Probit function - Revision history

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	* [[Event tree analysis]]		* [[Event tree analysis]]
	* [[Fire risk assessment]]		* [[Fire risk assessment]]
	* [[ALARP]]		* ALARP
	* [[Uncertainty analysis]]		* [[Uncertainty analysis]]

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	The same handbook names probit functions, together with the FED, among the more complex human exposure models, without describing them further. For the estimation of the time to compromised tenability in fires, it refers to ISO 13571 and ISO TR 13571-2, which give guidelines, a methodology and examples of tenability assessments.<ref name="AU-110" /> For buildings, it notes that simple tenability limits or FED models are generally adopted, so these simpler criteria are the ones most often used in building design.<ref name="AU-110" />		The same handbook names probit functions, together with the FED, among the more complex human exposure models, without describing them further. For the estimation of the time to compromised tenability in fires, it refers to ISO 13571 and ISO TR 13571-2, which give guidelines, a methodology and examples of tenability assessments.<ref name="AU-110" /> For buildings, it notes that simple tenability limits or FED models are generally adopted, so these simpler criteria are the ones most often used in building design.<ref name="AU-110" />
			[[File:fse-probit-function-v2.svg\|thumb\|upright=1.5\|alt=Three rising steps: simple tenability limits, fractional effective dose, and more complex models such as probit functions.\|Human exposure models from simple tenability limits to more complex models such as probit functions; for buildings, simple limits or FED models are generally adopted. Based on Australian Building Codes Board (2020b), p. 110.]]

	== Variation between people ==		== Variation between people ==
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	A table in the SFPE Handbook chapter on risk assessment in the chemical process industries lists effect levels based on observations of large fires.<ref name="B-186" /> At 37.5 kW/m² it gives 1 per cent lethality after 10 s and 100 per cent after one minute. At 12.5 kW/m² it gives 1 per cent lethality after one minute, and at 4.0 kW/m² pain after more than 20 s without likely blistering. At 1.6 kW/m² no discomfort is expected, even for a long exposure.<ref name="B-186" /> The two combinations for 1 per cent lethality, 37.5 kW/m² for 10 s and 12.5 kW/m² for one minute, deliver 375 and 750 kJ/m². The same tabulated effect thus corresponds to very different amounts of radiant energy.		A table in the SFPE Handbook chapter on risk assessment in the chemical process industries lists effect levels based on observations of large fires.<ref name="B-186" /> At 37.5 kW/m² it gives 1 per cent lethality after 10 s and 100 per cent after one minute. At 12.5 kW/m² it gives 1 per cent lethality after one minute, and at 4.0 kW/m² pain after more than 20 s without likely blistering. At 1.6 kW/m² no discomfort is expected, even for a long exposure.<ref name="B-186" /> The two combinations for 1 per cent lethality, 37.5 kW/m² for 10 s and 12.5 kW/m² for one minute, deliver 375 and 750 kJ/m². The same tabulated effect thus corresponds to very different amounts of radiant energy.
			[[File:fse-probit-function-v1.svg\|thumb\|upright=1.5\|alt=Log-log chart of exposure time against radiation intensity with points for pain, burns, injury and 1 and 100 per cent lethality.\|Exposure times for pain, burns, injury and death at different levels of thermal radiation, with the calculated time to pain. Based on Barry (2002), p. 5-186, and Beyler (2002a), pp. 3-309–3-310.]]

