Design of Attribute Control Chart Based on Regression Estimator

This paper presents a statistical analysis control chart for nonconforming units in quality control. In many situations the Shewhart control charts for nonconforming units may not be suitable or cannot be used. For many processes, the assumptions of binomial distribution may deviate or may provide inadequate model. In this Study we propose a new control chart, P r chart, which is based on regression estimator of proportion based on single auxiliary variable. The performance of the compared with P and Q charts with probability to signal as a performance measure. It has been observed that the proposed chart is superior to the P and Q charts. This study will help quality practitioners to choose an efficient alternative to the classical P and Q charts for monitoring nonconforming units in industrial process.

More details about the Shewart control chat can be seen in Niaki and Abbasi (2007), Mehmood et al. (2013) and Riaz et al. (2014).
According to Montgomery (2003), "in many quality control environments, the process or product under consideration has two or more correlated quality characteristics. For example, the quality of a chemical process may be a function of process temperature, pressure, and flow rates, all of which need to be monitored in a situation where some correlation may exist between any two of them. In these cases, if we want to monitor these quality characteristics separately, there will be some error associated with the out of control detection procedure". Riaz (2008) proposed the control charts using the regression estimator for variable sampling.
According to the best of the authors knowledge there is no work in the area of quality control using the proportion estimator for the attribute quality characteristics.
In this paper, we will propose attribute control chart using the proportion estimator given by Das (1982). We will develop the control limits by following Riaz (2008). We will propose a proportion control chart namely the Pr chart, based on Das (1982) based estimate of proportion.

Regression Estimator for Proportion:
Assuming bivariate normality of (x, y) a Shewhart-type process proportion of nonconforming control chart, namely chart (proposed which is based on the regression type estimator) of process proportion level. The regression estimator for proportion of Y using a proportion of single auxiliary variable X. I assume the numbers '1, 2 yx is the proportion of a bivariate random sample given as: Where the population proportion of Y, is the sample proportion of Y, is the population proportion of X and is sample proportion of X. Also, is the sample correlation between X and Y and and are corresponding sample standard deviations.

Proposed Control Charts and Construction of Control Limits
Assume that (P i , P j ) are bivariate normally distributed. Suppose the relationship between P i and Pr be defined by a random variable C as follows [Riaz (2008)].
The relationship defined in (3) helps in determining the parameters (i.e. centerline, lower and upper control limits) of the proposed Pr chart. Now, if the distributional behavior of C is known then the sample statistic Pr can easily be used for the testing of hypotheses about shifts in . When ( , ) follow bivariate normal distribution, the distributional behavior of C depends only on (the correlation between and ) and n. The distributional behavior of C, in terms of its proportion and quantile points, is required for the development of the Pr chart, and is explored in the following paragraphs when ( , ) follow a bivariate normal distribution.
By taking expectations of Eq. (3), we have Note that ( ) can be replaced by ̅ as discussed in Hillier (1969).
Then simplifying and rearranging equation (4), we get the following results: The regression estimator Pr is generally a biased estimator of the population proportion but the bias vanishes when the relationship between and is linear (see Sukhatme and Sukhatme, 1984, p. 238). So, for the case of bivariate normal distribution ( , ), Pr is unbiased for and hence E(C) =0. Thus, (5) results into the following: Replacing the estimate of in (6) we have the following results: For standard error, let the standard deviation of C be = It is not easy to get the analytical results for k because ( 2 ) is difficult to obtain analytically. So, simulation methods are often used to evaluate the expectation of a statistic, see Ross (1990).
By taking the variance on of C and by simplifying, we have the following results: represents the standard deviation of distribution of sample statistic Pr.
Using (8) in (9), rearranging and substituting the estimate for , we get the following: ̂=̂√ ⁄ An approximation for , when ( , ) follows a bivariate normal distribution, is given as (see Sukhatme and Sukhatme, 1984, p. 267): Consequently, Where ij  is the correlation coefficient between y and x .

Simulation Study
The quantile points of the distribution of C, let Ca represents the ath quantile point of the distribution of C (i.e. the point where C completes a% area). The analytical results for Ca are difficult to obtain; so, the simulation results are obtained for Ca. For a bivariate normal distribution of ( , ) the quantile points of the distribution of C depends entirely on and n. Using the bivariate normal distribution 10,000 simulated random samples, the results of Ca have been obtained.
Based on these results, the values of some commonly used quantile points, are provided for n=10, 20. . . 1000 in Appendix Tables A2-A11 at some representative values of . The similar results can easily be obtained for any combination of and n. These quantile points help determining the control limits and the power of the proposed Pr chart to detect shifts in process of proportion of the defectives.

Parameters of the Proposed Chart
Finally, the control limits for the proposed control chart is given as By using results The use of 3-sigma limits is based on the symmetric assumption of the plotted statistic, we will see that the distribution of Pr is not symmetric at least for small to moderate values of n. Hence there is a need to develop the probability limit structure for the proposed chart. Probability limits for the proposed chart can be computed by using quantile points of the distribution of C.
Let α be the specified probability of making Type-I error, denoting α-quantile of the distribution of C by . The probability limits based on Pr are given as: Where ∝=∝ +∝ and Pn represents the cumulative distribution function for a given value of n.
Now after simplification, we have the following results.
We need to find the results of the following by using simulation method.

Comparison
In this paper the performance of the Pr chart is compared with P (conventional Shewhart attribute control chart) and Q chart for binomial data developed by Quesenberry (1991).
The efficiency of Pr chart as compared to P and Q chart has been examined using power curves as a performance measure. Using their respective control structures, the power curves for different combinations of have been constructed. This show that the Pr chart has higher probability to signal shifts in process of nonconfirming items as compared to both P and Q chart.

Conclusion
The attribute control charts are particularly useful in the service industries and in nonconfirming quality improvement efforts. The classical application of P chart requires that the parameters of the distribution are known. In many situations the true fraction nonconforming, P, is unknown and need to be estimated. This study proposes an efficient control chart, namely Pr chart, to monitor the process proportion or the non-confirming items, based on proportion regression estimator. We derive the parameters of the proposed chart. The performance of the proposed chart is compared with the classical P and Q charts using OC curves. It has been shown that Pr chart more efficient to the P and Q chart. The design of the Pr chart is established and is shown to be more efficient as compared to the classical P chart and Q chart, particularly for bivariate data.