## ABSTRACT

The customary method of adjusting dosages of drugs and poisons as simple ratios of the body weight of the individual is questioned. Data recorded by F. L. Campbell on the survival time of silkworm larvae following the ingestion of sodium arsenate were used for analysis. By the method of covariance it is shown that there is a significant residual effect of individual body weight upon the rate of toxic action when the dosage is made proportional to the body weight. By graphic test the rate of toxic action seemed to be a linear function of the log of the dose per larva and the log of the body weight. Accordingly partial regression equations were computed separately for the 2nd to the 5th instars expressing the rate of toxic action in terms of these two dosage characteristics. The adjustment in dosage that was necessary to eliminate the effect upon the rate of toxic action of body size could be expressed as a size factor, where *w* is the body weight of the individual and *h* is determined from the partial regression coefficients of each equation. When based upon the reciprocal of the survival time, *h* increased from 1· 67 to 1· 96 from the 2nd to the 5th instars, but when similar values were computed for the logarithm of the rate, *h* was the same in all instars and averaged 1· 511 ± 0· 089. Also supporting the theoretical superiority of the log rate as a measure of toxic action was the fact that the data for the separate instars formed parts of a single straight line when log rate was plotted against the log of dose adjusted by the size factor w^{1}’^{6} and the statistical demonstration that the distribution of rates was significantly skewed and that of log rates symmetrical. If the adjusted dose is considered as a type of con-centration, the final equation for all four instars could be written as *C*^{0.49}*t* = 309, a type of formula having wide toxicological applicability. A method is proposed for computing the mean lethal dose from the rate of toxic action.

## I. INTRODUCTION

In order to obtain the same physiological response from animals of different size, when measuring the effectiveness of a drug or poison, it is the standard practice to use a dosage proportional to the body weight of each individual. This procedure implies that in any given case the constituents which are sensitive to the drug form a constant part of the total mass, a relationship that is assumed rather than demonstrated. Moore (1909) questioned the validity of this assumption and suggested that in many cases susceptibility was more likely to be proportional to the surface of absorptive tissue than to the mass (or volume) of the entire body. On the basis of laboratory and clinical experience with organic arsenicals, he proposed that dosages could be equalised if they were made proportional to the 2/3 power of the body weight rather than to the body weight directly. Although other pharmacologists (Durham, Gaddum, and Marchai, 1929) have appreciated that dosage often is not directly proportional to weight as is tacitly assumed, neither the 2/3 nor any other fractional power has come into general use. Moreover, it seems improbable that the different types of pharmacological and toxicological action should be describable in terms of a single exponent. A more rational procedure would be in each case to determine the size factor experimentally as one of the toxicological characteristics of the drug. In the absence of such a determination, there is no assurance that differences in pharmacological action which are now attributed to age or to some other factor correlated with the size of the animal may not be due instead to an inaccurate computation of the results of the experiment.

On critical examination the results of several tests on insects with poisons taken by mouth proved to be more variable than would be expected from the experimental techniques which were employed. In each case the dosage was expressed in milligrams of poison per gram of insect. It seemed possible, therefore, that the excessive variability might have been caused by the method of measuring dosage. Dr F. L. Campbell, of the United States Bureau of Entomology and Plant Quarantine, has lent me the original data of his experiments on the length of survival of silkworm larvae following the ingestion of known amounts of sodium arsenate. The sodium arsenate was administered as an aqueous solution in concentrations such that approximately the same amount of liquid was required at each dosage rate. To adjust for differences in individual body weight, each larva was weighed immediately before poisoning and the amount of a given solution varied according to its size. The doses were measured with a precision micro-burette and the larvae maintained under controlled environmental conditions both before and during the experiment. The survival time of each larva was measured from the time that the poison was imbibed to the failure of the proleg combs to retract upon stimulation, this last stage being determined under nearly continuous observation. A record was kept not only of the dosage rate, in terms of milligrams of poison per gram weight of insect, but also of the actual amount of each dose and of the weight of each larva, and it is upon this complete record that the present study is based. For further details of the experiment the reader is referred to the original papers (Campbell, 1926).