	In the Netherlands, the publication PGS 1, Part 1A, gives a relation between radiation dose and first-degree burns, second-degree burns and lethality. In practice, Dutch fire safety engineering mostly uses 2.5 kW/m² for escape and 1 kW/m² for long stays.<ref name="VR-7.7">Van Rossum (2027), §7.7.</ref> ''Introduction to Fire Safety Engineering'' stresses that the effect of radiation depends strongly on the duration of the exposure, which has to be taken into account when judging whether radiation actually causes problems.<ref name="VR-7.7" />		In the Netherlands, the publication PGS 1, Part 1A, gives a relation between radiation dose and first-degree burns, second-degree burns and lethality. In practice, Dutch fire safety engineering mostly uses 2.5 kW/m² for escape and 1 kW/m² for long stays.<ref name="VR-7.7">Van Rossum (2027), §7.7.</ref> ''Introduction to Fire Safety Engineering'' stresses that the effect of radiation depends strongly on the duration of the exposure, which has to be taken into account when judging whether radiation actually causes problems.<ref name="VR-7.7" />
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	== Criteria for explosion overpressure ==		== Criteria for explosion overpressure ==
	For explosions, effect levels are likewise stated as percentages of people affected.<ref name="B-186" /> At an overpressure of 2.5 to 5 psig, the table in the SFPE Handbook gives 1 per cent deaths from lung damage. At the same overpressure, more than half of the people suffer eardrum rupture, and more than half suffer serious wounds from flying objects. At 1 to 2.4 psig it gives 1 per cent eardrum rupture and 1 per cent serious wounds from flying objects.<ref name="B-186" /> With 1 psi equal to about 6.89 kPa, these ranges correspond to about 17 to 34 kPa and about 7 to 17 kPa.<ref name="A-12">DiNenno et al. (2002), p. A-12.</ref>		For explosions, effect levels are likewise stated as percentages of people affected.<ref name="B-186" /> At an overpressure of 2.5 to 5 psig, the table in the SFPE Handbook gives 1 per cent deaths from lung damage. At the same overpressure, more than half of the people suffer eardrum rupture, and more than half suffer serious wounds from flying objects. At 1 to 2.4 psig it gives 1 per cent eardrum rupture and 1 per cent serious wounds from flying objects.<ref name="B-186" /> With 1 psi equal to about 6.89 kPa, these ranges correspond to about 17 to 34 kPa and about 7 to 17 kPa.<ref name="A-12">DiNenno et al. (2002), p. A-12.</ref>
			[[File:fse-probit-function-v4.svg\|thumb\|upright=1.5\|alt=Two horizontal bars on an overpressure axis in kPa: 7 to 17 kPa for 1 per cent eardrum rupture, 17 to 34 kPa for 1 per cent deaths.\|Overpressure ranges with their effects on people: 1 per cent eardrum rupture and serious wounds at about 7 to 17 kPa, and 1 per cent deaths with more than half injured at about 17 to 34 kPa. Based on Barry (2002), p. 5-186.]]

	== Use in quantitative risk assessment ==		== Use in quantitative risk assessment ==
	In a quantitative risk assessment, risk is the product of the frequency of an undesired event and its expected consequences.<ref name="B-176">Barry (2002), p. 5-176.</ref> The assessment proceeds in steps: defining the objectives, identifying hazards, describing loss scenarios, assessing consequences and probabilities, presenting the risk and analysing ways to reduce it.<ref name="B-177">Barry (2002), p. 5-177.</ref> Consequence assessment estimates the harm to a target, usually expressed as effects on people, loss of property or business interruption. It combines the vulnerability of the target with the way the hazardous environment develops in intensity, distance and time.<ref name="B-182">Barry (2002), p. 5-182.</ref>		In a quantitative risk assessment, risk is the product of the frequency of an undesired event and its expected consequences.<ref name="B-176">Barry (2002), p. 5-176.</ref> The assessment proceeds in steps: defining the objectives, identifying hazards, describing loss scenarios, assessing consequences and probabilities, presenting the risk and analysing ways to reduce it.<ref name="B-177">Barry (2002), p. 5-177.</ref> Consequence assessment estimates the harm to a target, usually expressed as effects on people, loss of property or business interruption. It combines the vulnerability of the target with the way the hazardous environment develops in intensity, distance and time.<ref name="B-182">Barry (2002), p. 5-182.</ref>
			[[File:fse-probit-function-v3.svg\|thumb\|upright=1.5\|alt=Flowchart: release and fire or explosion, then exposure, then vulnerability of people, then individual and societal risk.\|Place of effect models for people in a quantitative risk assessment, from the fire or explosion to individual and societal risk. Based on Barry (2002), pp. 5-182, 5-186 and 5-210.]]

