**3. Control chart in analytical method**

#### **3.1 X-bar and MR-control chart**

*Cp* <sup>¼</sup> *USL* � *LSL*

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With respect to specification limits, we cannot apply Bouabidi et al.'s [8] proposed criteria, based on variations around the target value T. Other criteria could be to fix the specification limits equal to the control limits. In this case, the Cpk index was 1.03. To determine if the process meets the capability requirement, we must calculate the critical value Co for Cpk based on α risk, sample size, and C value (i.e., the required process capability) [25]. We find the critical value Co =1.095, based on C = 1.0, α = 0.05, and sample size n = 200, demonstrating that the process fails to

*The black-dashed line shows the specified limits (USL and LSL) established at* �*10% of mean value, whereas the red-dashed line corresponds to limits at* �*6% of the mean value. The black line is the process mean.*

**USL-LSL Cp Co Cpk Co Process capability** �3SD 1.02 1.081 1.03 1.095 Inadequate �6% of x-bar 0.73 1.081 0.73 1.095 Inadequate �10% of x-bar 1.21 1.081 1.21 1.095 Capable

*Cp and Cpk values as a function of the specification limit (USL-LSL). The process capability is based on the*

meet the capability requirements (**Table 1**).

**Figure 5.**

**Table 1.**

**220**

*critical values (Co) according to Pearn et al. [26].*

<sup>6</sup>*<sup>σ</sup>* (9)

The main objective of any validation process is to check the maintenance of validation conditions in the laboratory over a long time period. In this second example, we used the insulin peak area expressed as concentration (U/mL) as control parameters [12]. For this, a standard solution with a nominal concentration of 100 U/mL was analyzed each working day (n = 144). The predicted concentration for the standard solution was obtained from the method calibration. This value is not independent due to the measurement errors which depend on various factors related to the method and its validation but not on the analyst [29].

Histogram and normal probability plots show that the collected data follow the normal distribution (**Figure 1**). The Shapiro-Wilk test confirmed this assumption. Therefore, control charts can be used to obtain the method requirements.

The method mean was estimated to be 100.227 U/mL from the x-bar control chart (**Figure 6**), while the method standard deviation was estimated to be 0.60 U/ mL from the MR-chart. The control limits were estimated using the "qcc" package from the R-program [15].

The x-bar control chart shows that all plotted values fall within the control limits (98.40, 102.06), and therefore, the method is in statistical control. In addition, there is no evidence of cyclical or periodic behavior. However, the sample (#74) was outside of the control limits, but the cause of this was attributable to introducing a new column, whereas the sample #86 was related with the presence of "eight consecutive points plot on one side of the center line" [1]. The application of decision rules for detecting nonrandom patterns on control charts indicates that, in this situation, the method is out of control. However, the use of these rules allows enhancing the sensitivity of control charts against only criterion of control limit violation.

In such situation, it is necessary to search the cause and take corrective action. In the first case, the cause was assignable with a column change, whereas the second one was due to the presence of "more eight consecutive point plots on one side of

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control…*

The data were analyzed using the Cusum and EWMA control charts.

The EWMA control chart shows that all plotted values fall within the control limits (**Figure 7**) using a smoothing constant of 0.2 (λ = 0.2) and control limit

Cusum control chart, with a shift detection fixed at 1 SD, shows four points beyond boundaries, all of them greater than the upper control limit. The first alteration is located around the samples #84–85, whereas the second one appears close to the end of the process (#130–131). In addition, if both alterations presented an upward tendency, it indicates the process average changes, which requires a search to determine the causes. When we fixed the shift detection at 1.5 SD, the situation is totally different, all points fall within control limits (data not shown), and the data describe a random way with an average of zero, since the points show

*EWMA control chart for HPLC method. All points fall within control limit for λ = 0.2 and L = 3.054.*

that any value exceeds the upper control limit was 0.0053.

width fixed at three standard deviations (L = 3.054).

no evidence of an upward or downward tendency (**Figure 8**).

