The written directives of a quality control program are a necessary, but not a sufficient condition for obtaining and maintaining a state of statistical control. Although quality control directives explain how to conduct an analysis, they do not indicate whether the system is under statistical control. This is the role of quality assessment, the second component of a quality assurance program.
The goals of quality assessment are to determine when an analysis has reached a state of statistical control, to detect when an analysis falls out of statistical control, and to suggest possible reasons for this loss of statistical control. For convenience, we divide quality assessment into two categories: internal methods coordinated within the laboratory, and external methods organized and maintained by an outside agency.
Internal Methods of Quality Assessment
The most useful methods for quality assessment are those coordinated by the laboratory, which provide immediate feedback about the analytical method’s state of statistical control. Internal methods of quality assessment include the analysis of duplicate samples, the analysis of blanks, the analysis of standard samples, and spike recoveries.
Analysis of Duplicate Samples
An effective method for determining the precision of an analysis is to analyze duplicate samples. Duplicate samples are obtained by dividing a single gross sample into two parts, although in some cases the duplicate samples are independently collected gross samples. We report the results for the duplicate samples, X1 and X2, by determining the difference, d, or the relative difference, (d)r, between the two samples
\[d = X_1 - X_2 \nonumber\]
\[(d)_r = \frac {d} {(X_1 + X_2)/2} \times 100 \nonumber\]
and comparing to an accepted value, such as those in Table 15.3.1
for the analysis of waters and wastewaters. Alternatively, we can estimate the standard deviation using the results for a set of n duplicates
\[s = \sqrt{\frac {\sum_{i = 1}^n d_i^2} {2n}} \nonumber\]
where di is the difference between the ith pair of duplicates. The degrees of freedom for the standard deviation is the same as the number of duplicate samples. If we combine duplicate samples from several sources, then the precision of the measurement process must be approximately the same for each.
Table 15.3.1
: Quality Assessment Limits for the Analysis of Waters and Wastewaters
| analyte |
\((d)_r\): [analyte] < 20 \(\times\) MDL (\(\pm\)%)
|
\((d)_r\): [analyte] > 20 \(\times\) MDL (\(\pm\)%)
|
spike recovery limit (%) |
| acids |
40 |
20 \(\times\) MDL (\(\pm\)%)">20 |
60–140 |
| anions |
25 |
20 \(\times\) MDL (\(\pm\)%)">10 |
80–120 |
| bases or neutrals |
40 |
20 \(\times\) MDL (\(\pm\)%)">20 |
70–130 |
| carbamate pesticides |
40 |
20 \(\times\) MDL (\(\pm\)%)">20 |
50–150 |
| herbicides |
40 |
20 \(\times\) MDL (\(\pm\)%)">20 |
40–160 |
| metals |
25 |
20 \(\times\) MDL (\(\pm\)%)">10 |
80–120 |
| other inorganics |
25 |
20 \(\times\) MDL (\(\pm\)%)">10 |
80–120 |
| volatile organics |
40 |
20 \(\times\) MDL (\(\pm\)%)">20 |
70–130 |
|
Abbreviation: MDL = method's detection limit
Source: Table 1020.1 in Standard Methods for the Analysis of Water and Wastewater, American Public Health Association: Washington, D. C., 18th Ed. 1992.
|
To evaluate the precision for the determination of potassium in blood serum, duplicate analyses were performed on six samples, yielding the following results in mg K/L.
Table of potassium tests in blood serum
| duplicate |
\(X_1\) |
\(X_2\) |
| 1 |
160 |
147 |
| 2 |
196 |
202 |
| 3 |
207 |
196 |
| 4 |
185 |
193 |
| 5 |
172 |
188 |
| 6 |
133 |
119 |
Estimate the standard deviation for the analysis.
Solution
To estimate the standard deviation we first calculate the difference, \(d\), and the squared difference, \(d^{2}\), for each duplicate. The results of these calculations are summarized in the following table.
