Results of an investigation into the sensitivity to two types
of inspection error of link sampling procedures described by
Harishchandra and Srivenkataramana are reported. Relevant compound
distributions are derived. Some comparisons with results obtained
in a similar investigation for standard double sampling are also
given

Johnson, Norman L.

Kotz, Samuel

Rodriguez, Robert N.

Carolina Digital Repository

N-9Inspection Errors in Link SamplingbySamuel Kotz Norman L. JohnsonUniversity of Maryland University of North CarolinaCollege Park Chapel HillandRobert N. RodriguezSAS Institute, Cary, North Carolinay2 1983am December 1983" lhis documni-it hs beeri i; -for public release and sole; tsdistribution is unlimite.83 12 09 0974.. -.... ""<" ' , ...ABSTRACTResults of an investigation into the sensitivity to two typesof inspection error of link sampling procedures described byHarishchandra and Srivenkataramana are reported. Relevant compounddistributions are derived. Some comparisons with results obtainedin a similar investigation for standard double sampling are alsogiven.Key Words and Phrases: Quality control; Compound distributions;Binomial distribution; Hypergeometric distribution; Multivariatehypergeometric distribution; Two-stage sampling.MmI, *(- - ... I I I . . . . . .1. IntroductionIn two stage acceptance sampling, it is necessary to inspect furtheritems from a lot when the evidence provided by the first sample is incon-clusive. Commonly the second sample is twice as large as the first, thoughthis is by no means universal. In order to save sampling costs, sometimesthe additional evidence is provided from the results of standard ('firststage') inspection on the immediately preceding and following lots. Thereis, of course, an implicit assumption that these neighboring lots are ofabout the same quality as the lot under examination!In this paper we will derive some formulas for the properties of thiskind of acceptance sampling procedure, for inspection by attributes wheninspection is not perfect, there being a probability, p, of (correctly)declaring a defective item to be defective, and a probability, p', of(incorrectly) declaring a nondefective item to be defective.2. Link SamplingA procedure described by Harishchandra and Srivenkataramana (1982)(referred to as HS in the sequel) is as follows:Routine sampling takes random samples of size n (without replacement)from lots of size N. Denoting the number of items classified as defective(whether correctly or not) in the l-th lot by Zi;if Zi S a1  the lot is acceptedif Zi > a2  the lot is rejectedif aI < Zi & a2, the quantity Zi1 +Z+Z l+l - that is, the total numberof items found defective in the random samples of size n from the(l-l)-th, i-th, and (l+l)-th lots - is calculated, and-1I.9 ,if Zi.