A Bayesian Control Chart for the Coecient of Variation in the Case of Pooled Samples

Transcription

1 A Bayesian Control Chart for the Coecient of Variation in the Case of Pooled Samples R van Zyl a,, AJ van der Merwe b a PAREXEL International, Bloemfontein, South Africa b University of the Free State, Bloemfontein, South Africa Abstract By using the data results obtained by Kang, Lee, Seong, Hawkins [7], a Bayesian procedure is applied to obtain control limits for the coecient of variation Reference probability matching priors are derived for the coecient of variation in the case of pooled samples By simulating the posterior predictive density function of a future coecient of variation it is shown that the control limits are eectively identical to those obtained by Kang et al [7] This article illustrates the exibility unique features of the Bayesian simulation method for obtaining posterior distributions, predictive intervals run lengths in the case of the coecient of variation Keywords: coecient of variation, control charts, reference prior, probability-matching prior Introduction The monitoring of variability is a vital part of modern statistical process control SPC) Shewart control charts are widely used SPC tools for detecting changes in the quality of a process In most settings where the process is under control the process have readings that have a constant mean µ) constant variance σ ) In such settings the X chart is usually used to monitor the mean, the R S control charts the variance of the process In practice there are some situations though where the mean is not a constant the usual SPC control reduces to the monitoring of the variability alone As a further complication it sometimes happens that the variance of the process is a function of the mean In these situations the usual R S charts can also not be used The proposed remedy depends on the nature of the relationship between the mean the variance of the process One common relationship that we Corresponding author Principal corresponding author address: ruaanvanzyl@parexelcom R van Zyl) Preprint submitted to Statistical Planning Inference 3rd April 04

2 FREQUENTIST METHODS will look at is that the mean stard deviation of the process is directly proportional so that the coecient of variation γ = σ µ ) is a constant According to Kang, Lee, Seong, Hawkins [7] this is often the case in medical research By using frequentist methods they developed a Shewart control chart, equivalent to the S chart, for monitoring the coecient of variation using rational groups of observations The chart is a time-ordered plot of the coecient of variation for successive samples It contains three lines: A center line; The upper control limit UCL); The lower control limit LCL) By using the predictive distribution, a Bayesian procedure will be developed to obtain control limits for a future sample coecient of variation These limits will be compared to the classical limits obtained by Kang et al [7] Bayarri García-Donato [] give the following reasons for recommending a Bayesian analysis: Control charts are based on future observations Bayesian methods are very natural for prediction Uncertainty in the estimation of the unknown parameters is adequately hled Implementation with complicated models in a sequential scenario poses no methodological diculty, the numerical diculties are easily hled via Monte Carlo methods; Objective Bayesian analysis is possible without introduction of external information other than the model, but any kind of prior information can be incorporated into the analysis, if desired Frequentist Methods Assume that X i i =,,, n) are independently, identically normally distributed with mean µ variance σ X = n n i= X i is the sample mean S = n n i= Xi X ) is the sample variance The sample coecient of variation is dened as W = S X Kang et al [7] suggested a control chart for the sample coecient of variation, similar to that of the X, R S charts They proposed two methods in developing these charts:

3 FREQUENTIST METHODS 3 The use of the non-central t distribution; The use of the canonical form of the distribution of the coecient of variation It can be noted that n X T = = nw S follows a non-central t distribution with n ) degrees of freedom noncentrality parameter, n γ The cumulative distribution function of the coecient of variation can therefore be computed from the non-central t distribution In what follows, a more general distribution than the canonical form of Kang et al [7]) will be given for W = S X Using a Bayesian procedure this distribution will be used for prediction purposes: ) Aw) n I n+fw ) f+ f, w 0 γn+fw ) 05 f w γ) = ) n I f, w < 0 ) f Aw) n+fw ) f+ γn+fw ) 05 ) where γ = σ µ, f = n, { f ) } f nw f A w γ) = ) nfw π exp f ) Γ γ n + fw ) ) n I f γ n + fw ) 05 = ˆ is the Airy function Iglewicz [6]) 0 f { [ ]} q f exp n q dq γ n + fw ) The Data The example used by Kang et al [7] was that of patients undergoing organ transplantation, for which Cyclosporine is administered For patients undergoing immunosuppressive treatment, it is vital to control the amount of drug circulating in the body For this reason frequent blood assays were taken to nd the best drug stabilizing level for each patient The dataset consist of m = 05 patients the number of assays obtained for each patient is n = 5 By doing a regression test they conrmed that there is no evidence that the coecient of variation depends on the mean which means that the assumption of a constant coecient of variation is appropriate They used the weighted root mean square estimator m ˆγ = m i= w i = 05 = 0075 to pool the samples for estimating γ By substituting ˆγ in equation ) by calculating the lower upper 740 percentage points, they obtained a LCL = 008 UCL = The chart was then applied to a fresh data set of 35 samples from a dierent laboratory

