Phase-I design structure of Bayesian variance chart

Abstract: This article develops a new design structure for S2-Chart, namely Bayesian variance chart, in Phase-I analysis assuming the normality of the quality characteristic to incorporate the parameter uncertainty. Our approach consists of two stages: (i) construction of the control limits for S2-Chart and (ii) performance evaluation of the proposed control limits. The comparison of the proposed design structure with the frequentist design structure of S2-Chart is examined in terms of (i) width of control region and (ii) OC curves when the process variance goes out of control. It is observed that the proposed Phase-I S2-Chart is more efficient than the frequentist S2-Chart in discriminatory power of detecting a shift in the process dispersion. When the process variance is in-control (after implementation of Bayesian variance chart), then the control limits for X̄-Chart using in-control standard deviation are also given here for monitoring unknown mean under unknown standard deviation case.


Introduction
There is a large literature on the process variability control charts. To develop a variability control chart, a basic assumption is that the underlying distribution of the quality characteristics should be normal. Here, we assume that the lot-to-lot quality (process standard) observed after a fixed time interval remains constant throughout. The constant environmental stress on the operating conditions of the process over a long period leads to an unduly restrictive and unrealistic assumption about the constant standards of the process. The situation becomes alarming when one is going for quality control of the process of the same nature accomplishing the same task in varying conditions. Obviously, for overcoming the situation, it seems logical to assume variations in process standard represented by known suitable prior distribution. More so, the process control (PC) is a continuous quality valuation process and, as such, in all PC techniques, a strong prior information representing variations in quality is available as discussed by Sharma, Singh, and Geol (2007).

PUBLIC INTEREST STATEMENT
The products are produced every day in industry to fulfill the requirement of the common people. The most important issue of any produced products is its quality that is the major concern of any buyer. The quality control is an important field in industrial engineering. This work helps quality engineers to meet the challenges of buyers in the market. This work gives a very useful method to control the quality of outgoing item after meeting the specification. | | 2 ) n 0 +n (m 0 −x) 2 and the values of v 1 , m 1 , and n c are defined in Appendix A.
The lower control limit, central line, and upper control limit are the three parameters of a Shewharttype control chart. Assuming the measurable quality characteristic, X, to be normally distributed with unknown mean μ and unknown variance σ 2 , the control limits for the usual S 2 -Chart for retrospective analysis are defined as: where α is the level of significance or false alarm rate in quality terminology, S 2 x is the average of variance of k-subgroups each of size n, are the quantile points of the chi-square distribution as defined in Montgomery (2004, p. 249).
Now in a situation where variations in process variance σ 2 are assumed to be represented by a prior distribution given in (5) and using the sample information variation, the process variance can be updated in the form of posterior distribution as given in (7). The usual three-sigma control limits, for S 2 -Chart using the updated posterior distribution, are given by: The distribution of sufficient statistic S 2 x is not a symmetric even for moderate to large samples. Therefore, the true probability limits for S 2 -Chart using posterior distribution following Chhikara and Guttman (1982) are given by: is as given in (7) and α is the level of significance, which is 0.0027 in Shewhart-type control charts. We are updating the control limits of S 2 -Chart for monitoring the unknown process variance. So, the plotted statistic is the sample variance S 2 x because it is an unbiased estimator of σ 2 which is a variable of interest.

Control limits for X -Chart based on non-informative prior
In this subsection, we have proposed control limits for S 2 -Chart based on non-informative prior distribution. We have considered uniform prior and Jeffrey's prior as non-informative prior.

Control limits based on Jeffrey's prior
In this subsection, we have proposed control limits for S 2 -Chart based on Jeffrey's prior for unknown mean and variance of the normal distribution, so that we are then relying primarily on the likelihood involved for our inference following the work of Box (1980), Chhikara and Guttman (1982), and Gelman (2006).
Consider the sampling distribution of X, likelihood function of X, and sampling distribution of (X, S 2 ) as defined in Equations (1), (2), and (6). The Jeffrey's prior for unknown mean and variance as discussed in Banerjee and Bhattacharyya (1979) and Gelman (2006) is: where σ −2 is the precision of normal distribution. Then, following the methodology of Menzefricke (2002,2007,2010), the posterior distribution of f(μ, σ 2 ) given (x, s 2 x )is defined as: x is inverse-gamma distributed random variable with parameters = n−1 2 , = ( n−1 2 S 2 x ). The derivation of (12) is given in Appendix A. Now in a situation where variation in process variance σ 2 is assumed to be represented by a prior distribution given in (11) and using the sample information variation, the process variance can be updated in the form of posterior distribution as given in (12). The true probability control limits for S 2 -Chart using posterior distribution using Jeffrey's non-informative prior are given by: is as given in (12) and α is the level of significance, which is 0.0027 in Shewhart-type control charts.

