Response-scale dependence of dye-loss interactions

A dye-loss interaction is a departure from additivity on a specified response scale. Changing from fractional loss to survival, logarithmic survival or a fitted rate can change that departure. The contrast must therefore be defined before it is interpreted as evidence about coupled chemical mechanisms.

For a crossed initiation/illumination experiment, write fractional losses $L_{ij}$ at a common observation time and defineAn original example takes losses $(L_{00},L_{10},L_{01},L_{11})=(0,0.20,0.30,0.44)$. The contrast is $I_L=-0.06$. Yet the surviving fractions obey $S_{11}=0.56=(0.80)(0.70)=S_{10}{S}_{01}$, with $S_{ij}=1-L_{ij}$. They are exactly multiplicative on the survival scale.

For first-order survival $S_{ij}(t)=e^{-k_{ij}t}$, transform to $Y_{ij}=-\log S_{ij}/t$. The rate-scale interaction isThe example gives $I_Y=0$. More generally, additive rates $k_{ij}=k_0+i\alpha +j\beta$ yieldwhich is negative for positive $\alpha ,\beta ,t$. The fractional-loss ceiling creates the contrast without an additional rate interaction.

The transformation also changes noise. First-order uncertainty propagation gives $u(Y)\approx u(L)/(t(1-L))$, so near-complete loss can strongly amplify uncertainty. In addition, the mean of a nonlinear transformation need not equal the transformation of the mean; Jensen's inequality makes that distinction explicit. Averaging transformed replicates and transforming an average need not produce the same result.

The STY-BODIPY readout is a coupled oxidation signal. Its primary method distinguishes radical-trapping kinetics from stoichiometry.^[ref] A rate transformation requires a justified survival model after product absorption and background have been addressed. Report both the response definition and the uncertainty model; a transformed zero is conditional evidence about that model, rather than a unique identification of the chemistry.