Summary
A quantitative analysis of the toxicity of drugs or poisons applied jointly requires that they be administered at several dosages in mixtures containing fixed proportions of the ingredients. From a study of the dosage-mortality curves for several such mixtures, preferably in comparison with equivalent curves for the isolated active ingredients, most cases of combined action can be classified into one of three types:

(1) The first type is that in which the constituents act independently and diversely, so that the toxicity of any combination can be predicted from that of the isolated components and from the association of susceptibilities to the two components. The coefficient of association can be measured experimentally and should be constant at all proportions of the ingredients. When high, the toxicity of the mixture is reduced. The form of the dosage-mortality curve has been examined for several hypothetical mixtures. Whenever the curves for the two constituents were assumed to differ in slope, there was a relatively abrupt bend in the curve for the mixture, the rectilinear segments above and below the break approaching in slope the values for the original constituents. This observation indicates that in homogeneous populations the slope of a dosage-mortality curve is of toxicological significance. Since the same numerical relations would be expected if a single poison were to have two independent lethal effects within the animal, there is theoretical basis for fitting the linear segments of a dosage-mortality curve separately when a break occurs after transformation to probits and logarithms. This argument has been extended to time-mortality experiments to explain the smoothly concave curves characteristic of natural mortality.

(2) The second type of joint action is that in which the constituents act independently but similarly, so that one ingredient can be substituted at a constant ratio for any proportion of a second without altering the toxicity of the mixture. With homogeneous populations, dosage-mortality curves for the separate ingredients and for all mixtures should be parallel. Although by hypothesis the susceptibility to one ingredient is completely correlated with that to the other, mixtures in this category are more toxic than in the preceding class where association may vary from 0 to 1. The numerical relations have been illustrated by an experiment on the toxicity to the house-fly of solutions containing pyrethrin and rotenone. A mixture with a little less than four equitoxic units of pyrethrin to one of rotenone agreed closely with the definition but one in which the ingredients were about equally balanced showed a significantly greater toxicity than expected on the hypothesis of independent action, indicating the presence of synergism.

(3) Synergism forms the third type of joint action, characterized by a toxicity greater than that predicted from studies on the isolated constituents. It is the reverse of antagonism, which has not been considered directly. Two methods are proposed for the analysis of synergism. The more direct is to relate equitoxic dosages of mixture to its percentage composition in terms of the more active ingredient. When both are in logarithms the relation is linear over a useful range of compositions. This procedure preserves the original structure of the experiment, can be extended readily to three or more ingredients and leads to a convenient practical result. Theoretically it is less satisfactory than a second method in which for equitoxic dosages of each mixture the content of one ingredient (A) is related to the content of the other (B.) The equation which satisfies this relation most completely is (1 +k1A) Bi=k2, where the three constants are computed from the experimental data. When the exponent i is equal to 1, only two constants need be determined and their product, k1k2, is proposed as a measure of the intensity of synergism.

The synergism between a nitro-phenol and petroleum oil has been computed by both methods. For mixtures containing from 0·5 to 5% of the nitrophenol, the deposit of mixture (Dc) killing 98 % of the eggs of a plant bug could be expressed adequately in terms of the percentage of the phenol (Q) as log Dc= 0·687-0·307 log Q, for 98% of overwintering San Jose scale as log Dc= 0·472 -0·363 log Q. All observations, including those for a 0·1 % mixture and for oil alone which were omitted in the first method, could be fitted satisfactorily in terms of the separate ingredients. For plant bug eggs at LD50, (1+25·6A) B= 4·29 and for San Jose scale (1 + 66·7 A) B= 2·73: in both cases i= 1 and the intensity of synergism 110 and 182 respectively.

