CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION

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1 CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha Uiversity, Egypt & Maagemet ad Marketig Departmet, College of Busiess, Uiversity of Bahrai P.O. Box 32038, Kigdom of Bahrai shabib40@gmail.com ABSTRACT The classical Shewhart p-chart that costructed based o the biomial distributio is iappropriate i moitorig over-dispersio ad correlated biary data where it teds to overestimate or uderestimate the dispersio ad subsequetly lead to higher or lower false alarm rate i detectig out-of-cotrol sigals. Cosequetly, the p-chart is recommeded based o a multiplicative-biomial distributio that cout for depedet biary data. A test for idepedet amog biary data is proposed based o this distributio. Moreover, it used to costruct a oe-sided p-chart with its upper cotrol limit ad the sesitivity aalysis of this chart based o average ru legth is preseted. Applicatios are give that illustrates the beefits of the proposed chart. Keywords: Average ru legth; Correlated biary data; Over-dispersio; Statistical process cotrol. 1. INTRODUCTION A cotrol chart is a importat tool i moitorig the productio process i order to detect process shifts ad to idetify abormal coditios i the process. This makes possible the diagosis of may productio problems ad ofte reduces losses ad brigs substatial improvemets i product quality; see, [10] ad [3]. The use of attribute cotrol charts arises whe items are compared with some stadard ad the are classified as to whether they meet that stadard or ot; see, [6], [11] ad [9]. I 1924, Walter Shewhart desiged the first cotrol chart ad proposed the followig geeral model for cotrol charts. Let w be a sample statistic that measures some quality characteristic of iterest, ad suppose that the mea of w is μ w ad the stadard deviatio of w is σ w. The the ceter lie (CL), the upper cotrol limit (UCL) ad the lower cotrol limit (LCL) i the relevat cotrol chart are defied as follows: UCL = μ w + kσ w CL = μ w LCL = μ w kσ w where k is the distace of the cotrol limits from the ceter lie, expressed i stadard deviatio uits; see, [6]. The p-chart is a type of cotrol chart used to moitor the umber of ocoformig uits i a sample that is assumed to have a biomial distributio where the ispectio is doe idepedetly. The biomial distributio is P d = d pd (1 p) d, d = 0,1,, Where E d = μ w = p ad σ d = σ w = p(1 p). The biomial assumptio is the basis for the calculatig the upper ad lower cotrol limits. The cotrol limits are calculated as: UCL = p + k p(1 p) CL = p LCL = p k p(1 p) where is the umber of ispected items ad p is the proportio of defective items. Also, zero could serve as a lower boud o the LCL value. Whe the biomial assumptio is ot valid, the practitioer should seek alterative charts for moitorig the radom process. As a geeralizatio for the biomial distributio Loviso [5] had derived the distributio of the sum of depedet Beroulli radom variables as a alterative of Altham's multiplicative-biomial distributio [1] from Cox's log-liear represetatio [2] for the joit distributio of biary depedet resposes. The Loviso s multiplicative-biomial distributio (LMBD) is characterized by two parameters ad provides wider rage of distributios tha are provided by the biomial distributio (BD) where it icludes uder-dispersio, over-dispersio models ad icludes the biomial distributio as a special case. 156

