Although the relationship between observed probability of detection and the residue concentration is nonlinear, a generalized
linear modeling technique can be applied to these data. The logistic-regression technique fits the observed data with a linear
model, the parameters for which are estimated using the maximum likelihood technique. Next, logistic regression transforms
this linear model into a nonlinear logistic curve also known as an S-shaped or sigmoid curve. Logistic regression can therefore
be seen as the conversion of a linear model into a nonlinear model that is naturally suited to the description of a binary
response variable (13). The link function, commonly known as logit (the logarithm of odds), is used for converting the linear
model to nonlinear logistic model and vice versa.
The logistic regression model is represented by the following equation (13):
in which P(Y =1) represents the predicted probability of response being equal to 1 (i.e., the predicted probability of detection);
e is the exponent function; β0, β1, β2, ... βk are coefficients estimated from the data (obtained using the method of maximum likelihood); x
2, ... x
k are independent variables; and k is the number of independent variables.
The term β0 + β1
1 + β2
2 ... + βk
k is the logit function and is defined as a natural logarithm of odds (e.g., the probability that an observer detects the residue
divided by the probability that he or she does not detect it). Odds are expressed by the following equation:
For the data presented in Table III, which involves only one independent variable (i.e., residue concentration), logit = β0 + β1
1. Once a meaningful relationship is defined between spiked residue concentration and probability of detection, VRL could easily
be obtained from the regression model.