The full procedure has also been applied to the constituents of seven samples of Derris root. One sample gave an unaccountably low toxicity and was omitted. The log LD 50 of ether extract for the remaining six was related to the percentage composition of two components in the extract, rotenone (A) and dehydro mixture (B.) Since the toxicity of extract could be expressed almost entirely in terms of these particular two constituents, they were then related to each other by the second method. None of the samples contained a very small proportion of one ingredient, so that several equations were equally applicable, one of them being (1+0–714A) B= 56·1, from which the intensity of synergism was 40.

The problem of measuring synergism in fumigants has been discussed briefly.

The toxicity of poisons applied jointly1

The study of animal function is organized more or less under three heads, which in everyday language are, as applied to a machine, what it can do, how it works, and what makes it go. Insofar as fields of study can be classified in biology these divisions of the subject are ordinarily considered to be autecology, physiology, and biochemistry, with a great deal of individual taste governing the label any particular worker may choose for himself. The chapter discusses what fish can do in relation to their environment and therefore largely autecology. The chapter focuses on the action of the environment on metabolism and the effects of this action on the activity of the organism.

1 – The Effect of Environmental Factors on the Physiology of Fish

1. Theoretical Discussion. The purpose of this note is to summarize the transformations which have been used on raw statistical data, with particular reference to analysis of variance. For any such analysis the usual purpose of the transformation is to change the scale of the measurements in order to make the analysis more valid. Thus the conditions required for assessing accuracy in the ordinary unweighted analysis of variance include the important one of a constant residual or error variance, and if the variance tends to change with the mean level of the measurements, the variance will only be stabilized by a suitable change of scale. If the form of the change of variance with mean level is known, this determines the type of transformation to use. Suppose we write

The use of transformations.

The standard method of dealing with sensitivity of dosage-mortality data is the probit technique developed by Bliss and Fisher. This paper provides an alternative technique based on a special system for obtaining such data. It has some advantages when observations must be taken on individuals rather than groups of individuals, and it may be preferred in certain other situations. * This paper is in part an adaptation of a memorandum submitted to the Applied Mathematics Panel by the Statistical Research Group, Princeton University. The Statistical Research Group operated under a contract with the Office of Scientific Research and Development, and was directed by the Applied Mathematics Panel of the National Defense Research Committee.

https://www.jstor.org/stable/info/2280071

A Method for Obtaining and Analyzing Sensitivity Data

Tables for Convenient Calculation of Median-Effective Dose (LD50 or ED50) and Instructions in their Use.

Summary. 

The sigmoid dosage-mortality curve, secured so commonly in toxicity tests upon multicellular organisms, is interpreted as a cumulative normal frequency distribution of the variation among the individuals of a population in their susceptibility to a toxic agent, which susceptibility is inversely proportional to the logarithm of the dose applied. In support of this interpretation is the fact that when dosage is inferred from the observed mortality on the assumption that susceptibility is distributed normally, such inferred dosages, in terms of units called probits, give straight lines when plotted against the logarithm of their corresponding observed dosages. It is shown that this use of the logarithm of the dosage can be interpreted in terms either of the Weber-Fechner law or of the amount of poison fixed by the tissues of the organism. How this transformation to a straight regression line facilitates the precise estimation of the dosage-mortality relationship and its accuracy is considered in detail. Statistical methods are described for taking account of tests which result in 0 or 100 per cent, kill, for giving each determination a weight proportional to its reliability, for computing the position and slope of the transformed dosage-mortality curve, for measuring the goodness of fit of the regression line to the observations by the X2 test, and for calculating the error in position and in slope and their combined effect at any log. dosage. The terminology and procedures are consistent with those used by R. A. Fisher, who has contributed an appendix on the case of zero survivors. Except for a table of common logarithms, all the tables required to utilise the methods described are given either in the present paper or in Fisher's book. A numerical example selected from Strand's experiments upon Tribolium confusum with carbon disulphide has been worked out in detail.

