Quality control charts for short or long runs without a training phase. Part 1. Performances in state of control and in the presence of outliers

Abstract

Sometimes analytical laboratories receive requests with a small number of determinations and/or a small number of samples, or outside the typical scope of analytical services. As a result, they may not have historical data on the performance of analytical processes and/or appropriate reference materials. Under these conditions it is difficult or uneconomical to use traditional or classic quality control charts. This is the so-called start-up problem of these charts. The Q charts seem appropriate charts under these conditions because they do not need any prior training or study phase. The fundamentals and the algebraic expressions of Q charts for the mean (four cases) and for the variance (two cases) are offered. This experimental study of Q charts for individual measurements was done with data from quality control for the evaluation of mass fraction of Ni and Al₂O₃ in a laterite CRM by ICP-OES. The performance of these Q charts is discussed where the analytical process is in the state of statistical control and in the presence of outliers at the start-up. In the first situation performance of Q charts are quite satisfactory and they behave properly. When outliers are collected at the beginning, the deformation of some charts is evident or the charts become useless. Severe outliers will corrupt the parameter estimates and the subsequent plotted points, or the charts will become insensitive and useless. The practitioner should take extreme care to assure that the initial values are obtained in the state of statistical control to have adequate sensitivity to detect parameter shifts.

Introduction

In a laboratory, quality control activities are mandatory according to the international standard ISO/IEC 17025 [1]. Therefore, laboratories should have procedures for ensuring validity of results they issue. Data from monitoring activities shall be analyzed, used to control and, where appropriate, improve the activities of the laboratory. If the results of the analysis of data from surveillance activities are found to be outside predefined criteria, appropriate action shall be taken to prevent incorrect results from being reported. There are clear guidelines and techniques for routine labors, such as control charts, replicate tests using the same or different methods, retesting of retained items, use of reference materials or quality control materials, and some others. Among these techniques, control charts have received a much attention in analytical laboratories [2,3,4].

Perhaps, due to cost, time and other practical reasons, from the variety of available types of control charts, charts for the individual measurements and moving range of two seem the most used in analytical chemistry. However, classical or standard control charts (as mentioned above) require a first phase or stage of training or study and a second phase of control (or process monitoring), properly. The first phase is mandatory if the analytical process has not been previously validated or verified with the use of reference values. Thus, in this first phase, it is necessary to evaluate the central tendency (by means of bias evaluation) and the dispersion of data (this latter usually expressed as standard deviation). The bias evaluation requires a reference value of a certified reference material (CRM) or a surrogate [3]. If this information has been previously obtained, the laboratory can directly apply the control chart in the second phase.

However, sometimes laboratories receive requests that are outside the scope of their routine activities. Requests such as the use of an analytical procedure out of its predefined scope or in a completely different concentration range or matrix are typical. In such a case, the laboratory needs to do it before or train the control chart to develop the first phase. Usually, this solution is not economic.

A practical solution in such situations could be the use of so called Q charts by Quesenberry [5,6,7]. These charts do not need a first phase of study or training. Only one variant or case of these charts has been mentioned in the analytical chemistry literature [8, 9], although their capabilities have not yet been widely tested in analytical practice [9]. They are promising, especially for the short run situation. Conventional charts, as classical Shewhart charts, do not work well in a short run situation due to:

• (1) Requests with a small number of determinations and/or a small number of samples, without historical data for verification and/or validation of the required analytical process. Thus, the process variability is unknown, and/or;

• (2) A control material, such as a CRM or appropriate surrogate is unavailable. Thus, it is not possible to evaluate the measure of location (or central tendency) of the process.

However, Quesenberry’s Q charts are promising in these respects. The main objective of this series of publications is to test the Quesenberry’s Q charts for individual measurements in the analytical practice, to demonstrate their capabilities and to discuss their applicability in different circumstances encountered in laboratory tasks. This study is intended to provide the analytical community with these important tools, appropriate when it is necessary to apply non-validated analytical processes and/or CRM are not available to the laboratory. One of the most important characteristics of Q charts is they do not require a previous training or study step, prior to the quality control step. This important feature is very useful in the above circumstances, especially in ad hoc tasks. Q charts are the most-well known and popular self-starting scheme to monitor a process in short runs [10]. These charts can avoid the start-up problems when information on central tendency and/or dispersion is not available [5].

The main objectives of this first part of the series of publications on Q charts for individual measurements is to test their performances for: (1) a case where the analytical process is in state of statistical control and (2) where there are outliers at the beginning of the charts. Their performance is compared in the same conditions with the Shewhart charts for individual measurements and the moving range of two.

Fundamentals of Q charts

Quesenberry [5] defined Q statistics that can be computed in a number of situations, assuming that the normal process distribution is stable, i.e., that the process mean, μ, and standard deviation, σ, are constant. These statistics all have distributions that are either exactly or approximately standard normal distribution. Moreover, they are either exactly or approximately independent (not autocorrelated). These statistics can be handled as Shewhart charts with action limits ± 3 and a center line equal to zero. Therefore, we can know exactly the type of point pattern that represents a stable process on charts of these statistics. Consequently, if either the process mean or standard deviation is not constant during data collection, this will generally result in anomalous patterns that are different from those of a stable process.

