Control Charts for Lognormal Data

Transcription

1 Tamkang Journal of Science Engineering, Vol, NO, pp -7 ( Control Charts for ognormal Data Smiley W Cheng Hansheng Xie Department of Statistics, niversity of Manitoba, Winnipeg, Manitoba, Canada RT N smiley_cheng@umanitobaca Abstract In dealing with positively-skewed distributed data, the direct logarithmic transformation may result in a control chart with inappropriate control parameters in the application of quality control When a specific interval for the lognormal mean is given, a new method is introduced to set up two control charts these two charts can monitor a process for which the underlying distribution of the quality characteristic is lognormal Key Words: lognormal distribution, control chart, quality control, mean, variance, normal distribution Introduction Since many kinds of data in real life have a positively-skewed distribution, the lognormal distribution has been widely applied in many areas Morrison [] first applied the lognormal distribution to quality control, proposed a modified quality control scheme that can process skewed data in the original scale of measurement when the assumption of normality can not be made Ferrell [] also suggested using this control scheme for computing plotting control charts when data are from a badly skewed distribution which can be approximated by a lognormal distribution Based on the basic relationship between normal lognormal distributions, Morrison derived control limits for the lognormal variable from the corresponding control limits for a normal variable, using the exponential transformation However, because of the complexity of the lognormal distribution, its application to quality control cannot be referred to that of the normal distribution by simply taking the direct transformation, which may result in a control chart with inappropriate control parameters For simplicity, in the following a lognormal process refers to a process in which its characteristic follows a lognormal distribution a normal process refers to a process in which its characteristic follows a normal distribution In this article, quality control schemes for lognormal processes based on direct transformations, either the exponential transformation or the logarithmic transformation, are critically examined, a new quality control scheme is proposed To monitor a lognormal process, the corresponding normal process is obtained through the logarithmic transformation When it is given that the lognormal process mean lies in a specific interval, then two control charts are set up for the lognormal process The control of a complex lognormal process is simplified to that of a normal process, for which good control schemes are available it is much easier to implement Properties of the new control charts are discussed, an example is given to illustrate the implementation of the new charts The Direct Transformations The Exponential Transformations Suppose that Xij, i,, j,,, n represent the quality characteristic of a process, follow a lognormal distribution with parameters, ; ie, N(, et ij lnxij, then ij, i,, j,,, n, follow a normal distribution N(,, where is the normal process mean is the normal process stard deviation In the modified quality control scheme proposed by Morrison [], the statistics for a lognormal process the corresponding control limits can be obtained from the following derivations Notice that

2 Smiley W Cheng Hansheng Xie && E( && P && ( && exp( && exp( + && P exp( ( && X X exp( + && P exp( i( i(n ( where & is the sample midrange for a normal process, Xi( Xi(n are the minimum maximum of the i th sample, respectively Similarly, R i E(R i P R i P exp d d exp(r exp ( (( (( d + d X i(n P exp( ( d d exp( ( d + d X i( ( where Ri is the sample range for a normal process, d d are control chart constants Thus, from (, the geometric sample mean X i( X i ( n is used as a measure of the lognormal process mean the exact control limits are exp( ± & With the available sample results, the control limits are estimated by i ± A m A X i(n exp(r & ± X ( X (nm m i X i( ( where X( X(nm are the minimum maximum in nm sample values for a lognormal process, r is the average ratio of the maximum to the minimum from m samples of size n for a lognormal process, A is a control chart constant Similarly, from (, the sample ratio X i(n X i( is used as a measure of the lognormal process variability the exact control limits are exp[ (d ± d ], which are estimated by r D m X m i X 4 m D X i(n 4 r m i X i( where D D4 are control chart constants i(n i( D D (4 (5 From the derivations, it is seen that the incontrol probability of the derived statistics for a lognormal process is the same as that for its normal counterpart, but the control limits for lognormal control charts are inaccurate Because the control parameters for lognormal normal processes are different, the direct exponential transformations may not assure that the statistical state of a lognormal process is the same as that of the corresponding normal process Morrison's chart actually sets the target as the normal process mean, therefore a normal process is monitored through its lognormal counterpart Moreover, because the parameter estimators of (, (4 (5 resulting from the exponential transformation are biased, the control charts neither have proper probability nor proper -sigma control limits The ogarithmic Transformation In statistical analysis, a logarithmic transformation is often applied to a set of positively-skewed distributed data before proceeding with the analysis This approach works well for usual statistical analysis However, the direct logarithmic transformation may result in a control chart with inappropriate control parameters in the application of quality control et * be the lognormal process mean * be the lognormal process stard deviation For a lognormal process, it is of interest to control the parameters * *, while, for a normal process, are of interest It can be shown that, for a specified significance level α, the control limits for individual measurements of a lognormal process is different from those of the corresponding normal process Without loss of generality, assume that X ~ N(,, then ln(x ~ N(, For α 7, it follows from P( X * * > x5 5 that the upper percentile x5 can be found as x5 exp(5 Hence, the upper control limit for the X chart is C X * + x 5* exp(5 + exp(5 ln(cx 8 exp( 456