-+Zi +Zi+l < ai , the lot is acceptedif Zil +Zi +Z i+1 > aj , the lot is rejected.The numbers al,a2 and ai are integers chosen to give some desired probabilitiesof acceptance for specified numbers of defectives among the N items in thei-th lot. Commonly, but not necessarily, a2 = a'.3. DistributionsThe variables Zi l,zi,Z i+l are mutually independent. GenerallyZ. -(Bin(Y.,p) * Bin(n-Yjp')) A Hypg(n,Dj,N) (1)Yjwhere - means "is distributed as"* denotes "convolution"'A is a "mixing" or "compounding" symbol indicating the Y has theYdistribution following the symbol;0 is the number of defective items in the j-th lot;Y. is the number of (really) defective items in the sample of size nfrom the j-th lot;Bin(g,h) denotes the binomial distributionPr[X-x] = (x)hX(l-h)g ' x (0 < h < 1; x=O,l,...,g);and Hypg (g,h,k) denotes the hypergeometric distributionPr[X=xJ = (h X / k) (max(O,g-k+h) <_ x s. min(g,h))4. Calculation of Acceptance ProbabilitiesFrom (1) we can obtain the explicit expressionI;i* . . . .... .-. . . . .. . I I I I I-3-Pr[Zj = = z uN y y n.y uOu)n 2u(2)= P(zID), say. (0 . z S n)The probability of acceptance of the i-th lot isPr[Z i S a,] + PrC(a 1 < Zi s a2 ) (Zi.1 +Zi +Zi+ 1 :S ak)]I Z P(zilD i) + I P(zilD i) 11 P(zi- 1 IDi-l)P(zi+llli+l) (3)zijaI  al<zi<a2 Zi 'a+Z i+l -ziFor the link sampling method to be useful, it is necessary that Di.Di and Di+ l do not differ too greatly. In the case of binomial sampling(corresponding to N - -, the case considered in HS), if the lot proportiondefective is the same in all three lots, the acceptance probabilities forlink sampling are the same as they would be for regular two-stage sampling(with the same values of al,a 2 and aj) with the second sample (of size 2n)being chosen, when needed, from the i-th lot. A similar result is not validwhen N is finite (as it is in this paper), even when Di l=Di=D+1 (=D, say)because the convolutionHypg (n,D,N) * Hypg (n,DN) (4)is not the same as the distributionHypg. (2n,D,N)(or, indeed, as Hypg (2n,2D,2N).)5. Partial Link Sampling ProceduresIt may sometimes be a drawback, in the link sampling procedure, thatit is necessary to wait for the results of inspection of the (i+l)-th lotbefore reaching a decision on the i-th lot, if the sample from the i-th9X; a, -I i9-4-lot is inconclusive. One way around this difficulty would be to replace(Zi1 +Zl+Z +1 ) by (Z i.+Z i), which can be calculated immediately. It may befelt that this is straining the assumption of (roughly) constant 0 valuesunduly. In order to meet this difficulty HS propose the use of a partiallinking sampling procedure in which a second sample of size n (not 2n) istaken from the i-th lot and used in place of the sample from the (i+l)-thlot is reaching a decision. This means that (Zi 1 +Zi+Zi+1) is replaced by(Zi_+Zi+ZL), where Z! denotes the number of items found to be defective(whether correctly or not) is the second sample from the i-th lot. Theacceptance probability isPr[Z i a1) + Pr[(a1 < Zi j a2) n (Zil+Zi+Z :. aj)] (5)Now, of course, Zi and ZI are not independent, though Zi. I is independentof (Zi,Z!). The joint distribution of Zi and Z!, obtained from equation(4) of Kotz and Johnson (1983) by putting n1 = n2 = n, is[Z)] _Bin(Ylp *Bin(n.Yl,p') Mult. Hypg(O;n,n;N) (6)Z! Bin(Y ,p) *Bin(n-Y2,P')JY 1,Y2Here Mult. Hypg (D;n,n;N) is a multivariate hypergeometric distribution, withPrE(y y1) n (y2=Y2 ) = (n) (y )(I..y 2)/ ) (7)(0 -' y,y2 . n; D-N+2n -' y1+y2 s D)[ N N! .