The Q statistics for the process mean of individual measurements to obtain normal process mean are given for four cases of μ and σ known and unknown are defined as follows:

Case KK: μ and σ known ($\mu ={\mu }_{0}$ and $\sigma ={\sigma }_{0}$)

$$Q_{r} \left( {X_{r} } \right) = \frac{{X_{r} - \mu_{0} }}{{\sigma_{0} }},\;r = 1, 2, \ldots$$

(1)

This case is trivial. If both, the process mean and standard deviation are known, classical Shewhart control charts should be used, directly in the control phase. Therefore, this Q chart is interesting from the conceptual point of view, but its practical application is not so important. It could be used when a CRM is available (and used as a control material) and the analytical process has been previously validated or verified (thus, the precision in the given measurement condition is known). It is used in this report as a basis for testing the performance of the remaining cases.

Case UK: μ unknown and σ known ($\sigma ={\sigma }_{0}$)

$$Q_{r} \left( {X_{r} } \right) = \left( {\frac{r - 1}{r}} \right)^{\frac{1}{2}} \frac{{\left( {X_{r} - \overline{X}_{r - 1} } \right)}}{{\sigma_{0} }},\,r = 2, 3, \ldots$$

(2)

In this case,

$$\overline{X}_{r - 1} = \frac{1}{r - 1}\mathop \sum \limits_{j = 1}^{r - 1} X_{j}$$

(3)

This case is appropriate when a CRM (or a suitable surrogate) is not available as a control material, and the analytical process has been previously validated or verified.

Case KU: μ known and σ unknown ($\mu ={\mu }_{0}$)

$$Q_{r} \left( {X_{r} } \right) = \Phi^{ - 1} \left\{ {G_{r - 1} \left( {\frac{{X_{r} - \mu_{0} }}{{s_{r - 1} }}} \right)} \right\},\;r = 2, 3, 4, \ldots$$

(4)

In the above expression ${G}_{r-1}\left(\bullet \right)$ is the Student t distribution function with $r-1$ degrees of freedom and ${\Phi }^{-1}\left(\bullet \right)$ is the inverse of the standard normal distribution function. Also,

$$s_{r - 1} = \sqrt {\frac{1}{r - 1}\mathop \sum \limits_{j = 1}^{r - 1} \left( {X_{j} - \overline{X}_{r - 1} } \right)^{2} }$$

(5)

The above expression is a modification given in [6], with respect to the original report on Q charts [5]. This should be used instead of using ${s}_{o,r-1}$ as the sequential standard deviation with the expression

$$s_{0,r - 1} = \sqrt {\frac{1}{r - 1}\mathop \sum \limits_{j = 1}^{r - 1} \left( {X_{j} - \mu_{0} } \right)^{2} }$$

(6)

The use of the expression (5) was suggested by del Castillo and Montgomery [11] and accepted by Quesenberry [12], although it has advantages and disadvantages. This case KU is appropriate when a CRM (or an suitable surrogate) is available as control material, and the process has not been previously validated or verified (i.e., σ is unknown in the given measurement condition).

Case UU: μ unknown and σ unknown

$$Q_{r} \left( {X_{r} } \right) = \Phi^{ - 1} \left\{ {G_{r - 2} \left[ {\left( {\frac{r - 1}{r}} \right)^{\frac{1}{2}} \left( {\frac{{X_{r} - \overline{X}_{r - 1} }}{{s_{r - 1} }}} \right)} \right]} \right\},\;r = 3, 4, 5, \ldots$$

(7)

In this expression, the symbols have the same meaning as before. This case is applied when a CRM (or a suitable surrogate) is unavailable as a control material, and the process has not been previously validated or verified (then, σ is unknown).

In addition, the Q statistics for the process variance of individual measurements are given by the following expressions:

Given a range as

$$R_{r} = X_{r} - X_{r - 1}$$

(8)

Case K: μ unknown and σ known

$$Q\left( {R_{r} } \right) = \Phi^{ - 1} \left\{ {H_{1} \left( {\frac{{R_{r}^{2} }}{{2\sigma_{0}^{2} }}} \right)} \right\},\;r = 2, 4, 6, \ldots$$

(9)

In this expression ${H}_{1}\left(\bullet \right)$ is chi-square distribution function with one degree of freedom, and the rest of symbols has the same meaning as before. This case is applied when the analytical process is validated or verified (then σ is known), but not data concerning the mean of the process are needed.

