Control Charts for Joint Monitoring of the Lognormal Mean and Standard Deviation

: The Shewhart X - and S -charts are most commonly used for monitoring the process mean and variability based on the assumption of normality. However, many process distributions may follow a positively skewed distribution, such as the lognormal distribution. In this study, we discuss the construction of three combined X - and S -charts for jointly monitoring the lognormal mean and the standard deviation. The simulation results show that the combined lognormal X - and S -charts are more effective when the lognormal distribution is more skewed. A real example is used to demonstrate how the combined lognormal X - and S -charts can be applied in practice.


Introduction
Control charts are widely used in statistical process control (SPC) for monitoring and detecting out-of-control processes. The research on constructing control charts for monitoring normal processes has been extensively studied. Most control charts are designed to monitor either the process mean or the process variability, but it is usually desirable to simultaneously monitor the process mean and the process variability because both may change at the same time. A change in the standard deviation usually leads to out-of-control signals on the mean chart. When the distribution of quality characteristics is normal, the Shewhart X-chart [1] is one of the most commonly used control charting techniques for monitoring the process mean, while the Shewhart S-chart is commonly used to monitor the process variability. However, in many manufacturing applications, the quality variable typically follows a positively skewed distribution, such as the lognormal distribution. For example, the percent viscosity increase (PVI) of an engine oil after it has been put to an accelerated aging test for a specific period of time is a critical quality dimension of engine oil in the automotive industry. Engineering experience indicates that the PVI follows a lognormal distribution. In this case, it is very important to simultaneously monitor the mean and the standard deviation of the PVI based on a lognormal distribution.
In general, the implementation of a control chart is done in two stages, also known as Phase I control and Phase II monitoring. In Phase I control, in order to evaluate the variation of the process over time, assess the process stability, and estimate the in-control process parameters, one collects and analyzes certain amounts of historical data. In Phase II monitoring, one collects data sequentially and monitors the process in real time to quickly detect changes in the process parameters.
In the literature, there have been several studies on constructing control charts for monitoring the lognormal mean or the lognormal standard deviation. In monitoring the lognormal mean, a modified control chart using the sample ratio was proposed by Morrison [2]. A control chart for monitoring the "geometric midrange" of a lognormal distribution was developed by Ferrell [3]. A control chart for sequentially testing the arithmetic mean of a lognormal distribution was constructed by Joffe and Sichel [4]. A simple heuristic method for constructing the Xand Rcharts using the weighted variance (WV) method with no assumption on the form of the distribution was proposed by Bai and Choi [5]. Castagliola [6] proposed a newX control chart devoted to the monitoring of skewed populations. Huang et al. [7] discussed the control charts for the lognormal mean based on the confidence intervals of the lognormal mean. In monitoring the standard deviation, Abu-Shawiesh [8] presented a simple approach for robustly estimating the process standard deviation based on the median absolute deviation. Adekeye and Azubuike [9] derived the limits for control charts using the median absolute deviation for monitoring non-normal processes. Adekeye [10] proposed modified control limits based on the median absolute deviation. Huang et al. [11] proposed a control chart for monitoring the standard deviation of a lognormal process based on an approximate confidence interval of the lognormal standard deviation. Karagöz [12] proposed an asymmetric control limit for a range chart under a non-normal distributed process. Liao and Pearm [13] presented a modified weighted standard deviation index for the capability of a lognormal process. Shaheen et al. [14] presented a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. Omar et al. [15] proposed an efficient approach for monitoring a positively skewed process. The control charts for jointly monitoring the mean and the standard deviation of a lognormal distribution are not as well established as those for a normal distribution. McCracken and Chakraborti [16] gave an overview of control charts for joint monitoring of the mean and variance. Yang [17] proposed a single-average loss control chart to monitor a process's mean and variability. Chen and Lu [18] proposed a new sum-of-squares exponentially weighted moving average (SSEWMA) chart using auxiliary information-called the AIB-SSEWMA chart-for jointly monitoring the process mean and variability .
In this study, we discuss three combined Xand S-charts for jointly monitoring the mean and the standard deviation of a lognormal process: (1) The first combined charts are the conventional combined Shewhart Xand S-charts. (2) The second combined charts are constructed based on the median absolute deviation method. (3) The third combined charts are the combined lognormal Xand S-charts based on the methodologies studied in Huang et al. [7] and Huang et al. [11], respectively. The performances of these combined control charts are evaluated and compared in terms of the average run length (ARL), where the run length is defined as the number of samples taken before the first out-of-control signal alerts on a control chart [19].
The rest of this paper is organized as follows. The aforementioned combined Xand S-charts for jointly monitoring the lognormal mean and standard deviation are discussed in Section 2. Section 3 is devoted to assessing the performance of the combined Xand S-charts. A real example from the automotive industry is given in Section 4 to demonstrate how the aforementioned combined Xand S-charts can be used in practice. Concluding remarks are given in Section 5.