The logical and statistical principles which have been followed in the analysis of the problem and of these records are those described by Prof. R. A. Fisher in chapter ix of *The Design of Experiments* (1935). For the methods of computation the reader is referred to *Statistical Methods for Research Workers* by the same author (Fisher, 1934).

## II. INADEQUACY OF THE SIMPLE DOSAGE RATIO

The dosage ratio is assumed to be a correction for the individual variation in size. If the dosage administered to each individual were completely equalised by using amounts of poison proportional to its mass, body weight would then have no influence upon survival time within a given dosage ratio. However, if survival time should still depend upon body weight after this correction had been made, then the total mass of the individual would not in itself be an adequate size factor. This was tested with the data from six experiments, each based upon ten 5th instar larvae that had been poisoned at a single dosage rate. None of these larvae survived the treatment, although the last individuals to die were timed less exactly than the others. The observed survival periods varied more widely after smaller than after larger amounts of arsenic, but when the survival period was first transformed to its reciprocal, which measures the rate of toxic action, the variation in rate was found to be approximately the same at all dosages. For this reason the transformation facilitated the combination of the results from the several dosage ratios and was used in the subsequent analysis.

The method of covariance served to determine whether the weight of the individual larva modified the rate of toxic action significantly after allowance had been made for differences in the simple dosage ratio. The complete analysis is given in Table. That part of the variance and covariance in larval weight (*w*) and in rate of toxic action (*y*) contributed by differences in the dosage ratio is given in the first line of the table and the variation still remaining in the second or “error” line. The original variation in the rate of toxic action of 81ŀ9717, as measured by the sum of *y*^{2}, has been reduced to 15·2180 by eliminating the differences in rate which were contributed by using six instead of one dosage ratio. This residual sum of squares may be reduced still further by eliminating differences in the rate of toxic action associated with differences in body weight, or by subtracting (I degree of freedom) from 15·2180 (54 degrees of freedom) to give 11·5584 (53 degrees of freedom) as the sum of squares adjusted for inequalities in body weight. To determine whether this adjustment is of importance, the variance due to size (3·6596) may be compared by the *z* test with that due to “error”, giving *z* = 1·410 and *P*<0·01, and showing that an additional adjustment for size beyond the simple dosage ratio is of distinct statistical significance.

The direction that this correction must take is given by computing also from the second line of Table I the average change in the rate of toxic action per unit change in the weight of 5th instar larvae. The result,—0·538, shows that larger larvae died more slowly than did the smaller larvae which had been given the same amount of arsenic per gram of body weight. If the survival time is a valid measure of poisoning, the larger larvae were underdosed in comparison with smaller larvae in the same instar. The standard procedure of stating dosage as a ratio of gross body weight was not adequate.

## III. DERIVATION OF AN IMPROVED SIZE FACTOR

The size factor may be defined as a function of size determining the amount of poison which must be administered to each individual in order to produce equivalent toxic action. In the present case body weight *(w)* is the measure of size and survival time (*t*) the measure of toxic action. Instead of assuming that the size factor is the simple body weight, as is usually done, the factor should be computed directly from the experimental data. For this purpose the net relation is required of some function of survival time, *f*(*t*), to some function of the mass of arsenic per larva, *f* (m), when individual body weight is held constant, and of *f*(*t*) to some function of the individual body weight, *f* (*w*), when the dose of arsenic per larva is held constant. By selecting functions such that the relation between these three variables was linear, the problem could be solved by computing the multiple regression equation: *f(t) = a + b*_{1}*(m)—b*_{2}*f(w)*. When measured by the percentage mortality it has been shown (Bliss, 1934) that toxicity frequently is proportional to the logarithm of the dose. This general rule was applied here and log *m* substituted for *f(m)* in the above equation. For the derivation of a size factor the body weight was also transformed to logarithms as the simplest corresponding function. For the survival time Campbell has used two different transformations as measures of toxic action, the rate and the logarithm of the rate. If the equation were shown to be rectilinear with one of these functions, the effect of substituting the other could be considered separately.

