Probit functions

For single exposures, the probit (probability unit) method provides a transformation method to convert the dose-response curve into a straight line. The probit variable Y is related to the probability P by (Finney, 1971)  

For spreadsheet computations, a more useful expression for performing the conversion from probits to percentage is given by. 

Probit equations for the probit variable, Y, are based on a causative variable, V (representing the dose), and at least two constants. These equations are of the form, 

Probit equations are available for a variety of exposures, including exposures to toxic materials, heat, pressure, radiation, impact, and sound, to name a few. For toxic exposures, the causative variable is based on the concentration for 

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FIGURE 8.4 Cumulative size distribution of gelatin (lime-cured type B, bloom number 225) microparticles prepared at high pH obtained using a Coulter Counter and plotted as a log-probit function. (From Lou, Y. and Groves, M.J. (1995). J. Pharm. Pharmacol., 47, 97-102. With permission.)... 

Given that an individual is exposed to a dose D, the respective conditional probability of consequence Po (e.g., death, several types of injuries etc.), can be calculated through the corresponding "probit functions (Papazoglou etal, 1992 TNO, 1989) as follows ... 

The value of probit function (in the case where the probit function was used) ... 

The overview of acute toxicily limits was focused on the most often evaluated dangerous substances in the Czech Republic ammonia and chlorine (Fig. 4). The most often nsed limits of acute toxicity are LC50, ERPG-1, 2, 3 and IDLH. Specific group is presented by probit functions. These are used for evaluating mortahty percentage in exposure area. 

Probit functions were evaluated as the most often used tools for determining the effects of major accidents, probably due to the fact that these functions are included in the EFFECTS software. Furthermore, it was determined that the values of the limits of acute toxicity are not always the same when applying different methods. For example the value of IDLH for chlorine differs in order of tens. The ERPG has the lowest difference in the values in the order of units. This can be caused by the absence of literature sources. 

In section 4 we indicate how the probit function (lognormal distribution), the normal and WeibuU distribution function parameters are obtained from experimental data. We use the graphical method, point estimator and momentum method within a resambling approach. In section 5 we summarize and conclude. 

The probit distribution for a quantity z is formulated most often using the probit function... 

By definition the cumulated distribution function (8) increases (decreases) when the probit function (7) increases (decreases). 

Probit functions for damage analysis of blast effects in terms of the scaled distance... 

Using data from [12] and the relations (4) and (5), the probit function... 

Table 4. Probit functions for explosion damage for humans [12].

Table 5 gives an overview of literature using free parameterizations and the probit function approach. There were no other distribntions that allowed an interpretation as one-dimensional probability distribution function other than the probit approach. 

As Table 5 shows a lot of anthors nse probit functions to describe the hazard of explosion effects. Most of the authors use a probit function that can be inter-pretedd according to the first line of (14). As we have seen in section 3 we can derive from snch probit functions also probit functions in terms of the scaled distance that can be interpreted according to (15). 

In [14] different models (including the probit function) are applied to fit different data from animals (rats, dogs) and humans being exposed to radiation... 

In the case of biological effects probit functions have been used for example to analyse the activity of a virus, depending on the surrounding temperature, infecting a bug and leading to its death [15]. 

The probit function approach can be used for fast damage assessment of blast effects of high explosive events. In the literature is the only parameterization of blast effects that can be interpreted as a probability distribution function in terms of physical parameters. In this sense we found only one probability distribution that is used in literature for the description of blast effects. 

Two interpretations of the probit function approach are appropriate in the case of blast effects. In the first case we cumulate the damage starting from a small... 

Resampling also allows the quantification of imcer-tainty of the parameters and the final probability estimates. Future work could focus on the description of the uncertainty using the distributions of the parameters as obtained from resampling. The statistical parameter distributions can be compared to confidence intervals obtained from classical parameter estimation. Uncertainty should also be considered when damage analysis results are communicated because to our knowledge most probit functions are based on very few data points with large error. 

Probit functions Derivation, evaluation and use in risk analysis... 

ABSTRACT A procedure was set up to derive, evaluate and formalize piobit functions for use in quantitative risk analysis in the framework of external safety in the Netherlands. Probit functions for substances are either derived under the authority of RIVM or by an interested party such as industry. The method for deriving human probit functions from animal toxicity data was laid down by a newly installed scientific expert panel. Proposed probit functions are subject to a puhUc comment period. Next, the probit functions are scientifically evaluated by the expert panel. If a probit function is approved on scientific grounds, the Ministry of Housing, Spatial Planning and the Environment decides if and when it will be implemented, taking the results of a consequence analysis into account. This paper presents the current state of affairs regarding the derivation, evaluation and formalization of new and revised probit functions. 