The calculation of the dosage-mortality curve

SUMMARY
When the result of a toxicity test is measured in terms of the reaction time, the data can be plotted so as to show the percentage of animals which has reacted at different times from the beginning to the end of the experiment. This may be called a “time-mortality” curve and with most animals is sigmoid in its original form. On the hypothesis that it measures the individual variation in susceptibility, it frequently can be plotted as a straight line by converting the percentages to probits and the observed time to logarithms or to rates. When not based upon this type of graphic analysis, time-mortality measurements are often of indeterminate reliability, represent different degrees of effectiveness, conceal changes in the nature of the response necessary to its understanding, or cannot be reduced to a consistent biological formulation covering both partially and fully effective levels of dosage.

The preparation of original data for plotting depends in part upon whether they have been grouped during the experiment or afterwards in preparing the frequency distribution. In the first case equally spaced observations usually lead to unequal grouping intervals when converted to the function of time that is distributed normally, but for either case methods are given by which the loss of information due to grouping can be measured and minimized. Small distributions of individual reaction times can be plotted without grouping. The same procedures are available when the distribution is truncated, either artificially because of the experimental technique or biologically at a given level of susceptibility due to a change in the nature of the response or to its complete cessation. In either case graphic analysis leads directly to consistent and comparable approximate estimates of the mean and standard deviation and of their variances.

The time-mortality curve is computed directly from the non-cumulative frequency distribution of the rectified reaction times rather than from the cumulative curve that is plotted. The effect of grouping upon these calculations is discussed with particular reference to developing an efficient experimental design. Sheppard's correction of the variance for grouping is given in a form applicable to both unequal and equal grouping intervals. From the parameters of the time-mortality curve, the mean and the standard deviation, the reaction time for any given proportion of the population between 0 and 100% can be computed. The accuracy of the time-mortality curve is measured by the errors of random sampling of the mean and of the standard deviation. For the determination of the latter a newly computed table and a corrected formula are provided. Errors in both parameters reduce to a measurable degree the accuracy of an estimated reaction time earlier or later than that for 50% of the population. The agreement of a time-mortality curve with the hypothesis upon which it has been computed may be tested by means of the statistics g1 and g2. The first measures the asymmetry or skewness of the supposedly normal distribution and determines whether the main trend of the points in the transformed cumulative curve is really rectilinear; the second shows whether or not the secondary trends and twists about the rectilinear curve are statistically significant.

By means of the truncated time-mortality curve the toxicological value of studies on the reaction time can be extended considerably. Sometimes graphic analysis will supply sufficiently accurate estimates from an incomplete curve of its mean and standard deviation and of their standard errors, but when the data are less regular it is desirable to compute corrections for these graphic solutions. A new method for computing the truncated normal distribution by successive approximations has been developed by W. L. Stevens and is described by him in an appendix. On the basis of Stevens's method tables have been prepared for computing time-mortality curves that are truncated at their lower ends, a form which covers most cases. Usually the first approximation is sufficiently precise.

The different procedures are illustrated by numerical examples.

The calculation of the time‐mortality curve

Summary. 

The theoretical and practical applications of the dosage-mortality curve usually involve either a comparison of similar series of records to determine whether they differ significantly, or the estimation of the dosage corresponding to a selected or observed mortality and the error of this estimate. In an earlier paper we have considered the methods appropriate for computing the dosage-mortality curve as a straight line and for measuring its error of estimation. The present paper is an extension of these methods to cover some of the more frequent applications of the curve.



In measuring the degree of agreement between different series of dosage-mortality data, it is convenient to refer to one series, which has been transformed into a straight regression line of known accuracy, as the standard, and to determine the agreement of similar or smaller groups of data with this standard. When the standard is of absolute accuracy, experimentally the exceptional case, only the second group of observations is subject to sampling error and the X2 test is applied in a form appropriate for the binomial distribution. When the standard curve has been determined from homogeneous data that involve no errors other than those of the random sampling of test animals which vary in their inherent susceptibility, two forms of the X2 test are available, both based primarily upon the weighting coefficients used in computing the dosage-mortality regression lines. In one case the additional observations are combined with the original ones for a recomputation of the standard curve and of the X2 measure for the agreement of this second curve with the pooled data, the X2 for the discrepancy of the additional observations (or of their separately fitted curve) being derived by difference. In the second case the X2 s for these discrepancies are computed directly and, with two or more observations in the second series, separate the difference between the two series of data in position from that in slope. When the standard curve has been determined from heterogeneous data and the observed variation is statistically in excess of that attributable to errors of sampling, the X2 test is no longer applicable, and the discrepancy between the standard and one or more other observations is compared with the observed errors of estimation by means of the statistic t. Several of these procedures are illustrated by the numerical example taken from the preceding paper.



