Acceptance Probabilities for Link Sampling

 /****************************************************************/
 /*          S A S   S A M P L E   L I B R A R Y                 */
 /*                                                              */
 /*    NAME: IELINK                                              */
 /*   TITLE: Acceptance Probabilities for Link Sampling          */
 /* PRODUCT: QC                                                  */
 /*  SYSTEM: ALL                                                 */
 /*    KEYS: Inspection Sampling,                                */
 /*   PROCS: TABULATE                                            */
 /*    DATA:                                                     */
 /*                                                              */
 /*    MISC:                                                     */
 /*                                                              */
 /*   NOTES: This program tabulates the acceptance probability   */
 /*          for link sampling from a multiple lots under an     */
 /*          imperfect inspection model.                         */
 /*                                                              */
 /*          Notation:                                           */
 /*                                                              */
 /*          nlot    = size of lot                               */
 /*          d1      = number of nonconforming items in lot i-1  */
 /*          d2      = number of nonconforming items in lot i    */
 /*          d3      = number of nonconforming items in lot i+1  */
 /*                                                              */
 /*          z1      = number of apparent defectives in sample   */
 /*                    from (i-1)st lot                          */
 /*          z2      = number of apparent defectives in sample   */
 /*                    from ith lot                              */
 /*          z3      = number of apparent defectives in sample   */
 /*                    from (i+1)st lot                          */
 /*                                                              */
 /*          nsample = sample size                               */
 /*                                                              */
 /*          c1      = acceptance number                         */
 /*          c2      = rejection  number                         */
 /*          c2p     = acceptance number                         */
 /*                                                              */
 /*          p       = Pr[ nonconforming item is classified      */
 /*                        as nonconforming ]                    */
 /*          pprime  = Pr[ conforming item is classified as      */
 /*                        nonconforming ]                       */
 /*                                                              */
 /*          accprob = Pr[ acceptance ]                          */
 /*                                                              */
 /*                                                              */
 /*          General Procedure:                                  */
 /*                                                              */
 /*          Take a random sample of size n from the ith lot.    */
 /*          Record the apparent number Zi of defective items.   */
 /*                                                              */
 /*          (a) If Zi <= c1 then accept.                        */
 /*                                                              */
 /*          (b) If Zi > c1 then reject.                         */
 /*                                                              */
 /*          (c) If c1 < Zi <= c2, then if                       */
 /*                                                              */
 /*              Zi-1 + Zi + Zi+1 <= c2p,                        */
 /*                                                              */
 /*              then accept.                                    */
 /*                                                              */
 /*          (d) If c1 < Zi <= c2, then if                       */
 /*                                                              */
 /*              Zi-1 + Zi + Zi+1 > c2p,                         */
 /*                                                              */
 /*              then reject.                                    */
 /*                                                              */
 /*          Note that in the program, Zi-1 is denoted by z1,    */
 /*          Zi by z2, and Zi+1 by z3.                           */
 /*                                                              */
 /*                                                              */
 /*     REF: Johnson, N. L., Kotz, S., and Rodriguez, R. N.      */
 /*          (1986), Statistical Effects of Imperfect Inspection */
 /*          Sampling:  II. Double Sampling and Link Sampling,   */
 /*          Journal of Quality Technology 18, 116-138.          */
 /*          See Table 4.                                        */
 /*                                                              */
 /*          Johnson, N. L., Kotz, S., and Wu, X. (1991).        */
 /*          Inspection Errors for Attributes in Quality         */
 /*          Control.  London:  Chapman & Hall.  See Chapter 4.  */
 /*                                                              */
 /****************************************************************/

data table;

   keep nlot d1 d2 d3 nsample c1 c2 c2p p pprime accprob;

   label nlot    = 'N (lot)'
         d1      = 'D(i-1)'
         d2      = 'D(i)'
         d3      = 'D(i+1)'
         nsample = 'n'
         c1      = 'c1'
         c2      = 'c2'
         c2p     = 'c2'''
         p       = 'p'
         pprime  = 'p'''
         accprob = 'Pr[ Accept ]';

   format zprob   6.4
          accprob 6.4 ;

   /*---set main parameters---*/
   nlot    = 100;
   nsample = 20;
   d1      = 15;
   d2      = 10;
   d3      = 15;
   c1      = 1;
   c2      = 5;
   c2p     = 5;

   /*---loop over p values---*/
   do p = 0.75, 0.90, 0.95, 0.98, 1.00;

      /*---loop over pprime values---*/
      do pprime = 0.0, 0.01, 0.02, 0.05, 0.10;