4 3 BAYESIAN PROCEDURE 4 3 Bayesian Procedure Since ve observations per patient is quite small, groups of ve patients will be pooled together to implement the Bayesian procedure Based on similar means, as presented in AppendixA, the results of the rst ve patients will therefore be pooled together, similar the results of the second ve patients so forth k = new groups are therefore formed By assigning a prior distribution tot he unknown parameters the uncertainty in the estimation of the unknown parameters can adequately be hled The information contained in the prior is combined with the likelihood to obtain the posterior distribution of γ By using the posterior distribution the predictive distribution of a future coecient of variation can be obtained The predictive distribution on the other h can be used to determine the distribution of the run length Determination of reasonable non-informative priors is however not an easy task Therefore, in the next section, reference probability matching priors will be derived for the coecient of variation in the case of pooled samples 4 Reference Probability-Matching Priors for the Coecient of Variation in the Case of Pooled Samples As mentioned the Bayesian paradigm emerges as attractive in many types of statistical problems, also in the case of the coecient of variation Prior distributions are needed to complete the Bayesian specication of the model Determination of reasonable non-informative priors in multi-parameter problems is not easy; common non-informative priors, such as the Jereys' prior can have features that have an unexpectedly dramatic eect on the posterior Reference probability-matching priors often lead to procedures with good frequency properties while returning to the Bayesian avour The fact that the resulting Bayesian posterior intervals of the level α are also good frequentist intervals at the same level is a very desirable situation See also Bayarri Berger [] Severine, Mukerjee, Ghosh [3] for a general discussion 4 The Reference Prior In this section the reference prior of Berger Bernardo [3] will be derived for the coecient of variation in the case of pooled samples In general, the derivation depends on the ordering of the parameters how the parameter vector is divided into sub-vectors As mentioned by Pearn Wu [] the reference prior maximizes the dierence in information entropy) about the parameter provided by the prior posterior In other words, the reference prior is derived in such a way that it provides as little information possible about the parameter of interest The reference prior algorithm is relatively complicated, as mentioned, the solution depends on the ordering of the parameters how the parameter vector is partitioned into sub-vectors In spite of these diculties, there is growing evidence, mainly through examples that reference priors provide sensible answers from a Bayesian point of view that frequentist

5 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 5 properties of inference from reference posteriors are asymptotically good As in the case of the Jereys' prior, the reference prior is obtained from the Fisher information matrix In the case of a scalar parameter, the reference prior is the Jereys' prior Berger, Liseo, Wolpert [4] derived the reference prior for the coecient of variation in the case of a single sample From the medical example given in Kang et al [7] it is clear that the stard deviation of measurements is approximately proportional to the mean; that is, the coecient of variation is constant across the range of means, which is an indication that the a reference prior for a pooled coecient of variation should be derived Theorem Let x pl N µ, σ ) where p =,,, p, l =,,, l, =,,, k the coecient of variation is γ = σ µ = σ µ = = σ k µ k The reference prior for the ordering { γ; σ, σ,, σ k)} is given by p R γ, σ, σ,, σk ) γ γ + Proof The proof is given in AppendixB Note: The ordering { γ; σ, σ,, σ k)} means that the coecient of variation is the most important parameter while the k variance components are of equal importance, but not as important as γ Also, if k =, equation B) simplies to the reference prior obtained by Berger et al [4] 4 Probability-Matching Priors The reference prior algorithm is but one way to obtain a useful non-informative prior Another type of non-informative prior is the probability-matching prior This prior has good frequentist properties Two reasons for using probabilitymatching priors are that they provide a method for constructing accurate frequentist intervals, that they could be potentially useful for comparative purposes in a Bayesian analysis There are two methods for generating probability-matching priors due to Tibshirani [4] Datta Ghosh [5] Tibshirani [4] generated probability-matching priors by transforming the model parameters so that the parameter of interest is orthogonal to the other parameters The prior distribution is then taken to be proportional to the square root of the upper left element of the information matrix in the new parametrization Datta Ghosh [5] provided a dierent solution to the problem of nding probability-matching priors They derived the dierential equation that a prior must satisfy if the posterior probability of a one-sided credibility interval for a parametric function its frequentist probability agree up to O n ) where n is the sample size k = σ