	Vulnerability can be assessed at two levels.<ref name="B-186" /> The first level uses general tables of damage criteria, such as the tables for radiation and overpressure described above. This gives a good first review but deals poorly with the duration of exposure, so many projects use generic threshold levels. The second level applies detailed heat transfer or structural models where the time or the specific conditions justify the extra effort.<ref name="B-186" />		Vulnerability can be assessed at two levels.<ref name="B-186" /> The first level uses general tables of damage criteria, such as the tables for radiation and overpressure described above. This gives a good first review but deals poorly with the duration of exposure, so many projects use generic threshold levels. The second level applies detailed heat transfer or structural models where the time or the specific conditions justify the extra effort.<ref name="B-186" />

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Mark: New FSE wiki article: Probit function (written by Claude Opus 5.5)

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New FSE wiki article: Probit function (written by Claude Opus 5.5)

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A '''probit function''' is one of the more complex models of human exposure that can be used in fire safety engineering to estimate how people are affected by a fire.<ref name="AU-110">Australian Building Codes Board (2020b), p. 110.</ref> Models of this kind range from simple tenability limits to more detailed approaches such as the [[fractional effective dose]] (FED) and probit functions. For fire safety engineering in buildings, either simple limits or FED models are generally adopted.<ref name="AU-110" /> In [[quantitative risk assessment]] (QRA) of fires and explosions, the effect of an exposure on people is expressed as the likelihood of death and of different degrees of injury at different levels of exposure.<ref name="B-186">Barry (2002), p. 5-186.</ref>

The approach starts from the observation that people do not all respond in the same way to the same exposure. Their responses are spread around a most frequent value, and some groups, such as elderly people and people with asthma, are more sensitive than others.<ref name="P-90">Purser (2002), p. 2-90.</ref> Effect data for thermal radiation, for example, have been stated as the exposure that causes 1 per cent, about 50 per cent or nearly 100 per cent fatalities.<ref name="Eis">Eisenberg et al. (1975), as cited in Beyler (2002a), p. 3-310.</ref>

== Human exposure models ==
In its simplest form, a human exposure model is a tenability limit, such as a minimum visibility or a maximum thermal FED.<ref name="AU-110" /> The FED itself adds up the dose received over time until it reaches 1.<ref name="P-88">Purser (2002), p. 2-88.</ref> In the ISO definition quoted by the Australian Building Codes Board, the FED compares the dose received with the dose expected to produce a specified effect. The reference is an exposed subject of average susceptibility.<ref name="AU-140">Australian Building Codes Board (2020b), p. 140.</ref> The FED is therefore aimed at a person of average susceptibility.

The toxicity chapter of the SFPE Handbook explains what such a typical value represents. A tenability dose or concentration is the most frequent value in a statistical distribution of responses, and the responses of individuals are spread around it in a probability curve.<ref name="P-90" /> The data behind these values mostly come from healthy young adult animals or humans. Groups with weaker hearts or lungs, such as the elderly and the roughly 15 per cent of children and 5 per cent of adults with asthma, may be affected at much lower exposures.<ref name="P-90" /> The handbook gives such a spread for hydrogen chloride, as the proportions of a population expected to suffer severe irritancy, impaired escape and incapacitation at different concentrations.<ref name="P-120">Purser (2002), p. 2-120.</ref>

