Attention to the measured parameter, such as charting the results obtained by analysis of a control sample, can signal a change in performance that requires adjustment of the analytical system. Method Validation. All methods are appropriately validated as specified under Validation of Compendial Methods A validated method may be used to test a new formulation such as a new product, dosage form, or process intermediate only after confirming that the new formulation does not interfere with the accuracy, linearity, or precision of the method.
It may not be assumed that a validated method could correctly measure the active ingredient in a formulation that is different from that used in establishing the original validity of the method. Thus, the measured value differs from the actual value because of variability inherent in the measurement. If an array of measurements consists of individual results that are representative of the whole, statistical methods can be used to estimate informative properties of the entirety, and statistical tests are available to investigate whether it is likely that these properties comply with given requirements.
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The resulting statistical analyses should address the variability associated with the measurement process as well as that of the entity being measured. Statistical measures used to assess the direction and magnitude of these errors include the mean, standard deviation, and expressions derived therefrom, such as the coefficient of variation CV, also called the relative standard deviation, RSD. The estimated variability can be used to calculate confidence intervals for the mean, or measures of variability, and tolerance intervals capturing a specified proportion of the individual measurements.
The use of statistical measures must be tempered with good judgment, especially with regard to representative sampling.
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Most of the statistical measures and tests cited in this chapter rely on the assumptions that the distribution of the entire population is represented by a normal distribution and that the analyzed sample is a representative subset of this population. The normal or Gaussian distribution is bell-shaped and symmetric about its center and has certain characteristics that are required for these tests to be valid. If the assumption of a normal distribution for the population is not warranted, then normality can often be achieved at least approximately through an appropriate transformation of the measurement values.
For example, there exist variables that have distributions with longer right tails than left. Such distributions can often be made approximately normal through a log transformation. When the objective is to construct a confidence interval for the mean or for the difference between two means, for example, then the normality assumption is not as important because of the central limit theorem. However, one must verify normality of data to construct valid confidence intervals for standard deviations and ratios of standard deviations, perform some outlier tests, and construct valid statistical tolerance limits.
In the latter case, normality is a critical assumption. Simple graphical methods, such as dot plots, histograms, and normal probability plots, are useful aids for investigating this assumption. A single analytical measurement may be useful in quality assessment if the sample is from a whole that has been prepared using a well-validated, documented process and if the analytical errors are well known.
The obtained analytical result may be qualified by including an estimate of the associated errors. There may be instances when one might consider the use of averaging because the variability associated with an average value is always reduced as compared to the variability in the individual measurements. The choice of whether to use individual measurements or averages will depend upon the use of the measure and its variability. For example, when multiple measurements are obtained on the same sample aliquot, such as from multiple injections of the sample in an HPLC method, it is generally advisable to average the resulting data for the reason discussed above.
Variability is associated with the dispersion of observations around the center of a distribution. The most commonly used statistic to measure the center is the sample mean bar x :. Method variability can be estimated in various ways. The most common and useful assessment of a method's variability is the determination of the standard deviation based on repeated independent 1 measurements of a sample.
The sample standard deviation, s, is calculated by the formula:. The relative standard deviation RSD is then calculated as:. If the data requires log transformation to achieve normality e. A control sample is defined as a homogeneous and stable sample that is tested at specific intervals sufficient to monitor the performance of the method for which it was established. Test data from a control sample can be used to monitor the method variability or be used as part of system suitability requirements.
A control chart can be constructed and used to monitor the method performance on a continuing basis as shown under Appendix A. A precision study should be conducted to provide a better estimate of method variability. The intermediate precision studies should allow for changes in the experimental conditions that might be expected, such as different analysts, different preparations of reagents, different days, and different instruments. To perform a precision study, the test is repeated several times. Each run must be completely independent of the others to provide accurate estimates of the various components of variability.
In addition, within each run, replicates are made in order to estimate repeatability. See an example of a precision study under Appendix B. A confidence interval for the mean may be considered in the interpretation of data. Such intervals are calculated from several data points using the sample mean bar x and sample standard deviation s according to the formula:.
