Sharpening the Boundaries of the Sequential Probability Ratio Test

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1 Wester Ketuck Uiversit TopSCHOLAR Masters Theses & Specialist Projects Graduate School 5-22 Sharpeig the Boudaries of the Sequetial Probabilit Ratio Test Elizabeth Kratz Follow this ad additioal works at: Part of the Statistics ad Probabilit Commos Recommeded Citatio Kratz, Elizabeth, "Sharpeig the Boudaries of the Sequetial Probabilit Ratio Test" (22). Masters Theses & Specialist Projects. Paper This Thesis is brought to ou for free ad ope access b TopSCHOLAR. It has bee accepted for iclusio i Masters Theses & Specialist Projects b a authorized admiistrator of TopSCHOLAR. For more iformatio, please cotact topscholar@wku.edu.

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3 SHARPENING THE BOUNDARIES OF THE SEQUENTIAL PROBABILITY RATIO TEST A Thesis Preseted to The Facult of the Departmet of Mathematics ad Computer Sciece Wester Ketuck Uiversit Bowlig Gree, Ketuck I Partial Fulfillmet Of the Requiremets for the Degree Master of Sciece B Elizabeth Kratz Ma 22

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5 ACKNOWLEDGEMENTS I would like to express m deepest gratitude ad appreciatio to Dr. Melaie Auti for guidig me i this process. If it were ot for her, I would have ever pursued a iterest i statistics ad completed this thesis. She has ivested coutless hours assistig me i ever wa, ad oe of this would have happeed without her. I am trul grateful for her as ot ol as a advisor but also as a metor. Also, I would like to express m gratitude to Dr. David Neal ad Dr. Ngoc Ngue for givig of their valuable time to read ad give correctios to this thesis. Additioall, I would like to ackowledge Dr. Do Edwards for sharig i his itellectual propert. I must also ot forget the people who urged me to pursue m dreams. I am so thakful for Cor as he has stood beside me ad provided moral support eve whe he has bee usure of what he was supportig. Also, I am thakful for m Mom, Dad, Jammie, ad Jimm for their uedig suppl of ecouragemet, support, ad faith i me. iii

6 TABLE OF CONTENTS List of Figures List of Tables Abstract v vi vii Chapter : Itroductio Chapter 2: Sequetial Probabilit Ratio Test 3 2. Wald s Sequetial Probabilit Ratio Test Migoti s Expasio of SPRT Iglewicz s Expasio of SPRT McWilliams Expasio of SPRT 2.5 Igatova, Deutsch, ad Edwards Expasio of SPRT Chapter 3: A Method for Sharpeig the Boudaries 8 3. Creatig the Bouds Results for the New Bouds 24 Chapter 4: Simulatio Studies Simulatio Method Simulatio Results 35 Chapter 5: Coclusio ad Future Work 47 Appedix 49 A. Boudaries for T = 4 49 A.2 R Code 6 A.2. check.step() 6 A.2.2 create.bouds() 6 A.2.3 alpha.bouded() 64 A.2.4 errors.bouded() 66 A.2.5 sim.sprt() 69 A.2.6 Wald.bd() 73 A.2.7 Wald.step.bd() 74 Refereces 75 iv

7 LIST OF FIGURES Figure 2. Full Pascal s Triagle Example 2 Figure 2.2 Modified Pascal s Triagle Method 3 Figure 2.3 Wald Boudar Example 7 Figure 3. Stadard Hpothesis Test 9 Figure 4. Biomial T = Simulatio with π =. ad π =.7 42 Figure 4.2 Biomial T = Simulatio with π =. ad π =.9 43 Figure 4.3 Biomial T = Simulatio with π =.2 ad π =.8 44 Figure 4.4 Biomial T = Simulatio with π =.2 ad π =.9 45 Figure 4.5 Biomial T = Simulatio with π =.3 ad π =.9 46 v

8 LIST OF TABLES Table 3. Possible Sigle Boudaries for T = 4 2 Table 3.2 Total Number of Potetial Boudaries for T 24 Table 3.3 Biomial Results for T = 26 Table 3.4 Biomial Results for T = 2 27 Table 3.5 Biomial Results for T = 3 28 Table 3.6 Biomial Results for T = 4 29 Table 3.7 Hpergeometric Results for T = 3 Table 3.8 Hpergeometric Results for T = 2 3 Table 3.9 Hpergeometric Results for T = 3 3 Table 3. Hpergeometric Results for T = 4 3 Table 4. Hpergeometric Simulatio Results for T = 36 Table 4.2 Hpergeometric Simulatio Results for T = 2 36 Table 4.3 Hpergeometric Simulatio Results for T = 3 37 Table 4.4 Hpergeometric Simulatio Results for T = 4 37 Table 4.5 Biomial Simulatio Results for T = 38 Table 4.6 Biomial Simulatio Results for T = 2 39 Table 4.7 Biomial Simulatio Results for T = 3 39 Table 4.8 Biomial Simulatio Results for T = 4 4 vi

9 SHARPENING THE BOUNDARIES OF THE SEQUENTIAL PROBABILITY RATIO TEST Elizabeth Kratz Ma Pages Directed b: Dr. Melaie Auti, Dr. David Neal, ad Dr. Ngoc Ngue Departmet of Mathematics & Computer Sciece Wester Ketuck Uiversit I this thesis, we preset a itroductio to Wald s Sequetial Probabilit Ratio Test (SPRT) for biar outcomes. Previous researchers have ivestigated was to modif the stoppig boudaries that reduce the expected sample size for the test. I this research, we ivestigate was to further improve these boudaries. For a give maximum allowable sample size, we develop a method iteded to geerate all possible sets of boudaries. We the fid the oe set of boudaries that miimizes the maximum expected sample size while still preservig the omial error rates. Oce the satisfig boudaries have bee created, we preset the results of simulatio studies coducted o these boudaries as a meas for aalzig both the expected umber of observatios ad the amout of variabilit i the sample size required to make a decisio i the test. vii