For irritant gases the spread is wide. The handbook expects most people to be affected by hydrogen chloride at about 200 parts per million (ppm), and some already at about 100 ppm. Others tolerate more than 500 ppm without serious effects on escape.<ref name="P-120" /> The sensitivity to irritants is probably distributed on a logarithmic scale, in humans as in rodents. The proportions of the population affected at each concentration are based partly on human and animal data and partly on the judgement of the author.<ref name="P-120" />

== Population statistics ==
Effect data from animal tests are commonly reported as an LC<sub>50</sub>, the concentration that kills half of the exposed animals.<ref name="P-87">Purser (2002), p. 2-87.</ref><ref name="N-B66">Iqbal and Salley (2004), p. B-66.</ref> NUREG-1805, the fire hazard analysis guide of the U.S. Nuclear Regulatory Commission, explains this as the concentration that causes a 50 per cent probability of lethality. The value is only meaningful together with the exposure time, which is generally 30 minutes, followed by an observation period of 14 days.<ref name="Bab">Babrauskas et al. (1991), as cited in Iqbal and Salley (2004), p. B-66.</ref>

The spread of a quantity over a population is often described by the normal distribution, the familiar bell-shaped curve, which is defined by its mean and its standard deviation.<ref name="H-186">Hall (2002a), p. 1-186.</ref> The probability concepts chapter of the SFPE Handbook illustrates this with the body widths of adults. A door as wide as the mean width accommodates 50 per cent of the population, a door one standard deviation wider accommodates 84.1 per cent, and two standard deviations wider 97.7 per cent.<ref name="H-187">Hall (2002a), p. 1-187.</ref> To fit all but one person in 20,000, the door must be 3.87 standard deviations wider than the mean. When the logarithm of a quantity is normally distributed, the quantity itself is called lognormally distributed.<ref name="H-187" />

== Thermal radiation and burns ==
Thermal radiation from a fire can burn bare skin if the radiation is intense enough and lasts long enough.<ref name="BY-308">Beyler (2002a), p. 3-308.</ref> Burns form a continuous range of severity. A first-degree burn reddens the skin without blisters, a second-degree burn forms blisters, and a third-degree burn destroys all layers of the skin.<ref name="BY-309">Beyler (2002a), pp. 3-308–3-309.</ref> Healthy adults and teenagers can normally be expected to survive if less than 20 per cent of the body surface has second- and third-degree burns. Survival becomes unlikely when more than 50 per cent of the body is covered by full-thickness burns, even with intensive medical care.<ref name="BY-309" />

Pain and tissue damage are both related to the heating of the skin.<ref name="BY-309b">Beyler (2002a), p. 3-309.</ref> Pain is felt when the skin at a depth of 0.1 mm reaches about 45 °C. The time to pain can be estimated from the incident radiation with a simple correlation:<ref name="BY-309b" />
<math display="block">t_p = \left( \frac{35}{\dot{q}''} \right)^{1.33}</math>

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>t_p</math> || time until pain is felt || s
|-
| <math>\dot{q}''</math> || incident thermal radiation on the skin || kW/m²
|}

No pain was found below 1.7 kW/m², however long the exposure lasted.<ref name="BY-309b" /> Blisters form when the skin surface reaches about 55 °C, and the severity of the burn then depends on the energy absorbed after that moment. About 41.8 kJ/m² gives pain or a mild second-degree burn, a further 83.6 kJ/m² a severe second-degree burn, and more than 162.2 kJ/m² a severe third-degree burn.<ref name="BY-309b" /> These effects are described in more detail under [[effects of heat on people]].