It was convenient to test graphically for the rectilinearity of the proposed transformation. For each of the sixty larvae of the 5th instar used in the analysis of covariance, the rate of toxic action (1000/minutes survival =*y*) has been plotted against the logarithm of the milligrams of arsenic per larva x 1000 *(m)* in Fig. 1. Although the circles on the diagram, each representing a single larva, were scattered rather widely, they did not show any marked curvature from the straight line which has been fitted to them by least squares. This line has the equation *y =–* 3 · 923 + 3 · 00 log *m*, and from this equation an “expected” *y* could be computed for every observed value of *m*. The computed or “expected” *y* has been subtracted from each observed *y* and the differences plotted on the ordinate of Fig. 2 against the log of the body weight in centigrams *(w)* on the abscissa. These points agreed much more closely with the computed straight line and again there was no marked curvature. Since on these co-ordinates the rate of toxic action (y) seemed to be a linear function of log *m* and log *w*, the net relation between these variables has been determined by computing the multiple regression equation :

In order to correct the effect upon the rate of toxic action of differences between larvae in body weight, the required size factor was *w*^{1}·^{96} instead of the customary *w*^{1} or the *w*^{0}·^{67} suggested by Moore. The larger larvae required relatively more arsenic than that indicated by the ratio of their body weights to kill them in the same time as the smaller individuals. From the observed *m* and w an adjusted dose in logarithms has been computed for each 5th instar larva by the formula: log adjusted dose = log *m*—1·957 log w, and the rate of toxic action has been plotted against these values in the lower right chart of Fig. 3. The distribution of observed points about the computed straight line is apparently random.

## IV. COMPARATIVE SIZE FACTORS BASED UPON THE RATE OF TOXIC ACTION

Experiments similar to those described for larvae in the 5th developmental stage were also made upon larvae in the 2nd, 3rd, and 4th instars. The size factor was determined separately for each of these when based upon the rate of toxic action. There were several objectives in view. Not only did the various instars represent different ages but they varied greatly in body weight, much more so than did the individuals within a developmental stage, and it was desirable to test the method of computation on material that differed widely in body weight. In certain cases Campbell observed a variation from day to day in the susceptibility of these larvae, so that it was necessary to detect and eliminate fluctuations of this type. In each series some larvae were given so little arsenic that none or very few died at these dosage ratios. Although not included in computing the size factor, it was of interest to determine whether the reaction of these individuals was consistent with that of the more heavily dosed larvae all of which died from arsenic poisoning. Finally it might be expected that the size factor would be characteristic of the response of a species to a given poison, and would not change from one developmental stage to another. If it should change, it may indicate that the rate of response from which it has been computed was not a satisfactory biological measure of toxic action.

The size factors have been computed for the 2nd, 3rd, 4th, and 5th instars in the form *w*^{h} the several values differing in the exponent *h*. These values of *h* and of other statistical constants are given in Table II, where toxic action is measured by 1000/minutes survival time. Although computed in logarithms, the mean body weight and its standard deviation and the mean dose of arsenic and the dosage range have been transposed to their original units for listing in the table. The rate of toxic action *(y)* in terms of the logarithm of milligrams x 1000 of arsenic per larva *(m)* and the logarithm of the body weight in centigrams (*w*) has been computed as:

The standard error of the rate of toxic action given in the table is that for the variation about the line defined by each of these equations and plotted in Fig. 3. The part of each equation in brackets is the log of the dose adjusted for the size factor.