**3.2 Cusum and EWMA control chart**

*DOI: http://dx.doi.org/10.5772/intechopen.91354*

In this case, the obtained ARL value for MR-control was 189 since the probability

the centerline."

**Figure 7.**

**223**

#### **Figure 6.**

*Control charts for HPLC method used for the insulin quantification in pharmaceutical preparations: (upper) x-bar control chart and (lower) MR-chart. (UCL = upper control limit; LCL = lower control limit; CL = mean or average range for the MR-chart).*

The ARL was 370 since the probability that any point exceeds the control limits is 0.0027.

The MR-control charts exhibit one point above the upper control limits (UCL = 2.214) as well as other forms of nonrandom variation, around the sample #40, and therefore the method should be considered out of control (**Figure 6**).

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control… DOI: http://dx.doi.org/10.5772/intechopen.91354*

In such situation, it is necessary to search the cause and take corrective action. In the first case, the cause was assignable with a column change, whereas the second one was due to the presence of "more eight consecutive point plots on one side of the centerline."

In this case, the obtained ARL value for MR-control was 189 since the probability that any value exceeds the upper control limit was 0.0053.

#### **3.2 Cusum and EWMA control chart**

The data were analyzed using the Cusum and EWMA control charts.

The EWMA control chart shows that all plotted values fall within the control limits (**Figure 7**) using a smoothing constant of 0.2 (λ = 0.2) and control limit width fixed at three standard deviations (L = 3.054).

Cusum control chart, with a shift detection fixed at 1 SD, shows four points beyond boundaries, all of them greater than the upper control limit. The first alteration is located around the samples #84–85, whereas the second one appears close to the end of the process (#130–131). In addition, if both alterations presented an upward tendency, it indicates the process average changes, which requires a search to determine the causes. When we fixed the shift detection at 1.5 SD, the situation is totally different, all points fall within control limits (data not shown), and the data describe a random way with an average of zero, since the points show no evidence of an upward or downward tendency (**Figure 8**).

**Figure 7.** *EWMA control chart for HPLC method. All points fall within control limit for λ = 0.2 and L = 3.054.*

The ARL was 370 since the probability that any point exceeds the control limits

*Control charts for HPLC method used for the insulin quantification in pharmaceutical preparations: (upper) x-bar control chart and (lower) MR-chart. (UCL = upper control limit; LCL = lower control limit; CL = mean*

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

The MR-control charts exhibit one point above the upper control limits (UCL =

2.214) as well as other forms of nonrandom variation, around the sample #40, and therefore the method should be considered out of control (**Figure 6**).

is 0.0027.

**222**

*or average range for the MR-chart).*

**Figure 6.**

### **3.3 Process capability and specification limits**

Since the analytical method is in control and stable, capability indices can be computed. **Table 2** shows the estimated Cpk and Cpm indices to analyze the capability of our analytical method. The index Cpm, sometimes called the Taguchi index, adequately reveals the ability of the method to cluster around the target. This reflects the degree of method targeting (centering). For this, Cpm incorporates the variation in the method with respect to the target value and the specification limits preset by the analyst/customer [26]. This index conveys critical information regarding whether a method (or process) is capable of reproducing items satisfying a requirement that would be preset by the analyst [30]. If the prescribed minimum capability fails to be met, the method is considered incapable.

the mean and target concentration (<sup>μ</sup> T)<sup>2</sup> was 0.052 U/mL, whereas the precision calculated from the MR-chart was 0.36 U/mL. The overall uncertainty, calculated as the sum of the uncertainty of each component's contribution (precision and accuracy), was 0.41 U/mL. The expanded uncertainty was 0.82 U/mL, using a coverage

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control…*

If the specification limits are set at 5% of the T value, according to Bouabidi et al. [8] criteria, the Cpm index was 2.60 (n = 100) with a lower confidence bound C on Cpm (i.e., the value used to measure method capability) of 2.48 (**Figure 9**). We therefore conclude that the true value of the method capability Cpm is no less than 2.48 with a 95% level of confidence. This result indicates that the method is "super"