3 Control Charts for ognormal Data The upper control limit for the chart is C +z 5 Obviously, ln(cx is not equal to C so that the direct logarithmic transformation may result in a different control state for corresponding normal process Therefore, some restrictions have to be given in order to guarantee the control state of the corresponding process is equivalent to that of the original lognormal process New Control Charts for ognormal Processes It is difficult to directly construct a control chart for a lognormal process since sampling properties associated with the lognormal statistics are not easy to derive Making use of the relationship between normal lognormal distributions having been given a specific interval for the lognormal mean of a lognormal process, a new method is proposed to avoid the complexity of the lognormal distribution Two control charts for lognormal distribution can be constructed to monitor a lognormal process A Specific Interval for the ognormal Process Mean Suppose that m samples are romly drawn from a lognormal process, * is known to lie in an interval: (*,* it is given for the process according to technical specifications It could be either a given margin of error or specification limits for a single measurement The margin of error is given by a X ij * a (6 i,,, m; j,,, n where a a are known positive constants Equation (6 can be rewritten as a + X ij * a + X ij i,,, m; j,,, n which implies X (mn a * X ( + a X (mn X ( a + a Hence, an interval for possible values of * is * X ( + a (7 * X (mn a (8 If specification limits are available, the upper lower specification limits can be used as * *, respectively Derivation of Intervals for Parameters The control parameters for a lognormal process are * * The control parameters for the corresponding normal process are The parameters * * are functions of given by * exp( + 5 (9 * exp( + [exp( ] ( m preliminary samples collected from the incontrol process can be used to estimate, ie, ˆ S From (9, an interval for is obtained as below: ln( 5ˆ ( * ln( * 5 ( From (, an interval for * is obtained as below: * exp( + ˆ [exp(ˆ ] ( * exp( + ˆ [exp(ˆ ] (4 Because normal distribution is symmetric about mean, the target for the corresponding normal process is 5( + 5ln( * * 5ˆ (5 which implies that the target for the lognormal process is the geometric mean of * *, ie, * * * Constructing Control Charts for ognormal Processes ˆ

4 4 Smiley W Cheng Hansheng Xie When m preliminary samples are taken from a lognormal process, the logarithms of each observation form the m initial samples of the corresponding normal process To determine whether the process variability is stabilized, an S chart can be set up with control limits: C S χ α4 S n c 4 (6 χ α C S S n c 4 (7 where α α 4 are type-i error probabilities for lower upper tails respectively If all the stard deviations of these samples plot inside the control limits, then the process variability appears to be in control Otherwise, each of the out-of-control points for which assignable causes can be found is discarded the control limits are recalculated Then these control limits can be used for controlling current or future production is estimated from the formula ˆ S The percentiles for chart can be obtained by setting z α (8 z α (9 Then, z α ( z α ( where α α are type-i error probabilities for lower upper tails respectively A chart can be set up with the following control limits: C ( zα C + ( + n n n ( C zα + ( n n + n (4 4 Properties of the New Control Charts When a specific interval for * is given, the derivations of control limits for the control charts monitoring the two related processes are reversible hence the statistical control state of the lognormal process can refer to that of the corresponding normal process As a result, it is necessary to study properties of the two control charts for the normal process effects of normal parameter changes on the lognormal parameters 4 The Calculations of the Average Run ength (AR Assume that ij ~N(, independently, where i,,, m; j,,, n Suppose that the normal process mean changes from to + a (a the normal process stard deviation changes from to b (b > The probability of type-ii error for chart can be computed from β P ( C C + a ; b Φ ln * * a n * Φ ln a n * (5 When a, the probability of type I error for chart is α β Φ[ ln * ] * (6 from which it is noted that α is a function of * To achieve a small α, * are * * usually not larger for high precision products so that the process variability has to be small, while, is allowed to be a little bit large for medium or low precision products

