Extreme Value Charts and Analysis of Means (ANOM) Based on the Log Logistic Distribution

Transcription

1 Joural of Moder Applied Statistical Methods Volume 11 Issue Article Extreme Value Charts ad Aalysis of Meas (ANOM) Based o the Log Logistic istributio B. Sriivasa Rao R.V.R & J.C. College of Egieerig, Gutur, Adhrapradesh, Idia J. Pratapa Reddy St. A's College for Wome, Gutur, Adhrapradesh, Idia G. Sarath Babu Chebrolu Haumaiah Istitute of Pharmaceutical Scieces, Gutur, Adhrapradesh, Idea Follow this ad additioal wors at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Rao, B. Sriivasa; Reddy, J. Pratapa; ad Babu, G. Sarath (01) "Extreme Value Charts ad Aalysis of Meas (ANOM) Based o the Log Logistic istributio," Joural of Moder Applied Statistical Methods: Vol. 11 : Iss., Article 0. OI: 10.7/jmasm/ Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at igitalcommos@wayestate. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of igitalcommos@wayestate.

2 Joural of Moder Applied Statistical Methods Copyright 01 JMASM, Ic. November 01, Vol. 11, No., /1/$95.00 Extreme Value Charts ad Aalysis of Meas (ANOM) Based o the Log Logistic istributio B. Sriivasa Rao J. Pratapa Reddy G. Sarath Babu R. V. R. & J. C. College of Egieerig, Gutur, Adhrapradesh, Idia St. A s College for Wome, Gutur, Adhrapradesh, Idia Chebrolu Haumaiah Istitute of Pharmaceutical Scieces, Gutur, Adhrapradesh, Idia A probability model of a quality characteristic is assumed to follow a log logistic distributio. This article proposes variable cotrol charts, termed extreme value charts, based o the extreme values of each subgroup. The cotrol chart costats deped o the probability model of the extreme order statistics ad the size of each subgroup. The aalysis of meas (ANOM) techique for a sewed populatio is applied with respect to log logistic distributio. Results are illustrated usig examples based o real data. Key words: ANOM, LL, i cotrol, equi-tailed, Q-Q plot. Itroductio The probability desity fuctio (PF) of a log logistic distributio (LL) with shape parameter b ad scale parameter σ is give by b 1 x b σ f ( x, b, o) =, b x σ 1+ σ x > 0, σ > 0, b > 1 (1.0.1) ad its cumulative distributio fuctio (CF) is B. Sriivasa Rao is a Associate Professor of Statistics i the epartmet of Mathematics ad Humaities. him at: boyapatisriu@yahoo.com. J. Pratapa Reddy is a Associate Professor of Statistics i the epartmet of Computer Applicatios. him at: jaampratapa@yahoo.co.i. G. Sarath Babu is a Assistat Professor of Statistics. him at: goratla.sarath@gmail.com. x σ F( x, b, σ ) =, b x 1+ σ x > 0, σ > 0, b > 1. b (1.0.) Whe σ = 1 ad b > 1 these equatios are termed stadard PF ad CF. I order to costruct a cotrol chart usig extreme observatios of a subgroup draw from a productio process with the quality variate followig a LL, the percetiles of extreme order statistics from LL samples are eeded. Specifically, the test statistic o the extreme value cotrol chart is the origial sample vector X = (x 1, x,,x ) from ogoig productio. I this chart all idividual sample observatios are plotted ito the cotrol chart without calculatig ay statistics. A corrective actio is tae after oe, or either, of the extreme values amely x (1) (sample miimum) ad x () (sample maximum) of the sample respectively fall above or below specified lies (limits); this is why the chart is called a extreme value cotrolled chart. The Shewart (1986) cotrolled chart is a commo quality cotrol statistical tool: Whe a Shewart chart idicate the presece of a assigable cause, a process adjustmet ca be made if the remedy is ow; otherwise the 9