If the specification limits are reduced to 3% of T value, the Cpm was 1.56 with a lower 95% confidence limit of 1.47. This result implies that the method is considered "satisfactory" (1.33 < Cpm < 1.50). The method is inadequate for specification limits lower than 2% of the T value, since the lower 95% confidence limit for the Cpm is less than 1. A similar result was obtained when the specification limits and the control limits were of equal width. Thus, the number of observations out of specification in the method was zero when the specification limits are greater than 3% of the T value, and the proportion of nonconforming results was less than 1, giving a process yield of 100%. When the reference limits and the specification

*The black-dashed line shows the specified limits (USL and LSL) established at 5% of T value, whereas the*

*blue-dashed line corresponds to 3%. The red line is the T value.*

factor of 2. The calculated concentration is thus 100 0.82 U/mL.

*DOI: http://dx.doi.org/10.5772/intechopen.91354*

(Cpm > 2.0), and no further stringent precision control is required.

**Figure 9.**

**225**

To calculate the Cpm, the method mean and variability must be estimated relative to the method target and specification limits [27]. In our case, the target value T corresponds to standard solution concentration (T = 100 U/mL). The analysis of the measurements during this period shows the accuracy; the average error between

#### **Figure 8.**

*Cusum control chart for HPLC method. The number of samples beyond limits was four for shift detection fixed at 1 SD with an upward tendency. The cause of this alteration is unknown.*


#### **Table 2.**

*Cpk and Cpm values as a function of the specification limit (USL-LSL). The method capability is based on the critical values (Co).*

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control… DOI: http://dx.doi.org/10.5772/intechopen.91354*

the mean and target concentration (<sup>μ</sup> T)<sup>2</sup> was 0.052 U/mL, whereas the precision calculated from the MR-chart was 0.36 U/mL. The overall uncertainty, calculated as the sum of the uncertainty of each component's contribution (precision and accuracy), was 0.41 U/mL. The expanded uncertainty was 0.82 U/mL, using a coverage factor of 2. The calculated concentration is thus 100 0.82 U/mL.

If the specification limits are set at 5% of the T value, according to Bouabidi et al. [8] criteria, the Cpm index was 2.60 (n = 100) with a lower confidence bound C on Cpm (i.e., the value used to measure method capability) of 2.48 (**Figure 9**). We therefore conclude that the true value of the method capability Cpm is no less than 2.48 with a 95% level of confidence. This result indicates that the method is "super" (Cpm > 2.0), and no further stringent precision control is required.

If the specification limits are reduced to 3% of T value, the Cpm was 1.56 with a lower 95% confidence limit of 1.47. This result implies that the method is considered "satisfactory" (1.33 < Cpm < 1.50). The method is inadequate for specification limits lower than 2% of the T value, since the lower 95% confidence limit for the Cpm is less than 1. A similar result was obtained when the specification limits and the control limits were of equal width. Thus, the number of observations out of specification in the method was zero when the specification limits are greater than 3% of the T value, and the proportion of nonconforming results was less than 1, giving a process yield of 100%. When the reference limits and the specification

**Figure 9.**

**3.3 Process capability and specification limits**

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

**Figure 8.**

**Table 2.**

**224**

*critical values (Co).*

capability fails to be met, the method is considered incapable.

Since the analytical method is in control and stable, capability indices can be computed. **Table 2** shows the estimated Cpk and Cpm indices to analyze the capability of our analytical method. The index Cpm, sometimes called the Taguchi index, adequately reveals the ability of the method to cluster around the target. This reflects the degree of method targeting (centering). For this, Cpm incorporates the variation in the method with respect to the target value and the specification limits preset by the analyst/customer [26]. This index conveys critical information regarding whether a method (or process) is capable of reproducing items satisfying a requirement that would be preset by the analyst [30]. If the prescribed minimum

To calculate the Cpm, the method mean and variability must be estimated relative to the method target and specification limits [27]. In our case, the target value T corresponds to standard solution concentration (T = 100 U/mL). The analysis of the measurements during this period shows the accuracy; the average error between

*Cusum control chart for HPLC method. The number of samples beyond limits was four for shift detection fixed*