6 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 6 According to Datta Ghosh [5] p θ) is a probability-matching prior for ] θ = [ γ, σ, σ,, σk the vector of unknown parameters, if the following differential equation is satised: where Υ θ) = k+ α= θ α {Υ α θ) p θ)} = 0 F θ) t θ) t θ) F θ) t θ) = [ Υ θ) Υ θ) Υ k+ θ) ] [ t θ) = θ t θ) θ t θ) θ k+ t θ) ] t θ) is a function of θ F θ) is the inverse of the Fisher information matrix Theorem The probability-matching prior for the coecient of variation γ the variance components is given by p M γ, σ, σ,, σk ) γ + γ ) k = = γ γ + σ k σ Proof The proof is provided in AppendixC From Theorems it is clear that the reference probability-matching priors are equal 43 The Joint Posterior Distribution By combining the prior with the likelihood the joint posterior distribution of γ, σ, σ,, σk can be obtained p γ, σ, σ,, σk data ) { k ) n σ exp [n σ x σ ) ]} + v s γ = γ + γ ) 3) The conditional posterior distributions are given by k = σ p γ σ, σ,, σk, data ) γ + γ ) exp { k σ = [n x σ γ ) ]} 4)

7 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 7 p σ, σ,, σk γ, data ) k ) σ n +) exp { [n σ x σ ) ]} + v s γ = 5) For the medical example, p = 5, l = 5, n = p l = 5 k = As mentioned the reason for the pooling is that ve observations per patient is quite small By using the conditional posterior distributions equations [4] [5]) Gibbs sampling the unconditional posterior distribution of the coecient of variation, p γ data) can be obtained as illustrated in Figure Figure : Histogram of the Posterior-Distribution of γ = σ µ mean γ) = 0075, median γ) = 00750, mode γ) = 00748, var γ) = 595e 6 95% equal-tail interval = 00705; 00800), length % HDP interval =007048; ), length From a frequentist point of view Kang et al [7] mentioned that the best way to pool the sample coecients of variation is to calculate the weighted root mean square ˆγ = m t i w i = ) = 0075 It is interesting to note that the weighted root mean square value is equal to the mean median) of the posterior distribution of γ By substituting each of the simulated γ values of the posterior distribution into the conditional predictive density f w γ) using the Rao-Blackwell procedure the unconditional posterior predictive density f w data) of a future sample coecient of variation can be obtained This is illustrated in Figure for n = 5

8 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 8 Figure : Predictive Density f w data) for n = 5 mean w) = 00705, median w) = 00686, mode w) = 00647, var w) = 66743e 6 95% equal-tail interval = 0059; 060), length = % HDP interval = 000; 007), length = % equal-tail interval = 00; 060), length = % HDP interval = 00086; 0546), length = 0460 Kang et al [7] calculated lower LCL=008) upper UCL=05957) control limits which are for all practical purposes the same as the 9973% equaltail prediction interval Kang et al [7] then applied their control chart to a new dataset of 35 patients from a dierent laboratory Eight of the patients' coecient of variation based on ve observations) lie outside the control limits Since the 9973% equal-tail prediction interval is eectively identical to the control limits of Kang et al [7] our conclusions are the same As mentioned the rejection region of size α α = 0007) for the predictive distribution is ˆ α = p w data) dw Rα) In the case of the equal-tail interval, R α) represents those values of w that are smaller than 00 or larger than 060 Assuming that the process remains stable, the predictive distribution can be used to derive the distribution of the run length or average run length The run length is dened as the number of future coecients of variation, r until the control chart signals for the rst time Note that r does not include the coecient of variation when the control chart signals) Given γ a stable Phase I process, the distribution of the run length r is geometric with parameter ˆ Ψ γ) = f w γ) dw Rα) where f w γ) is the distribution of the sample coecient of variation given γ as dened in equation )