3 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION suspected presece of assigable cause is regarded as a idicatio of heterogeeity of the subgroup statistic for which the cotrol chart was developed. For example, if the statistic is the sample mea, this leads to heterogeeity of the process mea ad idicates departures from the target mea. Such a aalysis is geerally carried out by dividig a collectio of a give umber of subgroup meas ito categories, such that meas withi a category are homogeous ad those betwee categories are heterogeeous. This procedure, developed by Ott (1967) is called aalysis of meas (ANOM). Whe usig the ANOM techique the cocept of a cotrol chart for meas is viewed differetly, groupig of plotted meas that fall withi or outside cotrol limits. For the homogeeity of the meas it is ecessary that all meas fall withi the cotrol limits. If (1 α) is the cofidece coefficiet, the the probability that all subgroup meas will fall withi the cotrol limits is (1 α). Assumig idepedece of the subgroup, the probability statemet becomes th power of the probability that a subgroup mea will fall withi the limits. This meas that, i the samplig distributio of x, the cofidece iterval for x to lie betwee two specified limits should be equal to (1 α) 1/. This same priciple is adapted through log logistic distributio i this study. This article explores ANOM usig cotrol limits of extreme value statistics cosiderig oly cotrol chart aspects. (See Rao (005) for a detailed descriptio of ANOM; other related wors iclude: Ramig, 198; Bair, 199; Berard & Wludya, 001; Wludya, et al., 001; Motgomery, 001; Nelso & udewicz, 00; Rao & Praumar, 00; Farum, 00; Guirguis & Tobias, 00; Sriivasa Rao & Katam, 01.) Extreme Value Charts The give sample observatios are assumed to follow log logistic mode. The cotrolled lies are determied by the theory of extreme order statistics based o a half logistic model. The cotrolled lies are determied i such a way that a arbitrarily chose x i of X = (x 1, x,, x ) lies with the probability (1 α) 1/ withi the limits. This ca be formulated as a probability iequality as: P(x 1 L) = α/ ad P(x U) = α/. The theory of order statistics states that the cumulative distributio fuctio of the least ad highest order statistics i a sample of size from ay cotiuous populatio are [F(x)] ad 1 [1 F(x)], respectively, where F(x) is a cumulative distributio fuctio (CF) of the populatio. If 1 α is desired at 0.997, the α would equal Taig F(x) as the CF of a stadard log logistic model results i solutios of the equatios 1 [1 F(x)] = ad [F(x)] = which, i tur, ca be used to develop the cotrolled limits of a extreme value chart. The solutios for the two equatios for = (1) 10 with b =,, ad 5 are show i table.1 ad deoted as Z = Z (1) ad Z = Z () The values show i table.1 idicate the followig probability statemets: Z() < Z ( ), i < Z P = i, 1,,..., = (.0.) ad σ Z < x <σz P i, = 1,,..., () i ( ) =, (.0.) x x x x Taig,, ad as a ubiased estimates of σ whe b =, b =, b = ad b = 5, respectively, the equatio becomes x < xi < Z( ),. P = i, = 1,,..., (.0.5) where, for b = : ad () Z 1 (0 0015) = 9

4 SRINIVASA RAO, REY & BABU For b = : ad For b = : ( ) Z ( ) = () Z 1 (0 0015). = ( ) Z ( ) = () Z 1 (0 0015). = ad For b = 5: ad ( ) Z ( ) = () Z 1 (0 0015) = ( ) Z ( ) = Thus, ad costitute the cotrol chart costats for the extreme value charts (see Table. for = (1)10). Table.1: Cotrol Chart Limits of Extreme Value Charts b= b= b= b=5 Z Z Z Z Z Z Z Z