Data on deaths from thermal radiation are much scarcer than data on pain and burns. The only large source is the analysis of fatalities from nuclear weapons, where exposure times were very short and interpretation is somewhat subjective.<ref name="BY-310">Beyler (2002a), p. 3-310.</ref> Eisenberg and co-workers used these data to relate radiation intensity and exposure time to four levels of harm. These are a threshold for significant injury, 1 per cent lethality, about 50 per cent lethality and nearly 100 per cent lethality.<ref name="Eis" /> On logarithmic axes of time and incident flux, each level appears as a straight line, and the time needed for a given level falls as the flux rises. Based on these data, an exposure of about 40 s at about 10 kW/m² can be taken as a threshold for fatality.<ref name="BY-310" />

A table used in the chemical process industries gives similar levels based on observations of large fires.<ref name="B-186" /> At 37.5 kW/m² it lists 1 per cent lethality after 10 s and 100 per cent after one minute. At 12.5 kW/m² it lists 1 per cent lethality after one minute, and at 4.0 kW/m² pain after more than 20 s without likely blistering. At 1.6 kW/m² no discomfort is expected, even for a long exposure.<ref name="B-186" /> In the Netherlands, the publication PGS 1, Part 1A, gives a relation between radiation dose and first-degree burns, second-degree burns and lethality. In practice, Dutch fire safety engineering mostly uses 2.5 kW/m² for escape and 1 kW/m² for long stays.<ref name="VR-7.7">Van Rossum (2027), §7.7.</ref>

== Explosion overpressure ==
For explosions, effect levels are likewise stated as percentages of people affected.<ref name="B-186" /> At an overpressure of 2.5 to 5 psig, the table in the SFPE Handbook gives 1 per cent deaths from lung damage. At the same overpressure, more than half of the people suffer eardrum rupture, and more than half suffer serious wounds from flying objects. At 1 to 2.4 psig it gives 1 per cent eardrum rupture and 1 per cent serious wounds from flying objects.<ref name="B-186" /> With 1 psi equal to about 6.89 kPa, these ranges correspond to about 17 to 34 kPa and about 7 to 17 kPa.<ref name="A-12">DiNenno et al. (2002), p. A-12.</ref>

== Use in quantitative risk assessment ==
In a quantitative risk assessment, risk is the product of the frequency of an undesired event and its expected consequences.<ref name="B-176">Barry (2002), p. 5-176.</ref> The assessment proceeds in steps: defining the objectives, identifying hazards, describing loss scenarios, assessing consequences and probabilities, presenting the risk and analysing ways to reduce it.<ref name="B-177">Barry (2002), p. 5-177.</ref> Consequence assessment estimates the harm to a target, usually expressed as effects on people, loss of property or business interruption. It combines the vulnerability of the target with the way the hazardous environment develops in intensity, distance and time.<ref name="B-182">Barry (2002), p. 5-182.</ref>

Vulnerability can be assessed at two levels.<ref name="B-186" /> The first level uses general tables of damage criteria, such as those for radiation and overpressure. This gives a good first review but deals poorly with the duration of exposure, so many projects use generic threshold levels. The second level applies detailed heat transfer or structural models where the time or the specific conditions justify the extra effort.<ref name="B-186" />

The results feed two common measures of human risk. [[Individual risk]] is the probability of injury or death to a person at a specified location relative to the source of the fire or explosion.<ref name="B-210">Barry (2002), p. 5-210.</ref> Societal risk takes into account the likelihood of multiple casualties and is often shown as a curve of the frequency of exceeding a given number of fatalities.<ref name="B-210" /> Most of this quantification is based on fatalities. There is uncertainty about what exactly constitutes a fatal dose of thermal radiation, blast or toxic chemicals, so all assumptions and judgements have to be documented.<ref name="B-211">Barry (2002), p. 5-211.</ref>

== Exposure, time and dose ==
Many effect models use a dose, the combination of exposure level and duration. For toxic gases the simplest form is [[Haber's rule]], which treats the product of concentration and time as constant for a given effect.<ref name="P-87" /> Lethal concentrations from animal tests are often converted to a common 30-minute exposure with this rule.<ref name="P-92">Purser (2002), p. 2-92.</ref>

For thermal radiation the relation between intensity, time and effect is not that simple. In the table quoted above, 37.5 kW/m² for 10 s and 12.5 kW/m² for 60 s both correspond to 1 per cent lethality.<ref name="B-186" /> The first combination delivers 375 kJ/m² and the second 750 kJ/m², so the same effect follows from half the radiant energy when the intensity is three times higher. The time to pain shows the same pattern: according to the correlation above, pain starts after about 13 s at 5 kW/m² and after about 5 s at 10 kW/m². Doubling the intensity therefore shortens the time to pain by more than half.