10 Chapter : Itroductio I statistical aalsis we are cocered with data-drive decisio makig, ad oe aveue that we pursue to test cojectures is hpothesis testig. I hpothesis testig, we test H, the ull hpothesis, versus H, the alterative hpothesis. I a stadard hpothesis test, we take a sample of size, ad make a decisio based o the etire collectio of data. The sequetial probabilit ratio test (SPRT) istead requires that each item be sampled oe at a time. After each item, the researcher either makes a decisio i the hpothesis test or decides to cotiue samplig b selectig aother item. Thus, the SPRT ca result i a sigificatl smaller sample size tha a stadard hpothesis test. I practice, SPRT teds to be more efficiet tha stadard hpothesis testig ad is more cost effective i idustr. For example, SPRT has bee used i detectig Medicare fraud; istead of examiig ever case that a particular doctor turs i for Medicare to pa, the aalst looks at oe case at a time ad either makes a decisio that the case is or is ot fraudulet. From that, the ivestigator ma be able to make a decisio about the doctor without lookig at all of his cases. Thus, moe ad time are possibl saved because the testig will likel termiate before all cases have bee examied. Additioall, i qualit cotrol for idustr, istead of examiig a etire productio lie of goods, a compa could fid it more ecoomical to look at oe product at a time ad determie whether or ot the item is defective ad, i tur, make a decisio about the etire product lie before samplig the etire group of products. Thus,

11 compaies ca icrease profit b spedig less time ad moe o testig for defects i productio. The ca also more quickl idetif ad correct productio problems. For the purpose of this thesis, we are iterested i formulatig a possible method that ca reduce the expected umber of observatios ecessar to make a decisio i the sequetial probabilit ratio test. Some research i this area has bee coducted, but we set out to build upo ad improve the methods that alread exist. I Chapter 2, we first look at the origial SPRT developed b Wald ad the cotiue b lookig at expasios o SPRT b other researchers. Next, Chapter 3 will detail the process of creatig ew sets of boudaries as a exploratio i sharpeig the boudaries of the SPRT. I Chapter 4, we provide a descriptio of simulatio studies that were coducted as a wa to aalze the ew sets of boudaries. Fiall, we summarize the research i Chapter 5 ad examie some improvemets that could be made i the future. 2

12 Chapter 2: Sequetial Probabilit Ratio Test 2. Wald s Sequetial Probabilit Ratio Test I 947, Abraham Wald published a book etitled Sequetial Aalsis [6], i which he details the process for coductig the sequetial probabilit ratio test for qualit cotrol. We defie a radom variable X o the items beig ispected b lettig X = if the item observed is foud to be defective ad X = if the item observed is o-defective. Defie f(x,π) to be the probabilit distributio fuctio of the radom variable X where π is the proportio of defective items i a (possibl ifiite) collectio of size N. Whe cosiderig biar outcomes (defective vs. o-defective), the first successive observatios of x will be desigated as x, x 2,, x, where x i idicates whether the item is foud to be defective or o-defective. It is of iterest to test two competig values of π. We let H be the hpothesis that π = π ad H be the hpothesis that π = π, where π < π. Thus, f(x,π ) is the distributio fuctio of X whe the ull hpothesis is true, ad f(x,π ) is the distributio fuctio of X whe the alterative hpothesis is true. We let = x i i = be the total umber of defective items i the sampled items. For a positive value, the probabilit that a sample x,, x is i, i= observed is determied b the likelihood fuctios L(, π ) Π f ( x π ) is true ad L( π ) Π f ( x, ), i π i = = whe H = whe H is true. The sequetial probabilit ratio test (SPRT) for testig H agaist H is determied b the followig process. 3

13 Two positive costats A ad B are chose such that B < A. At each stage of the test ( th trial), compute Λ L = L (, π ) (, π ). If B Λ A, cotiue the test b takig aother observatio. If A < Λ, the test is eded, ad the ull hpothesis is rejected. If Λ < B, the test is eded, ad we fail to reject the ull hpothesis. For practicalit i computatio, l(λ ) is ofte used over Λ sice l(λ ) ca be writte as the sum of terms, i.e l( Λ ) f = l f ( x, π) ( x, π ) f l f ( x ), π ( x, π ) The i th term of the sum is deoted as z i. Thus, at each stage of the test ( th trial). z i i = the cumulative sum is computed. If l( B) zi l( A), i = cotiue the test b z i i = takig more observatios. If ( A) < l, the test is eded, ad H is rejected. If i = z i < l( B), the test is eded, ad H is ot rejected. Assumig that the test has ot termiated prior to a give step, we cosider a sample (x,, x ). We sa that the sample is of tpe if A < Λ, which leads to rejectio of H at step. We sa that the sample is of tpe 2 if Λ < B, which leads to failig to reject H at step. For a tpe sample, the probabilit of achievig such a sample uder H is at least A times as large as the probabilit of achievig such a sample uder H, as see b L, π ) A L(, ). Therefore, the probabilit that the ( π sequetial test will termiate with rejectio of H is also at least A times as large uder H as it is uder H. Whe H is true, the probabilit that the test will 4