5 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION Aalysis of Meas (ANOM): Log Logistic istributio Whe the data variate follows log logistic distributio, suppose x 1, x,, x are arithmetic meas of subgroups of size draw from a log logistic model. The subgroups meas are used to develop cotrol charts to assess whether the populatio from which these subgroups are draw is operatig with admissible quality variatios. epedig o the basic populatio model, cotrol chart costats may be used. I geeral, the process may be said to be i cotrol if all subgroup meas are withi the cotrol limits; otherwise the process is said to lac cotrol. If α is the level of sigificace of this decisio, the followig probability statemets apply: ( < i, 1,,..., < ) P LCL xi, = UCL = 1 α (.0.6) usig the otio of idepedet subgroups, (.0.6) becomes ( < < ) = ( 1 α ) 1 P LCL x UCL i / (.0.7) With equi-tailed probability for each subgroup mea, two costats, for example L ad U, may be foud such that 1 ( 1 ) ( ) ( ) 1 / α P xi < L = P xi > U = (.0.8) For sewed populatios, such as the LL, it is ecessary to calculate L, U separately from the samplig distributio of x i. Accordigly, these deped o the subgroup size ad umber of subgroups. The percetiles of the samplig distributio of i samples from a log logistic distributio for b =, b =, b = ad b = 5 with σ = 1 were calculated usig Mote-Carlo simulatios (see Tables.1,.,. ad.). Table.: Cotrol Chart Limits of Extreme Value Charts b = b = b = b =

6 SRINIVASA RAO, REY & BABU Table.1: Percetiles of Sample Mea i LL with b = Table.: Percetiles of Sample Mea i LL with b =

7 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION Table.: Percetiles of Sample Mea i LL with b = Table.: Percetiles of Sample Mea i LL with b =

8 SRINIVASA RAO, REY & BABU The percetiles show i Tables.1. are used i equatio (.0.8) for specified ad to determie L ad U for α = 0.05 (see Tables.5,.6,.7 ad.8). A cotrol chart for averages showig i cotrol coclusios idicates that all subgroups meas, though varyig amog themselves, are homogeous i some cells. This is the ull hypothesis i a aalysis of variace techique, hece, the costats show i tables ca be used as a alterative to aalysis of variace techiques. For a ormal populatio Ott s (1967) tables ca be used, ad for a LL the tables show herei ca be used. Table.5: LL Costats for Aalysis of Meas for b =, (1 α) =

9 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION Table.6: LL Costats for Aalysis of Meas for b =, (1 α) =

10 SRINIVASA RAO, REY & BABU Table.7: LL Costats for Aalysis of Meas for b =, (1 α) =

11 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION Table.8: LL Costats for Aalysis of Meas for b = 5, (1 α) =

12 SRINIVASA RAO, REY & BABU Example 1 (Wadsworth, 1986) The followig 5 observatios are from a metal product maufacturig site. Variatios i iro cotet were suspected i raw material supplied by 5 differet suppliers. Five igots were radomly selected from each of the suppliers. The followig table cotais the iro determiatios for each igot by weight from each of the 5 suppliers. Supplier 1 5 The followig cocetratios were obtaied: Catalyst Igot Iro Cotet (g) The goodess of fit of data, as revealed by a Q-Q plot (correlatio coefficiet), for the examples are summarized Table.9, which shows that the log logistic distributio is a better model tha the ormal because it exhibits a sigificat liear relatio betwee sample ad populatio quatiles. Example I a study of brads of batteries, it was suspected that the life (i wees) of the three brads was differet. Five of each brad of battery were tested with the followig results: Battery Life (wees) Brad 1 Brad Brad Table.9: Goodess of Fit ata from Q-Q Plot Example b LL Normal Example Four catalysts that may affect the cocetratio of a compoet i a three compoet liquid mixture were ivestigated. Treatig the observatios i the data as sigle samples, the decisio limits for the ormal ad the LL populatios were calculated ad are show i Tables.10 ad.11 respectively. 50