Case U: μ unknown and σ unknown

$$Q\left( {R_{r} } \right) = \Phi^{ - 1} \left\{ {F_{1,\nu } \left( {\frac{{\nu R_{r}^{2} }}{{R_{2}^{2} + R_{4}^{2} + \ldots + R_{r - 2}^{2} }}} \right)} \right\},\;r = 4, 6, 8, \ldots$$

(10)

where ${F}_{1,\nu }\left(\bullet \right)$ is the Snedecor-F distribution function with $\left(1, \nu \right)$ degrees of freedom, and $\nu =r/2-1$. This Q chart is useful when the analytical process has not been previously validated or verified, and a CRM (or a surrogate) is used to prepare the chart.

Expressions (9) and (10) apply to the even-numbered indices of the sequence of points. The $Q\left({R}_{r}\right)$ values can be calculated for all measured values (odd indices included). The sequences of $Q\left({R}_{r}\right)$ will be normally distributed in the state of statistical control, but they are not independent. For this reason, it is recommended to calculate these values only for measurements with even, and to plot them in the Q charts.

Nowadays, with the availability of PCs in most laboratories, the above Q statistics are easily calculated with a common spreadsheet or another standard application intended to simple calculations. Therefore, calculations are not a problem at present. Nevertheless, for a quick and approximate estimation of Q values, this report is accompanied with a pdf file with graphical aids for this purpose. It is a supplemental electronic material containing four plots of isopleths intended for visual estimation of Q values to an accuracy sufficient for manual plotting of Q control charts. The cases included are:

• Fig. 1 Isopleths for Q charts for the mean of individual observations (unknown process mean and variance, case UU).

• Fig. 2 Isopleths for Q charts for the variance of individual observations (unknown process mean and variance, case U).

• Fig. 3 Isopleths for the Q charts for the mean of individual observations (known process mean and unknown process variance, case KU)

• Fig. 4 Isopleths for the Q charts for the variance of individual observations (unknown process mean and known process variance, case K).

The other two cases (cases KK and UK for the mean) are trivial and the use of a hand calculator is sufficient. The graphs are complemented with explanations of their construction and examples. This supplementary electronic material can be printed to have it at the bench level.

Due to the characteristics of these statistics, the derived control charts are symmetrical Shewhart charts, but it is not necessary to develop a training phase to obtain the initial results. Thus, the start-up problem is avoided. However, some practical issues should be taken into account. The best way to handle the charts is to draw the control limits defined by ± 3, ± 2 and ± 1. These limits define 6 zones. The upper half of the chart is referred to as A (outer third), B (middle third) and C (inner third). The lower half is the mirror image. The tests for special causes, as applied to Shewhart control charts [13], are intended to detect an extraneous influence that causes an out-of-control signal.

Materials and methods

Certified reference material, analytical procedure and measurement conditions

Data from the determination of mass fractions of Ni and Al₂O₃ by ICP-OES, in the laterite L1 CRM were used. This CRM was developed by COMECON, Project L1, with the participation of more than 30 laboratories from Cuba, Soviet Union, GDR, Czechoslovakia, Bulgaria, Hungary, Romania, France and some other countries [14]. An update of the CRM was issued in 2015. The validity period was extended to May, 2030 [14]. The certified values for Ni and Al₂O₃ are (11.48 ± 0.087) mg/g and (37.0 ± 0.47) mg/g, respectively, where the numbers following the symbol ± are the uncertainty of the reference value and not a confidence interval.

Mass fractions were evaluated using a laboratory developed standard operating procedure for eight elements [15]. The dry test sample is automatically melted with lithium metaborate in Pt-Au crucibles and dissolved in dilute HCl. Test solutions were properly diluted if necessary. The solution is introduced with a peristaltic pump in a Horiba, model Activa-M, sequential optical atomic emission spectrometer with inductively coupled plasma. The used spectral lines (in nm) were Ni I 352.454 and Al I 308.215, with a power of 1 200 W. Calibrators were prepared from commercial standard solutions (Merck KGaA). Online instrumental quality control was performed with reference solutions to monitor bias, precision and signal drift of instrumental measurements.

Data for quality control were obtained in the [time + operator]-different intermediate precision condition of measurement [16, 17], over several weeks. Operator is understood here a given combination of bench analyst and instrument operator.

Preliminary statistical analysis of raw data

The raw data of the selected elements were statistically analyzed using Microsoft Excel and Statgraphics Centurion XV, version 15.2.05 [18]. In this research, Q values were calculated with Microsoft Excel and introduced in Statgraphics Centurion XV. The calculated statistical tests and control charts prepared with Excel were validated with the statistical application. For each element, the first 25 values were used for data validation, in order to determine if the values were obtained under the state of statistical control. Outlier plots and box and whisker plots were applied as initial analysis. Dixon’s and Grubb’s tests were used to search using box and whisker plots. Normal probability plots, Shapiro–Wilk test for normality, Kolmogorov–Smirnov and Anderson–Darling goodness of fit tests were also applied. Autocorrelation was tested using intervals for autocorrelations coefficients and partial autocorrelations between values of the variable at different time lags. Randomness was examined using points above and below the median test, the runs up and down test and the Box-Pierce test to verify whether the autocorrelation is equal to 0 based on the first 8 correlations. All statistical tests were performed at a significance level of 0.05. If outliers were found or some value resulted in an alarm signal in the charts for individual measurements or the moving average of two, they were discarded from the data sets and data validation was performed with new values to complete the set of 25 values.

