In many effectiveness evaluation tests of insecticides or acaricides for their target pests the same dose is usually given to each pest individual, as if its action were independent of body size,. Alternatively, in various pharmacometric experiments of drugs or poisons in vivo, doses proportioned directly to the body weight of animals are given to each individual. Instead of an arbitrary correction such as thelatter, a size factor for equalizing individual differences in body weight can be determined experimentally for each drug or poison by multiple regression. This approach was taken by Bliss for analysing a controlled experimental data on the rate of toxic action of sodium arsenate in silkworm larvae.
When the response y is a measurement an empirical size factor w^<h*> can he estimated from the multiple regression of y upon the dose per animal(x_1 = log d_*) and its body weight(x_2 = log w) by transforming the basic regression equation
Y = a'+b_1x_1+b_2x_2
or
Y = a'+b_1 (log d_*+(b_2)/(b_1) log w)
to
Y = a'+b_1 log (d_*w^<h*>),
where h^* = (b_2)/(b_1), and b_2 is almost always negative.
In the present experment, each 4th instar gypsy moth larva was weighed in mg in advance of the application of chemical. Five μ1 of DDVP dissolved in acetone at a prescribed concentration was topically applied by Arnold hand microapplicator to the dorsal side of larva. The time of its surrender was measured in minutes as judged by the turning over on its back from the intoxication of chemical. For the 68 larvae in Table 1, x_1 = log μg is the log dose of DDVP per larva, x_2 = log(grams×10) is the body weight, and y = log(1,000/mimtes survived) is the log rate of toxic action. A larva with a prolonged non-reacted period No.18 has been omitted as an outlier in the calculation.
The multiple regression equation
Y = 1.86179 + 1.23916x_l - 1.89270x_2
was given for the remaining 67 larvae. From the ratio of the two partial regression coefficients, the size factor w^<h*> for equalizing the effective dose of DDVP in the gypsy moth larvae differing in body weight has been estimated within 95% confidence limits of -2.3051 and -1.0325, with its most probable value at -1.5274. It could be concluded that k times larger larvae required relatively k^< 1.5274> times more DDVP than that indicated by the ratio of their body weight to kill them in the same time as the smaller larvae. Here, h^* = -1 means that the dose should be proportioned directly to the body weight and h^* = 0 means that doses are independent of body weight. For graphic test of linearity, the log doses adjusted for differences in body weight have been computed as z^* = x_1-1.5274 x_2 for each larva and listed in Table 1, then log rate y for each larva has been so plotted against z^* in Fig. 1. It has been fitted with the line
Y = 1.8616+ 1.2392 z^*,
the slope differing from the original b_1 by rounding error. As shown in Table 2, the scatter about the predicted line gave no indication of curvature. Since the coefficient of determination R^2 = 0.756, three-quarters of the total variation in y could be attributed to dose and weight. The latter variable related little more to the response than the former. The slope coefficient C = 1.074215, (1 < C < 2), indicates that the result of the present experiment is sufficiently useable for bioassay purpose. The larva No.18 which responsed very slowly to DDVP has been omitted in computing the multiple regression equation. The test for this suspected outlier showed that No.18 does not belong in the equatron.