The Calculation of the Time‐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 g 1 and g 2 . 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.