The data were collected retrospectively from the laboratory records as a first step in implementing a quality control system. They did not necessarily defined laboratory results.

Application of control charts

Shewhart control charts

Four types of Shewhart control charts were used in this study: the control chart for individual measurements, the moving range of two chart and the Q charts for process mean and process variance. In the control charts for individual measurements, the center line was defined by the certified values. The standard deviation of the validated data was used to evaluate the control limits. In the moving range of two control charts, the center line was defined by the mean of the moving ranges of the validated data. All the charts were run in the control phase. There was no need to train any chart. According to accepted guidelines for quality control charts, the control chart for the moving range of two and the control chart for individual measurements were applied and analyzed in this order [19].

In the charts for individual measurements and the moving range of two six control limits were set: in the upper part above the center line and descendent order upper action limit, outer upper warning limit and inner upper warning limit. Below the center line to lower values, inner lower warning limit, outer lower warning limit and lower action limit. Defined zones: A) between upper action limit and outer upper warning limit; B) between outer upper warning limit and inner upper warning limit; and C) between inner upper warning limit and center line. They are labeled as A, B and C, respectively. In descending order, zones are defined as a mirror: C) between center line and inner lower warning limit; B) between inner lower warning limit and outer lower warning limit; and A) between outer lower warning limit and lower action limit, labeled C, B and A, respectively (See Fig. 1). However, in the moving ranges of two control charts, the last two of the above control limits are zero, and they do not appear on the chart (See Fig. 2). These zone definitions follow recognized recommendations [13].

Fig. 1

Control chart for individual measurements for the first 25 results for the determination of the mass fraction of Ni in the laterite L1 CRM by ICP-OES. Solid line represents the center line and dashed lines represent the six control limits: in the upper part above the center line the upper action limit, the outer upper warning limit and the inner upper warning limit. Below the center line are the inner lower warning limit, the outer lower warning limit and the lower action limit. Zones between outer action limits and outer warning limits, outer warning limits and inner warning limits, and inner warning limits and center line are labeled as A, B and C, respectively. The chart has no alarm signal

Fig. 2

Control chart of the moving range of two for the first 25 results for the determination of the mass fraction of Ni in the laterite L1 CRM. The solid line represents the center line and the dashed lines represent the control limits: in the upper part above the center line are the upper action limit, the outer upper warning limit and the inner upper warning limit. Below the center line the inner lower warning limit. The outer lower warning limit and the lower action limit are zero. Zones between the outer action limits and the outer warning limits, the outer warning limits and the inner warning limits, and the inner warning limits and the center line are labeled as A, B and C, respectively. The same applies below center line, but only zone C is labeled because the other lower limits are zero. A thick dashed line represents the median of the data. The chart has no alarm signal

The Q charts have a center line at zero and the respective limits are define as ± 3, ± 2 and ± 1. They have the same bilateral zones (A, B and C). This organization of the Shewhart charts is used because it is easier to apply the tests for special causes [13]. The tests for special causes (or decision rules) used along this work were:

• (1) One point beyond zone A.

• (2) Eight points in a row in zone C or beyond.

• (3) Two out of three points in a row in zone A or beyond.

• (4) Four out of five points in a row in zone B or beyond.

Different sources suggest slightly different sets of tests [13, 18,19,20], but several tests are common. In our particular experience, listed above are the more important in analytical chemistry. These patterns represent the most common behaviors that may be encountered in practice. Although some other patterns may arise in the analytical practice, they are rare. Nevertheless, a careful inspection of the charts was applied to identify some others non-random patterns as cyclic patterns, alternative runs down and up, and some others seemingly non-random patterns [19]. For example, during this study six points in a row steadily increasing or decreasing was found one time. The probability of this behavior when the process is under control is very low. Therefore, it was interpreted as a loss of statistical control.

In general, care should be exercised when using multiple decision rules simultaneously, as they are not statistically independent. If too many tests or decision rules are applied, the frequency of false alarms increases and the average run length (ARL) of the chart is unnecessarily reduced [19]. In addition, the false alarm rate also increases and charts become useless and difficult to use.

Q charts for the process mean (cases KK, UK, KU and UU) and process variance (cases K and U) were prepared taking into account or ignoring the certified value of the CRM and/or the standard deviation (or range) from data validation, depending on the given algorithm (Eqs. (1, 2, 4, 7, 9, 10)).

Exponentially weighted moving average charts (EWMA charts)

The use of the EWMA charts is a complement to Shewhart charts. This type of chart is well established in the scientific literature. Therefore, its fundamentals will not be explained here because they are given elsewhere [21, 22]. It has been shown, for example, that EWMA charts are much more effective than Shewhart charts in detecting small and moderate shifts in the parameters of the probably distribution of a quality characteristic [19]. The EWMA charts were used here as a supplement to the charts for the individual measurements and the Q charts.