**USL-LSL Cpm Co Cpk Co Method capability** 3SD 0.95 1.08 0.79 1.095 Inadequate 2.5% 1.30 1.08 1.26 1.095 Capable 3% 1.56 1.44 1.54 1.45 Satisfactory 5% 2.60 2.16 2.65 2.18 Super

*Cpk and Cpm values as a function of the specification limit (USL-LSL). The method capability is based on the*

*at 1 SD with an upward tendency. The cause of this alteration is unknown.*

*The black-dashed line shows the specified limits (USL and LSL) established at 5% of T value, whereas the blue-dashed line corresponds to 3%. The red line is the T value.*

limits are of equal width, only 0.27% of the expected observations will be out of specification in the long term, and the process yield is 100%.

**Conflict of interest**

**Author details**

**227**

Alexis Oliva\* and Matías Llabrés

Department of Chemical Engineering and Pharmaceutical Technology, School of

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Pharmacy, University of La Laguna, La Laguna, Tenerife, Spain

\*Address all correspondence to: amoliva@ull.es

provided the original work is properly cited.

The authors declare no conflict of interest.

*DOI: http://dx.doi.org/10.5772/intechopen.91354*

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control…*

The results obtained showed that we cannot use the control limits as specification limits, since the method is considered "inadequate" according to the criteria proposed by Pearn and Shu [24]. However, the control limits can be used in the development of the specification limits. In this example, a level of variability 2.5% of T can be enough to declare the method capable (1.0 < Cpm < 1.33), as seen in **Table 2**.

All these aspects were also analyzed using the Cpk index. The results are summarized in **Table 2**. To determine if the method meets the capability requirement, we must calculate the critical value Co for Cpk based on α risk, sample size, and C value (i.e., the required method capability) [25]. We find the critical value Co = 1.081, based on C = 1.0, α = 0.05, and sample size n = 200. When the specification limits are set at 2.5% of T value, the estimated Cpk value is 1.30 and exceeds the critical value of 1.081, demonstrating that the method meets the capability requirements. Furthermore, if the limits are increased to 5% of the T value, the Cpk increases to 2.65 with a critical value of 2.18 (based on C = 2.00; α = 0.05; n = 200), indicating that the capability is "super." The obtained quality requirements were similar in both cases.

The use of Cpk is clearly preferable when the limits are not equidistant, whereas the Cpm index can be overly conservative in this scenario. In our case, the target is at the center of the specification range, and if the aim of our method is to achieve a measure close to the target value with minimum variation, then Cpm is the most sensitive capability index. Given two analytical methods with different performances (i.e., precision and accuracy) and the same method departure, a simple comparison between both Cpm is sufficient to select the better, although similar results were obtained with the Cpk index. This fact was analyzed by Oliva and Llabrés [29].

### **4. Conclusions**

The traditional x-chart and moving range chart represent the first option to monitor the analytical method or process over a long time. The EWAMA and Cusum are two good alternatives in those situations where it is important to detect small process shifts. The capability indices are calculated to evaluate whether the process or analytical method under study is able to provide sufficient conforming results when it is operating under statistical control. To calculate the PCIs, it is necessary to know the actual performance of the process or analytical method (using the control chart) and the customer requirements in terms of specification limits. The specification limits should be determined externally from previous knowledge of inherent process variability. However, different criteria have been proposed to fix these limits. The Cpm and Cpk indices were used as part of the control strategy. Cpk is the best option because it is not dependent on the process or method being centered. However, Cpm is more sensitive to departure from the method target than Cpk is. Independent of the criteria used to establish the specification limits, computation of the capability indices depends on the analyzed response, and their application is limited to each particular situation and is not general.

#### **Acknowledgements**

The authors wish to thank Dr. José B. Fariña from University of La Laguna, Tenerife, Spain, for providing the compaction process data. This research was financed by Instituto de Salud Carlos III, Ministerio de Ciencia, and Innovación y Universidades y FEDER as part of Project PI18/01380.

*Combining Capability Indices and Control Charts in the Process and Analytical Method Control… DOI: http://dx.doi.org/10.5772/intechopen.91354*
