measurement noise.

Bounding examines results at the extremes of an input's considered range. It can sometimes show that further detailed treatment of that uncertainty is unnecessary for the decision. That conclusion requires the examined bounds to capture the relevant response. A complex non-monotonic relationship cannot be assumed harmless merely because two endpoint calculations appear acceptable.

Changing one input at a time isolates its influence near the chosen baseline. This makes interpretation straightforward, but leaves interactions with other inputs largely unexplored. Notarianni notes that fire models can be nonlinear, multivariate and discontinuous. A single-factor study should therefore be described by its actual scope, rather than as complete coverage of uncertainty.

Correlations between inputs also require attention. Joint variations should represent combinations that are meaningful for the scenario. Treating related quantities as unrelated can produce a different uncertainty problem from the one intended. The chapter identifies correlated inputs as a feature that complicates analysis of large fire models.

The baseline should be documented alongside the alternatives. It gives meaning to relative changes and provides a reproducible reference. However, it is not made more credible merely by being called the baseline. The selected value and the range around it need the same scrutiny as other modelling inputs.

Local measures and wider model behaviour

A local sensitivity describes the change in an output near a particular input state. Notarianni discusses partial derivatives as measures of response to model parameters. These can be calculated directly for some equation systems. They describe a local relationship and do not necessarily extend to a wider range where the governing response changes.

A numerical comparison can approximate such a derivative by perturbing an input on either side of a baseline. For an output time t and an input width W, the central difference below estimates the local slope. It is a numerical approximation to the derivative concept discussed by Notarianni. Its value depends on the chosen perturbation and the response over that interval.

S_W ≈ [t(W + ΔW) − t(W − ΔW)]/(2ΔW)

Symbol	Meaning in the illustrative width study	Unit
SW	Estimated sensitivity of passage time to clear width	s/m
t	Calculated passage time	s
W	Baseline clear width	m
ΔW	Selected width perturbation on either side	m

The sign identifies the direction of the local response. A negative slope means that increasing the selected width reduces the calculated time under the retained assumptions. The magnitude has units and depends on how the input and output are expressed. Raw slopes for unrelated quantities cannot be ranked meaningfully without considering units and relevant variation ranges.

Relative changes can provide another useful description. An input change can be stated as a percentage of its baseline and compared with the output's percentage change. This describes responsiveness over the selected interval. It still does not assign a probability to that interval or determine how uncertain the input actually is.

Discontinuities require particular care. A change in model state, route availability or a decision threshold can alter the form of the response. A small local derivative on one side may miss the consequence of crossing that change. The chapter therefore cautions that differential information can be useful locally without supporting a global interpretation.

Sensitivity can also vary with time. An input important during early fire growth may have a different influence later. The relevant output should therefore be defined as an instantaneous value, a maximum, a time to an event or another specified measure. Combining these different questions into one unexplained sensitivity number loses information needed for interpretation.

Worked doorway-width study

Consider a deliberately simplified passage calculation for 90 people. Select a constant specific flow of 1.3 persons/(s·m) and apply the cited doorway boundary allowance of 0.15 m on each side. The baseline clear width is selected as 1.20 m. These inputs illustrate sensitivity and do not define a compliant doorway or a complete evacuation scenario.

The effective width is the clear width less the combined boundary allowance. Total flow is specific flow multiplied by effective width. Dividing the population by that flow gives the duration for the group to pass the selected point. The relationship assumes a sustained supply of occupants and the stated constant flow conditions.

t = P/[F_s(W − b)]

Symbol	Description	Unit
t	Passage duration at the selected doorway	s
P	Population passing the doorway, selected as 90	persons
Fs	Selected constant specific flow	persons/(s·m)
W	Clear doorway width	m
b	Combined boundary allowance, 0.30 m here	m

At the baseline width, effective width is 0.90 m and the calculated duration is 77.0 seconds when rounded upwards to three significant figures. Reducing clear width to 1.10 m gives 0.80 m effective width and 86.6 seconds on the same rounding basis. Increasing clear width to 1.30 m gives 1.00 m effective width and 69.3 seconds on that basis. All other selected inputs remain unchanged.

Equal width perturbations give unequal percentage changes in the selected inverse-flow relationship. Based on Nelson and Mowrer (2002), pp. 3-369–3-371, and Notarianni (2002), p. 5-50.

The clear-width perturbations are approximately 8.33 percent below and above the baseline. Using unrounded durations, the resulting time changes are a 12.5 percent increase and a 10.0 percent decrease. Equal input steps therefore do not produce equal-and-opposite percentage time changes. The inverse relationship between time and effective width explains the asymmetry.

Applying the central-difference expression to the unrounded alternative durations gives approximately −86.5 s/m. This is a slope estimated over the selected 0.20 m interval. It is not a universal doorway sensitivity and should not be extrapolated indiscriminately. Population, specific flow, baseline width and the boundary convention all belong to the result.