EWMA Q charts are calculated from the sequence of associated Q statistics. For the Q charts the EWMA statistic ${Z}_{t}$ is given by

$$Z_{t} = \lambda Q_{t} + \left( {1 - \lambda } \right)Z_{t - 1} , \ldots t = 1, 2, \ldots$$

(11)

with ${Z}_{0}=0$, defining the center line. ${Z}_{t}$ may be ${Z}_{tm}$ in the case of the Q chart for the process mean or ${Z}_{tv}$ for the Q chart for the process variance. Also, λ is the smoothing constant ($0<\lambda \le 1$) and t is the sample number. These values are plotted on a chart with action limits at

$$U_{AL} , L_{AL} = \pm K\sqrt {\frac{\lambda }{{\left( {2 - \lambda } \right)}}\left[ {1 - \left( {1 - \lambda } \right)^{2t} } \right]}$$

(12)

where ${U}_{AL}$ is the upper action limit and ${L}_{AL}$ is the lower action limit, respectively. K is the selected process’ shift by the user to be detected in terms of σ. Observe these limits are not constant at the beginning of the chart, where they are closer to the center line. In this work, the guidelines of Crowder [22] were applied to detect a shift of 1.5 standard deviations in a normal mean. Note that this shift is in the mean of the ${Q}_{t}$’s and not in the process mean. The interpolated values were $\lambda =0.25$ and $K=2.90$, which give action limits at ± 1.096 for the EWMA charts associated to the Q charts. The same values were used for all the Q charts in this series of publications. The test for special causes applied is: if ${Z}_{t}>{U}_{AL}$, an increase in μ or σ is signaled and if ${Z}_{t}<{L}_{AL}$ a decrease is signaled. These values of $\lambda$ and $K$ give an in-control ARL of 372.6 and an ARL of 5.18 to detect a shift of 1.5 standard deviations. This proposal is coincident with the recommendations of the Quesenberry’s seminal article [7].

It is possible to use the CUSUM chart instead of the EWMA chart. However, the overall performance of EWMA and CUSUM charts was comparable in one of the first reports of Quesenberry on Q charts, and his personal preference was the EWMA chart [6]. The authors of this report also prefer the EWMA chart because it seems more easy to handle and more practical. Thus, only the EWMA chart was used in this series.

The preliminary statistical analysis was applied to validate the raw data, as expressed above. Then, the classical Shewhart charts used for quality control (the chart for individual measurements and the moving average of two) are used as a basis for studying the performance of Q charts. In this study, the objective is to study the performance of Q charts, not to apply them for quality control. Therefore, in the presence of alarms, signaled values (points) were usually not rejected and subsequently replaced. This was done in order to study the Q charts in different circumstances and patterns of the represented points. This reasoning is applied to all the control charts discussed in this report.

It is necessary to point out that the charts for the cases where parameters are known and when they are unknown are not competing procedures. Each is intended to achieve the same goal, but under different circumstances (CRM available or unavailable (or appropriate surrogate) and/or validated (or verified) or not analytical process). Nevertheless, in this report sometimes different charts with the same data appear in the same figure. This is a way to compare the performance, advantages and limitations of different cases.

Results and discussion

Analytical process in the state of statistical control

To study the performance of Q charts in the state of statistical control, 25 values from quality control for the determination of mass fraction of Ni in the laterite L1 CRM were used. From the preliminary statistical analysis of these raw data, it was concluded the values were obtained in the state of statistical control. No outliers were detected on the data set. The Student’s t-test failed to reject the hypothesis of absence of systematic error. Normal probably plots, the Shapiro–Wilk test and the goodness of fit tests failed to reject the hypothesis of normality. Autocorrelation tests also failed to reject its absence, and randomness tests showed the values were randomly sampled. So, the set of values was validated.

Figure 1 shows the chart for individual measurements for Ni. The chart has no alarm that can be assigned to a special cause. Additionally, Fig. 2 displays the moving range of two chart for the same data. It also shows no alarm signal. Both facts confirm the preliminary validation of data. Therefore, the analytical processes were developed in the state of statistical control in the intermediate precision measurement condition along several weeks.

The EWMA chart associated to the chart for individuals for Ni (not shown here) does not show any alarm signal, i.e., neither point is above the upper action limit or below the lower action limit. Nevertheless, most of the points in the EWMA chart are below the center line. This is attributed to the fact that the chart for individual measurements shows 17 points below the center line and 8 above.

The different cases of the Q chart for the mean, (cases KK, μ known and σ known; UK, μ unknown and σ known; KU, μ known and σ unknown; and UU, μ unknown and σ unknown) are shown in Fig. 3. As stated before, the case KK is trivial because its expression contains the process mean and the process standard deviation, which should be known. If the relevant information is available, classical Shewhart control charts should be used directly in the control phase. Nevertheless, in this study we use this chart as a reference, in order to analyze the performance of the other cases. The case KK of the Q chart for the mean has an identical pattern to the chart for individuals. The associate EWMA Q chart also has an identical pattern to the EWMA chart associated to the chart for individual measurements.