9 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 9 The value of γ is of course unknown the uncertainty is described by the posterior distribution The predictive distribution of the run length or the average run length can therefore be easily simulated The mean second moment about zero of r given γ are given by E r γ) = Ψ γ) E r γ ) = Ψ γ) Ψ γ) The unconditional moments E r data), E r data ) V ar r data) can therefore easily obtained by simulation or numerical integration For further details see Menzefricke [8, 9, 0, ] In Figure 3 the predictive distribution of the run length is displayed in Figure 4, the distribution of the average run length is given Figure 3: Predictive Distribution of the Run Length p r data) for n = 5 E r data) = 39749, Median r data) = 65, V ar r data) = 6379e 5 95% Equal-tail Interval = 8; 487), Length = % HDP Interval = 0; 96), Length = 96 As mentioned for given γ, the run length r is geometric with parameter Ψ γ) The unconditional run length displayed in Figure 3 is therefore obtained using the Rao-Blackwell method, ie, it is the average of the conditional run lengths

10 4 REFERENCE AND PROBABILITY-MATCHING PRIORS 0 Figure 4: Distribution of thee Expected Run Length Mean = , Median = , Variance = 49863e 3 95% Equal-tail Interval = 374; 4956), Length = % HDP Interval = 695; 49774), Length = 3479 From Figure 3 it can be seen that the expected run length, E r data) = 3974, is somewhat larger than the ARL of 370 given by Kang et al [7] The median run length Median r data) = 65 is smaller than he mean run length This is clear from the skewness of the distribution In the case of the HDP limits, R α) represents those values of w that are smaller than larger than 0546 The predictive distribution of the run length is illustrated in Figure 5 while the distribution of the average run length is given in Figure 6 Figure 5: Predictive Distribution of the Run Length in the Case of HDP Limits E r data) = 45847, Medianr data) = 677, V ar r data) = 300e 5 95% Equal-tail Interval = 835; 7500), Length = % HDP Interval = 0; 36308), Length = 36308

11 5 CONCLUSION Figure 6: Distribution of the Expected Run Length for HDP Limits Mean = 4673, Median = , Variance = 70e 4 95% Equal-tail Interval = 8074; 807), Length = % HDP Interval = 6939; 78976), Length = 6037 A comparison of Figure 3) Figure 5) show that the median run length for equal tail HDP limits are more or less the same 5 Conclusion This paper develops a Bayesian control chart for monitoring the coecient of variation in the case of pooled samples In the Bayesian approach prior knowledge about the unknown parameters is formally incorporated into the process of inference by assigning a prior distribution to the parameters The information contained in the prior is combined with the likelihood function to obtain the posterior distribution By using the posterior distribution the predictive distribution of a future coecient of variation can be obtained Determination of reasonable non-informative priors in multi-parameter problems is not an easy task The Jereys' prior for example can have a bad eect on the posterior distribution Reference probability matching priors are therefore derived for the coecient of variation int he case of pooled samples The theory results are applied to a real problem of patients undergoing organ transplantation for which Cyclosporine is administered This problem is discussed in detail by Kang et al [7] The 9973% equal tail prediction interval of a future coecient of variation is eectively identical to the lower upper control chart limits calculated by Kang et al [7] The example illustrates the exibility unique features of the Bayesian simulation method for obtaining posterior distributions, prediction intervals run lengths