5 Control Charts for ognormal Data 5 The AR's for chart can be easily obtained from AR β (7 When α is fixed, AR will decrease as a n increase The AR's for S chart can be computed from AR S β S (8 where β S H χ α 4,n H χ α,n b b α S α +α 4 4 Effects of Changes in Parameters When there are changes in the normal process mean /or process variability, the lognormal parameters will be changed to * * exp[ exp[( + a where a b > + 5(b + a + (b ] ] [exp((b ] Because the derivatives of * with respect to a b are * a * > (9 * b b * > ( Notice that * is a monotone increasing function of a b, * can be written as a monotone increasing function of * : * * exp((b ( since exp((b > Then * is also a monotone increasing function of a b Thus, the direction of an out-of-control signal from a lognormal process can be identified from the corresponding normal process 5 Charting Procedure Example The steps to set up the two charts are summarized below: Determine the values of * * (a se values provided by technical specifications, or if not available, (b use * * obtained from preliminary data as follows: if the overall range of the data is less than or equal to a + a, calculate * * ; however, if the overall range is greater than a + a, remove the possible outliers X(nm, X(,, until the overall range is less than or equal to a + a, then calculate * * Transform data using ln(x Construct an S chart estimate by S when the process variability is in control 4 Compute *, *,, * * 5 Construct a chart 6 For a sample point that plots outside one of the control limits, calculate ˆ ˆ using i as the estimate of S i as the estimate of Plot 'm+' or 'm ' against sample number if only ˆ > * or ˆ < * ; plot 'v+' ' or 'v ' against sample number if only ˆ > * or ˆ < * ; plot 'm+v+', 'm+v ', 'm v+' or 'm v ' against sample number according to the sources the directions of an out-ofcontrol signal 7 Examine the assignable cause(s An example is given to illustrate how to apply the new control scheme to lognormal distributed data The data, consisting of 4 samples of size 5, are given in Table The first samples are taken from Morrison [], where it was stated that they were collected from a process in the valves industry The last 4 samples are added to simulate an out-of-control process For the measurement of individual values, the upper lower specification limits are A probability plot of the real data in Figure suggests that the observations do not behave as though arising from a normal distribution To

6 6 Smiley W Cheng Hansheng Xie adjust for non-normality, lognormal transformation is applied to the original data A probability plot of the transformed data in Figure shows that a lognormal distribution curve can be fitted quite well, suggesting that lognormal quality control scheme should be employed in this case Suppose that the first samples in Table are used as preliminary samples After applying logarithmic transformation, an S chart is set up with α S 7 it is shown in Figure When the sample stard deviations are plotted on this chart, there is no indication of an out-of-control condition Then is estimated by S 6 Figure The probability plot for the valve data mean shift on the first subsequence sample is 96, the probability of detecting the variability change on the first subsequence sample is 798 Hence, the expected number of samples taken before the shift is detected is 87, the expected number of samples taken before the change is detected is 57 When the last 4sample means are plotted on the chart shown in Figure 5 the last 4 sample starddeviations are plotted on the S chart shown in Figure 6, it is seen that the last 4 points are above at least one of the C's This indicates that the lognormal process is out of control with an increase in * * To identify the sources of these out-of-control signals, calculated It is found that * greater than *, ˆ *, exceed ˆ * ˆ ˆ * ˆ * ˆ *i are ˆ are ˆ *4, which is equal to 465 This diagnosis is supported by reference back to the individual measurements of the last 4 samples, since there are individuals exceeding * in sample greater variability within sample,, 4 It should be noted that, for the last sample, only the lognormal process variability is out of control although both of the corresponding normal process mean variability are out of control Figure The probability plot for the logarithm of the valve data Since z α 449 z α 449, a chart can be set up with α it is shown in Figure 4 When the sample means are plotted on this chart, there is also no indication of an out-of-control condition Since the S charts constructed using the first samples indicate that both the process variability the process mean are in control, the control limits obtained can be used in on-line statistical process control Assuming that there is a shift in the process mean a times change in the process stard deviation, the probability of detecting the Figure The first S chart for the valve data

7 Control Charts for ognormal Data 7 Table Valve data 6 Conclusions Our discussion shows that direct data transformation methods may be inappropriate for the control of a lognormal process if no constraints are applied to the lognormal process This clarifies some confusion in lognormal quality control When a specific interval for the lognormal mean is given, our new method enables the control of a lognormal process through that of its normal counterpart, avoiding the complexity of the lognormal distribution A detailed analysis of the two new control charts is presented In general, the new control charts are applicable to a lognormal process whenever a reasonable interval for the lognormal mean can be found based on the nature of the lognormal data Acknowledgment Figure 4 The first -bar chart for the valve data The research was partially supported by a Nature Sciences Engineering Research Council, Canada grant The first author would also like to express the gratitude to the Department of Statistics Actuarial Science, niversity of Waterloo, Waterloo, Ontario, Canada for providing facilities while he is on Administrative leave there References Figure 5 The second S chart for the valve data [] Ferrell, E D, Control Charts for ognormal niverse, Industrial Quality Control, V 5, pp 4-6 (958 [] Morrison, J, The ognormal Distribution in Quality Control, Applied Statistics, V 7, pp 6-7 (958 [] Xie, H, Contributions to Qualimetry PhD dissertation, niversity of Manitoba, Winnipeg, Canada, (999 Figure 6 The second -bar chart for the valve data