Probit transform

Figure 2-10 The probit transformation converts the sigmoidal response vs. log dose curve into a straight line when plotted on a linear probit scale. Source D. J. Finney, Probit Analysis, 3d ed. (Cambridge Cambridge University Press, 1971), p. 24. Reprinted by permission.

Graphical evaluation the distribution of the frequency of perceptions as a function of dilution has to be determined. The evaluation can be performed using frequencies before or after probit transformation. 

Sigmoidal curves of the concentration-response data are presented in Figure 1, and their probit transformations in Figure 2, where a log-normal model is assumed. Here, the log-concentration transformation (pT-scale) is paired with the probit parameter, which is indicative of the proportion of percentage inhibition . 

Figure 2. Probit transformation of the sigmoidal concentration-effect curves of Figure 1 for calculation of effective concentrations as IC50, IC20, or IC19.

Brought to logical conclusion, it was shown that the probit transformations of bacterial growth inhibition and inhibition of DNA biosynthesis by Nitroakridin 3582 were superposable while the same functions for inhibition of RNA and protein biosyntheses were superposable upon each other but indicated a lesser susceptibility of the test organism. This led to the conclusion that the mode of antibacterial action of the nitroacridine was its inhibition of DNA biosynthesis27. ... 

The concentration of choice for mode of action studies is the lowest drug concentration which inhibits growth entirely. This can either be estimated by extrapolation of the probit-transformed log dosage response correlation or by determination of the MIC, ie. minimal inhibitory concentration, by the method of serial twofold dilution of drug-containing medium in test tubes, inoculation of these media and visual observation of growth after incubation overnight. 

Figure 5.7 The effect of probit transformation. The normal sigmoid curve of Figure 5.6 is transformed to a straight line when ordinates are measured on a linear scale of probits instead of percentages. (From Finney, D.J., Probit Analysis, 2nd ed., Cambridge University Press, Cambridge, 1964. With permission.)...

The most frequent difficulty in environmental toxicology is to demonstrate a true effect at very low concentrations of the substance. It is relatively easy to draw so-called dose-effect relationship for SO2 (Rondia, 1970), even if the ordinate of the effects is somewhat subjective and if mathematical artifice (log or probit transformation) has been used to make the relationship a nearly straight line. We meet no problem in the high concentrations zone we can drive experiments and repeat them showing that increasing concentrations of SO2, preferably associated with an inert aerosol (sodium chloride, haematite, etc—) results in correlated increases in lung or bronchial irritation (Amdur, 1968). 

The probit method is perhaps the most widely used method for calculating toxicity vs. concentration or dose. As its name implies, the method used a probit transformation of the data. A probit is a unit of divergence from the mean of a normal distribution equal to one standard deviation. The central value of a probit would be 5.0, representing the median effect of the toxicity test. A disadvantage of the method is that it requires two sets of partial kills. However, a confidence interval is easily calculated and can then be used to compare toxicity results. There are several programs available for the calculation, and as discussed below, they provide comparable results. 

The hermetic dose response can be tested because its low-dose response starts immediately to the left (in the dose-response space) of any hypothetical threshold. Recollecting that the threshold model is the linearized form of the S-shaped toxicological cumulative distribution of responses, this response is generally not within the observations (it is an extrapolation via a probit transformation from the experimental results to a dose intercept). On the other hand, the hormetic dose-response can be either validated or rejected with normal testing protocols, provided that a sufficient number of experimental results are available (five or more). 

The LD50 may be derived graphically from a plot of the probit transform of the incidence of death against the logarithm of the dose administered. This is the classical quantal assay. 

The basis of the probit transform lies in the observation that, for a wide range of toxic compounds, if the incidence of mortality is plotted against the logarithm of the dose of the test... 

The arithmetical treatment of data from both quantal and continuous response assays are essentially the same, the difference lying in that whereas in continuous response assays the variance of a single response is established from the data itself (current or from previous experience), in quantal response assays the variance is established with infinite precision, and in some treatments (probit transformation) varies according to the level of the response. The arithmetic will be greatly reduced if the numbers in each dosage group are equal and the logarithmic interval between the dose levels the same for standard and test. 

For the purpose of example the treatment of a quantal response assay by probit transformation using two doses of standard and two of the test preparation will be considered. The data used are taken from the assay for insulin in which the number of mice displaying hypoglycaemic convulsions is taken as the response. 

The scheme shown can be used for all quantal response assays if probit transformations are to be used. When three doses of standard or test are used and the doses are equally spaced on the logarithmic scale, scores 1,... 

Reference has been made to the essential similarity in the treatment of continuous and quantal response assays. When the response groups are equal in size the treatment as described for the use of angular transformations may be applied using the mean response for the group. In such assays an estimate of variance is established from the data presented and t has the value appropriate to the degrees of freedom with which it is established. When the groups are not equal in size the more extended method used for probit transformations may be used. Each response could... 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

Probit Equation The probit equation has been used in an attempt to quantitatively correlate hazardous material concentration, duration of exposure, and probability of effect/injury, for several types of exposures. The objective of such use is to transform the typical sigmoidal (S-shaped) relationship between cause and effect to a straight-line relationship (Mannan, Lees Loss Prevention in the Process Industries, 3d ed., p. 9/68, 2005). 

There is also a special subset of statistical techniques that is part of both the second and third functions of statistics. This is data transformation, which includes such things as the conversion of numbers to log or probit values. 

Another reason is to linearize the relationship between a paired set of data, such as dose and response. This is the most common use in toxicology for transformations and is demonstrated in the section under probit and logit plots. 

Our approach to the first strategy requires that we construct a table with the pairs of values of x, and yt fisted in order of increasing values of T, (percentage response). Beside each of these columns a set of blank columns should be left so that the transformed values may be fisted. We then simply add the columns described in the linear regression procedure. Log and probit values may be taken from any of a number of sets of tables and the rest of the table is then developed from these transformed x and j/- values (denoted as x and y ). A standard linear regression is then performed. 

Figure 11.3 The dose-response relationship, (a) Five segments of the sigmoidal dose-response curve as described in the text, (b) Linearized dose-response relationship through log (dose)-probit (effect) transformations. Locations of the LD50 and LD05 are depicted.

Figure 2 represented a log-probit plot of the observed inhibition of purified bovine erythrocyte acetylcholinesterase as a function of concentration for several of the transformation products of aminocarb. The observation that these inhibition curves are parallel suggests a similar mechanism of interaction for the various derivatives. The parameter I5f. (the concentration of inhibitor required to achieve 50% inhibition oi the enzyme activity) for each of the inhibitors were calculated and are recorded in Table 1. These values are reported relative to the parent compound aminocarb = 1. Also included in Table 1 are the relative toxicities of several of these products to house crickets (Acheta domesticus). It had been our intention to develop bioassay tests using the target insect itself, the eastern spruce budworm (Choristoneura fumiferana). However, spray tower results were quite variable and it was considered that genetic variability of the stock culture made the production of uniform test batches difficult to achieve. Using the house crickets, an LD q of 130-155 ppm for aminocarb standard was observed over the course of more than 25 bioassays. Also included in Table 1 are observations by Abdel-Wahab and Casida (19) using human plasma or house fly head cholinesterases. 

Figure 2. Log-probit plot of acetylcholinesterase inhibition as a function of concentration of aminocarb and transformation products.

The result of this transformation is that the probit for 50% mortality is 5, for 16% mortality is 4, and for 84% mortality is 6. Probits for every percent mortality between 1% and 99.9% can be obtained from Table 5.1. 

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