13 EXTREME VALUE CHARTS AN ANOM BASE ON LOG LOGISTIC ISTRIBUTION Table.10: Normal istributio Examples [LL, UL] (Ott, 1967) Number of Couts I P = i/ Out Out/ 1: =5, =5, α=0.05 [.517,.879] : =5, =, α=0.05 [87.8, 95.5] : =, =, α=0.05 [6.1, 8.8] b = Examples 1 =5, =5, α=0.05 =5, =, α=0.05 =, =, α=0.05 b = 1 =5, =5, α=0.05 =5, =, α=0.05 =, =, α=0.05 b = 1 =5, =5, α=0.05 =5, =, α=0.05 =, =, α=0.05 b = 5 1 =5, =5, α=0.05 =5, =, α=0.05 =, =, α=0.05 Table.11: Log Logistic istributio Number of Couts [LL, UL] I P=i/ Out Out/ [1.55, 5.8] [.1805, 57.0] [.0, 0.95] [1.905, ] [5.510, 59.51] [8.57, ] [.1685, 7.966] [60.88, ] [.8685, ] [.99, 6.019] [65.987, ] [6.766, 98.1]

14 SRINIVASA RAO, REY & BABU Coclusio Ott s (1967) ANOM tables are desiged for ormal distributios, the umber of homogeous meas for each data set was,,, respectively, ad those ot homogeeous are, 1 ad, respectively. Whe the ANOM tables of the proposed model (LL) are used for the same data sets, the umber of homogeous meas are 5, ad, respectively, ad they do ot exhibit deviatio from homogeeity for values of b =, b =,b = ad b = 5. Use of the ormal model resulted i homogeeity for some meas ad deviatio for other meas, thus idicatig possible rejectio of those meas. The rejectio decisio is valid if a ormal distributio is a good fit for the data. However, by compariso, results show that the LL is a better model tha the ormal. Results are supported by the Q-Q plot correlatio coefficiet for each data set with the ormal ad with the LL. It is therefore assumed that more error is liely to be associated with the decisio process whe data are from a ormal distributio, thus, maig all the meas homogeous usig LL (see Table.11) is recommeded over usig the ormal- ANOM procedure. Refereces Bair, S. T. (199). Meas usig the ras for radomized complete bloc desigs. Commuicatios i Statistics-Simulatio ad Computatio,, Berard, A. J., & Wludya, P. S. (001). Robust I-sample aalysis of meas type radomizatio tests for variace. Joural of Statistical Computatio ad Simulatio, 69, Farum, N. R. (00). Aalysis of meas table usig mathematical processors. Quality Egieerig, 16, Guirguis, G. H., & Tobias, R.. (00). O the computatio of the distributio for the aalysis of meas. Commuicatio i Statistics- Simulatio ad Computatio, 16, Motgomery,. C. (000). esig ad Aalysis of Experimets, 5 th Ed. New Yor, NY: Joh Wiley ad Sos Ic. Nelso, P. R., & udewicz, E. J. (00).Exact aalysis of meas with uequal variaces. Techometrics,, Ott, E. R. (1967). Aalysis of meas: A graphical procedure. Idustrial Quality Cotrol,, Ramig, P. F. (198). Applicatios of aalysis of meas. Joural of Quality Techology, 15, Rao, C. V. (005). Aalysis of meas: A review. Joural of Quality Techology, 7, Rao, C. V., & Praumar, M. (00) ANOM-type graphical methods for testig the equality of several correlatio coefficiets. Gujarat Statistical Review, 9, Shewart, W. A. (1986). Statistical method from the view poit of quality cotrol, 1 th Ed. New Yor, NY: Mieola, over Publicatios. Sriivasa Rao, B., & Katam, R. R. L. (01). Extreme value charts ad aalysis of meas based o half logistic distributio. Iteratioal Joural of Quality, Reliability ad Maagemet, 9(5), Wadsworth, H. M., Stephes, K. S., & Godfrey, A. F. (001). Moder methods of quality cotrol ad improvemet. New Yor, NY: Joh Wiley ad Sos, Ic. Wludya, P. S., Nelso, P. R., & Silva, P. R. (001). Power curves for aalysis of meas for variaces. Joural of Quality Techology,,