The Combined Shewhart X-and S-Charts
The Shewhart Xand S-charts are based on the assumption that the distribution of the quality characteristic is normal. The upper control limit (UCL) and lower control limit (LCL) of the combined Shewhart Xand S-charts are given by respectively, where L x and L s are multipliers chosen to satisfy a specific in-control chart performance and c 4 = (2/(n − 1)) 1/2 [Γ(n/2)/Γ((n − 1)/2)] [1,19]. If the parameters θ and ξ are unknown, they can be estimated byθ andξ using data obtained from Phase I control data. Let X i and S X (i) be the sample mean and sample standard deviation of the ith sample, i = 1, 2, . . . , m, that is, X i = (∑ n j=1 X ij )/n and S X (i) = ∑ n j=1 (X ij − X i ) 2 /(n − 1), respectively. The sample grand mean is X = ∑ m i=1 X i /m, and the average of the m standard deviations isS X = ∑ m i=1 S X (i)/m. Then, the parameters θ and ξ are estimated byθ SW = X andξ SW =S X /c 4 , respectively. Therefore, the control limits of the combined Shewhart Xand S-charts are estimated by respectively, where L SW x and L SW s are multipliers that depend on n and the desired incontrol average run length (ARL 0 ). When Phase II joint monitoring begins, independent samples, each of size n, are repeatedly taken from the process. Assume that the true in-control parameter θ 0 and ξ 0 are equal toθ SW andξ SW , respectively, for each sample, X 1 , X 2 , . . . , X n ; one calculates X = (∑ n j=1 X j )/n and S X = ∑ n j=1 (X j − X) 2 /(n − 1) and plots X and S X against the sampling sequence, respectively. An out-of-control signal is detected when X is below LCL SW x or above UCL SW x , or when S X is below LCL SW s or above UCL SW s .

The Combined Median Absolute Deviation X-and S-Charts
The median absolute deviation (MAD), which measures the deviation of the data from the sample median, was first studied by Hampel [20]. It is a more robust scale estimator than the sample standard deviation and is often used as an initial value for computing more efficient and robust estimators. The MAD for a random sample, X 1 , X 2 , . . . , X n , is defined as follows: where b is a constant used to make the estimator consistent for the parameter of interest and MD = Median{X i } is the sample median of X 1 , X 2 , . . ., X n . If the sample observations are normally distributed, the constant b is equal to 1.4826, and the statistic b n MAD is an unbiased estimator of the standard deviation (Rousseeuw and Croux ) [21], where b n is a function of the sample size n; the values of b n were derived and tabulated in Abu-Shawiesh [8].
Based on the conventional Shewhart principle, when the parameters θ and ξ are unknown, they can be estimated byθ andξ using data obtained from Phase I control data. Let MAD = ∑ m i=1 MAD i /m be the average median absolute deviation, where MAD i is the median absolute deviation of the ith sample, i = 1, 2, . . . , m. The parameter θ and ξ can be estimated byθ MAD = X andξ MAD = b n MAD, respectively. Hence, the control limits of the combined Xand S-charts based on MAD are estimated by respectively, where L MAD x and L MAD s are multipliers that depend on n and the desired ARL 0 . When Phase II joint monitoring begins, independent samples, each of size n, are repeatedly taken from the process. For each sample, X 1 , X 2 , . . . , X n , one calculates X = (∑ n j=1 X j )/n and b n MAD and plots X and b n MAD against the sampling sequence, respectively. An out-of-control signal is detected when X is below LCL MAD x or above UCL MAD x , or when b n MAD is below LCL MAD s or above UCL MAD s .

