Limits, action confidence

FIGURE 11.22 Control charts and outliers, (a) pEC50 values (ordinates) run as a quality control for a drug screen over the days on which the screen is run (abscissae). Dotted lines are the 95% c.l. and the solid lines the 99.7% c.l. Data points that drift beyond the action lines indicate significant concern over the quality of the data obtained from the screen on those days, (b) The effect of significant outliers on the criteria for rejection. For the data set shown, the inclusion of points A and B lead to a c.l. for 95% confidence that includes point B. Removal of point A causes the 95% limits to fall below points B, causing them to be suspect as well. Thus, the presence of the data to be possibly rejected affects the criteria for rejection of other data. 

Certain limitations of this study are worth reviewing. The FDA standard or action level for PCBs in fish sold in interstate commerce was used even though the river is not commercially fished.. In any event, a study of contamination in fish which will result in comparisons with a regulatory action level for justifying a remedial action must provide results which retain some degree of confidence. 

A process, such as a manufacturing process, may also be monitored with a control chart. In this case, the desirable value, warning limits, and action limits for the product of the manufacturing process is determined over time using materials and equipment that the scientists and engineers are confident provide an accurate picture of the product. 

Few data are available to guide this decision beyond clinical experience. There is a modest amount of evidence that nonresponders to TCAs, principally desipramine or imipramine, alone may respond to a SSRI alone and vice versa (136). No compelling evidence exists showing that nonresponders to one SSRI as a result of a lack of efficacy will respond to a second trial with another SSRI. There is limited confidence in the results of studies that have been done switching nonresponders from one SSRI to another for two reasons. First, virtually all have been open label and, second, most were conducted by the manufacturer of the second SSRI. Until there is more substantive evidence that switching from one SSRI to another is worthwhile, it may be more prudent to switch to a class of antidepressants with a different putative mechanism of action. 

The planning team is now prepared to assign decision error limits to false acceptance and false rejection decision errors. A decision error limit is the probability that an error may occur when making a decision based on sample data. The probability curve tells us that the highest probability of error exists in the gray region this error goes down as the mean concentrations move away from either side of the action level. The probability curve reflects our level of tolerance to uncertainty associated with a decision or, conversely, level of confidence with which a decision will be made. 

Excavated soil suspected of containing lead has been stockpiled. We may use this soil as backfill if the mean lead concentration is below the action level of lOOmg/kg. To decide if the soil is acceptable as backfill, we will sample the soil and analyze it for lead. The mean concentration of lead in soil will represent the statistical population parameter. The theoretical decision rule, the baseline and the alternative conditions, and the null and alternative hypotheses have been stated in Examples 2.1 and 2.2. The assigned probability limits are a —ft — 0.05. This means that the false acceptance error rate is 0.05. The probability of making a correct decision is 0.95 or the confidence level is 95 percent. 

The underlying assumptions of the Student s t-test include simple random and systematic sampling and a normal distribution of the sample mean. The upper limit of the confidence interval for the mean concentration is compared to the action level to determine whether solid waste contains a contaminant of concern at a hazardous level. (The calculation is conducted according to Equation 10, Appendix 1.) A contaminant of concern is not considered to be present at a hazardous level, if the upper limit of the confidence interval is below the action level. Otherwise, the opposite conclusion is reached. Example 5.13 demonstrates the application of this test for deciding whether the waste is hazardous or not. 

In Example 2.3, we have calculated that 14 samples are needed to reach the decision with a 95 percent level of confidence. To be on the safe side, we collected and analyzed 20 samples. The collected samples have the concentrations of lead ranging from 5 to 210mg/kg the mean concentration is 86 mg/kg the standard deviation is 63 mg/kg and the standard error is 14mg/kg. From Appendix 1, Table 2 we determine that the t-value for 19 degrees of freedom (the number of samples less one) and a one-sided confidence interval for a — 0.05 is 1.729. Entering these data into Equation 10, Appendix 1, we calculate the 95 percent confidence interval of the mean 86 24 mg/kg. The upper limit of the confidence interval is 110 mg/kg and it exceeds the action level. Therefore, the null hypothesis Hq p > lOOmg/kg, formulated in Example 2.2 is true, as supported by the sample data. Based on this calculation we make a decision not to use the soil as backfill. 

For a 90 percent confidence level, the upper limit of the confidence interval is 104 mg/kg, still exceeding the action level. 

Student s t-test is frequently used in statistical evaluations of environmental chemical data. It establishes a relationship between the mean (x) of normally distributed sample measurements, their sample standard deviation (,v), and the population mean (p). Confidence intervals may be calculated based on Student s t-test (Equation 10). The upper limit of the confidence interval is compared to the action level to determine whether the sampled medium contains a hazardous concentration of a pollutant. If the upper confidence limit is below the action level, the medium is not hazardous otherwise the opposite conclusion is reached. 

Statistical tests may be one-sided (one-tailed) or two-sided (two-tailed). One-sided confidence intervals are used for testing the data that are compared to action levels to determine whether the mean concentration is greater or lower than the action level. Two-sided confidence limits are used for comparing two sets of data to each other to establish whether they differ, for example, for comparing sample concentrations to background concentrations. One-sided and two-sided confidence intervals are illustrated in Figure 2. 

Using 3 doses of test compound and standard, activity ratios with confidence limits can be determined after each time interval with the 3 + 3 point assay giving evidence for the duration of action (Vogel 1969a,b). 

The action limits are the boundaries on the control chart at which the user must take action if any points exceed the defined limit. They are usually set at an appropriate confidence level, equivalent to the sample standard deviation value obtained from precision studies multiplied by a number. Warning limits are normally set at the 95% confidence level ( = 2 x sample standard deviation)... 

Within each grid cell we could take three random samples and calculate an overall average for the target area (Figure 2.2) from the analysis results for the 75 samples. This stratified random sampling design allows statistical analysis of the results. We can calculate an upper confidence limit of the overall average concentration and test the null hypothesis that the soil is contaminated above the action threshold of 1%. 

If the average concentration, at the 95% higher confidence limit, is below the action threshold for treatment, the target area can be... 

When the selected squares have been treated, the pollution in the target area might be as shown in Figure 2.3. If the remaining estimated average overall concentration of PCBs is below the action threshold at the 95% confidence limit, then no further treatment is needed. 

Quality Control Data. Data obtained from assays of blood gas and pH control materials may be handled in the same way as data from other clinical chemistry determinations (i.e., mean, SD, and coefficient of variation, and control and confidence limits for construction of Levey-Jennings plots). As stability of commercial aqueous control materials is generally several months, vendors often provide data reduction programs that standardize and simplify documentation. However, the resulting reports are temporally delayed and are most useful for meeting accreditation requirements as opposed to real-time corrective or preventive action. They are however useful to compare long-term performances with other laboratories. Equally important features of quality assurance to an active blood gas service are the sixth sense of practiced operators for detecting subtle manifestations of deterioration of instrument performance and the suspicion of trouble expressed by clinicians. 

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