During these experiments Campbell observed an inconsistency in the response of 4th instar larvae treated on different days and suggested that a similar error might have been present in the record for other instars as well. In order to measure and eliminate any systematic change in susceptibility from one day to another, the different days in the experiment have been numbered consecutively from 1 to 18 and introduced as a third variable (*D*) in computing a series of partial regression equations similar to those given above. The regression coefficients representing the change in *y* with date were tested for their significance by the *t* test. Only a single day’s observations were used in computing the equation for the 2nd instar; the susceptibility of 3rd and 5th instar larvae did not change significantly during the period of the test; but the susceptibility of 4th instar larvae did differ significantly. The foregoing size factors and their equations for the 2nd, 3rd, and 5th stages have been computed with two independent variables, that for the 4th instar with three variables. The additional term in the equation for instar 4 provides a correction for this difference from day to day. By setting *D* equal to its mean (9’73) this term vanishes, and the remainder of the equation represents the net effect of dosage and body weight upon rate of toxic action when variations due to differences between days are eliminated. In the diagram of the 4th instar in Fig. 3 this error has been removed by adding a correction determined from the last term of the equation to each observed value of *y*. Two months later a larger number of 4th instar larvae (141), selected for uniformity in size and handling, were tested at a single dosage rate over a period of 5 days. There was no significant change in susceptibility during this period (*t*= 1·062, *P*=0·29). The average rate of toxic action for these individuals has been plotted in Fig. 3 as a single larger circle, using the size factor determined from the earlier test.

In computing the size factors, only dosage rates were included which were fatal to all larvae in a given instar. All instars were tested, however, at a lower dosage rate (0.01 mg. of arsenic per gram body weight) which was fatal to only four 2nd instar larvae. Individuals which survived poisoning and moulted to the succeeding stage have been considered to have an infinite survival time in respect to the poison and in consequence the rate of toxic action was o. In Fig. 3 they have been plotted on the same diagram as the larvae which received a fatal dose. If these co-ordinates (rate of toxic action and log of the adjusted dose) were biologically precise and arsenic poisoning followed the same course at both low and high dosages, then at any given dosage the individual rates of toxic action would be normally distributed about curves similar to those drawn in Fig. 3. Conversely, a statistical determination that the rates in fact were normally distributed would support the validity of this transformation. If experimental test were to supply this confirmation, one-half of the larvae would be more susceptible or die more rapidly and one-half would be less susceptible or die less rapidly than the average value given by the curve itself. At the point of intersection with the base line of o rate of poisoning, one-half of the larvae would die from the arsenic and be plotted as points lying above the line, and one-half of them would be more resistant and survive the dosage corresponding to this point on the abscissa. The point thus established would be the median lethal dose. If, on the other hand, the rates of toxic action should prove not to be normally distributed, the median would not coincide with the mean and in consequence the point of intersection with the base would be at a mean lethal dose but not at the median lethal dose, although the difference between them would not be large.

In the present experiments the distribution of the rates of toxic action proved to be asymmetrical, as is shown later, so that the mean lethal dose rather than the median was estimated. Curves computed exclusively from larvae which ingested a fatal dose have been extrapolated in Fig. 3 to zero rate of poisoning and in the 2nd, 3rd, and 5th instars the point of intersection was in close agreement with the record of larvae which survived the treatment. Mean lethal doses have been computed for three of the instars from the equations for these curves by putting *y = o* and w = mean body weight and solving for *m*. The results have been listed in the last column of Table II. The computed median lethal dose for the 4th instar (0-232) was obviously in error, since as shown in Fig. 3 the computed curve if extrapolated would intersect the base at a point much lower than dosages from which all individuals recovered. The failure in this case probably can be attributed to experimental error because (1) the observations varied more widely from the computed curve than in the other instars (Table II), (2) the susceptibility changed during the course of the experiment, and (3) a later experiment with 141 individuals at a single dosage rate (the large circle in Fig. 3) indicated that the larvae treated in the first test with a dose equal to or lower than this had been abnormally susceptible. The possible position of the 4th instar curve if it had been retested at all dosages is suggested by the dotted line in the lower left diagram of Fig. 3, and the mean lethal dose given by this assumed line is bracketed in Table II.