The Combined Lognormal X-and S-Charts
Based on the conventional Shewhart X-chart, if the parameters θ and ξ are unknown, they can be replaced by the estimatorsθ andξ obtained from Phase I in-control data.
According to Huang et al. [7], let Y i = ∑ n j=1 Y ij /n and S Y (i) = ∑ n j=1 (Y ij − Y i ) 2 /(n − 1) be the sample mean and the sample standard deviation of the ith sample, i = 1, 2, . . . , m, respectively. The grand sample mean of Y is Y = ∑ m i=1 Y i /m, and the average of the m standard deviations isS Y = ∑ m i=1 S Y (i)/m. The parameters θ and ξ can be estimated bŷ , respectively. Therefore, the control limits of the lognormal X-chart are estimated using where L Log x is set to satisfy a desired ARL 0 . According to Huang, et al. [11], there are two cases for the standard deviation. For the first case of σ < 1, the control limits of the lognormal S-chart are estimated using where L Log s is a multiplier that depends on n and the desired ARL 0 . For the second case of σ > 1, the control limits of the lognormal S-chart are estimated using When Phase II joint monitoring begins, independent samples, each of size n, are repeatedly taken from the process. For each sample, Y 1 , Y 2 , . . . , Y n , in the case of σ < 1, one calculates and plots e Y+S 2 Y /2 and . An out-of-control signal is detected if the plotting statistic e Y+S 2 Y /2 is below LCL Log x or above UCL Log x , or when the plotting statistic Y + S 2 Y /2 + log(S Y ) falls below LCL Log s or above UCL Log s . For the case of σ > 1, one computes and plots e Y+S 2 Y /2 and Y + S 2 Y /2 against the sampling sequence. An out-of-control signal is detected if the plotting statistic e Y+S 2 Y /2 is below LCL Log x or above UCL Log x , or when the plotting statistic Y + S 2 Y /2 falls below LCL Log s or above UCL Log s .

Remark 1.
The population standard deviation σ is usually unknown and needs to be estimated in practice. It can be estimated by utilizing data collected from Phase I control, when the process was in control. Based on the estimate, one can then decide whether to use Case I or Case II to construct the lognormal S-chart.

Chart Performance Evaluations and Comparisons
In this section, we conduct a simulation study to compare the performance of the combined lognormal Xand S-charts with the combined MAD Xand S-charts and the conventional Shewhart Xand S-charts in terms of the ARL.