Table of potassium tests in blood serum
| duplicate |
\(d=X_{1}-X_{2}\) |
\(d^{2}\) |
| 1 |
13 |
169 |
| 2 |
–6 |
36 |
| 3 |
11 |
121 |
| 4 |
–8 |
64 |
| 5 |
–16 |
256 |
| 6 |
14 |
196 |
Finally, we calculate the standard deviation
\[s=\sqrt{\frac{169+36+121+64+256+196}{2 \times 6}}=8.4 \nonumber\]
To evaluate the precision of a glucometer—a device a patient uses at home to monitor his or her blood glucose level—duplicate analyses are performed on samples drawn from five individuals, yielding the following results in mg glucose/100 mL.
| duplicate |
\(X_1\) |
\(X_2\) |
| 1 |
148.5 |
149.1 |
| 2 |
96.5 |
98.8 |
| 3 |
174.9 |
174.5 |
| 4 |
118.1 |
118.9 |
| 5 |
72.7 |
70.4 |
Estimate the standard deviation for the analysis.
Answer
-
To estimate the standard deviation we first calculate the difference, d, and the squared difference, \(d^{2}\), for each duplicate. The results of these calculations are summarized in the following table.
| duplicate |
\(d=X_{1}-X_{2}\) |
\(d^{2}\) |
| 1 |
–0.6 |
0.36 |
| 2 |
–2.3 |
5.29 |
| 3 |
0.4 |
0.16 |
| 4 |
–0.8 |
0.64 |
| 5 |
2.3 |
5.29 |
Finally, we calculate the standard deviation.
\[s=\sqrt{\frac{0.36+5.29+0.16+0.64+5.29}{2 \times 5}}=1.08 \nonumber\]
Analysis of Blanks
We introduced the use of a blank in Chapter 3 as a way to correct the signal for contributions from sources other than the analyte. The most common blank is a method blank in which we take an analyte free sample through the analysis using the same reagents, glassware, and instrumentation. A method blank allows us to identify and to correct systematic errors due to impurities in the reagents, contaminated glassware, and poorly calibrated instrumentation. At a minimum, a new method blank is analyzed whenever we prepare a new reagent, or after we analyze a sample with a high concentration of analyte as residual carryover of analyte may produce a positive determinate error.
When we collect samples in the field, additional blanks are needed to correct for potential sampling errors [Keith, L. H. Environmental Sampling and Analysis: A Practical Guide, Lewis Publishers: Chelsea, MI, 1991]. A field blank is an analyte-free sample carried from the laboratory to the sampling site. At the sampling site the blank is transferred to a clean sample container, which exposes it to the local environment. The field blank is then preserved and transported back to the laboratory for analysis. A field blank helps identify systematic errors due to sampling, transport, and analysis. A trip blank is an analyte-free sample carried from the laboratory to the sampling site and back to the laboratory without being opened. A trip blank helps to identify systematic errors due to cross-contamination of volatile organic compounds during transport, handling, storage, and analysis.
A method blank also is called a reagent blank. The contamination of reagents over time is a significant concern. The regular use of a method blank compensates for this contamination.
Analysis of Standards
Another tool for monitoring an analytical method’s state of statistical control is to analyze a standard that contains a known concentration of analyte. A standard reference material (SRM) is the ideal choice, provided that the SRM’s matrix is similar to that of our samples. A variety of SRMs are available from the National Institute of Standards and Technology (NIST). If a suitable SRM is not available, then we can use an independently prepared synthetic sample if it is prepared from reagents of known purity. In all cases, the analyte’s experimentally determined concentration in the standard must fall within predetermined limits before the analysis is considered under statistical control.
Table 4.2.6 in Chapter 4 provides a summary of SRM 2346, a standard sample of Gingko biloba leaves with certified values for the concentrations of flavonoids, terpene ketones, and toxic elements, such as mercury and lead.
Spike Recoveries
One of the most important quality assessment tools is the recovery of a known addition, or spike, of analyte to a method blank, a field blank, or a sample. To determine a spike recovery, the blank or sample is split into two portions and a known amount of a standard solution of analyte is added to one portion. The analyte’s concentration is determined for both the spiked, F, and unspiked portions, I, and the percent recovery, %R, is calculated as
\[\% R=\frac{F-I}{A} \times 100 \nonumber\]
where A is the concentration of analyte added to the spiked portion.