Its values are obtained from published tables of the Student t -distribution. Note that it is important to define the population appropriately so that all relevant sources of variation are captured. Occasionally, observed analytical results are very different from those expected. Aberrant, anomalous, contaminated, discordant, spurious, suspicious or wild observations; and flyers, rogues, and mavericks are properly called outlying results. Like all laboratory results, these outliers must be documented, interpreted, and managed. Such results may be accurate measurements of the entity being measured, but are very different from what is expected.
Alternatively, due to an error in the analytical system, the results may not be typical, even though the entity being measured is typical. When an outlying result is obtained, systematic laboratory and process investigations of the result are conducted to determine if an assignable cause for the result can be established. If an assignable cause that is not related to a product or component deficiency can be identified, then retesting may be performed on the same sample, if possible, or on a new sample. The precision and accuracy of the method, the Reference Standard, process trends, and the specification limits should all be examined.
Data may be invalidated, based on this documented investigation, and eliminated from subsequent calculations. If no documentable, assignable cause for the outlying laboratory result is found, the result may be tested, as part of the overall investigation, to determine whether it is an outlier. However, careful consideration is warranted when using these tests. Two types of errors may occur with outlier tests: a labeling observations as outliers when they really are not; and b failing to identify outliers when they truly exist.
Any judgment about the acceptability of data in which outliers are observed requires careful interpretation. The selection of the correct outlier identification technique often depends on the initial recognition of the number and location of the values. Outlier labeling is most often done visually with graphical techniques.
When used appropriately, outlier tests are valuable tools for pharmaceutical laboratories. Several tests exist for detecting outliers. Choosing the appropriate outlier test will depend on the sample size and distributional assumptions. Many of these tests e.
If a transformation is made to the data, the outlier test is applied to the transformed data. Common transformations include taking the logarithm or square root of the data. Other approaches to handling single and multiple outliers are available and can also be used. These include tests that use robust measures of central tendency and spread, such as the median and median absolute deviation and exploratory data analysis EDA methods.
The use of such methods reduces the risks associated with both types of error in the identification of outliers. However, an outlier test cannot be the sole means for removing an outlying result from the laboratory data. An outlier test may be useful as part of the evaluation of the significance of that result, along with other data. Outlier tests have no applicability in cases where the variability in the product is what is being assessed, such as content uniformity, dissolution, or release-rate determination.
In these applications, a value determined to be an outlier may in fact be an accurate result of a nonuniform product. All data, especially outliers, should be kept for future review. Unusual data, when seen in the context of other historical data, are often not unusual after all but reflect the influences of additional sources of variation.
In summary, the rejection or retention of an apparent outlier can be a serious source of bias.
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The nature of the testing as well as scientific understanding of the manufacturing process and analytical method have to be considered to determine the source of the apparent outlier. An outlier test can never take the place of a thorough laboratory investigation. Rather, it is performed only when the investigation is inconclusive and no deviations in the manufacture or testing of the product were noted. Even if such statistical tests indicate that one or more values are outliers, they should still be retained in the record.
Including or excluding outliers in calculations to assess conformance to acceptance criteria should be based on scientific judgment and the internal policies of the manufacturer. It is often useful to perform the calculations with and without the outliers to evaluate their impact. Outliers that are attributed to measurement process mistakes should be reported i. When assessing conformance to a particular acceptance criterion, it is important to define whether the reportable result the result that is compared to the limits is an average value, an individual measurement, or something else.
If, for example, the acceptance criterion was derived for an average, then it would not be statistically appropriate to require individual measurements to also satisfy the criterion because the variability associated with the average of a series of measurements is smaller than that of any individual measurement.
It is often necessary to compare two methods to determine if their average results or their variabilities differ by an amount that is deemed important. The goal of a method comparison experiment is to generate adequate data to evaluate the equivalency of the two methods over a range of concentrations. Some of the considerations to be made when performing such comparisons are discussed in this section. Precision is the degree of agreement among individual test results when the analytical method is applied repeatedly to a homogeneous sample.