In expressing the relative susceptibilities of different biological races or species, or the relative potencies of toxic agents, the comparisons are in terms of dosages required to produce selected or observed mortalities rather than of mortalities produced by specified dosages. It is then important to meaaure the accuracy of such estimated dosages. When the mortality is chosen by the experimenter, the dosage estimated from the standard curve is subject only to the errors involved in its determination, and the relative merits of selecting different mortalities, such as at 5.0 and at 7.0 probits, as a basis for these comparisons is considered in the light of the use that is to be made of them. As an illustration, numerical data are given of one such application. When the mortality at which the dosages in a standard and a second series are to be related is determined by experiment, we have the conditions observed in the biological assay of drugs or other toxic materials. There is not only the error of the standard curve to be considered, but also the error in the mortality observed with the unknown which is being standardised. When the stock of test animals used in a particular biological assay is constant in respect to the standard preparation, both with regard to its average susceptibility and its relative susceptibility within the population, an unknown material can be standardised by direct comparison with a single curve of comparatively high accuracy, determined from many experiments with the standard preparation. When the population of test animals is not stable in these respects, each separate assay must be based upon parallel tests between the standard preparation and the unknown, and require, therefore, more observations to secure the same accuracy. A method is described for determining the ratio of potencies and the error of this ratio in both instances.

The comparison of dosage-mortality data

This chapter discusses slope-ratio assays. The response could be plotted as a straight line against the logarithm of the dose over an adequate dosage range. Parallel lines were fitted to concurrent tests with the standard and the unknown and the relative potency of the unknown determined on the log dose scale from the horizontal distance between them for a log ratio assay. Sometimes, the response is a linear function of the dose rather than of the log dose. The most important applications are in the microbiological assays where in many cases the response can be plotted linearly against arithmetic dosage units at the lower vitamin concentrations. Concurrent tests on two preparations that differ only in their content of a single active ingredient are fitted by two straight lines that converge at 0 dose. The potency of an unknown relative to the standard can then be determined from the ratio of the slopes of the fitted lines. Assays meeting these conditions have been designated as slope-ratio assays. The term has since been extended to assays where the response can be plotted linearly against some power of the dose. One of the first requirements for an efficient microbiological assay is a linear relation between some function of dosage and response over an effective range.

Slope-Ratio Assays

Synergism between nicotine and pyrethrum applied by injection to adult Oncopeltus fasciatus Dal. has been reported by Turner (1951).



When these insecticides were applied alone and as a mixture to adult Tribolium castaneum Hbst., using a dipping technique, the data indicated that independent joint action occurred. Similar action could be eliminated because the two insecticides had varying relative potency.



Since the effect of the pyrethrins has been short-lived in some insects, it was postulated that the absence of synergism might be caused by failure of the nicotine to reach a site of action while the pyrethrins were still acting.



Application of nicotine, followed later by treatment with pyrethrins, gave evidence of synergism. A test in which the interval between treatments was varied from ¾ to 6 hr. showed that toxicity was greatest with the shortest interval between applications, and evidence of synergism had practically disappeared after 6 hr. The maximum amount of synergism observed was about twofold.

C. I. Bliss

Papers

The calculation of the dosage-mortality curve

The calculation of the time‐mortality curve

The comparison of dosage-mortality data

Slope-Ratio Assays

Tests of synergism between nicotine and the pyrethrins