12 APPENDIXA DATA FOR MEDICAL EXAMPLE AppendixA Data for Medical Example m X W m X W m X W AppendixB Proof of Theorem Proof The likelihood function is given by

13 APPENDIXB PROOF OF THEOREM 3 L γ, σ, σ,, σ k data ) k = { ) n σ exp [n σ x σ ) ]} + v s γ where l p x = n l= p= x pl v s = l p l= p= x pl n x n = p l By dierentiating the log likelihood function, l, twice with respect to the unknown parameters taking expected values the Fisher information matrix can be obtained l = log L γ, σ, σ,, σ k data ) = n Therefore E l ) n = σ ) [ l σ ) ] = n σ σ k log σ = n x σ )3 + 3n x m 4γσ 5 k σ = v s σ )3 ) { + } where γ =,,, k Also [ l ] E σ = 0 where =,,, k, m =,,, k m σ m [n x σ ) ] + v s γ Further l γ) = k = n x m σ γ 3 ) 3n γ 4

14 APPENDIXB PROOF OF THEOREM 4 [ ] l E γ) = kn γ 4 If we dierentiate l with respect to σ γ we get l σ γ = n x γ σ 3 [ l ] E = n σ γ γ 3 σ The Fisher information matrix then follows as F γ, σ, σ,, σk ) = [ ] F F F F where F = n [ F = kn γ, 4 F = F = n γ 3 σ σ ) { } + γ 0 n σ n γ 3 σ n γ 3 σ k 0 0 ) { } + γ 0 n 0 0 σ k ] ) { } + γ To calculate the reference prior for the ordering { γ; σ, σ,, σ k)} we must rst calculate F then F Now F = F F F F = kn γ 4 kn 4γ γ 3 ) n γ + ) = kn γ γ + ) = h Also n F = p γ) h γ γ + { + }) k k ) γ σ = h =

15 APPENDIXC PROOF OF THEOREM 5 which means that p σ, σ,, σ k γ ) h k σ ) Therefore the reference prior for the ordering { γ; σ, σ,, σ k)} is p R γ, σ, σ,, σk) = p γ) p σ, σ,, σk γ ) γ γ + k = σ B) AppendixC Proof of Theorem Proof Using the previously derived Fisher information matrix we can calculate F F F 3 F,k+ F θ) = F γ, σ, σ,, σk) F F F 3 F,k+ = F k+, F k+, F k+,3 F k+,k+ Let t θ) = t γ, σ, σ,, σ k) = γ Since θ) = [ γ t θ) t θ) t θ) ] σ σ = [ 0 0 ] k we have that θ) = = [ [ F F F,k+ ] γ +γ ) n k γσ n k γσ n k γσ k n k ] t θ) F θ) t θ) = { γ + γ ) n k } Further Υ t θ) F θ) θ) = t θ) F θ) t θ) = [ Υ θ) Υ θ) Υ k+ θ) ]

16 APPENDIXC PROOF OF THEOREM 6 where Υ θ) = γ + γ ) n k), ) σ Υ θ) =, {n k + γ )} The prior ) σ Υ 3 θ) = {n k + γ )} ) σk Υ k+ θ) = {n k + γ )} p M θ) = p M γ, σ, σ,, σk ) γ + γ ) is therefore a probability-matching prior since γ {Υ θ) p M θ)} + σ k = σ C) {Υ θ) p M θ)} + + σk {Υ k+ θ) p M θ)} = 0 [] Bayarri, M, Berger, J, 004 The interplay of bayesian frequentist analysis Statistical Science 9 ), 5880 [] Bayarri, M, García-Donato, G, 005 A bayesian sequential look at u- control charts Technometrics 47), 45 [3] Berger, J, Bernardo, J, 99 On the development of reference priors Bayesian Statistics 4, 3560 [4] Berger, J O, Liseo, B, Wolpert, R L, 999 Integrated likelihood methods for eliminating nuisance parameters Statistical Science 4, 8 [5] Datta, G, Ghosh, J, 995 On priors providing frequentist validity for bayesian inference Biometrika 8 ), 3745 [6] Iglewicz, B, 967 Some properties of the coecient of variation PhD thesis, Virginia Polytechnic Institute

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