Control limits based on uniform prior
In this subsection, we have proposed control limits for S 2 -Chart based on uniform prior distribution for unknown mean and variance of the normal distribution, so that we are then relying primarily on the likelihood involved for our inference following the work of Box (1980), Chhikara and Guttman (1982), and Gelman (2006).
The uniform prior for unknown mean and variance is p(μ, σ 2 )∞1. The posterior distribution of f(μ, σ 2 ) given (x, s 2 x )is defined as: The control limits for S 2 -Chart using Bayesian inference following Chhikara and Guttman (1982) based on uniform prior distribution are given by: is as given in (14) and α is the level of significance, which is 0.0027 in Shewhart-type control charts.
The limits defined in Equations (10), (13), and (15) are based on updated information and can be used to monitor the unknown normal process variance. The successive sample of a given size n are observed if a sample variance is outside the control limits defined in Equations (10), (13), and (15) chart signals. If a sample variance or more sample variances falls outside the control limits, the trial control limits will be revised until all the sample variances lie within the control region. When the process is in-control, the future sample variances are generated from the posterior predictive distribution as proposed by Menzefricke (2010); otherwise, the use of these will be misleading as discussed by many authors including Montgomery (2004) and Saghir (2015). This work is related to the Phase-I monitoring of S 2 -Chart while the work of Menzefricke (2010) is for Phase-II monitoring. The Phase-I monitoring is necessary before applying the Phase-II study, see Montgomery (Box, 1980).

Evaluation of control limits
For the evaluation of the control limits obtained by frequentist (sampling) and Bayesian methods, we use OC function under the hypothetical situation that the variance of the normal distribution does not remain at level σ 2 following Sharma et al. (2007), Saghir (2007), and Menzefricke (2010). At this stage, we use sampling theoretical considerations approach following Menzefricke, (2002Menzefricke, ( , 2007Menzefricke, ( , 2010 and Saghir (2007Saghir ( , 2015. Let the sampling distribution for the next sample variance be IG( , 2 2 0 ) where α 2 is an amount of shift in the process variance 2 0 .
The OC function in the corresponding situations provides a measure of the sensitivity of the control limits, i.e. their ability to detect a shift in the mean of the process quality characteristic following Menzefricke (2002Menzefricke ( , 2010 and Sharma et al. (2007). The relative distribution of 2| | | S 2 x to shifted variance 2 2 0 is derived in Appendix A. The OC function for S 2 -Chart based on the control limits defined in (10) and relative distribution is defined as: The OC function for S 2 -Chart based on the control limits defined in Equation (13) and relative distribution is defined as: The OC function for S 2 -Chart based on the control limits defined in Equation (15) and relative distribution is defined as: The corresponding OC function for usual control limits of S 2 -Chart is: where "Ga" denotes gamma distribution with parameters

Simulation study and comparison
In this section, we have made a comparison among the proposed (informative and non-informative) control limits and usual control limits of S 2 -Chart. First, we have made comparison based on control region obtained by the proposed and usual limits following Chhikara and Guttman (Sharma et al., 2007) and then the comparison based on OC S 2 x function following Menzefricke (2002Menzefricke ( , 2007Menzefricke ( , 2010, Sharma et al. (2007), and Saghir (2007Saghir ( , 2015 is provided.

Control region-based comparison
In this subsection, a comparison of the proposed limits with the frequentist control limits of S 2 -Chart for the initials samples is made. The comparison based on width of the control limits is made for Phase-I process monitoring. Using Monte Carlo simulation technique, 10,000 random samples are drawn from standard normal process of size nine and sample variance along standard error is calculated and given in Tables 1-3. Let us assume that (i) α = 0.0027 and (ii) x = m 0 (without loss of generality). We have considered three situations when (i) S 2 x = S 2 0 , (ii) S 2 x < S 2 0 , and (iii) S 2 x > S 2 0 . The critical region defined in Equation (10) changes with the posterior degree of freedom v 1 which is a measure of the amount of uncertainty regarding the unknown process variance, σ 2 (see Menzefricke, 2002Menzefricke, , 2010. The control limits v 1 for S 2 -Chart for different values of v 1 based on these random samples and updated posterior distribution are calculated and given here in Tables 1-3 for comparison purposes.   Tables 1-3 give the following considerable points: (1) The width of the control region for the proposed limits decreases as posterior degree of freedom v 1 increases (see last column of Tables 1-3).
(2) Comparison across the informative prior distribution-based limits reveals that the control limits are more contract in terms of minimum width of the control region when S 2 x = S 2 0 than other choices (see Tables1-3).
(3) The control limits obtained by non-informative prior distributions, which do not incorporate the parameter uncertainty (Menzefricke, 2002(Menzefricke, , 2010, are wider than frequentist control limits (see first and last two rows of Tables 1-3).
(4) As the posterior degree of freedom v 1 (which is a quantitative assessment of how much certain we are about the accuracy of the prior variance "S 2 0 ", prior estimate of σ 2 , lower the variance,  more accurate our believe) increases, the width of the control limits obtained based on informative prior distribution decreases.