This example concerns passage at one point. It excludes detection, response before movement, approach travel and subsequent movement. It also holds specific flow constant rather than modelling changing arrivals or congestion. Sensitivity of the simplified passage expression cannot establish sensitivity of the complete evacuation process without those additional mechanisms.

Decision thresholds and changes in ranking

Notarianni uses the term switchover for a change large enough to alter the design decision. A prediction can vary without affecting that decision, or change only slightly while crossing an important threshold. The distinction depends on the chosen acceptance criteria. Sensitivity should therefore connect numerical response with the criterion actually used to judge the outcome.

For illustration only, impose an arbitrary 80-second limit on the doorway passage example. The baseline and wider case fall below it, while the narrower case exceeds it. Solving the same expression gives a clear-width crossing near 1.17 m when rounded upwards to three significant figures. This selected threshold is an explanatory device, not an egress requirement or recommended design limit.

A selected criterion can turn a continuous output change into a different decision. Based on Notarianni (2002), p. 5-50, and Nelson and Mowrer (2002), p. 3-371.

The threshold example separates two sources of decision dependence. One is the response of the passage model to width. The other is the selected criterion against which that time is compared. Altering either can change the outcome, so both need clear justification in a real assessment.

In performance-based design, sensitivity can also concern the ranking of a proposed solution against a reference design. The 2020 handbook from the Australian Building Codes Board (ABCB) specifically discusses checking whether that ranking changes across likely input values. This asks a comparative question, rather than merely whether each prediction moves by a small percentage. Both alternatives must be assessed on a consistent basis.

The ABCB data sheets similarly require examination of viable input variations in an evacuation-model comparison. They advise investigating unexpected responses instead of immediately assuming an error. An unexpected change may reveal an interaction or an implementation problem. Its explanation matters to the credibility of the comparative conclusion.

Sampling and uncertainty importance

Monte Carlo simulation can propagate mathematically described input uncertainty through repeated model runs. Notarianni explains that the resulting outputs approximate a distribution rather than one deterministic value. Sensitivity or importance measures can then examine how inputs contribute to its variation. This requires a justified description of the uncertainty being sampled.

More samples can improve the numerical representation of the selected uncertainty model. They do not make an unsupported input distribution correct. Nor do they remove uncertainty from a physical mechanism omitted by the underlying fire model. Sampling quality and physical-model adequacy are separate questions.

A response surface offers another approach: outputs from selected runs are used to fit a simpler relationship. That relationship can then be explored instead of repeatedly running the full model. The usefulness depends on how well it represents the relevant behaviour. Strong nonlinearity or discontinuities can limit a simple fitted approximation.

Importance rankings depend on the output, input ranges and dependence structure selected. A ranking for one scenario or time is not automatically transferable to another. It can nevertheless guide further work within the stated analysis. Improving the evidence for influential uncertain inputs may be more valuable than refining quantities with negligible decision impact.

A small sensitivity result also needs a bounded interpretation. It applies to the investigated parameter range and retained model structure. It cannot show that an omitted mechanism is unimportant, because that mechanism was never varied. Sensitivity analysis works on the represented problem and should be reported with that limit.

Reporting and using the findings

A reproducible account identifies the baseline, varied inputs, range basis and quantities held fixed. It states the output measure and the criterion used for interpretation. The resulting changes can then be traced to the selected perturbations. Without that information, a reported percentage sensitivity has little meaning.

The account should distinguish local exploration, bounding cases and probabilistic propagation. These methods answer related but different questions. A set of deterministic alternatives should not be labelled a confidence interval without an appropriate probabilistic basis. Likewise, one local derivative should not be presented as complete uncertainty in the final prediction.

The most consequential finding may be a condition that reverses the design decision. That condition should be made explicit, together with its evidential basis and whether it is plausible. Stable ranking across the examined range can also be useful evidence. Its scope remains the range, model and outputs actually examined.

Sensitivity analysis ultimately supports judgement about a calculation's dependence on information and assumptions. It can identify where better data, a different representation or further scenarios would affect the conclusion. It cannot determine the correct input merely by finding a favourable result. The defensible outcome is a transparent account of what changes, why it changes and whether that matters to the decision.

References

• Australian Building Codes Board. (2020a). Handbook: Fire Safety Verification Method (Version 1.1).
• Australian Building Codes Board. (2020b). Data sheets: Fire Safety Verification Method (Version 1.1).
• Nelson, H. E., & Mowrer, F. W. (2002). Emergency movement. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-367–3-380). National Fire Protection Association.
• Notarianni, K. A. (2002). Uncertainty. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 5-40–5-64). National Fire Protection Association.