Quality Improvement by Quantile Approach Gopal K. Kanji Sheffield Business School City Campus, Sheffield Hallam University Sheffield, UK.
[email protected] Osama Hasan Arif Computing Management Science City Campus, Sheffield Hallam University Sheffield, UK.
[email protected] 1. Abstract It is desirable for the data of a control chart to be normally distributed. If the data is not normal, then some kind of transformation is required, e.g. Box-Cox transformation, to produce a suitable control chart. In this paper to produce a control chart we will discuss a quantile approach to study the behaviour of the data. We will also provide examples to indicate how quantile approach could be used to construct a control chart for non-normal distribution. 2. Introduction In recent years customers have exerted enormous pressure on the organisation to improve the quality of their products and services. As a result, many organisation have implemented various quality improvement process as part of their every day business activity (see, for example, Reibstein, Washington, Levinson, and Shenitz (1992)). Some of these improvements are due to application of statistical process control and process improvement methods, (see, Blache, Stewart, Zimmerman, Shaull, Benner, and Humphrey (1988)). Due to complexity of the data, it is sometime difficult for people to interpret the various approaches of statistical process control specially when they are modified (See DuBois, Stephenson, Gibberd, and Shaw (1991)) with the help of various transformations. Successful quality improvement process must be based on proper interpretation of statistical data and quality improvement methods. To overcome such difficulties in this paper we will discuss quality improvement process through quantile method. In doing so we will first of all discuss quantile approach and then develop a quality control chart for this purpose. 3. Quantile Approach In 1960, Tukey has introduced a family of random variable defined by the transformation (1)
x p = [ p λ − (1 − p ) λ / λ ]
where p is uniformly distributed random variable on (0,1). It can be shown that the rectangular and logistic distributions are also members of the above family. For example a limiting form of (1) when λ → 0 is given by (2)
x p = ln p − ln(1 − p )
Where x p is known as the logistic distribution. Here, location and scale parameters could be added to obtain the generalised quantile distribution or lambda distribution for logistic distribution. One of the important aspect of the lambda family is that the percentage points are available directly (Joiner 1971). Various distribution about generalised Lambda Distributions can be found in Shapiro and Gross (1981) and Ramberg et al (1979). Further, a new quantile distribution can be obtained by using the inverse function of the generalised lambda distribution (GLD). 4. Control Chart Using Quantile Approach In this quantile approach the residuals has been transformed into an independent residuals which are in fact logistically distributed (i.e. logistic transformation), the limits which are used in the control chart can be derived from the logistic distribution. The logistic quantile distribution can be described as 1− δ 1+δ Q( p ) = λ + η[( ) ln( p ) − ( ) ln(1 − p )] 2 2
(3)
Where, the warning limits lies between Q (0.995) and Q (0.005), and the actions limits lies between Q (0.999) and Q (0.001), and the central point lie at Q (0.5). The chart below displays the control limits. Here, a quantile control chart has been constructed using Logistic Quantile Distribution as follows. Quantile Control Limits
Quantile Plot Exact Median Rankit
Warning limit at 0.005 & Action Limit at 0.001 5 Fit x
2
4
1.5 1 0.5 -1
0 -0.5 -0.5 0
0.5
1
-1
1.5
2
2.5
3
Simulate skew logistic x
Simulate skew ogistic x
2.5
3
Action Limit (4.65)
2
Warning Limit (3.56)
1
Central Limit (0.26) Warning Limit (-1.34)
0 -1
-0.5
0
0.5
1
-1
-1.5
1.5
2
2.5
Action Limit (-1.74)
-2
Q(p)
Q(p)
)LJXUH
)LJXUH
5. Conclusion Quantile approach (figure 1 & 2) help us to establish the suitable model for the data and it can be seen from the result that the quantile approach is better than the transformation (e.g. BoxCox transformation) used for non-normal data in order to obtain the control chart. 6. References Gilchrist, W. (1997). Modeling with Quantile Distribution Functions. Journal of Applied Statistics 24, 113-122. Tukey, J. W. (1960). The practical Relationship between the Common Transformation of Percentages of Counts and of Amounts. Technical Report 36, Statistical Techniques Research Group., Princeton University. 2
