Test for an Outlier

However, it is also useful to have a quick method to decide whether a particular value is an outlier or not. The method recommended by ISO is Grubbs s test, although many older texts still present Dixon s 

G can be compared to tables of critical values for G at a = 0.05, critical, calculated using equation 3.2 below. If G Gcvmca then the suspect point is rejected. Note that in the case of Grubbs s test, we compare with tabulated critical values simply because the calculation of the probability associated with the value of G is nontrivial. We do have a formula for the calculation of Gcriticai- 

In other significance tests we can calculate a probability associated with the parameter see the F test in example 3.4. 

The level of calcium in milk was determined using an EDTA titration method. Ten repeat measurements were performed with the following measured concentrations (units mgg ) 4.59, 10.00, 6.07, 4.73, 9.91, 5.28, 16.65, 5.17, 4.59, and 4.38. 

Determine whether there are any outliers in this data set. 

---

Never. You may decide not to use a value in the calculation of mean and standard deviation after performing a Grubbs s test for an outlier. (Section 3.5)... 

Testing for an outlier under the assumption of normal distribution can be carried out by the Dixon s test. This test uses the range of measurements and can be applied even in cases where only few data are available. The n measurements are arranged in ascending order. If the very small value to be tested as an outlier is denoted by Xi and the very large striking value by x, then the test statistics is calculated by... 

In order to use Grubbs test for an outlier, that is to test Ho all measurements come from the same population, the statistic G is calculated ... 

Before adopting statistical tests to assess the reliability of data, outliers should be first analyzed carefully to identify any anomaly in instrament fidelity, calibration, procedure, environmental conditions, recording, etc. The first objective is to reject an outlier based on physical evidence that the data point was unrepresentative of the sample population. If this exercise fails to identify probable cause (or if details on the experimental methods are unavailable), Chauvenet s criterion may be applied to assess the reliability of the data point (Holman 2001). Simply put, the criterion reconunends rejecting a data point if the probability of obtaining the deviation is less than the reciprocal of two times the number of data points—l/(2n). For an outlier smaller than the mean, the data point may be rejected when ... 

In the development of an electroanalytical protocol, comparisons need to be made with a well-established and accepted accurate or gold standard procedure. The two important statistical analysis methods to allow such comparisons are the t-test and F-t st. Before the application of these, the g-test needs to be performed, that is, the obtained data should be tested for potential outliers (anomalies), which are data values which appear to be unreasonably distant (or outlying) from the others comprising the data set, and to do this objectively, involves the implementation of the 2-test [5]. 

Statistical test for deciding if an outlier can be removed from a set of data. 

Dixon s Q-test statistical test for deciding if an outlier can be removed from a set of data. (p. 93) dropping mercury electrode an electrode in which successive drops of Hg form at the end of a capillary tube as a result of gravity, with each drop providing a fresh electrode surface, (p. 509)... 

If Q is greater than values from a table yielding Q values for 90% probability of difference, then the value may be removed from the data set (p<0.10). An example of how this test is used is given in Table 11.17a. In this case, the pKB value of 8.1 appears to be an outlier with respect to the other estimates made. The calculated Q is compared to a table of Q values for 90% confidence (Table 11.17b) to determine the confidence with which this value can be accepted into the data set. In the case shown in Table 8.17, Q<0.51. Therefore, there is <90% probability that the value is different. If this level of probability is acceptable to the experimenter, then the value should remain in the set. 

Example 53 If the standard deviation before elimination of the purported outlier is not much higher than the upper CLf method), as in the case = 0.358 < CL(/(0.3) 0.57 factor Chu/sx 1.9 for = 9, see program MSD), an outlier test should not even be considered both for avoiding fruitless discussions and reducing the risk of chance decisions, the hurdle should be set even higher, say at p < 0.01, so that CLu/sx > 2.5. 

Other authors used a simple 2 standard deviation criteria or an outlier test (F-test) to check for significant differences between within-bottle and between-bottle results (Martin-Esteban et al. 1997 Quevauviller et al. 1995). The degree of homogeneity of elements and compounds in the materials tested in these studies does not seem to be adequately described and, hence, the asigned uncertainties in the mean values may represent only the bias between the analytical methods used in the certification. 

For an acceptable accuracy test, the value of Vxo should be not more than 5%, and the intercept should not be significantly different from zero (p > 0.05). Kromidas [29] suggested that the maximum value of the intercept is about 2-5% of the target concentration and the maximum RSD of the response factor is 2.5%, while the maximum y-intercept of the impurity is <25% to its specification limit [30]. As a matter principle, calibration data should be free of outlier values, which can be proved by using a FJ or Ftest5 [28]. 