14 termiate with rejectio of H is. This icorrect decisio is a Tpe I error, where is the Tpe I error rate. Whe H is true, the probabilit that the test will correctl termiate with rejectio of H is deoted as. Thus we obtai, L(, π ) A L(, π ) A A, (2.) which determies a upper limit for A to be. I a similar maer, a lower limit for B ca be derived. For a give tpe 2 sample (x,, x ), the probabilit of achievig such a sample uder H is at most B times as large as the probabilit of achievig such a sample uder H, L(, π) B L(, π ). Therefore, the probabilit that the sequetial test will termiate with failig to reject H is also at most B times as large uder H as it is uder H. Whe H is true, the probabilit that the test will correctl termiate without rejectio of H is deoted. Whe H is true, the probabilit that the test will icorrectl termiate with failure to reject H is. This icorrect decisio is referred to as a Tpe II error, where is the Tpe II error rate. Thus we obtai, L(, π ) B L(, π ) B( ) B, (2.2) which determies a lower limit for B to be,. Cosequetl, upper limits are obtaied for ad based o the above iequalities: 5

15 ad B. (2.3) A We must also ow cosider the opposite approach i which we are ot give the values of A ad B. Previousl, we chose a give A ad B ad determied the Tpe I ad Tpe II error rates. I practice, testig will be coducted uder predetermied maximum allowable error rates ( ad ), ad A ad B must istead be determied. We would like to test with give stregth (, ); so, deote A(, ) ad B(, ) as the values of A ad B which satisf the predetermied (, ). From the previousl metioed iequalities, (2.) implies A (, ), ad (2.2) implies (, ) B. Wald [6] proposed to set A = = a(, ), which is i fact greater tha or equal to the exact value of A(, ), ad B = b(, ) = which is less tha or equal to the exact value of B(, ). Wald discusses the cosequeces that the assiged combiatios of A ad B have o the error rates ad. Whe A is a value greater tha or equal to A(, ) ad B is equal to B(, ), the Tpe I error rate is less tha the omial, but the Tpe II error rate is greater tha. If A is equal to A(, ) ad B is a value less tha or equal to B(, ), the the Tpe II error rate is less tha the omial, but the Tpe I error rate is greater tha. If A is a value greater tha A(, ) ad B is a value less tha B(, ), the the effect o the Tpe I ad Tpe II error rates is uclear. Wald thus proposes to deote ad as the respective error rates whe A=a(, ) ad B=b(, ). 6

16 7 From above, ( ) =, ' ' a ad ( )., ' ' = b Thus, upper limits for ad ca be foud to be ' ad. ' Through some arithmetic, it ca be show that. ' ' + + Thus, we set ) ( = A ad ) ( = B. We ow cosider the case of samplig with replacemet (or samplig without replacemet from a populatio of ifiite size). Sice the probabilit of selectig a defective item is π for each item ad observatios are idepedet, each observatio ca be cosidered a Beroulli trial. Thus, Wald s boudaries ca be foud as follows. I the likelihood ratio Λ, the biomial desit ca be substituted for the likelihood fuctios, givig ( ) ( ) ( ) ( ).,, L L = = Λ π π π π π π. (2.4) Usig the criterio B Λ A preseted b Wald, ( ) ( ) ( ) ( ).,, A L L B = π π π π π π (2.5) Notice that istead of computig this ratio at each observatio, we ca solve for as follows:

17 8 ( ) ( ) ( ) ( ) ( ). ) ( ) ( l ) ( l ) l( ) ( ) ( l ) ( l ) l( l l ) ( ) ( l ) l( ) l( ) ( ) ( l ) l( + π π π π π π π π π π π π π π π π π π π π π π A B A B A B (2.6) Notice that is ow bouded above ad below b parallel lies. Thus, if the total umber of defectives at the curret positio is betwee or o these lies, is said to fall i the zoe of idifferece. Although values of ma occur here, samplig cotiues withi this regio, ad we therefore do ot refer to these results as possible outcomes of the test. We refer to values that are above the upper boudar lie as beig i the rejectio regio, for these are the outcomes that would lead to rejectio of the ull hpothesis. Though we ever actuall accept H whe values of are below the lower boud, istead of referrig to the regio as the zoe of failig to reject, for simplicit we call it the acceptace regio; values of i this regio are the outcomes that would lead to failig to reject the ull hpothesis. Wald [6] has show that the sequetial probabilit ratio test will evetuall ed i a decisio beig made with a probabilit of, but i practice, the sequetial test is trucated at a maximum allowable sample size, T. If a decisio has ot bee made prior to reachig the T observatio, samplig stops

18 with the T observatio. The a decisio is made based o the value of T. We sa that sequetial testig is a closed procedure because testig is limited to a maximum allowable size. Research o SPRT has cotiued sice its first itroductio b Wald, ad selected examples are preseted i the followig sectios of this chapter. 2.2 Migoti s Expasio of SPRT Researchers have also bee iterested i determiig the effects that the choices of the hpothesized values, π ad π, have o the required sample size. Migoti [3] discusses the sample size eeded to perform Wald s sequetial probabilit ratio test whe items are geerated b a process for which the results of the ispectios are correlated. It was show that for values of π ad π which are close together, the sample size is larger tha that which are foud to be ecessar whe usig values of π ad π that are more distat. Additioall, whe cosiderig correlatio (ρ) betwee the results of the observatios with respect to a biar respose variable, the sample size icreases as the value of the correlatio coefficiet ρ icreases. I fact, whe comparig the correspodig expected sample size for whe ρ = to whe.5 < ρ <.7, the expected sample size is three times higher; it is six times higher whe ρ > Iglewicz s Expasio of SPRT It is most commo for sequetial procedures to be compared i terms of the expected sample size, which is also kow as the average sample umber 9

