Experiments on the biological measurement of toxicities, and others requiring the use of the probit transformation in the statistical analysis, have in die past been largely of a unifactorial nature. In order to reach a fuller understanding of the action of insecticides and other poisons it is necessary that experiments be performed in-which two or more factors relating to the dosage are subjected to variation. Bliss has recently described methods for the analysis of two-factor experiments, but his technique is not a direct extension of the simple case for the fitting of a probit regression line, nor will it generalize easily when more factors are involved.
In the present paper it is shown that probit regression planes and hyper-planes can be fitted in a way exactly analogous to that used for the regression line. Tests of heterogeneity and of parallelism, and standard errors of parameters are readily derived. The notion of relative potency of two poisons does not extend so easily, and it is suggested that the mean probit difference should be used in its stead as a measure of their relative effectiveness. As in the unifactorial analysis, the routine of fitting the. regression equations may be repeated until satisfactory accuracy in the estimation of the parameters is attained, but usually one, or at the most two, cycles of the process gives values sufficiently near to the maximum likelihood solution for practical purposes.
A more general class of equations is introduced for use when the probit mortality is linearly related to each dosage factor separately, but the individual effects are not additive. The suggestion made by Bliss (1040) for this case lacks symmetry and does not completely fulfil the linearity conditions, but the present proposal avoids these defects and permits the making of tests for the significance of the im-proval in fit effected by the additional parameters.
The arithmetical procedure for the fitting of a probit plane is described in detail for data on the toxicity to Tribolium castameion of a pyrethrin oil spray at various levels of concentration and deposit weight. Two methods of applying the poison were compared in this experiment, and the two series of results were excellently fitted by parallel probit planes. Direct spraying gave slightly, but not significantly, higher kills than the film method of application, and under the particular conditions obtaining, increasing the concentration of pyrethrin in the spray was a far more effective means of increasing kill than was a proportionate increase in the total amount of spray available to the insects. Numerical examples used by Bliss are also discussed in relation to die methods of fitting and forms of equation given in this paper.