A decrease in precision or increase in variability can lead to an increase in the number of results expected to fail required specifications. On the other hand, an alternative method providing improved precision is acceptable. One way of comparing the precision of two methods is by estimating the variance for each method the sample variance, s 2 , is the square of the sample standard deviation and calculating a one-sided upper confidence interval for the ratio of true variances, where the ratio is defined as the variance of the alternative method to that of the current method.
An example, with this assumption, is outlined under Appendix D. The one-sided upper confidence limit should be compared to an upper limit deemed acceptable, a priori, by the analytical laboratory. If the one-sided upper confidence limit is less than this upper acceptable limit, then the precision of the alternative method is considered acceptable in the sense that the use of the alternative method will not lead to an important loss in precision. Note that if the one-sided upper confidence limit is less than one, then the alternative method has been shown to have improved precision relative to the current method.
The confidence interval method just described is preferred to applying the two-sample F -test to test the statistical significance of the ratio of variances. To perform the two-sample F -test, the calculated ratio of sample variances would be compared to a critical value based on tabulated values of the F distribution for the desired level of confidence and the number of degrees of freedom for each variance. Tables providing F -values are available in most standard statistical textbooks. If the calculated ratio exceeds this critical value, a statistically significant difference in precision is said to exist between the two methods.
However, if the calculated ratio is less than the critical value, this does not prove that the methods have the same or equivalent level of precision; but rather that there was not enough evidence to prove that a statistically significant difference did, in fact, exist. Comparison of the accuracy see Analytical Performance Characteristics under Validation of Compendial Methods of methods provides information useful in determining if the new method is equivalent, on the average, to the current method.
A simple method for making this comparison is by calculating a confidence interval for the difference in true means, where the difference is estimated by the sample mean of the alternative method minus that of the current method. The confidence interval should be compared to a lower and upper range deemed acceptable, a priori, by the laboratory. If the confidence interval falls entirely within this acceptable range, then the two methods can be considered equivalent, in the sense that the average difference between them is not of practical concern.
The lower and upper limits of the confidence interval only show how large the true difference between the two methods may be, not whether this difference is considered tolerable. Such an assessment can only be made within the appropriate scientific context. The confidence interval method just described is preferred to the practice of applying a t -test to test the statistical significance of the difference in averages.
One way to perform the t -test is to calculate the confidence interval and to examine whether or not it contains the value zero. The two methods have a statistically significant difference in averages if the confidence interval excludes zero. A statistically significant difference may not be large enough to have practical importance to the laboratory because it may have arisen as a result of highly precise data or a larger sample size.
On the other hand, it is possible that no statistically significant difference is found, which happens when the confidence interval includes zero, and yet an important practical difference cannot be ruled out. This might occur, for example, if the data are highly variable or the sample size is too small. Thus, while the outcome of the t -test indicates whether or not a statistically significant difference has been observed, it is not informative with regard to the presence or absence of a difference of practical importance. Determination of Sample Size.
Sample size determination is based on the comparison of the accuracy and precision of the two methods 4 and is similar to that for testing hypotheses about average differences in the former case and variance ratios in the latter case, but the meaning of some of the input is different. The first component to be specified is , the largest acceptable difference between the two methods that, if achieved, still leads to the conclusion of equivalence. That is, if the two methods differ by no more than , they are considered acceptably similar. The comparison can be two-sided as just expressed, considering a difference of in either direction, as would be used when comparing means.
One consideration, when there are specifications to satisfy, is that the new method should not differ by so much from the current method as to risk generating out-of-specification results. One then chooses to have a low likelihood of this happening by, for example, comparing the distribution of data for the current method to the specification limits. This could be done graphically or by using a tolerance interval, an example of which is given in Appendix E. In general, the choice for must depend on the scientific requirements of the laboratory.
The next two components relate to the probability of error.