19 (ASN). This tpe of compariso does ot cosider sample size variabilit ad is ofte biased i favor of tests with large sample-size variabilit. I terms of practicalit, large sample-size variabilit is costl because it ofte leads to a large umber of observatios beig required before the process termiates. Thus, Iglewicz [] proposes a alterative measure for the ASN, D( π ) N E = σ ( π ) ( π ) where N is the umber of required observatios for a fixed sample size test, π is the true value of the parameter, E( π) is the average sample umber, ad σ (π) is sample-size stadard deviatio. D(π) is a measure of the gai obtaied b usig a sequetial procedure i place of a fixed sample size procedure. Whe D(π) is large, the sequetial test is efficiet, but whe D(π) is small, the desig is ot., 2.4 McWilliams Expasio of SPRT Due to the lack of published research regardig trucatio o the SPRT, McWilliams [4] presets two case studies that ispect the ifluece that the choice of trucatio parameters have o test performace. He examies both the choice of the trucatio sample size ad the choice of the trucatio rejectio value R. Whe the test reaches the trucatio sample size, R idetifies how ma defectives are required for rejectio. Through the explaatio of the two case studies, both choices appear have a deep impact o test performace. McWilliams compares the ASN uder both the ull hpothesis, π = π, ad the alterative hpothesis, π = π, sice the value of the ASN will differ depedig

20 upo the true value of π i the populatio. Additioall, the author examies the amout of variabilit i required sample size ad the value of the error rates. McWilliams cocludes that adoptig a aggressive trucatio strateg results i a cosequeces i terms of the error probabilities ad the ASN. The author does ot geeralize his results but expresses that, from his fidigs, the performace of the test over various values of R eeds to be cosidered istead of a arbitrar choice of the rejectio value. 2.5 Igatova, Deutsch, ad Edwards Expasio of SPRT While cosiderig closed sequetial or multistage samplig from a lot of N items, the goal is to make coclusios about the proportio of defectives, π, i the sample. Istead of usig asmptotic results, Igatova, et. al. [2] show that exact iferece about the proportio of defectives ca be calculated usig curret statistical software. While cosiderig to be the umber of defectives at stage, a decisio is made whether to cotiue samplig or to ed testig b comparig to a set of boudaries. Sice, i practice, sequetial testig is trucated at T, the trucated SPRT is the sole focus for the authors. For a give outcome,, there are sequeces of defectives ad ( ) o-defectives that result i a total of defectives i trials. As a example, cosider the case of T =, show i Figure 2.. Each umber represeted i Figure 2. represets the umber of was to get (o the -axis) defectives i trials (o the x-axis); we

21 refer to this as a path-cout. Thus, we ca cosider the umber of paths to a specific poit (, ) o the graph. Arbitraril chose upper ad lower boudaries are show as a referece, but o modificatios i the path couts have bee made as a result of outcomes fallig outside of the boudaries. The urestricted path-couts are foud via Pascal s triagle. Figure 2.: Full Pascal s Triagle Example 2

22 The boudaries must ow be take ito accout to determie the actual umber of paths to. A observatio occurrig outside of the boudaries results i samplig to ed; thus, the paths extedig from that poit o loger exist. Based o a modified Pascal s triagle algorithm, [2] utilizes a path-cout method to determie the various paths that lead to specific termiatio poits, while staig withi the boudaries. A visual example of the path-cout method ca be see i Figure 2.2. Figure 2.2: Modified Pascal s Triagle Method 3

23 Note that the solid dots that appear i Figure 2.2 are outcomes for which a decisio is made ad samplig eds. There are ideed values of which exist betwee the lower ad upper bouds, but o decisio is made i the test ad samplig cotiues. Notice that poits (4,), (6,), (9,2), ad (,) all occur outside of the upper ad lower boudaries show. Thus, paths extedig from (4,), (6,), ad (9,2) do ot exist, ad the path-cout for successive poits are altered. For example, the path-cout to poit (5,) reduced from five i the Pascal s triagle example (Figure 2.) to four i the modified Pascal s triagle method; the path-cout to poit (7,2) reduced from 2 to 4. Additioall, otice that some of the paths to the outcomes o the trucatio boudar are also reduced. For example poits (,3) ad (,5) reduced from 2 to 62 ad 252 to 242, respectivel. Also, otice that ot all path-couts were altered. Igatova, et. al. cosider this method for both the biomial ad hpergeometric cases. The have crossed ito ew territor b examiig the hpergeometric case, as o other authors have pursued it [2]. Now cosider the case of testig without replacemet from a populatio of a fiite size N. Thus, Wald s boudaries ca be determied where is a hpergeometric radom variable. Similar to what was doe i the biomial case i (2.4), the hpergeometric desit ca be substituted for the likelihood fuctios, givig 4

24 5 = = = Λ D N D D N D N D N D N D N D D L D L ), ( ), (. (2.7) where D is the umber of defective items i the populatio uder H ad D is the umber of defectives i the populatio uder H. Usig the criterio B Λ A preseted b Wald, we have. A D N D D N D B < < (2.8) I the pursuit of trig to isolate i (2.8), we begi trig to simplif, givig ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).!!!!!!!!!!!!!!!!!!!! D N D D N A D D N D D N D D N D D N B D A D N D D N D D N D D N D B < + + < < + + < (2.9) Although it was attempted, isolatio of could ot be determied at this time. The authors created a fuctio etitled seqbi, usig the statistical software package R [5], to determie sequetial acceptace samplig for biar outcomes from a populatio of size N while samplig with or without replacemet. Based o Wald s SPRT method, a trucatio positio T ad upper ad lower boudaries are give as argumets to the fuctio. The authors determied that Wald s boudaries ca be represeted as a step boudar