For small data sets (n < 10), which are often encountered in chemical analysis, a simple method to determine if an outlier is rejectable is the Q test. In this test, a value for Q is calculated and compared to a table of Q values that represent a certain percentage of confidence that the proposed rejection is valid. If the calculated Q value is greater than the value from the table, then the suspect value is rejected and the mean recalculated. If the Q value is less than or equal to the value from the table, then the calculated mean is reported. Q is defined as follows ... 

GRUBBY TEST for rejection of an observation is applied in order to determine if one of the observations should be rejected as being an outlier. The following equation was used for the test ... 

If the data as a whole appear normally distributed but there is concern that an extreme point is an outlier, it is not necessary to apply the Rankit procedure. The Grubbs s outlier test (1950) is now recommended for testing single outliers, replacing Dixon s Q-test. After identifying a single outlier, which, of course, must be either the maximum or minimum data value, the G statistic is calculated ... 

The organizing laboratory performs statistical tests on the results from participating laboratories, and how outliers are treated depends on the nature of the trial. Grubbs s tests for single and paired outliers are recommended (see chapter 2). In interlaboratory studies outliers are usually identified at the 1% level (rejecting H0 at a = 0.01), and values between 0.01 < a < 0.05 are flagged as stragglers. As with the use of any statistics, all data from interlaboratory studies should be scrutinized before an outlier is declared. 

Equation (4.20) was proposed by Hoskuldsson [65] many years ago and has been adopted by the American Society for Testing and Materials (ASTM) [59]. It generalises the univariate expression to the multivariate context and concisely describes the error propagated from three uncertainty sources to the standard error of the predicted concentration calibration concentration errors, errors in calibration instrumental signals and errors in test sample signals. Equations (4.19) and (4.20) assume that calibrations standards are representative of the test or future samples. However, if the test or future (real) sample presents uncalibrated components or spectral artefacts, the residuals will be abnormally large. In this case, the sample should be classified as an outlier and the analyte concentration cannot be predicted by the current model. This constitutes the basis of the excellent outlier detection capabilities of first-order multivariate methodologies. 

Because outliers are expected to occur on occasion with this testing (for example, due to an air bubble between the product sample and the membrane), a nonpara-metric method is proposed, whose performance tends to be resistant to the presence of outliers. 

Note The ASTM (American Society for Testing Material) uses a different test for rejection of an outlier, called the reduced central value z,- = (x, — x)/s, which has its own table of critical values. 

In very rare occasions and only if a full investigation has failed to reveal the cause of the OOS result, a statistical analysis (such as an outlier test) may prove valuable as one assessment of the probability of the OOS result as discordant and for providing aperspective on the result in the overall evaluation of the quality of the batch. 

Based on the survey of industry OOS practices mentioned earlier, many laboratories have begun using outlier testing for OOS results from chemical assays, but the test is being used cautiously and only to support solid analytical evidence that confirms the OOS was truly unrepresentative of the sample. It has been used only to augment evidence that an OOS result is invalid since it sheds no technical light on the possible cause of the aberrant result. 

A plot of this sort is useful for qualitative analysis, especially of outliers. A study of the test conditions (e.g., the activating systems used) the particular strain of test organisms (e.g., its sensitivity to frame shifts vs. base substitutions), or the chemical nature of the mutagen may reveal the basis for the discrepancy and ultimately yield deeper insights. For example, the discrepancy in the two tests for furylfuramide (the extreme outlier in Figure 9-2) could be due to different nitroreductase activities in the two systems, which would lead to differences in capacity to convert the chemical into an active form. 

TABLE 6.6. Critical Values for Dixon s Test. The Suspected Value is an Outlier if the Calculated 2-Statistic is Above the Critical Value... 

The definition of the limit of quantitation (LOQ) is handled quite differently. For example, an easy, often used approach is the definition of using the double of the mean blank value as LOQ. This definition sounds simple and a scientific theoretical justification seems not to be available. However, the thus defined LOQ is then a reasonable value when standard deviation (SD) of the blank values is low (< 10 %, according to own experience). More sophisticated is the following definition The mean blank value plus 3 times the standard deviation is required for a limit of detection (LOD) and the mean blank value plus 10 times the standard deviation is required for LOQ (see for instance Krull (1998)). Outliers can be identified for instance by the Grubbs test (for instance explained in www.graphpad.com). 

---