Fig. 3

Q control charts for the mean (the four cases: KK, UK, KU and UU) for the 25 results for the determination of the mass fraction of Ni. No alarms are detected in the chart. The central line, control limits and zones as detailed in Fig. 1

In Fig. 3, it can be observed that patterns of the other three cases of the Q control charts for the mean are similar to the pattern of the case KK. In the other cases, no alarms are signaled. In addition, their patterns are very similar to the pattern of the chart for the individual measurements. The observed small differences in the patterns of the four charts could be attributed to differences in the joint effect of the consecutive values of the sequential mean and standard deviation (Eqs. (3, 5)) on the expressions of the different cases (Eqs. (2, 4, 7)). The responses of the four charts for the mean with data in state of statistical control is satisfactory.

The four cases of the respective EWMA charts associated to the Q charts for the mean show a very similar pattern (See Fig. 4). There are no alarms on the charts.

Fig. 4

EWMA Q control charts for the mean (the four cases: KK, UK, KU and UU) for the 25 results for the determination of the mass fraction of Ni. The center line is zero and upper and lower action limits are shown by dashed lines. In the chart, there are no points beyond the upper and lower action limits, no alarms are detected

EWMA charts for cases KK and KU most of the points are below the center line. By contrast, for the other two cases, UK and UU, most of the points are above the center line. This difference between patterns depends on the number of points above and below the center line in the Q chart for the mean. On the one hand, the cases KK and KU of the Q chart for the mean have 8 points above the center line and 17 and 16 points below, respectively. On the other hand, the cases UK and UU have 13 points above the center line and 11 and 10 below, respectively. As explained before, the shift in these patterns could be attributed to the different joint effect of the consecutive random values of the sequential mean and standard deviation in the Eqs. (2, 4, 7). Nevertheless, the absence of alarm signals in the EWMA charts corroborates the state of statistical control of the data.

The Q charts for the variance (both cases: K, μ unknown and σ known; U, μ unknown and σ unknown) are shown in Fig. 5. As can be easily seen, no alarms signals were raised. Apparently, there were no noticeable modifications in the dispersion. The associated EWMA Q charts also show processes in the state of statistical control from the point of view of the dispersion, without no alarms signaled (Fig. 6).

Fig. 5

Q control charts for the process variance (two cases: K, μ unknown and σ known; U, μ unknown and σ unknown) for the 25 results for the determination of the mass fraction of Ni. The central line, control limits and zones as detailed in Fig. 1. No alarms are detected in the chart

Fig. 6

EWMA Q control charts for the process variance (two cases: K and U, mean unknown and standard deviation known and unknown) for the 25 results for the determination of the mass fraction of Ni. The central line and action limits as in Fig. 4. No alarms are detected in the chart

All the cases of the Q charts for the mean and the variance has shown a good performance when data are in state of statistical control. Their patterns are in agreement with the charts for individual measurements (Fig. 1) and the moving ranges of two (Fig. 2).

Influence of outliers at the beginning of the Q charts

The influence of outliers at the beginning of the Q charts was studied with 25 values from the quality control for the estimation of mass fraction of Al₂O₃ in the laterite L1 CRM. In the preliminary statistical analysis of the raw data, the outlier plot and the box and whisker plot clearly showed two potential outliers on the left at the beginning of the series. A normal probably plot showed important deviations of both points from the fitted straight line. The Shapiro–Wilk test of normality, and the Kolmogorov–Smirnov and the Anderson–Darling goodness-of-fit tests resulted in p-values less than 0.05. These facts are consistent with the results of Dixon’s test and the Grubb’s test. These tests identified two outliers on the left. Therefore, the two outliers were discarded from the set of values, substituted by the two following ones to obtain a new set of data, and the same statistical analysis was repeated. This procedure was used to compare performance in the presence and in the absence of outliers. With the new set of 25 values, no outliers were found with the Dixon’s test and the Grubb’s test and the hypothesis of normality of population was not rejected with the normality and goodness-of-fit tests. In addition, the autocorrelation tests failed to reject its absence, and the results of randomness tests did not reject the hypothesis on random sampling. Thus, no new outliers were found. However, one of the new set of values generated an alarm signal in the chart for the moving range of two (sample number 24). This value was again discarded and substituted by a new one. The statistical analysis described above was performed again, with the same results. This third set of values did not originate alarm signals in the charts for the individual measurements and the moving average of two. It was concluded the last set of 25 values were obtained in the state of statistical control, so this last set was validated. Therefore, the mean and the standard deviation of the analytical process from the set of values was considered good estimates of these parameters in the time-operator different intermediate precision condition of measurement. Subsequently, the systematic error was studied using the reference value of the CRM and bias became non-significant according to the Student’s t-test.