Simulated Settings
Since E(X) = e µ+σ 2 /2 , we set µ = −σ 2 /2 such that the mean of the lognormal distribution E(X) = 1 remains unchanged. Let ξ 0 = e σ 2 0 − 1 be the value of the in-control parameter. As discussed earlier, the control limits need to be determined using Phase I observations and will depend on σ 0 , the subgroup size n, and the number of Phase I samples m. Therefore, we approximate the control limits of the three combined Xand S-charts using simulations for various values of σ 0 and combinations of m = 50 and 100, as well as n = 5 and 10. The value of σ 0 is set to be between 0.2 and 2.0, with an increment of 0.2. The multipliers of the three combined control charts are calibrated to have an overall ARL 0 that is approximately equal to 370. Note that the considered ARL for the three combined control charts is conditional on the estimated UCL and LCL. Here, we describe how the simulation is conducted for the combined lognormal Xand S-charts, as it is similar for the other two combined charts. Given σ 0 , m, and n, the following steps are carried out: Step 1: Choose a value of L Log x and a value of L Log s , and generate m independent samples of n observations each from a lognormal distribution with a mean of 1 and a standard deviation of ξ 0 . Compute the UCL Log x and LCL Log x , and calculate the UCL Log s and LCL Log s using either Equation (1) if σ 0 < 1 or Equation (2) if σ 0 ≥ 1.
Step 2: Repeatedly generate samples of n observations each from a lognormal distribution with a mean of 1 and standard deviation of ξ 0 . For each sample, calculate the two plotting statistics for the X-chart and S-chart, respectively. Then, evaluate whether the plotting statistic for the X-chart exceeds UCL Log x or goes below LCL Log x ; next, evaluate whether the plotting statistic for the S-chart exceeds UCL Log s or goes below LCL Log s . Stop when it does, and denote the number of samples generated by RL i .
The multipliers of the three combined Xand S-charts are given in Tables 1 and 2 for different values of ξ 0 . Note that the multipliers of these three combined Xand S-charts increase when ξ 0 (σ 0 ) increases.
Denote the out-of-control process parameters by θ 1 = θ 0 + aξ 0 and ξ 1 = bξ 0 , where a > 0, b > 1. Given ξ 0 = e σ 2 0 − 1, we simulated the ARL 1 for the three combined charts in the same way as in Steps 2 and 3, which were mentioned earlier. The out-of-control ARLs (ARL 1 s) of the three combined Xand S-charts are summarized in Tables 3 and 4 (m = 50) and Tables 5 and 6 (m = 100).

Discussion of Results
According to the assessment of the numerical results summarized in Tables 3-6, the combined lognormal Xand S-charts perform better than the other two combined Xand S-charts when σ 0 > 0.6. Nevertheless, the ARL 1 s of the combined lognormal Xand Scharts are larger than those of the other two combined Xand S-charts when σ 0 < 0.6. Note that smaller values of ξ 0 correspond to smaller values of σ 0 , under which the lognormal distribution is more symmetric. Therefore, when σ 0 is small, which means that the data are more symmetric, the combined lognormal Xand S-charts are less effective than the combined Shewhart Xand S-charts and the combined MAD Xand S-charts. On the other hand, as σ 0 becomes larger, the lognormal distribution becomes more skewed, thus making the combined lognormal Xand S-charts more effective in detecting changes in θ and ξ than the other two combined control charts. Table 3. The ARL 1 s of the three combined Xand S-charts for different shift sizes a, b when the sample m = 50 with the subgroup size n = 5 under various values of in-control σ 0 (θ 1 = θ 0 + aξ 0 , ξ 1 = bξ 0 ).   Table 4. The ARL 1 s of the three combined Xand S-charts for different shift sizes a, b when the sample m = 50 with the subgroup size n = 10 under various values of in-control σ 0 (θ 1 = θ 0 + aξ 0 , ξ 1 = bξ 0 ).  Table 5. The ARL 1 s of the three combined Xand S-charts for different shift sizes a, b when the sample m = 100 with the subgroup size n = 5 under various values of in-control σ 0 (θ 1 = θ 0 + aξ 0 , ξ 1 = bξ 0 ).   Table 6. The ARL 1 s of the three combined Xand S-charts for different shift sizes a, b when the sample m = 100 with the subgroup size n = 10 under various values of in-control σ 0 (θ 1 = θ 0 + aξ 0 , ξ 1 = bξ 0 ).