( n n , 2 n ( n ) 1 3 2 ( N ' 2n ) !The expected number of items inspected in the i-th lot isn{l+Pr[a1 < Zi I a2]1 - i[n+n{1+2 Pr[a, < Z. a2]}1while with regular two-stage sampling (with second sample size 2n) it is,(- --5-n{l+2Pr[a1 < Z i  a2]}[HS also describe another modification in which Z! replaces Zi. ratherthan Zi+ I . The analysis is exactly similar to that set out above].6. TablesTable 1 gives acceptance probabilities for both link sampling and partiallink sampling for aI = 1, a2 = ai = 5 (the values used in HS); and N = 100,200; n = 20, 50; p = 1, 0.9. 0.75; p' = 0, 0.1 and four sets of values ofproportions defectiveDiI/N Di/N Di+I0.05 0.05 0.050.1 0.1 0.10.05 0.1 0.150.15 0.1 0.15These tables were computed by RNR. The multivariate hypergeometric distri-bution in (7) can be computed by calculating its logarithm, using the factthat Mn(k!) = tnr(k+l) and calling a log-gamma subroutine repeatedly. However,, i (this can result in a severe loss of significant digits, even when the log-gammafunction is evaluated in double precision.A better alternative (used in the construction of Table 1) is to re-express(7) as the product of two univariate hypergeometric probabilities:Prt(Yy) n (Ymy) F yl+y2 2n-(y1+y2) y2) -y 8l'YI 2n2)] N 2L -1 I L (8)I. ... ... ..I -6-The equivalence of (7) and (8) is not obvious, but it is easily establishedalgebraically or probabilistically. The advantage of computing (8) ratherthan (7) is that subroutines for the univariate hypergeometric distributionare available in a number of high-level computing languages.The values in the tables exhibit the very marked Influence of p' on theacceptance probabilities.Table 2 contains acceptance probabilities for standard double sampling,with the same parameter values (a1 = 1, a2 = aj = 5) as in Table 1. Thesize of the first sample is n = 20 and the second sample (taken from thesame (i-th) lot) is n" = 2n = 40. The probability of acceptance isPr[Z i  a,] + Pr[(al < Zi 5 a') n (Zi+Z s a2 (9)2 1 82)]where Z' denotes the number of item judged to be defective in the secondIsample (size 2n") from the i-th lot. The joint distribution of Yi and Y",1 Ithe actual numbers defective in the two samples is the multivariate hyper-geometricn n" {N-n'-n"Pr[(Y.=y) n Y=y")] = (y)(,), y"M)/(N) (10)I 1 yyD-y-y D(n+n" N n' n" N) 1 (n)(n")/(n+n")) (10)'y+yh (DyY,. )/(y)} y y y+y',[Given Yi and Y' Zi and Z 7 are distributed independently as in (Z)The figures in Table 2 should be compared with those in Table 1 for n = 20and Di1 /N=Di/N-Di+I/N (- 0.05, 0.1). As is to be expected the partiallink sampling acceptance probabilities fall between the values for link samp-ling and standard double sampling. It appears that they are closer to the linksampling values than to the standard values. The differences decrease as thelot size increases (and would be zero for infinite lot size).hti,,I , i ...-7-AcknowledgementSamuel Kotz's research was supported by the U.S. Office of NavalResearch under Contract NOO14-81-K-0301.REFERENCESHarishchandra, K. and Srivenkataramana, T. (1982). "Link sampling forattributes, Commun. Statist. Theor. Meth., 11, 185-1868.Kotz, S. and Johnson, N.L. (1984). "Effects of false and incompleteidentification of defective items on the reliability of acceptancesampling" (to appear in Operations Research.)I48Table 1: Probabilities of Acceptance forLink Sampling (LS) and Partial Link Sampling (PLS)Acceptance numbers a81 = 1; a 2 -a2 = 5Probability of AcceptanceN n D 11I/N D 1/N D 1~1 /N p pLS PLS (LS and PLS)100 20 0.05 0.05 0.05 1 0 .9572 .9752 .73951 0.1 .2312 .2302 .18180.9 0 .9733 .9854 .78020.9 0.1 .2598 .2591 .20060.75 0 .9887 .9943 .83790.75 0.1 .3067 .3065 .23080.1 0.1 0.1 1 0 .5366 .5464 .36301 0.1 .0808 .0797 .07350.9 0 .6385 .6535 .42970.9 0.1 .1043 .1031 .09220.75 0 .7837 .8012 .53960.75 0.1 .1504 .1494 .12690.05 0.1 0.15 1 0 .5348 .6866 .36301 0.1 .0806 .0880 .07350.9 0 .6373 .7788 .42970.9 0.1 .1040 .1144 .09220.75 0 .7835 .8879 .53960.75 0.1 .1502 .1652 .12690.15 0.1 0.15 1 0 .4037 .4547 .36301 0.1 .0746 .0760 .07350.9 0 .4939 .5585 .42970.9 0.1 .0948 .0973 .09220.75 0 .6474 .7202 .53960.75 0.1 .1343 .1396 .12690.05 0.05 0.05 1 0 .9503 .9589 .73721 0.1 .2376 .2371 .18680.9 0 .9680 .9741 .77630.9 0.1 .2649 .2645 .20460.75 0 .9857 .9889 .83270.75 0.1 .3102 .3101 .23360.1 0.1 0.1 1 0 .5504 .5550 .37821 0.1 .0872 .0867 .07890.9 0 .6456 .6522 .44120.9 0.1 .1102 .1097 .09720.75 0 .7833 .7910 .54570.75 0.1 .1553 .1548 .1308wat _9Probability of AcceptanceOn Ist SampleN n Di 1/N Di/N Di+ /N p LS PLS (LS and PLS)0.05 0.1 0.15 1 0 .5486 .6782 .37821 0.1 .0870 .0959 .07890.9 0 .6444 .7637 .44120.9 0.1 .1099 .1216 .09720.75 0 .7830 .8717 .54570.75 0.1 .1551 .1709 .13080.15 0.1 0.15 1 0 .4225 .4713 .37821 0.1 .0804 .0822 .07890.9 0 .5081 .5665 .44120.9 0.1 .1002 .1032 .09720.75 0 .6538 .7168 .54570.75 0.1 .1388 .1447 .1308100 50 0.05 0.05 0.05 1 0 .2517 .2041 .18111 0.1 .0018 .0018 .00180.9 0 .3701 .3543 .25120.9 0.1 .0029 .0029 .00290.75 0 .5804 .6163 .37860.75 0.1 .0052 .0052 .00520.1 0.1 0.1 1 0 .0079 .0078 .00781 0.1 .0001 .0001 .00010.9 0 .0195 .0191 .01910.9 0.1 .0002 .0002 .00020.75 0 .0631 .0594 .05810.75 0.1 .0007 .0007 .00070.05 0.1 0.15 1 0 .0079 .0078 .00781 0.1 .0001 .0001 .00010.9 0 .0195 .0192 .01910.9 0.1 .0002 .0002 .00020.75 0 .0629 .0679 .05810.75 0.1 .0007 .0007 .00070.15 0.1 0.15 1 0 .0078 .0078 .00781 0.1 .0001 .0001 .00010.9 0 .0191 .0191 .01910.9 0.1 .0002 .0002 .00020.75 0 .0582 .0582 .05810.75 0.1 .0007 .0007 .0007,," t, . . - -10Probability of AcceptanceOn Ist SampleN n DiI/N Di/N Di+I/N p p LS PLS (LS and PLS)200 50 0.05 0.05 0.05 1 0 .3250 .3198 .23681 0.1 .0028 .0028 .00280.9 0 .4273 .4290 .29910.9 0.1 .0039 .0039 .00390.75 0 .6051 .6180 .41080.75 0.1 .0062 .0062 .00620.1 0.1 0.1 1 0 .0197 .0195 .01941 0.1 .0002 .0002 .00020.9 0 .0368 .0361 .03540.9 0.1 .0004 .0004 .00040.75 0 .0895 .0872 .08070.75 0.1 .0010 .0010 .00100.05 0.1 0.15 1 0 .0197 .0207 .01941 0.1 .0002 .0002- .00020.9 0 .0367 .0408 .03540.9 0.1 .0004 .0004 .00040.75 0 .0892 .1119 .08070.75 0.1 .0010 .0010 .00100.15 0.1 0.15 1 0 .0194 .0194 .01941 0.1 .0002 .0002 .00020.9 0 .0355 .0355 .03540.9 0.1 .0004 .0004 .00040.75 0 .0810 .08i9 .08070.75 0.1 .0010 .0010 .0010,__ _ _ _ __ __ _Table 2: Acceptance Probabilities With Regular Double SamplingIn all cases n =20, n'11=40; . a 1 2=a 2 '=5N p p Di/N =0.05 Di/NO= .1100 1 0 1.0000 0.53051 0.1 0.2241 0.07690.9 0 1.0000 0.65650.9 0.1 0.2542 0.09970.75 0 1.0000 0.82020.75 0.1 0.3033 0.1458

Inspection errors in link sampling

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Inspection errors in link sampling

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