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Current methods for establishing visible residue limits (VRLs) are not statistically justifiable. The author proposes a method for estimating VRLs based on logistic regression.

Mar 2, 2010 By:
M. Ovais Pharmaceutical Technology
Volume 34,
Issue 3,
pp. 58-71

Table IV: Results of fitting the logistic regression model to the data presented in Table III.*

The logistic-regression model is applied to develop a relationship between the residue concentration and its probability of
detection. The summary of model parameters, obtained after fitting the data with logistic regression, is provided in Table
IV. With a 0.1-μg/cm^{2 } increase in the residue concentration on the model surface, the odds of detecting the residue increase by 42.710%. An increase
in concentration of this magnitude is likely to increase the probability of detection by 20.150–69.506%. Thus, at the 95%
two-tailed level of significance, the residue concentration does have an effect on the response variable. The statistical
significance of regression coefficients is tested using the Wald X^{2} statistic. Table IV shows that the estimated coefficient for residue concentration has a p value of less than 0.05, indicating that the coefficient is probably not zero using an α level of 0.05. Therefore, residue
concentration is a significant predictor of the probability of detection. The goodness-of-fit tests, with p values ranging from 0.979 to 0.999, indicate that the model fits the data adequately. In other words, the null hypothesis
of a good model fit to data is tenable.

Table V: Relationship between spiked residue concentration and predicted probability of detection based on logistic regression
model.

To understand fully the predictions made from the model, all values were transformed into direct measures of probability,
and confidence intervals for these probabilities were derived using the method of Fleiss et al. to describe the uncertainty
associated with fitting the model (14). Table V lists the logit and predicted probabilities of detection for each residue
concentration obtained after fitting the data with logistic regression. Logit and predicted probabilities, when plotted against
residue concentration, gave a straight line and an S-shaped curve, respectively, as may be seen in Figure 2. Table V shows
that the higher the predicted probability of detection for a residue concentration, the more likely an observer will visually
detect the residue. A comparison with the data in Table III implies that, based on the current method, the VRL should be 1.80
μg/cm^{2} because it is the lowest concentration of residue for which the observed probability of detection is equal to 1. However,
based on the logistic regression model (see Table V), the predicted probability associated with a residue concentration of
1.80 μg/cm^{2} is only 0.949 (i.e., approximately 95 out of 100 observers are predicted to detect the residue).

Figure 2: Logistic regression model of the relationship between residue concentration and (a) Logit and (b) probability of
detection. CI is 95% confidence interval, O is observed probability of detection, and P is predicted probability of detection.

Table V also compares observed and predicted probabilities and lists the expected number of detections for each residue concentration
calculated as the predicted probability multiplied by the number of observers in each category. In logistic regression, the
standard error and confidence interval for the model-based probabilities tend to be much smaller and narrower, respectively,
than the ones based on the sample proportion. Instead of using a sample of only five observations as the current method does,
the logistic-regression model uses information from all the observations in estimating the probabilities of detection. The
result is a more precise estimate. Using the logistic-regression model to estimate the probabilities of detection instead
of simply using observed proportions is therefore justifiable.