2 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio I this paper the p chart is proposed based o LMBD which accout for uder-dispersio, over-dispersio ad correlated biary data relative to biomial distributio. A test for idepedet biary data is suggested based o maximum likelihood ratio. Also, the LMBD is used to costruct a oe-sided p-chart with its upper cotrol limit. Moreover, sesitivity aalysis of this chart based o average ru legth is studied. LMBD is reviewed i Sectio 2. Test for idepedet biary data is proposed i Sectio 3. The cotrol chart based o LMBD is itroduced i Sectio 4. The sesitivity study usig ARL is preseted i Sectio 5. Applicatios are illustrated i Sectio LOVISON S MULTIPLICATIVE-BINOMIAL DISTRIBUTION Let Z be a biary respose that measures whether some evet of iterest is preset 'success' or abset 'failure' for sample uits,, ad D = i=1 Z i deotes the sample frequecy of successes. To accommodate for the possible depedece betwee Z i ad uder the assumptio that the uits are exchageable Loviso [5] had give the distributio of D as P D = d = d ψd 1 ψ d d( d), d = 0,1,, t=0 t ψt 1 ψ t t( t) 0 < ψ < 1 ad > 0 are the parameters. This distributio provides wider rage of distributios tha are provided by the biomial distributio; for more details, see, [5] ad [4]. Figure 1 the distributio of D for differet values of, ψ = 0.10 ad = 5. Figure 1 shows the distributio of D for differet values of, ψ = 0.10 ad = 5. It should be poited out that the LMBD is very easy to use as the mea ad variace are of closed form. The expected value ad the variace of D are give by μ = E D = ψ κ 1 ψ, κ ψ, = p, ad σ 2 =V D = 1 ψ 2 κ 2 ψ, κ ψ, κ a ψ, = a x=0 a x From the expected value the relatioship betwee p, ψ ad is + ψ κ 1 ψ, κ ψ, 2 ψ 2 κ 1 ψ, κ ψ, ψ x 1 ψ ( a x) p = ψ κ 1 ψ, κ ψ, a x (x+a) 2 157

3 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio A graph 2 shows this relatio. It may explai p as a probability of success ad ψ = exp 2v /(1 + exp 2v ), < v <, as a probability of a particular outcome weighted by itra-uits associatio measure = exp 2λ, λ > 0, goverig the depedet betwee the biary resposes of the uits. If = 1, the p = ψ represets the probability of success of the uits where they are idepedet. If 1, the biary uits are ot idepedet ad therefore p ψ. Cosequetly to accout for depedet amog biary uits the ψ must be weighted by to give the probability of success p. For example if < 1, the values of ψ < p, the ψ must be weighted by to equalize p. Figure 2 the relatioship betwee p, ψ ad from LMBD with = 25. The variace ca be re-writte as V D = 1 p 1 + p 1 p = p 1 p + 1 (p 1 p 2 ) Therefore, V D = V D b + 1 Cov Z k, Z h Where the variace of the biomial is V D b = p 1 p The covariace of Z k ad Z h is Cov Z k, Z h = p 1 p 2 where p = ψ κ 1 ψ, κ ψ, ad p 1 = ψ 2 κ 2 ψ, κ ψ, Therefore, the variace of the LMBD is equal to the variace of biomial whe Cov Z k, Z h = 0, more tha the variace of biomial whe Cov Z k, Z h > 0 ad less tha the variace of biomial whe Cov Z k, Z h < 0; see, Figure 3. The ratio of the variace to the mea is τ = V D E D = p 1 p + 1 (p 1 p 2 ) = 1 p + 1 (p 1 p 2 ) p p 158

4 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio The biomial distributio is obtaied for = 1 with E D = p ad V D = p(1 p). Figure 3 the variace for LMBD for differet chocices of ψ ad ad = 25. From Habib [4] the first two derivatives of log likelihood for LMBD are l ψ = df d ( d)f d Rq 1ψ, ψ 1 ψ q 1 where ad q 1ψ = q 1 = t=0 t=0 q 1 = l = t t t=0 d( d)f d q 1R q 1 ψ t 1 ψ ( t) t t t t ψ 1 ψ, t t ψ t 1 ψ t t t 1, t ψ t 1 ψ ( t) t t The maximum likelihood estimates of ψ ad ca be foud by solvig l ψ = 0 ad l = TESTING FOR INDEPENDENT BINARY UNITS Whe the LMBD fit the data, the likelihood ratio test ca be used to test for appropriateess of the biomial distributio ad therefore the idepedet betwee uits biary data. Cosider a LMBD distributio with likelihood fuctio L(, ψ) ad ψ, the maximum likelihood (ML) estimates. The likelihood ratio test 159