25 because as icreases b oe, the value of stas the same (if that observatio is o-defective) or icreases b oe (if that observatio is defective). Samplig cotiues if is betwee or o the boudaries ad < T. Usig the boudar-modified Pascal algorithm, the seqbi fuctio detects all possible outcomes (ad the umber of paths possible to reach that outcome) where is less tha the lower boud or greater tha the upper boud (i.e., where the test does ot stop before reachig T ) or where is o the trucatio boudar. Additioall, the seqbi fuctio calculates the umber of defectives sampled ad calculates the probabilit distributio of all possible outcomes. We ow cosider a example of Wald s trucated SPRT for testig the ull hpothesis that π =.2 versus the alterative hpothesis that π =.8, with T =, =.5, ad =.. Usig A = ( )/ ad ( ) / as defied i sectio 2., we have A = 8, ad B.53. From (2.6), the lower boudar lie is give b l , ad the upper boudar lie is give b u , idicated b the dotted lies i Figure 2.3. The blue dashed lie ad red lie represet step-boudar represetatios of the Wald upper ad lower boudaries, respectivel. The poits i Figure 2.3 are the possible outcomes; otice that poits exist i both the rejectio regio ad acceptace regio. The poits that occur at = T = represet the outcomes that occur whe samplig has ot eded before reachig the trucatio boudar. It is ecessar that each of these poits be icluded i either the rejectio regio or the acceptace regio sice a decisio eeds to be made i the test. 6

26 Figure 2.3: Wald Boudar Example 7

27 Chapter 3: A Method for Sharpeig the Boudaries 3. Creatig the Bouds Based o the work previousl completed b others, the goal of this thesis is to determie if there are better boudaries available tha the oes which have bee curretl foud i [6] ad [2]. We would like to cosider all possible sets of boudaries that exist for a give T. We will the use certai criteria to determie which of these sets of boudaries is the best. We first cosider the set of boudaries that are equivalet to the stadard hpothesis test: the lower boudar is the lie =, ad the upper boudar is the lie =. I other words, the lower boud icludes successive observatios of all o-defective items ad the upper boud icludes successive observatios of all defective items. Observatios here are alwas i the zoe of idifferece, so the sample size will alwas reach the trucatio value of T. This gives boudaries that form a triagle, as ca be see i Figure 3. for the case whe T = 4. 8

28 Figure 3.: Stadard Hpothesis Test Startig with this set of triagle boudaries, we will fid all possible sets of boudaries for a give T. We the wat to determie all sets of upper ad lower boudaries that have both the Tpe I ad Tpe II error rates bouded. The we would like to examie uder various criteria which set of boudaries is ideed the best. For a give T, we must first geerate ever possible set of boudaries. Whe a sample is take at = (i.e., ol the first item is sampled), either zero defective items ca be observed (x = = ) or oe defective item is observed (x = = ); thus, ever upper ad lower boudar is restricted to begiig at either = or =. Next, it is required that the step size is o-decreasig ad of at most oe because as the umber of observatios icreases b oe, the value of the umber of defectives ( ) ca ol either sta the same or icrease 9

29 b oe. Additioall, the upper ad lower boudaries are created such that the ca touch oe aother but ma ot cross. To create all sets of possible boudaries, we first create all possible sigle boudaries. Oce this has bee doe, we combie all possible boudaries such that we have lower ad upper boudaries that do ot cross. To create all of the sigle boudaries, we first desigate the iitial boudar as (,,,,), a boudar of T zeros (i.e., all T sampled items are o-defective). For a idex i iitiall set equal to zero, we add to the T - i positio i the boudar, as we keep the bouds that are o-decreasig with a step-size of at most. After ca o loger be added to the T - i colum due to the step-size requiremet, i is icremeted b oe, ad the process cotiues of addig to the ext positio to the left util the step-size requiremet is o loger fulfilled, ad i is agai icremeted. The process cotiues successivel util the boudar that represets a sample of T defective items, (,2,3,, T ), is obtaied. Workig backwards, we the desigate that the boud begis as (,2,3,, T ) ad for a idex i iitiall set equal to zero, we subtract from the T - i positio i the boudar as we keep the bouds that are o-decreasig with a step-size of at most. After ca o loger be subtracted from the T - i positio due to the step-size requiremet, i is icremeted b oe, ad the process cotiues of subtractig from the ext positio to the left util the step-size requiremet is o loger fulfilled, ad i is agai icremeted. The process cotiues util the (,,,,) boudar is obtaied. We the combie all bouds that have bee foud ad delete a duplicate boudaries. Table 3. displas the boudaries 2

30 geerated for T = 4 via the above described process (before deletig duplicate boudaries). Table 3.: Possible Sigle Boudaries for T = 4 Buildig Up Buildig Dow We the combie this master set of boudaries ito sets of lower ad upper boudaries, keepig ol the boudaries which do ot cross oe aother. A R [5] fuctio called create.bds has bee created to implemet this procedure (see Appedix A.2.2). A example of all possible sets of boudaries for T = 4 ca be foud i Appedix A.. Oce all of the possible sets of boudaries are created, we are left to determie which set of boudaries is the best. I practical testig, ou will be give a maximum allowace for Tpe I ad Tpe II error rates. Thus, we must reduce the possible set of boudaries b first determiig which sets of boudaries preserve, the predetermied maximum allowable Tpe I error rate. To determie if is preserved, each set of upper ad lower T boudaries is cosidered idividuall. A give set of bouds is first evaluated usig the seqbi 2

