Figure 3: Point of intersection for the logistic curve and given acceptance criterion. CI is confidence interval.
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Irrespective of which method is used, two aspects of considerable importance in determining VRLs are the acceptance criterion
for establishing VRLs and the number of observers participating in the VC verification studies. The acceptance criterion indicates
the probability of detection (observed or predicted) at which a specific residue concentration could be regarded as the VRL.
For the current method, the acceptance criterion is 1, which is based on the idea that if all the observers are able to detect
a specific residue concentration during the VC verification phase, then the same residue concentration could also be detected
by an observer in future situations. However, the same acceptance criterion could not be used for the logistic-regression
model because the probabilities that it predicts can neither be less than 0 nor greater than 1.
Table VI: Predictions (point estimates) derived from the logistic regression model.
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To estimate VRLs at different acceptance criteria, the model was inverted to estimate the concentrations that yield a certain
response probability. The residue concentration at which the given acceptance criterion intersects with the logistic curve
(see Figure 3) was determined. Table VI shows the residue concentrations thus obtained. The notation P
x (e.g., P
50) denotes the residue concentration that would give a response of x% according to the model (e.g., the probability that the residue would be detected by 50% of the observers). Confidence intervals
for P
x were then derived using Fieller's theorem.
These point estimates provided a framework for evaluating the reliability of the logistic model. The logistic models and associated
point estimates could be considered reliable if the observed probability of detection was found to be consistent with the
predicted probability of detection. If one assumes 0.999 to be approximately equal to 1, then the VRL for the given residue
should be 2.921 μg/cm2, which is larger than the one obtained with the current method (see Table VI). Thus, based on the logistic-regression model,
the residue concentration at 2.921 μg/cm2 is predicted to be detected by all the observers with a 95% confidence interval of 2.266–4.761. Because the probability
of detection increases significantly with an increase in spiked-residue concentration, setting higher acceptance criteria
would give larger VRLs. Table VI shows that as the acceptance criterion approaches 1, the relationship requires a larger change
in the explanatory variable to have the same effect as a smaller change in the explanatory variable at the middle of the curve.
For example, a change in the predicted probability of detection from 0.9 to 0.99 requires a larger change in residue concentration
(i.e., a change of 0.674 μg/cm2) than does a change in the probability from 0.5 to 0.6 (i.e., a change of 0.114 μg/cm2). Similarly, the confidence interval for these VRLs would tend to be wider as the acceptance criterion increases. Logistic
regression may provide a much larger VRL than the current method. However, manufacturers may achieve lower VRLs by adjusting
the acceptance criterion.
Unlike continuous responses, binary responses require a large number of observations. The more trials are attempted, the more
accurate the estimated probability is. For VC verification studies with a small number of observers, a large number of observations
with some replicates at each spiking level is recommended. However, for an accurate estimation of sample size, one may use
the formula proposed by Hsieh et al. (15).
Logistic regression, as previously described, can be generalized to incorporate more than one explanatory variable, which
may be continuous or categorical. However, care should be taken when interpreting and reporting results from multiple logistic-regression
models. To correctly interpret the results from a multiple logistic-regression analysis and arrive at meaningful conclusions,
appropriate steps must be taken to incorporate statistical interaction or curvilinear effects properly (e.g., including additional
x
1 × x
2 or polynomial terms such as x
1
2 in the systemic component of the model) (13). If the logistic coefficient for the product or polynomial term is not statistically
significant, then the interaction or curvilinear effect is not statistically significant. One problem that may arise while
modeling multiple explanatory variables is that sometimes the value of one or more independent variables may raise the probability
of the dependent variable close to 1, therefore the effects of other variables cannot have much influence. In that case, such
variables should be excluded from the model or individual VRLs should be determined for the most appropriate viewing conditions.
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