The data could lead to a conclusion of similarity when the methods are unacceptably different as defined by. This is called a false positive or Type I error. The error could also be in the other direction; that is, the methods could be similar, but the data do not permit that conclusion. This is a false negative or Type II error. With statistical methods, it is not possible to completely eliminate the possibility of either error.
However, by choosing the sample size appropriately, the probability of each of these errors can be made acceptably small. The desired maximum probability of a Type II error is commonly denoted by. In the context of equivalency testing, power is the probability of correctly concluding that two methods are equivalent. The protocol for the experiment should specify , , and power.
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The sample size will depend on all of these components. An example is given in Appendix E. Although Appendix E determines only a single value, it is often useful to determine a table of sample sizes corresponding to different choices of , , and power. Such a table often allows for a more informed choice of sample size to better balance the competing priorities of resources and risks false negative and false positive conclusions.
Figure 1. Individual X or individual measurements control chart for control samples. In this particular example, the mean for all the samples bar x is These moving ranges are averaged bar MR and used in the following formulae:. For the example in Figure 1, the bar MR was 1. Table 1 displays data collected from a precision study. This study consisted of five independent runs and, within each run, results from three replicates were collected. Because there were an equal number of replicates per run in the precision study, values for Variance Run and Variance Rep can be derived from the ANOVA table in a straightforward manner.
Estimates can still be obtained with unequal replication, but the formulas are more complex. Studying the relative magnitude of the two variance components is important when designing and interpreting a precision study. For example, for these data the between-run component of variability is much larger than the within-run component.
This suggests that performing additional runs would be more beneficial to reducing variability than performing more replication per run see Table 2. Table 2 shows the computed variance and RSD of the mean i. For example, the Variance of the mean, Standard deviation of the mean, and RSD of a test involving two runs and three replicates per each run are 0.
Where As illustrated in Table 2, increasing the number of runs from one to two provides a more dramatic reduction in the variability of the reportable value than does increasing the number of replicates per run. No distributional assumptions were made on the data in Table 1 as the purpose of this Appendix is to illustrate the calculations involved in a precision study. Given the following set of 10 measurements: This is a modified version of the ESD Test that allows for testing up to a previously specified number, r, of outliers from a normally distributed population.
Let r equal 2, and n equal The maximum value is larger than the tabled value and is identified as being inconsistent with the remaining data. Sources for -values are included in many statistical textbooks. Caution should be exercised when using any statistical table to ensure that the correct notations i. Again, find the mean and standard deviation Table 3, right two columns , normalize each value, and take the absolute value of these results.
13.2.1 Theory and Practice
The maximum value is not larger than the tabled value. Dixon-Type Tests. Similar to the ESD test, the two smallest values will be tested as outliers; again assuming the data come from a single normal population. The same r 11 equation is used, but a new critical r 11 , 0. This stepwise procedure is not an exact procedure for testing for the second outlier as the result of the second test is conditional upon the first.
Because the sample size is also reduced in the second stage, the end result is a procedure that usually lacks the sensitivity of the exact procedures that Dixon provides for testing for two outliers simultaneously; however, these procedures are beyond the scope of this Appendix. Hampel's Rule. However, instead of subtracting the mean from each data point and dividing the difference by the standard deviation, the median is subtracted from each data value and the resulting differences are divided by MAD see below.
The calculation of MAD is done in three stages. First, the median is subtracted from each data point. Next, the absolute values of the differences are obtained. These are called the absolute deviations. Finally, the median of the absolute deviations is calculated and multiplied by the constant 1. In the first part, the text covers the historical aspects of chemical.
The book then proceeds to tackling methods for analysis in which the final measurement is preceded by one or more chemical reactions. The first two chapters of the second part discuss distillation and chromatography, respectively. Next, the title details the physical methods that only occasionally and incidentally need to be preceded by chemical reactions. The text will be of great use for students, researchers, and practitioners of chemistry. We are always looking for ways to improve customer experience on Elsevier.
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