Figure 7 shows the control charts for individual measurements for Al₂O₃. The center line and control limits were calculated using the reference value and the last preliminary statistical analysis of data. The two outliers and the third discarded value were not taken into account when plotting the control limits, although they were included in the charts (labeled as OV: outlier values included, diamonds in Fig. 7). The two outliers at samples 2 and 3 are clearly visible. However, the chart stabilizes quickly around the center line after about the fifth sample. The remaining points are not alarms. If the two outliers and the third discarded value are ignored, the new chart (labeled in Fig. 7 as VV: validated values, circles) clearly shows a pattern corresponding to the state of statistical control.

Fig. 7

Control charts for individual measurements for the results for the determination of the mass fraction of Al₂O₃ in the laterite L1 CRM by ICP-OES. The center line, control limits and zones as described in Fig. 1. OV: series with outlier values. VV: series of validated values. Crosses represent alarms. For OV, the second and third points are below the lower action limit. They are outliers that should be discarded from the rest of results. However, to study the characteristics of the raw data in order to test the performance of Q charts in the presence of outliers at the start-up, these outliers are represented in this particular chart. The validated series of values (VV) shows no alarms

The EWMA chart associated to the control chart for individual measurements (not shown here) shows the second, the third and the fourth points below the lower action limit. The position of the fourth point below the lower action limit is explained due to the weight given to older data in the EWMA chart. Therefore, this is further evidence of the presence of the two outliers. However, at the beginning of the Q charts, the analyst sometimes does not know these points are outliers, and may not have information about the central tendency and/or the dispersion of the process. Therefore, he/she frequently continues the control in the search of new results to plot on the charts. The EWMA chart associated to the chart for individual measurements has no other point beyond the action limits. The EWMA chart without the inclusion of outliers has no point beyond the action limits.

In the control chart of the moving range of two (Fig. 8), with the outliers values (labeled OV, diamonds) there is only one alarm at sample number 4. After sample 5 it seems the moving range stabilizes. It gives the impression the precision becomes under control after this sample number. Validated values without the inclusion of outliers (Fig. 8, circles) show no alarm.

Fig. 8

Control charts of the moving range of two for the determination of the mass fraction of Al₂O₃ in the laterite L1 CRM. The center line, control limits and zones as describe in Fig. 2. Crosses represent alarm signals. OV: series with outlier values. VV: series of validated values. For OV, the point of the fourth sample number is above the upper action limit

As expressed above, Q chart for the mean, case KK, is trivial. Nevertheless, when applied to the full set of Al₂O₃ data with outlier values, (see Fig. 9, diamonds) it shows a pattern that is virtually identical to the chart for individual measurements shown in Fig. 7. Only the second and the third samples are labeled as alarms. Immediately, beyond the fifth sample the chart stabilizes around the center line. In addition, the EWMA chart associated to this Q chart, case KK, (Fig. 10, diamonds) shows points labeled as alarms (second, third and fourth points). The occurrence of the fourth point is explained by the weight given to older data in the EWMA charts. The pattern of this chart is identical to the pattern of the EWMA chart associated to the chart for individual measurements (this latter not presented here). Using the Q chart, case KK, and its associated EWMA chart is very easy to identify the outliers.

Fig. 9

Q control charts for the mean (cases KK, UK, KU and UU) for the 25 results for the determination of the mass fraction of Al₂O₃ including the outliers (non-validated data). The center line and control limits as in Fig. 1. Alarms are signaled by crosses. According to the preliminary statistical analysis, the second and the third points are outliers. It is possible to see the influence of the outliers (or their absence) in the general patterns of different cases

Fig. 10

EWMA Q control charts for the mean (cases KK, UK, KU and UU) for the determination of the mass fraction of Al₂O₃.in the presence of two outliers at the beginning. The center line and actions limits as in Fig. 4. Alarms are indicated by a cross. According to the preliminary statistical analysis, the second and the third points are outliers. It is possible to see the influence of both outliers in the general patterns of different cases

In addition, Fig. 9 shows the other cases of the Q control chart for the mean (cases UK, KU and UU) for the 25 results in the determination of the mass fraction of Al₂O₃. In case UK (Fig. 9, triangles), there are no alarms at sample numbers 2 and 3, but there are alarms at samples sample numbers 4, 5 and 9, because these last points are beyond the upper action limit. In addition, the points at sample numbers 6, 7 and 10 also show alarm signals (two out of three points in a row in zone A or beyond). It is evident the deformation caused by the outliers at the beginning of the chart. If this chart is used in a short run (say, up to 10th sample), it becomes obvious that something is wrong. The analyst should analyze the process and identify the cause. However, according to data validation, points at sample numbers 4 to 7, 9 and 10 should not be discarded. In addition, when Grubbs and Dixon tests for outliers are applied to the first 10 values, no outliers are identified for a significance level of 0.05. The points more or less stabilize around the center line at sample number 11, but the chart is deeply affected by the outliers for the initial 10 samples. In this particular short run situation the chart was not sensitive enough to diagnose the presence of outliers. In conclusion, the Q chart for the mean, case UK, was not able to identify the outliers, although its deformation makes one to think something is wrong with the data. The EWMA Q chart associated to the Q chart for the mean, case UK, (see Fig. 10, triangles) shows alarms at samples numbers 6, 7, 9, 10 and 11. This chart is also strongly affected by the outliers in the short run situation.

