Statistically Justifiable Visible Residue Limits - Pharmaceutical Technology

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

Pharmaceutical Technology
Volume 34, Issue 3, pp. 58-71

Statistical limitations of the current method

One statistical limitation of the current method is that the VRL is determined based on observed data without describing a relationship between observations and the experimental parameters. Suppose that in a VC verification study, a residue is spiked at levels of 0, 0.5, 1.0, 2.0, 3.0, and 4.0 μg/cm2. If all four inspectors detect residue at 2.0 μg/cm2 and only three inspectors detect residue at 1.0 μg/cm2, then the VRL would be 2.0 μg/cm2, assuming that the residue levels between 1.0 and 2.0 μg/cm2 would not be detected by all inspectors. The VRL is inappropriate because a residue level between 1.0 and 2.0 μg/cm2 could possibly have been detected by all inspectors.

To predict the number of observers that would detect residue at levels other than those spiked (e.g., 1.5 μg/cm2), the observed data must be incorporated into a reasonable model that describes a relationship between an outcome and a set of independent variables. The results obtained from spiking studies for verifying the VC criterion are binary (i.e., only two values are possible) rather than continuous. The regulatory guidelines and the available literature do not explain how to establish a modeling procedure, based on these discrete responses, that could be used to derive VRLs, however.

Another important parameter in determining appropriate VRLs is the sample size (i.e., number of inspectors and total number of observations) for VC verification studies. Most published studies are based on relatively small sample sizes. Forsyth's studies are based on only four observers (4, 10, 11). Because the VRL depends on the proportion of detection (i.e. the number of detections of residue to the number of inspections), a small sample size increases the width of the confidence interval and the margin of error. Thus, an 0.8 proportion of detection with a sample size of five would result in a 95% exact-confidence interval of 0.2836–0.9949 and an approximately 35.57% margin of error. At the same proportion of detection, a sample size of 25 would result in a 95% exact-confidence interval of 0.5930–0.9317 and an approximately 16.94% margin of error. Because no consensus has been established about the appropriate number of observers for VC verification studies, the sample size of the study could cause over- or underestimation of VRLs.

Logistic regression

The objective of VC verification studies is to prove that the VC criterion would ensure cleanliness if implemented in the manufacturing setting. During VC verification, spiking studies are performed and inspectors state whether they can detect residue visually under controlled viewing conditions. VC verification studies thus provide a basis for the establishment of VRLs and the determination of appropriate viewing conditions. The lowest concentration of residue that is visually detected by all the observers is then used as VRL. In the current method, VRL could be defined mathematically as the lowest residue concentration for which the ratio of the number of observers able to detect the residue to the total number of observers is equal to 1. As discussed earlier, any knowledge about the outcome in future situations could not be obtained from the observed data unless the data were fitted with the most conservative model that explains the data.

One of the most common examples of modeling is the linear-regression technique. However, linear regression is not suitable for binary data. If we represent the binary responses "Yes" and "No" with values of 1 and 0, respectively, then the mean is the proportion of cases with a value of 1 and can be interpreted as proportion or probability of detection. Although the proportions and probabilities cannot exceed 1 or fall below 0, fitting the data with linear regression could give predicted values of the response variable above 1 and below 0. Clearly, linear regression is not appropriate when the data must lie between 0 and 1 because predictions from the model are not similarly constrained. Other problems that arise when fitting binary data with linear regression are that the variance of the error term is not constant and that the error term is not normally distributed.

Table III: The outcome of visually clean verification studies.
The most suitable modeling technique that could be applied to describe a relationship between explanatory variables (e.g., experimental parameters such as residue concentration, viewing distance, viewing angle, and light intensity) and the binary-response variable is logistic regression. Logistic regression allows scientists to predict probabilities of detection of residue based on experimental variables, which is an advantage compared with other prediction techniques. Logistic regression is a flexible and easily applied modeling technique that can be used effectively to model data with continuous or discrete explanatory variables. In addition, the technique accommodates response variables that are not normally distributed.

Figure 1: Plot of probability of detection against the residue concentration for the data presented in Table III.
The modeling procedure based on logistic regression is explained here using a hypothetical data set (see Table III). This data set may represent an ideal case and is certainly within the experience of those involved in conducting VC verification studies (4, 11, 12). The response variable is binary and indicates two different outcomes (i.e., "Yes" or "No") based on the detection of residue by the observers at a specific viewing condition. The continuous explanatory variable is the measure of the theoretical residue concentration spiked on the model surface. At each spiking level, five replicates indicate the total number of observers used for the visual inspection of the surface. The observed proportion and probability of detection for each spiking level is the ratio of observers that detected the residue to the total number of observers. Figure 1 shows a plot of these observed probabilities of detection. It suggests that the probability of detection increases with the spiked residue concentration. The relationship is nonlinear, however, and the probability of detection changes little at the high extreme of spiked residue. This pattern is typical because proportions and probabilities cannot lie outside the range of 0 to 1.


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