*h*in the size factors for the several instars may be compared. In all except the 4th instar

*h*was significantly greater than 1. The standard errors of

*h*have been computed by an equation derived from the usual formula for the variance of a ratio and the standard errors of the partial regression coefficients. Without showing its derivation in detail, the equation may be written as where

*V*(

*h*) is the mean variance of

*h*or the square of its standard error,

*b*

_{1}and

*b*

_{2}a re the partial regression coefficients of y upon log

*m*and

*y*upon log

*w*respectively,

*s*

_{v}is the standard error about the regression line, and c

_{11}, c

_{12}, and

*c*

_{22}are the determinants of the simultaneous equations obtained in solving for

*b*

_{1}and

*b*

_{2}. The computation of the last four terms is described in Section 29 of

*Statistical Methods for Research Workers*(Fisher, 1934). Although none of the exponents differed significantly from one another, they increased so consistently from the 2nd to the 5th instar that it is difficult to believe that this was due purely to chance. Apparently the size factor determined for larvae of one average weight and stage was not applicable to those of markedly different weight and stage. It has been suggested that the size factor should be constant within a species for a given poison and method of administration. When toxic action was measured by the reciprocal of survival time, the size factor varied systematically. Moreover, when plotted each against its own adjusted dose on a single graph (the curves which have been plotted separately in Fig. 3), the results for the several instars formed sections of a curve which was concave upwards or apparently logarithmic.

## V. SIZE FACTORS BASED UPON THE LOGARITHM OF THE RATE OF TOXIC ACTION

The above calculation of the size factors for each instar has been based upon the rate of toxic action as the measure of arsenical poisoning. With some other measure, such as the logarithm of the rate, the size factors determined from the same data might be quite different. An apparent rectilinear relation within an instar between the rate of poisoning and its adjusted dose is not sufficient proof of its adequacy, as the computed constants are those which minimise the variance about the straight line connecting these two variables. Since the inconsistency between instars in the foregoing dosage factors may have been due to the use of the rate as a measure of toxic action, the computations have been repeated with the logarithm of the rate. The results are given in Table III and in the following equations :

The change in susceptibility from day to day was corrected as before. Since this source of error had not been anticipated in planning the experiments, the dosage rates were not distributed at random from one day to the next. In consequence, the partial regression term by which this trend was eliminated depended in part upon the function of survival time used as a measure of toxic action. As judged by the *t* test, the trend was just significant in the 3rd instar (*P*=0·05), highly significant in the 4th instar (*P*<0.01), and moderately significant in the 5th instar (*P*=0.01). In plotting the individual observations in Fig. 4, a correction factor, computed as before, has been added to each observed *y’*, so that the result in each instar was made equivalent to a single day’s test. The percentage reduction in the total variance of *y’* listed in Table III, as is the equivalent value for *y* in Table II, is that due exclusively to the factors for size of dose and body weight after the portion attributable to daily changes in susceptibility had been eliminated.

On the basis of the following four comparisons the log of the reciprocal of survival time proved to be a more effective measure of the toxic action of arsenic than the reciprocal.

The size factor was here constant for all four stages of development, giving a weighted mean value for

*h*of 1 ·511 ± 0 ·089. If,**as**is proposed, this is a biological constant characteristic for a given species and a particular type of toxic action, the correct method of computation would be that which gives it a value independent of size and age.- When an adjusted dose was computed for each larva from the mg. of arsenic ingested divided by the (body weight in grams)
^{1-5}, the logarithm of the rate of response for all four instars could be plotted as a single straight line against the logarithm of the adjusted dose (Fig. 4). The mode of action of arsenic was the same in all four stages, so that the data summarised in Tables II and III for 268 individuals have been supplemented by the 141 4th stage larvae (the two larger black circles in Fig. 4) used as a separate uniformity test for computing the equation of the curve in Fig. 4:Disregarding differences in instar and date, 82-8 per cent, of the total variation iny’ was accounted for by this equation and the standard error of

*y’*about the curve was o-ioo in terms of log of rate. It should be noted that the distribution of points in Fig. 4 supports the previous conclusion that the discrepant values at the lower dosages for the 4th instar represented an experimental error. With the log of the rate there was better agreement in the 2nd, 3rd, and 4th instars between the observations and the fitted curves as measured by the percentage reduction in the variance of