== Worked example ==
The United States federal safety standards for liquefied natural gas facilities suggest 5 kW/m² as an acceptable level for direct exposure of people.<ref name="BY-310" /> This level can be used as a criterion for injury.<ref name="BY-310" /> With the pain correlation, the time to pain at this level is:
<math display="block">t_p = \left( \frac{35}{5} \right)^{1.33} = 7^{1.33} \approx 13\ \mathrm{s}</math>

This agrees with the handbook, which gives about 13 s before unbearable pain and about 40 s before second-degree burns at 5 kW/m².<ref name="BY-310" /> At 2.5 kW/m², the level used for escape in the Netherlands, the same correlation gives about 33 s.<ref name="VR-7.7" /> At 10 kW/m², the fatality threshold for an exposure of about 40 s, pain starts after about 5 s. A person who notices pain after 5 s and needs another 35 s to get out of the radiation would therefore reach the fatality threshold. At 5 kW/m², the same 40 s would cause second-degree burns.<ref name="BY-310" />

== Limits and uncertainty ==
The data behind effect levels for fatalities are limited. For thermal radiation they come mainly from nuclear weapons, with exposures much shorter than in most fires.<ref name="BY-310" /> Tables of damage criteria are weak on the effect of exposure time, and the exact fatal dose of radiation, blast or toxic chemicals is uncertain.<ref name="B-186" /><ref name="B-211" /> The pain correlation is not valid below 1.7 kW/m², where no pain occurs at all, and different studies diverge at high fluxes, although the correlation predicts the time to pain with reasonable accuracy.<ref name="BY-309b" />

For toxic and irritant gases, the spread of sensitivity within a population is partly based on expert judgement, because experiments with humans at dangerous exposures cannot be done.<ref name="P-120" /> Sensitive groups may respond far below the typical values.<ref name="P-90" /> For buildings, fire safety engineering generally uses simple tenability limits or FED models.<ref name="AU-110" />

== See also ==
* [[Societal risk]]
* [[Event tree analysis]]
* [[Fire risk assessment]]
* [[ALARP]]
* [[Uncertainty analysis]]

== Notes ==
<references />

== References ==
* Australian Building Codes Board. (2020b). ''Handbook: Fire Safety Verification Method'' (Print version 1.1).
* Barry, T. F. (2002). Quantitative risk assessment in chemical process industries. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 5-176–5-213). National Fire Protection Association; Society of Fire Protection Engineers.
* Beyler, C. L. (2002a). Fire hazard calculations for large, open hydrocarbon fires. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 3-268–3-314). National Fire Protection Association; Society of Fire Protection Engineers.
* DiNenno, P. J., Drysdale, D., Beyler, C. L., Walton, W. D., Custer, R. L. P., Hall, J. R., Jr., & Watts, J. M., Jr. (Eds.). (2002). ''SFPE handbook of fire protection engineering'' (3rd ed.). National Fire Protection Association; Society of Fire Protection Engineers.
* Hall, J. R., Jr. (2002a). Probability concepts. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 1-182–1-192). National Fire Protection Association; Society of Fire Protection Engineers.
* Iqbal, N., & Salley, M. H. (2004). ''Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program'' (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
* Purser, D. A. (2002). Toxicity assessment of combustion products. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 2-83–2-171). National Fire Protection Association; Society of Fire Protection Engineers.
* Van Rossum, E. (2027). ''Introduction to Fire Safety Engineering''. Fire Safety Academy.

[[Category:Fire Risk Analysis]]