5 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio statistic(λ) of the hypothesis H 0 : = 1 agaist H 1 : 1 equals 2lL betwee the full model ad the reduced model as Λ = 2 ll ψ, ll(1, ψ ) = 2 l ψ, l(1, ψ ) ψ is the ML estimate of the reduced model (biomial model). From Habib [4] the logarithm of likelihood of the sample ca be writte as The model uder the alterative hypothesis is The model uder the ull hypothesis is Hece, Therefore, l(ψ, ) = logr! + f d logp d; ψ, log f d! l(ψ, ) = logr! + f d logp d; ψ, log f d! l(ψ, 1) = logr! + f d logp d; ψ, 1 log f d! l ψ, l 1, ψ = f d logp d; ψ, f d logp d; ψ, 1 Λ = 2 f d logp d; ψ, logp d; ψ, 1 Uder the ull hypothesis, the test statistic Λ approximately follows chi-square distributio (χ 2 ) with oe degree of freedom, see, [8]. See applicatios below for examples. 4. DEVELOPMENT OF THE CONTROL CHART Whe the cout data ca be modeled by LMBD, the p-chart ca be obtaied as UCL = p + k p 1 p + 1 (p 1 p 2 ) CL = p ad LCL = p k p 1 p + 1 (p 1 p 2 ) The estimate limits ca be obtaied usig maximum likelihood estimates for ψ ad. For differet ψ ad values, the LCL would ot be positive values especially for ear-zero defect process. a. Upper limit chart I rare health evets ad ear-zero-defect maufacturig eviromet, may samples will have o defects. If the LMBD provides a good fit to the data, the upper cotrol limit ca be determied by d ψd 1 ψ d d( d) = α d=ucl t=0 t ψt 1 ψ t t( t) where α is the probability of false alarm or type Ierror. If we caot fid exact value of UCL that gives α we could obtai approximate value of UCL as followig: 1. For UCL 1 ad UCL 2 fid the correspodig α 1, α 2 that iclude the required α. 2. The approximate UCL = UCL 1 + α 1 α UCL α 2 α 2 UCL 1 1 Example Let α = 0.005, = 45, ψ = 0.25 ad =.95. The above upper limit does ot give the exact α. The approximated UCL is 1. at UCL 1 = 7 the value of α 1 = , ad at UCL 2 = 8 the value of α = the approximated UCL at α = is 160

6 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio UCL 7 + ( AVERAGE RUN LENGTH (ARL) It is importat to study the sesitivity of a cotrol chart based o the LMBD. The average ru legth will be studyig here. As the LMBD cotais two parameters, each could have a differet impact o the alarm probability. A sigle cotrol limit is cosidered ad the study the effect of the chage of each model parameter from the assumed stable level; see, [7]. Also, the average ru legth (ARL) is the average umber of samples eeded to obtai a poit out of cotrol limit. For the LMBD it is give by 1 ARL = P(poit is out of cotrol) = 1 d ψd 1 ψ d d( d) d=ucl t=0 t ψt 1 ψ t t( t) Some umerical values of ARL are give i Tables 1 ad 2 for ψ 0 = 0.10, 0 = 0.85, = 25 ad α = ad ψ 0 = 0.10, 0 = 1.10, = 25 ad α = Table 1 Numerical values of ARL for ψ 0 = 0.20, 0 = 0.95, = 25 ad α = ψ Table 2 Numerical values of ARL for ψ 0 = 0.026, 0 = 1.03, = 50 ad α = ψ From Tables 1 ad 2, the ARL idicates that the cotrol chart is relative sesitive to the values of ψ ad. This is a importat i statistical process cotrol as ψ is related to the occurrece of defects uits ad is the parameter that cotrol depedet i biary data. Whe icreases, the average umber of items to be ispected for a alarm decreases rapidly. Whe decreases, the average umber of items to be ispected for a alarm icreases rapidly. However, this is the same whe ψ icreases whe it has icreased beyod a certai value, the average ru legth will ot decrease ay further depedig o the value of. 6. APPLICATIONS a. Applicatio 1 Table 3 lists the umber of defectives foud i 100 samples of size = 50 take every hour from a process producig polyurethae foam product. The estimated frequecy usig LMBD (E. LMBD) ad biomial distributio (E.BD) are also reported. 161