In the Netherlands, probit functions are used in a quantitative risk analysis (QRA) to predict the number of acute fatalities caused by an accidental release of toxic substances (RIVM 2009). A probit function describes the lethality rate as a function of any given combination of the exposure concentration and the duration of exposure for a specific substance. Hmnan probit functions are derived from acute inhalation toxicity data obtained in animals. These toxicity data are also used in other frameworks, including (1) the determination of intervention values for dangerous substances, (2) the classification of substances for transport QRAs, and (3) the determination of threshold values for the selection of plants relevant to a QRA. 

For a nmnber of substances, probit functions have been formally implemented. These are included in the Reference Manual Bevi Risk Assessments (RIVM 2009). For any other substance a probit function may be developed existing probit functions may be revised if necessary. 

In order to make the QRA calculation more transparent, a procedure was recently introduced that describes in detail the process resulting in the formal implementation of a probit function. 

For harmonization purposes, any decision by the Ministry of VROM with regard to probit functions is prepared by a committee of representatives from several other departments, including the ministries of Transport, Public Works and Water Management Economic Affairs Social Affairs and Employment and Interior and Kingdom Relations. 

A five-member expert panel, consisting of toxicologists from both public and private parties, was installed by the Ministry of VROM. The primary task of the expert panel is the scientific evaluation of probit functions for substances. The panel defined the method for deriving human probit functions from animal toxicity data (see paragraph 3.2). The expert panel ensures that the method is kept up-to-date. Probit functions are discussed by the panel in meetings that take place approximately four times a year. 

The Netherlands National Institute for Public Health and the Environment (RIVM) hosts the technical secretariat that controls the derivation, evaluation and formalization procedures. Further, RIVM supervises the development of probit functions derived through the public route (see paragraph 3.1), and the performance of consequence analyses (see paragraph 3.5). Finally, RIVM is responsible for the completeness check of all substance dossiers submitted. 

The procedure comprises the derivation, evaluation and formalization of probit functions. A new probit function can be derived for a substance currently without a formalized probit function, or an existing probit function as included in RIVM (2009) can be revised. Such a revision may be initiated when, for example, new toxicological data or procedures have become available. 

There are two possible starting points for the procedure a public and a private route. The public route is intended for substances for which the implementation of a probit function is considered of general interest. For these substances, that are placed on a priority list authorized by the Ministry of VROM, the probit function is derived imder the authority of RIVM. The private route is intended for any other substance, for which an applicant (e.g. industry) is interested in having a probit function implemented. An applicant may request inclusion of a substance of interest on the priority list. If such a request is granted, the substance... 

The substance document, drafted according to a template provided by RIV M, shall contain all relevant information, including a clarification of the steps leading to the derivation of the draft probit function and a rationale for the choices made. 

The methodology for the derivation or revision of probit functions is described in voliune 1, part 4 of the Dutch PubHcation Series on Dangerous Substances (Ministry of VROM 2005). The method was recently updated by the scientific expert panel, in order to meet current scientific standards and insights. The method makes use of various assessment factors, to accoimt for e.g. data quahty, sensory irritation, and inter- and intra-species variability. The method is not described in further detail here. 

The substance dossier is checked for completeness by RIVM. If a substance dossier is judged to be complete, the probit function for that substance is assigned the status proposed , and the substance is included in the plaiming of the expert panel meetings. 

Prior to discussion of a substance document in the expert panel meeting, the docinnent is published on a designated website and subject to a pubhc comment period. Up to 6 weeks after pubhcation of the substance document, interested parties can submit written comments on the derivation of the probit function. Comments received are evaluated in the expert panel discussion. 

Proposed probit functions are evaluated in a meeting of the expert panel. Meetings of the panel are closed, but parties that have provided comments may provide oral clarification prior to the meeting. 

The discussion focuses on two elements (1) assessment of the quality of the substance document including the studies used to derive the probit function, and (2) discussion of the derivation of the probit function. Consequently, a substance document, including the proposed probit function, may be either approved or rejected by the expert panel on scientific grounds. Approval or rejection is realized by means of majority voting by the panel members. 

Approval of a probit function results in its status being raised from proposed to interim . A rejected... 

If a probit function is formalized its status is raised from interim to established . Decisions on the formalization of probit functions are published on the designated website the Reference Manual Bevi Risk Assessments (RIVM 2009) and the External Safety Order (Ministry ofVROM 2004) are updated accordingly. 

Interim probit functions that are not formalized by the Ministry may still be used in QRAs, and RIVM will at all times advise to do so. This does not hold for revisions of already formalized probit functions if such a revision is not formalized the existing probit function remains vahd. 

Table 1. Overview of the current derivation, evaluation and formalization status of probit functions.

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