31 fuctio. Usig this, we the determie which of the outcomes are i the rejectio regio, which are i the acceptace regio, ad which are o the trucatio boudar. The, assumig that the ull hpothesis is true (π = π ), the outcomes i the rejectio regio, if the exist, are used to calculate. For each outcome i the rejectio regio, we cosider the probabilit of that outcome occurrig, P (( )) out, out, assumig π = π where out ad out deote the umber of items sampled ad the umber of defectives, respectivel, for a give outcome. (( )) P, out out is foud b takig the product of the probabilit mass fuctio ad the proportio of paths staig withi the boudaries leadig to (, out out), as show b [2]. The, is calculated b takig the sum of (( )) P, out out for each outcome i the rejectio regio. If o outcomes appear i the rejectio regio, = iitiall. It is importat to ote that at this stage, the outcomes o the trucatio boudar are ot et cosidered. Oce has iitiall bee calculated for a give set of bouds, we determie if this iitial value of is ideed bouded b the pre-specified value. All sets of boudaries which preserve the give Tpe I error rate are retaied, ad the remaiig sets of boudaries are elimiated. A R [5] fuctio called alpha.bouded has bee writte to determie the sets of boudaries for which the Tpe I error rate is preserved (see Appedix A.2.3). From the remaiig sets of boudaries, we must the determie which of those sets of bouds also maitai, the predetermied maximum allowable Tpe II error rate. To determie if is maitaied, each set of upper ad lower 22

32 -bouded boudaries for T must be cosidered. First, a give set of - bouded bouds is evaluated usig the seqbi fuctio. Usig this, we the determie which of the outcomes are i the rejectio regio, which are i the acceptace regio, ad which are o the trucatio boudar. The, assumig that the ull hpothesis is true (π = π ), the outcomes i the rejectio regio, if the exist, are used to calculate as explaied above. If o outcomes appear i the rejectio regio, = iitiall. Now the outcomes that occur o the trucatio boudar eed to be cosidered, ad a decisio i the test must be made. Begiig with the outcome o the trucatio poit closest to the rejectio regio, we add that outcome to the rejectio regio as log as remais bouded b our desired sigificace level. Each successive outcome o the trucatio poit is added to the rejectio regio util is beod our desired boud. Thus, the remaiig trucatio outcomes are the added to the acceptace regio. Assumig that the alterative hpothesis is true (π = π ), is calculated usig the sum of P( ( )) for all of the ( ) out, out outcomes i the out, out acceptace regio. All sets of boudaries which preserve the give Tpe II error rate are retaied, ad the remaiig sets of boudaries are elimiated. A R [5] fuctio called errors.bouded has bee writte to determie the sets of boudaries for which both the Tpe I ad II error rates are preserved (see Appedix A.2.4. Oce the sets of boudaries that preserve both ad have bee collected, we begi trig to discover which set of boudaries is the best for give values of T, π, ad π. Oe wa to defie which set of boudaries is the 23

33 best is to idetif which set of boudaries miimizes the expected umber of observatios required to make a decisio i the hpothesis test. Sice the true value of π is ukow, we use a fie grid of π-values from to i steps of. to assist i determiig the expected umber of total observatios required to make a decisio i the test, deoted as E(). Usig the value of at each outcome, out, ad the probabilit of out over all outcomes, we are able to determie the expected umber of observatios to be E( ) = P( ). out (3.) out all out Oce E() is calculated usig π equal to each of the values, we determie the maximum expected umber of observatios, maxe()}, for each set of - ad -bouded boudaries. We the determie the best set of boudaries to be that which miimizes maxe()}. 3.2 Results for the New Bouds The method described i sectio 3. has bee coducted for T =, 2, 3, ad 4 i both the biomial ad hpergeometric cases with all combiatios of π =.,.2, ad.3 ad π =.,.2,.3,.4,.5,.6,.7,.8, ad.9. The determied total umber of possible sets of lower ad upper boudaries (disregardig coservatio of the error rates) is show i Table 3.2. Table 3.2: Total Number of Potetial Boudaries for T T Number of Boudar Sets 4, , , ,893 24

34 I the case of the biomial distributio, for each combiatio of T, π, ad π the origial Wald boudar has also bee calculated as a compariso to determie if this ew method is ideed a improvemet. R [5] fuctios writte to fid the Wald boudaries ca be foud i Appedices A.2.6 ad A.2.7. Because the likelihood ratio i the hpergeometric case caot be simplified to have parallel upper- ad lower-boudar lies as i the biomial case, determiig a closed form for the Wald upper ad lower boudaries has limitatios as see i (2.9). Thus, the Wald boudaries for the hpergeometric case have ot bee cosidered at the preset time. I Tables 3.3 through 3., the results for the best ew boudaries appear for both the biomial ad hpergeometric distributios. I the biomial case, the correspodig Wald results also appear. For each π ad π combiatio that has bee cosidered, the value of the miimized maxe()} is give alog with the correspodig values of ad. The combiatios of π ad π which appear as a shaded box are ot possible due to the coditio that π < π. The combiatios of π ad π which are empt idicate that o boudaries where both ad are bouded were foud. 25