A spike recovery for the analysis of chloride in well water was performed by adding 5.00 mL of a 250.0 ppm solution of Cl– to a 50-mL volumetric flask and diluting to volume with the sample. An unspiked sample was prepared by adding 5.00 mL of distilled water to a separate 50-mL volumetric flask and diluting to volume with the sample. Analysis of the sample and the spiked sample return chloride concentrations of 18.3 ppm and 40.9 ppm, respectively. Determine the spike recovery.
Solution
To calculate the concentration of the analyte added in the spike, we take into account the effect of dilution.
\[A=250.0 \mathrm{ppm} \times \frac{5.00 \mathrm{mL}}{50.0 \mathrm{mL}}=25.0 \mathrm{ppm} \nonumber\]
Thus, the spike recovery is
\[\% R=\frac{40.9-18.3}{25.0} \times 100=90.4 \% \nonumber\]
To test a glucometer, a spike recovery is carried out by measuring the amount of glucose in a sample of a patient’s blood before and after spiking it with a standard solution of glucose. Before spiking the sample the glucose level is 86.7 mg/100 mL and after spiking the sample it is 110.3 mg/100 mL. The spike is prepared by adding 10.0 μL of a 25 000 mg/100mL standard to a 10.0-mL portion of the blood. What is the spike recovery for this sample.
- Answer
-
Adding a 10.0-μL spike to a 10.0-mL sample is a 1000-fold dilution; thus, the concentration of added glucose is 25.0 mg/100 mL and the spike recovery is
\[\% R=\frac{110.3-86.7}{25.0} \times 100=94.4 \% \nonumber\]
We can use a spike recovery on a method blank and a field blank to evaluate the general performance of an analytical procedure. A known concentration of analyte is added to each blank at a concentration that is 5 to 50 times the method’s detection limit. A systematic error during sampling and transport will result in an unacceptable recovery for the field blank, but not for the method blank. A systematic error in the laboratory, however, affects the recoveries for both the field blank and the method blank.
Spike recoveries on a sample are used to detect systematic errors due to the sample’s matrix, or to evaluate the stability of a sample after its collection. Ideally, samples are spiked in the field at a concentration that is 1 to 10 times the analyte’s expected concentration or 5 to 50 times the method’s detection limit, whichever is larger. If the recovery for a field spike is unacceptable, then a duplicate sample is spiked in the laboratory and analyzed immediately. If the laboratory spike’s recovery is acceptable, then the poor recovery for the field spike likely is the result of the sample’s deterioration during storage. If the recovery for the laboratory spike also is unacceptable, the most probable cause is a matrix-dependent relationship between the analytical signal and the analyte’s concentration. In this case the sample is analyzed by the method of standard additions. Typical limits for spike recoveries for the analysis of waters and wastewaters are shown in Table 15.3.1
.
Figure 15.4.1, which we will discuss in the next section, illustrates the use of spike recoveries as part of a quality assessment program.
External Methods of Quality Assessment
Internal methods of quality assessment always carry some level of suspicion because there is a potential for bias in their execution and interpretation. For this reason, external methods of quality assessment also play an important role in a quality assurance program. One external method of quality assessment is the certification of a laboratory by a sponsoring agency. Certification of a lab is based on its successful analysis of a set of proficiency standards prepared by the sponsoring agency. For example, laboratories involved in environmental analyses may be required to analyze standard samples prepared by the Environmental Protection Agency. A second example of an external method of quality assessment is a laboratory’s voluntary participation in a collaborative test sponsored by a professional organization, such as the Association of Official Analytical Chemists. Finally, an individual contracting with a laboratory can perform his or her own external quality assessment by submitting blind duplicate samples and blind standards to the laboratory for analysis. If the results for the quality assessment samples are unacceptable, then there is good reason to question the laboratory’s results for other samples.
See Chapter 14 for a more detailed description of collaborative testing.