Example from the Automotive Industry
In this section, we present a real example to illustrate the applicability of the combined lognormal Xand S-charts. The ASTM D7320 Ref Oil Data were provided by the Test Monitoring Center [22]. In order to assess the engine oil quality, especially for new vehicles, the percent viscosity increase (PVI), which follows a lognormal distribution, needs to be tested. In this dataset, the quality of three reference oils-Ref Oils 434, 435, and 438-needs to be tested. We collected 50 samples, each of size 10, for each of these three reference oils, and used them to construct the Phase I control charts.
The multipliers of the three combined Xand S-charts were calibrated to have a Type I error approximately equal to 0.0027 in order to have a fair comparison. Summarized in Table 7 are the upper and lower control limits of the three combined Xand S-charts, the estimated means, and the estimated standard deviations of the PVI for the three reference oils. Figures A1-A3 show the three combined Xand S-charts for Ref Oils 434, 435, and 438, respectively. Table 7. The control limits, estimated means, and estimated standard deviations of percent viscosity increase (PVI) for the three reference oils in the Phase I control.

Reference
(θ 0 ,ξ 0 ) Method Neither the combined Shewhart Xand S-charts nor the combined MAD Xand S-charts showed any out-of-control samples for Ref Oil 434 ( Figure A1). On the other hand, sample 9 was outside the control limits of the lognormal S-chart. Consequently, the control limits were recalculated without sample 9 for the Phase II joint monitoring of the combined lognormal X-  Figure A3), none of the three combined X-charts and S-charts showed any out-of-control samples.

X-Chart S-Chart
The sample mean and the sample standard deviation of the PVI calculated based on the Phase I samples were used as the "true mean" and the "true standard deviation", respectively, in the simulation process because the true population mean and population standard deviation were unknown. The mean and the standard deviation of the PVI were respectively assumed to be 154. 747 20 new samples with a subgroup size of 10 were generated when θ changed from θ 0 to θ 1 = θ 0 + ξ 0 and ξ changed from ξ 0 to ξ 1 = 2.5ξ 0 . Note that the 20 new samples were computer simulated, not obtained from additional samples using the ASTM test method. All combined Xand S-charts were tuned to produce an overall ARL 0 that was approximately equal to 370. The resulting control charts for Ref Oils 434, 435, and 438 are shown in Figures A4-A6, respectively.
For Ref Oil 434 ( Figure A4), the combined lognormal Xand S-charts detected out-ofcontrol signals at samples 5 and 4 in the Xand S-charts, respectively, while the combined Shewhart Xand S-charts triggered at samples 8 and 11 in the Xand S-charts, respectively.
In addition, out-of-control signals were detected at sample 8 in both the MAD Xand S-charts. For Ref Oil 435 ( Figure A5), out-of-control signals were detected on sample 6 in both the Shewhart Xand S-charts and both the MAD Xand S-charts. At the same time, out-of-control signals also appeared at sample 6 in both the lognormal Xand S-charts. As for Ref Oil 438 ( Figure A6), the combined Shewhart Xand S-charts detected out-of-control signals at samples 6 and 2 in the Xand S-charts, respectively, while the combined MAD Xand S-charts detected at sample 6 in both charts. In addition, out-of-control signals were detected at sample 6 in both the lognormal Xand S-charts.

Conclusions
In this study, we discuss the construction of three combined Xand S-charts for jointly monitoring the mean and the standard deviation of the lognormal distribution. The simulation studies show that the combined lognormal Xand S-charts have good performance when the underlying lognormal distribution is more skewed. The practical application of the combined lognormal Xand S-charts is also demonstrated in a real example.
The numerical results of the current work indicate that for skewed non-normal processes, it is possible to construct more effective control charts for monitoring the process mean, process variability, or both based on the actual process distribution. This is at least the case for lognormal processes. It would be worth it to investigate how to construct more effective control charts for other skewed non-normal processes.

Remark 2.
As for other skewed non-normal processes, the process mean or process variability may need to be approximated first, and these can be estimated using Phase I in-control data. Based on the conventional Shewhart X-and S-charts, one can construct the control limits of X-and S-charts for monitoring the process mean or process variability under other skewed non-normal processes.