7 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio Table 3 The umber of defectives foud i samples of size 50 polyurethae foam products. # of defective Freq E. LMBD E. BD From Habib [4] the maximum likelihood estimates are ψ = ad = The ML estimate of p is To test the appropriateess of biomial distributio the Λ = 2 f d logp d; ,1.04 logp d; ,1 = d =0 Comparig with χ 2 with df = 1 idicates that the biomial distributio is appropriate for the data. The upper cotrol limit based o α = usig biomial distributio is UCL BD = 4.33 ad usig LMBD is UCL LMBD = 4.2. Figure 4 LMBD chart i compariso with BD chart usig = 50 ad subgroups 100. A graph 4 shows that the biomial distributio ad LMBD idicate i-cotrol process. This chart could be used i the secod phase. b. Applicatio 2 Table 4 lists the umber of defectives foud i 200 samples of size = 50 take every hour from a process producig ballpoit pe cartridges. The estimated frequecy usig LMBD (E. LMBD) ad biomial distributio (E.BD) are also give. 162

8 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio Table 4 The umber of defectives foud i samples of size 50 ballpoit cartridges. # of defective Freq E. LMBD E. BD From Habib [4] the maximum likelihood estimates are ψ = ad = the ML estimates of p is p = To test the appropriateess of biomial distributio the Λ = 2 f d logp d; 0.424,0.93 logp d; ,1 = Comparig with χ 2 with df=1 idicates that the biomial distributio is ot appropriate for the data. The upper cotrol limit based o α = usig biomial distributio is UCL BD = 5.5 ad usig LMBD is UCL LMBD = 6.2. Figure 5 LMBD chart i compariso with BD chart usig = 50 ad subgroups 200. A graph 5 shows that the biomial distributio idicates out-of-cotrol process. O the other had the LMBD chart 2 2 shows i-cotrol process ad fit the data χ LMBD = 2.68 < χ 2,0.05 = This may idicate that the chart based o biomial distributio produces more false alarms. 163

9 Habib Cotrol Chart Based o a Multiplicative-Biomial Distributio 7. CONCLUSION Whe aalyzig defect data with correlated biary data, the chart based o biomial distributio has drawback i producig may false alarms which results i high cost of ispectio ad frequet stoppig of the maufacturig processes. LMBD istead of biomial distributio is suitable i this situatio where more appropriate upper cotrol limit ca be derived. Average ru legth approach is used to evaluate the performace of the proposed chart. The mai advatage of this chart is that if the biary data are idepedet the LMBD reduces to biomial distributio. REFERENCES [1]. Altham, P. (1978) Two geeralizatios of the biomial distributio. Applied Statistics, 27, [2]. Cox, D.R. (1972) The aalysis of multivariate biary data. Applied Statistics, 21, [3]. Elamir, E ad Seheult, A. (2001) Cotrol charts based o liear combiatios of order statistics. Joural of Applied Statistics, 28, [4]. Habib, E (2010) Estimatio of log-liear-biomial distributio with applicatios. Joural of Probability ad Statistics, 10, [5]. Loviso, G. (1998) A alterative represetatio of Altham's multiplicative-biomial distributio. Statistics & Probability Letters, 36, [6]. Motgomery, D.C. (2005) Itroductio to Statistical Quality Cotrol. 5 th e,, Wiley, New York. [7]. Ott, E.R., Schillig, E.G. ad Neubauer, D.V. (2005) Process quality cotrol: troubleshootig ad iterpretatio of data. 4 th ed. Quality Press, ASQ. [8]. Severii, T.A. (2000). Likelihood methods i statistics. (1 st Ed.), Oxford Uiversity Press. [9]. Sim, C.H. ad Lim, M.H. (2008) Attribute charts for zero-iflated processes. Commuicatios i Statistics: Simulatio ad Computatio, 34, [10]. Woodal, W.H. (2006) The use of cotrol charts i health-care ad public-health surveillace. Joural of Quality Techology, 38, [11]. Xie, M., He, B. ad Goh, T.N. (2001) Zero-iflated Poisso model i statistical process cotrol. Computatioal Statistics & Data Aalysis, 38,