*y*and*y’*respectively. The poorer fit in the record for the 5th instar was attributable to two of the sixty observations that were made at the lowest dosage rate (Fig. 4). A recomputation of the 5th instar results with these two omitted showed an improvement in the fit for*y’*to 89-2 per cent, reduction in the variance, which was greater than the equivalent value for*y*.The final comparison was to test the normality of the survival time of 141 4th instar larvae when the time was transformed to the rate and to the logarithm of the rate respectively. These larvae had been selected for uniform size and condition and were all given 0 ·10 mg. of arsenic per gram of body weight. Without correcting for the small differences in dosage resulting from the use of an inaccurate exponent

*(h)*in the size factor, the skewness of the two measures has been computed as follows:

The distribution of the reciprocal of the survival time of these larvae was significantly asymmetrical, but the logarithm of the rate was normally distributed, the measure of its asymmetry being less than the standard error of this measure. It frequently has been shown that the variation in susceptibility to a poison is normally distributed when an appropriate measure of susceptibility is used. The mathematical function (or functions) in terms of which the distribution is normal is considered to be more directly representative of the biological processes upon which this susceptibility depends and alternative transformations, which are not normally distributed, to be one or more steps removed. On this basis the logarithm of the rate (or the logarithm of the survival time directly with a reversal of the curves) is a more suitable measure of toxicity than the rate itself.

In terms of log rate and log of the adjusted dose, the mode of toxic action was the same in all instars. However, the younger instars were able to survive a larger adjusted dose than the later instars. The upper range of adjusted doses which were survived by larvae of each stage has been indicated in Fig. 4 by a heavy horizontal line. We may note, for example, that adjusted doses (mg. arsenic per larva/w^{1.5}) which were invariably fatal to the 5th instar were survived by 2nd, 3rd, and 4th stage larvae. The relative susceptibility of the several instars, therefore, depended upon the criterion by which susceptibility was measured. In terms of the conventional dosage ratio of mg. of arsenic per gram of body weight there was a constant decrease in susceptibility during development as has been shown with these data by Campbell; in terms of the logarithm of the rate of toxic action with an adjusted dose susceptibility remained constant during development ; in terms of the maximum adjusted dose which the larvae could survive, there was a continuous increase in susceptibility during development.

The toxicological processes indicated by the characteristic curve for log rate of toxic action (Fig. 4) seemed to be modified by secondary reactions at dosages in the zone of incomplete mortality. At the smallest fatal dosages for 2nd and for 5th instar larvae a few individuals (4 and 2 respectively) were excessively slow in dying. It is not likely that these exceptions represented experimental errors, for a similar discontinuity has been observed in the critical partially lethal zone of dosages in timing other toxicological processes (Broadbent and Bliss, 1935). The method proposed in the foregoing section for computing the mean lethal dose assumed a continuity of lethal action over this range of dosages. The observed discontinuity does not necessarily invalidate the method. As zero rate of dying is approached, large differences in survival time and in the log of the rate have a relatively small effect upon the rate itself, as is apparent from a comparison of the positions of the above-mentioned observations in Figs. 3 and 4. This form of biological discontinuity, therefore, may quite disappear on the rate chart.

The mean lethal dose computed from the intersection of the rate curve with the ordinate for zero rate of toxic action was here less than the median lethal dose. Since the rate was not normally distributed, the mean rate at any given dose was slightly larger than the median and in consequence the fitted curve would intersect the base at a lower adjusted dose than the median lethal dose. This systematic difference probably was less than the standard error of any given determination.

## VI. DISCUSSION

In these experiments 8-12 parts of arsenic to one million parts of silkworm larva constituted a mean lethal dose. Obviously if so small an amount of poison was effective, it could only be because it combined with and inactivated cell constituents which comprised a similarly small part of the total body weight. The relation of these vital constituents to the gross weight of the body is not likely to be simple and direct as is implied when dosage is made proportional to body mass. The size factor, *w*^{h}, is proposed as a biological constant which makes it possible for the relation between constituent and total mass to be determined from the data rather than be imposed by the experimenter. In the present case the size factor is *w*^{1.5}, indicating that relatively more arsenic is required in larger than in smaller individuals to obtain the same toxicological effect. Presumably, therefore, the one or more cell constituents with which the arsenic combines, whether glutathione (Fink, 1927) or some other substance, must be present in greater concentration in larger than in smaller individuals. Not only may different modes of poisoning be characterised by different size factors, but they may also differ between species since the relative growth rates of different structures often show wide specific variation. It would be hazardous to suggest a more detailed interpretation, therefore, of the fact that the exponent *h* in the size factor is here equal to 3/2 until the method has been applied more widely.