35 π..2.3 New Wald New Wald New Wald Table 3.3: Biomial Results for T = π

36 π..2.3 New Wald New Wald New Wald Table 3.4: Biomial Results for T = 2 π

37 π..2.3 New Wald New Wald New Wald Table 3.5: Biomial Results for T = 3 π *

38 π..2.3 New Wald New Wald New Wald Table 3.6: Biomial Results for T = 4 π *

39 π..2.3 New New New Table 3.7: Hpergeometric Results for T = π π..2.3 New New New Table 3.8: Hpergeometric Results for T = 2 π *

40 π..2.3 New New New Table 3.9: Hpergeometric Results for T = 3 π * π..2.3 New New New Table 3.: Hpergeometric Results for T = 4 π *

41 I Tables 3.3 through 3., results marked with a asterisk (*) have multiple sets of boudaries with the same maxe()} ad value whe testig π vs. π. Thus, the result preseted i the table is the set of boudaries that has the smallest value of. Takig ote from the results preseted i Tables 3.3 through 3., it ca be see that for T =, the ew boudaries foud for testig π =. vs. π =.7, π =. vs. π =.9, π =.2 vs. π =.8, π =.2 vs. π =.9, ad π =.3 vs. π =.9 all reduced the miimized maxe()} whe compared to the Wald boudaries for the biomial case. For T =, the ew boudaries foud for testig π =. vs. π =.7 reduces the expected umber of observatios b.6875, which was greatest differece observed. It was ot util close to the ed of coductig this research that the Wald boudaries were calculated ad aalzed. Ol after aalzig this summar iformatio did we otice a discrepac betwee the Wald boudaries ad the ew boudaries. Because we set out to determie all sets of possible boudaries for a give T, the Wald boudaries should have bee icluded i our possible set of boudaries with both ad bouded. Thus, the ew best boudaries should be o worse tha the Wald boudaries. Whe aalzig the results, we should have see that if the miimized maxe()} of the ew boudaries was ot smaller tha the maxe()} of the Wald boudaries, the the miimized maxe()} should have at most bee equal to that of the Wald boudaries. 32

42 After retracig our steps, it was discovered that although our R fuctios correctl reduce a give collectio of lower/upper boudaries to those for which both ad are bouded, the process i which we geerated all possible boudaries was ot exhaustive for all T. The idea of creatig all possible bouds for T uder the coditios that the lower ad upper bouds must both begi at zero or oe, the lower ad upper bouds must have a o-decreasig step size of at most oe, ad the lower ad upper bouds ma touch oe aother but ot cross is correct. Thus, we discovered that ot all possible boudaries uder the three coditios were geerated usig the algorithm that we created; ol a subset of the possible boudaries was geerated. Ufortuatel, as T grows larger, it appears that more ad more boudaries were missed. Whe creatig the method for geeratig all possible boudaries for T, we tested the method b had for several small T values ad icorrectl assumed it would work for all T values. Sice T = is the smallest T value icluded i this research, we were able to fid ew boudaries with smaller miimized maxe()} tha their correspodig Wald boudaries because eough of the possible boudaries were geerated. Thus, we are optimistic that this method actuall would sharpe ma more of the boudaries for the SPRT if all possible boudaries are geerated. 33

43 Chapter 4: Simulatio Studies 4. Simulatio Method To test the performace of the boudaries, a simulatio stud was performed for each set of ew T -boudaries that miimized the maxe()} for a specified π so that we could observe the value of the required sample size ad the decisio made i each simulatio. Because we have two values beig tested of the kow ukow parameter π, π = π ad π = π, we performed each simulatio stud for a set of bouds uder both assumptios idividuall, similar to what was doe i [4]. For each set of lower ad upper bouds which miimize the maxe()}, the rejectio ad acceptace regios are determied ad decisios are made regardig the outcomes o the trucatio boudar. Whe samplig from a ifiite populatio (or samplig with replacemet), sample data is created with the specified probabilit of success, π = π or π = π, b usig the biomial radom variable geerator i R with T trials [5]. The total umber of defectives at stage is calculated for T. Whe samplig without replacemet, a populatio with a specified size of, is created with the correct proportio of defective ad o-defective items. From the created populatio we radoml sample T items from the populatio without replacemet (resultig i a hpergeometric radom variable), ad the umber of defectives at stage is calculated for T. The SPRT is the performig usig this geerated data ad the give set of boudaries. For each simulatio settig (a give value of π, 34

44 a give set of boudaries, ad a give distributio),, samples were simulated ad aalzed with SPRT. For each set of ew boudaries that were determied to miimized the maxe()}, simulatios were ru for both π = π ad π = π. We also performed simulatios for the boudaries where the maxe()} was withi oe of the miimized maxe()}. Additioall, whe samplig with replacemet, the Wald boudaries were also studied via simulatio. Sice the simulatios were coducted before the results were recogized as icomplete, 53 Wald boudaries, 26 miimized maxe()} boudaries, ad,26 boudaries with a maxe()} withi oe of the miimized maxe()} were aalzed. The R [5] fuctio created for the simulatio stud ca be foud i Appedix A Simulatio Results Although we kow that we do ot have all possible boudaries, we also do ot have the Wald boudaries with which to compare i the hpergeometric case. Thus, for each combiatio of π ad π we preset maxe()} (calculated as described i sectio 3.), the simulated average required umber of observatios whe π = π, deoted as Ê ( ) umber of observatios whe π = π, deoted as ˆ ( ), ad the simulated average required E. These results ca be see i Tables 4. through 4.4. For maxe()} we otice that for π =., as π icreases, the value of maxe()} either stas the same or decreases; this is cosistet with the results foud i [3]. Cosiderig π =.2 ad the lowest satisfig π value, maxe()} starts at a higher value tha the lowest 35