The cases KU and UU for the Q chart for the process mean are completely different (Fig. 9, circles and squares, respectively). They are insensitive to the initial outliers. They show no alarm signal and have a very similar pattern. They are clear examples of the importance of obtaining the initial values in state of statistical control with the Q charts. Many analytical processes are not stable or in control at the start. The data from out-control period (e.g. due to insufficient warm-up period of instruments, initial use of personal without sufficient training in the analytical process, lack of analytical process standardization, etc.) should be avoided at the beginning of the Q charts. Data from the initial out-of-control period should not be used in the calculation of estimates of process parameters estimates for plotting later points. These above ideas for cases KU and UU are supported by the corresponding EWMA charts (see Fig. 10, circles and squares, respectively). No alarms appeared on the charts. They are insensitive to detect outliers at the beginning.

Figure 11 shows the Q control charts for the variance (cases K, diamonds and U, squares). Case K clearly shows an alarm at the fourth sample number, due to the effect of the two outliers. The associate EWMA Q chart (not shown here) also has a point above the upper action limit at sample number 4, but the rest of points are between the two action limits. However, the Q chart for the variance, case U, is insensitive to the presence of outliers. It has no alarms, but the first point is above the center line, and the following 6 points are below. The inclusion of the two outliers introduces biases in the estimation of the successive sequential standard deviations. Similarly, the EWMA chart associated to the case U does not show any alarm signal. This is a consequence of the lack of essential information compared to case K. However, most of the points are below the center line.

Fig. 11

Q control charts for the variance (cases K and U) for the determination of the mass fraction of Al₂O₃ with inclusion of outliers. The center line and control limits as in Fig. 1. For case K, an alarm is signaled by a cross at the fourth sample number

As a general conclusion, the sensitivity of Q charts for the mean to detect the presence of outliers at the beginning, decreases in the series of cases, KK, UK, KU and UU. Except for the case KK, outlier points at the beginning had a deep impact on the overall performance of the Q control charts. In the case UK, the detection of this fact can be delayed after several samples. This fact limits its use in short runs. In the other two cases, the charts became useless for short and long runs. Although the EWMA control charts associated to the Q charts are generally very useful, they were not a solution of these problems.

After discarding the outliers and the additional value, the Q charts for the process mean (all cases) show patterns coherent with the state of statistical control (see Fig. 12). No alarms were observed. Recall that the final series after data validation showed good patterns in the chart for the individual measurements (Fig. 7, labeled VV, circles), and in the chart for moving range of two (Fig. 8, with the same label, circles). The corresponding EWMA charts (not shown here) have all points between the upper and lower action limits.

Fig. 12

Q control charts for the mean (cases KK, UK, KU and UU) for the determination of the mass fraction of Al₂O₃, using validated data (without the initial outliers). The center line and control limits as in Fig. 1. No alarms were signaled. Compare with Fig. 9

Similarly, the Q charts for the process variance (cases K and U) are in agreement with the control of the dispersion. Figure 13 shows no alarms. The associated EWMA charts (not presented here) have a good pattern with no points outside the upper or lower action limits.

Fig. 13

Q control charts for the variance (cases K and U) for the determination of the mass fraction of Al₂O₃, after rejecting the two outliers and the third discarded value. The center line and control limits as in Fig. 1. No alarms were reported. Compare with Fig. 11

The above results are consistent with the following concepts. If a special cause results in out-of-control points, especially severe outliers, these points will vitiate the parameter estimates and therefore the plotted points, unless they are removed from the data sequence and not used in subsequent calculations. If outliers are automatically removed and not used in subsequent computations of Q statistics, then the sensitivities to detecting parameter shifts will be improved [7]. Thus, the above results provide experimental confirmation for these ideas.

Conclusions

The Q charts for the process mean and the process variance showed an appropriate performance in the state of statistical control. The four cases of Q charts for the mean were consistent with the results of the chart for individual measurements and reproduced its general pattern. The Q charts for the variance also demonstrated a satisfactory pattern without any alarm signal.

Except of the case KK, the presence of outliers at the beginning deeply affects the rest of cases for the Q chart for the mean. The case UK of the Q chart for the mean becomes completely distorted in the presence these outliers. Cases KU and UU become useless. The practitioner should take extreme care to assure that the first values are obtained in the state of statistical control. The so-called “warm-up period” should be avoided. The importance of appropriate values at the beginning was pointed out in the Quesenberry’s seminal article [7]. This fact has been experimentally confirmed in this research.

The performance of Q charts in the presence of a persistent systematic error, simultaneous small drifts of mean and variance, a gradual increase of the mean and in the presence of a slight autocorrelation of the raw data will be discussed in subsequent part of this series.