OC curve comparison
In this subsection, we evaluate the proposed control limits of S 2 -Chart using OC curve as performance measures. Using the control limits, calculated in Section 4.1, OC function is calculated for different amounts of shifts (i.e. different values of α 2 ), α = 0.0027, n = 9, and OC curves are made and provided here in Figures 1-3. where "F" denotes OC function based on usual control limits, "JNI" OC function based on Jeffrey's non-informative prior distribution limits, "UNI" OC function based on uniform non-informative prior distribution limits, and "10, 15, 20, 40" based on informative prior distribution degrees of freedom limits.
From Figures 1-3, it is obvious that the proposed control limits based on informative prior distribution perform better than the frequentist control limits of S 2 -Chart in the sense that the discriminatory power of detecting a shift in the parameter of interest is high for proposed control limits than the existing control limits of S 2 -Chart. The higher power of detecting a shift results in low value of OC function; therefore, the curve of OC function for the proposed control limits is less than the frequentist control limits as it is clear from Figures 1-3. The performance of non-informative priors-based control limits of S 2 -Chart to detect a shift in the parameter is also better than the frequentist control limits as it is obvious from above figures but less than informative prior-based control limits' performance. Therefore, control limits of S 2 -Chart based on informative prior perform better to detect a shift in a parameter of the continuous process for larger value of posterior degree of freedom v 1 . The more the posterior degree of freedom v 1 , more the power of detecting a shift for informative prior distribution limits, as it can be seen from Figures 1-3. A similar behavior has been observed for other choices of sample sizes and in-control false alarm rate α.

Control limits for X -Chart when variance is unknown
In this section, we are constructing and evaluating the control limits of Bayesian X -Chart based on posterior distribution for Phase-I process monitoring following Saghir (2015).

Construction of the limits
The control limits for Bayesian X -Chart, based on the updated posterior distribution defined in Equation (7), when process mean as well as process variance is unknown, are: where Z (α/2) is a (α/2)th quantile point of the standard normal distribution. For the usual choice of incontrol probability of false alarm rate α = 0.0027, Z (α/2) = 3.00 and control limits defined in Equation (20) are know as 3σ-control limits. As the process standard deviation or variance is unknown, therefore, we replace unknown standard deviation by any unbiased estimator like ̂=̄R d 2 or ̂=̄S c 4 given in many textbooks including Montgomery (2004). We are updating the control limits of X -Chart for monitoring the unknown process mean. So, the plotted statistic is the sample average X because it is an unbiased estimator of unknown population mean μ, which is the variable of interest to be monitored.

Evaluation of the limits
For the evaluation of the proposed control limits of X -Chart, we have used OC function as performance measures under the hypothetical situation that the mean of the normal distribution does not remain at level μ following Saghir (2007Saghir ( , 2015. Let the sampling distribution for the next sample . The OC function in the corresponding situations provides a measure of the sensitivity of the control limits, i.e. their ability to not detect a shift in the mean of the process quality characteristic following Menzefricke (2002Menzefricke ( , 2007Menzefricke ( , 2010, Saghir (2007), and Saghir (2015). The OC function for X -Chart based on the control limits defined in B I N = Pr [not detecting a shift in the sample mean (using informative Prior) on the first sample following the shift] For the given values of b and n, the OC function decreases with prior sample size n 0 and approaches 0 as n 0 → ∞. A larger value of n 0 implies more precise knowledge about m 0 and produces thus a narrower control region.

Conclusions and recommendations
The proposed posterior distribution-based design structure of S 2 -Chart, which incorporates the parameter uncertainty by considering a suitable prior distribution of unknown parameter, is more efficient than the frequentist design structure, which ignores this uncertainty and assumes that σ 2 = S 2 , with reference to the width of the limits, lowest type-I error, and more power of detecting a shift in the parameter. Larger values of posterior degree of freedom v 1 provided more efficient control limits in terms of lowest width of control region as well as more discriminatory power of detecting a shift in the parameter when actually the shift occurs in the parameter. The control limits of S 2 -Chart based on informative prior are more efficient than non-informative prior-based control limits and usual control limits. The performance of the usual control limits is least among the compared control limits. These control limits must be calculated for Phase-I data and when the process variance is statistically in-control, the control limits proposed by Menzefricke (2010) should be used. The control limits of X -Chart are also constructed when mean and standard deviation of the normal process are unknown. The constructed control limits are evaluated and it has been observed that a larger value of n 0 implies more precise knowledge about m 0 . When the process mean is statistically in-control for Phase-I samples based on these Bayesian limits, the Phase-II monitoring of the sample mean using Menzefricke (2002) control limits could be used. (22)