Another point of interest is the linear relation between the logarithm of the rate of toxic action and the logarithm of the adjusted dose. Dosage may be looked upon as a form of concentration, the concentration of arsenic per unit of the cell substances with which it reacts. Time-concentration curves have been determined for many drugs and poisons upon a wide range of organisms and Clark (1933) has shown that in most cases there is a linear relation between the logarithm of the concentration (C) and the logarithm of the rate of reaction (I/*t*) such that *C*^{n}*t =* constant, where *n* is given by the slope of the log-log curve. Fig. 4 is consistent with these findings. If in the equation for the line in Fig. 4, *y’* is equated to log 1000/*t* and *m/w*^{.5 ·517} to 0.1C, it may be rewritten as C^{0.49}*t* = 309. This value of *n* (0.49) is similar to that obtained for the toxic action of several other metallic poisons. It should be noted in the present case that this relationship has been fitted to a range of concentrations (adjusted dosages) varying from 1 to 170 or much greater than the range in many time-concentration experiments. While there is a rapid and relatively abrupt transition from this curve to survival as the dosage is reduced within an instar, the combined evidence shows that the fundamental relation between the two factors does not, in consequence, break down for the species.

The size factor can be determined for measurements of size other than weight and for measurements of toxicity other than survival time. As an alternative to weight the volume of the individual may be determined and the size factors computed from such measurements should parallel those for weights. In metabolic experiments, however, dosage frequently is adjusted to surface area, since energy production and dissipation may be considered as proportional to the surface exposed to radiation, and dosage factors may correspondingly be determined from surface measurements. As a measure of toxic action survival time has the advantage of enabling a separate observation to be made upon each organism which dies. For many purposes, however, percentage mortality is a more satisfactory initial criterion of toxicity, and dosage mortality curves have been determined for arsenicals with the dosage in terms of mg. of arsenic per gram of insect. The excessive variability sometimes observed in determinations of this type is probably due in part to the use of an inaccurate size factor. Since percentage mortality can be transformed to probits and plotted as straight lines against the logarithm of the dosage (Bliss, 1935), a size factor can be determined from experiments of this type if each percentage is based upon individuals of the same known size and several different sizes are included in the experiment. The probit mortality is substituted for the log rate of toxic action in the foregoing analysis, each value is assigned its appropriate weighting coefficient, and the size factor is computed otherwise in the same way.

When the number of animals that can be handled experimentally is limited, the mean lethal dose computed from the rate of toxic action is probably more accurate for many poisons than the median lethal dose determined from the dosage-mortality curve, especially if the size factor is unknown. With suitable data a correction doubtless could be computed for converting mean dosages to median dosages. Furthermore, there may be cases in which the rate of toxic action rather than the log of the rate is normally distributed and in such instances the point of intersection of the calculated curve with the ordinate for zero rate would give on the abscissa both the mean and the median lethal dose. For the comparison of insecticides the mean lethal dose alone should be a better criterion than areas under freehand curves which have been proposed for this purpose. In order to minimise the errors of estimation, the determination for each developmental stage should be made separately, each dosage range should include the mean lethal dose and adjusted doses approximately ten times as large, and all individuals which survive an adjusted dosage that is fatal to one-third or more of the number treated at the same or a lower dose should be included in the computation at zero rate of toxic action.

## ACKNOWLEDGMENTS

It is a pleasure to acknowledge my indebtedness to Dr F. L. Campbell for the loan of the original data upon which this paper is based, to Prof. R. A. Fisher for his inspiration and aid in its preparation, to Dr A. E. Brandt for helping me to compute the initial constants by the Hollerith system of punched cards, and to other friends who have read the original manuscript for me.

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