45 combiatio of π =. ad π, but the patter does cotiue that as π ad π become more distat, the value of maxe()} either decreases or stas the same. However, i simulatio, this patter also came ver close to holdig true for Ê ( ) ad ˆ ( ) E. If we had icreased the umber of simulatios, it is possible that the patter would have alwas held for Ê ( ) ad ( ) Ê. Table 4.: Hpergeometric Simulatio Results for T = E } Ê π π max ( ) Ê ( ) ( ) Table 4.2: Hpergeometric Simulatio Results for T = 2 E } Ê π π max ( ) Ê ( ) ( ) *

46 Table 4.3: Hpergeometric Simulatio Results for T = 3 E } Ê π π max ( ) Ê ( ) ( ) * Table 4.4: Hpergeometric Simulatio Results for T = 4 E } Ê π π max ( ) Ê ( ) ( ) *

47 I Tables 4.2 through 4.4, results marked with a asterisk (*) i the maxe()} colum have multiple sets of boudaries with the same maxe()} whe testig π vs. π. Thus, the result preseted i the table is for oe such set of boudaries. Sice we had the available Wald boudaries for compariso i the biomial case, we also cosidered maxe()}, Ê ( ), ad Ê ( ) for each combiatio of π ad π usig the Wald boudaries, as see i Tables 4.5 through 4.8. I Tables 4.7 ad 4.8, results marked with a asterisk (*) i the maxe()} colum have multiple sets of boudaries with the same maxe()} whe testig π vs. π. Thus, the result preseted i the table is for oe such set of boudaries. Table 4.5: Biomial Simulatio Results for T = Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald Ê ( ) 38

48 Table 4.6: Biomial Simulatio Results for T = 2 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald Ê ( ) Table 4.7: Biomial Simulatio Results for T = 3 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald * Ê ( ) 39

49 Table 4.8: Biomial Simulatio Results for T = 4 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald * Ê ( ) We observed that the same patter described i [3] was exhibited i the biomial case for both the ew boudaries ad the Wald boudaries, as it was i the hpergeometric case. After closer observatio, we also foud that for a few of the ew boudaries, Ê ( ), ad Ê ( ) were smaller tha that of their Wald couterparts. These istaces are highlighted i Tables 4.5 through 4.8. To be exact, Ê ( ) was foud to be better o four occurreces, ad Ê ( ) was foud to be better o three occurreces, all occurrig whe T =. These istaces correspod to the ew boudaries that were foud to be better tha the Wald boudaries, described i sectio 3.2. For a additioal qualificatio of what could be cosidered as the best boudar, we have also cosidered the boudaries that i simulatio miimize 4

50 the variabilit i required sample size, similar to []. Sice ol five ew sets of boudaries were foud to have miimized maxe()} i compariso to the Wald boudaries, we ol preset further results for the five cases where a improvemet was foud. However, ma more simulatio results are available from the author. Figures 4. through 4.5 detail the results for the combiatios of π ad π for which a improvemet i maximum expected sample size was foud (with the ew boudaries labeled as Mi ). For each set of boudaries, the horizotal dotted lie exteds oe stadard deviatio above ad below Ê ( ), which is idicated b the ope circle. Similarl, the solid horizotal lie exteds oe stadard deviatio above ad below Ê ( ), which is idicated b the ope circle. Additioall, the maxe()} is preseted as a solid circle. Sice we also coducted simulatios for the sets of boudaries whose maxe()} is withi oe of the miimized maxe()} value, we preset results for those observatios as well. These boudaries are labeled as Mi + i the figures, ad the subscript serves as a sigifig idex for each cosidered set of boudaries for the hpotheses beig tested. 4

51 Figure 4.: Biomial T = Simulatio with π =. ad π =.7 I Figure 4., we otice that uder the ull hpothesis (π =.), based upo the simulatio performed, all ew boudaries cosidered ted to require a smaller sample size to make a decisio i the test tha Wald s boudaries; however, less variatio is observed i the ew boudaries uder the alterative hpothesis (π =.7). To the cotrar, uder the alterative hpothesis, the use of Wald s boudaries teds to require a smaller sample size tha the ew boudaries (except for the ew best boudar), ad the ull case exhibits less 42

52 variatio for the Wald boudaries. Additioall, all ew maxe()} (except for Mi + ) are less tha maxe()} for Wald. Figure 4.2 demostrates that although maxe()} has bee reduced with the ew boudaries, uder simulatio Wald s boudaries result i more cosistec i that it has small variatio uder both the ull (π =.) ad alterative hpotheses (π =.9). I simulatio usig the ew best bouds ad the bouds whose maxe()} are withi oe, uder the ull hpothesis all but oe (Mi + 7 ) has less variatio tha uder Wald s simulatio. Figure 4.2: Biomial T = Simulatio with π =. ad π =.9 43

53 Figure 4.3 shows that although maxe()} is reduced, Wald appears agai to be more cosistet uder both the ull (π =.2) ad alterative (π =.8) hpotheses. Also, maxe()} for the ew bouds is the ol maxe()} value which is less tha Wald s. Figures 4.4 ad 4.5 demostrate that uder the alterative hpothesis (π =.9), much more variabilit is show i the ew best bouds tha i Wald s boudaries. Figure 4.3: Biomial T = Simulatio with π =.2 ad π =.8 44

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