         /*---compute term1---*/
         link first;

         /*---computer term2---*/
         link second;

         accprob = term1 + term2;

         output;

         end;  /* finish loop over pprime values */

      end;  /* finish loop over p values */

   return;  /* finish main program */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability Pr[ Z2 <= c1 ]        */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    c1      = acceptance value for first sample             */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    n1      = first sample size                             */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    term1   = Pr[ Z2 <= c1 ]                                */
   /*                                                            */
   /*------------------------------------------------------------*/
   first:

   term1 = 0.0 ;
   do z2 = 0 to c1 by 1;

      /* nlot, nsample, p, pprime are globally defined */
      z = z2;
      d = d2;
      link uncond;

      term1 = term1 + zprob;

      end;

   return;  /* finish first */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability                       */
   /*                                                            */
   /*    term2   = Pr[ c1 < Z2 <= c2, Z1 + Z2 + Z3 <= c2p ]      */
   /*                                                            */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    c1      = acceptance value                              */
   /*    c2      = acceptance value                              */
   /*    c2p     = acceptance value                              */
   /*    nlot    = lot size                                      */
   /*    d1      = number of defectives in lot i-1               */
   /*    d2      = number of defectives in lot i                 */
   /*    d3      = number of defectives in lot i+1               */
   /*    nsample = sample size                                   */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /*                                                            */
   /*------------------------------------------------------------*/
   second:

   term2 = 0.0 ;
   zimax = max( c1, c2, c2p );

   do z1 = 0 to zimax by 1;

      do z2 = 0 to zimax by 1;

         do z3 = 0 to zimax by 1;

            if ( c1 < z2 ) & ( z2 <= c2 ) & ( z1 + z2 + z3 <= c2p )
            then do;

               /* nlot, nsample, p, pprime are globally defined */
               z = z1;
               d = d1;
               link uncond;
               product = zprob;

               z = z2;
               d = d2;
               link uncond;
               product = product * zprob;

               z = z3;
               d = d3;
               link uncond;
               product = product * zprob;

               term2 = term2 + product;
               end;

            end;

         end;

      end;

   return;  /* finish second */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability Pr[ Z = z ], where Z  */
   /* is the number of items classified as defective.            */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    z       = number of items classified as defective       */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    nsample = sample size                                   */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    zprob   = Pr[ Z = z ]                                   */
   /*                                                            */
   /*------------------------------------------------------------*/
   uncond:

   /*---used for roundoff---*/
   fuzz = 0.0001 ;

   /*---lower and upper limits for y---*/
   miny = max( 0, nsample + d - nlot );
   maxy = min( nsample, d );

   /*---initialize probability to zero---*/
   zprob = 0.0 ;

   /*---Case I: p = 0 ---*/
   if p = 0 then do;

      /*---Ia: pprime = 0 ---*/
      if pprime = 0 then do;

         if z = 0 then zprob = 1 ;

         end;  /* finish Ia */

      /*---Ib: pprime = 1 ---*/
      else if abs( pprime - 1 ) < fuzz then do;

         minz = max( 0, nsample - d );
         maxz = min( nsample, nlot - d );

         if ( minz <= z ) & ( z <= maxz ) then do;

            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = nsample - z;
            link hypergmt;

            zprob = hypprob;

            end;

         end;  /* finish Ib */

      /*---Ic:  0 < pprime < 1 ---*/
      else do;

         /* Note: minz =  0 */
         maxz = nsample - max( 0, nsample + d - nlot );

         if ( z <= maxz ) then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            n_ = nsample - y;
            k_ = z;
            p_ = pprime;
            link binomial;

            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish Ic */

      end;  /* finish Case I */


   /*---Case II:  p = 1 ---*/
   else if ( abs( p - 1 ) < fuzz ) then do;

      /*---IIa:  pprime = 0 (perfect inspection) ---*/
      if pprime = 0 then do;

         minz = max( 0, nsample + d - nlot );
         maxz = min( nsample, d );

         if ( minz <= z ) & ( z <= maxz ) then do;

            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = z;
            link hypergmt;

            zprob = hypprob;

            end;

         end;  /* finish IIa */

      /*---IIb:  pprime = 1 ---*/
      else if ( abs( pprime - 1 ) < fuzz ) then do;

         if z = nsample then zprob = 1 ;

         end;  /* finish IIb */

      /*---IIc:  0 < pprime < 1 ---*/
      else do;

         minz = max( 0, nsample + d - nlot );
         maxz = nsample ;

         if ( minz <= z ) & ( z <= maxz ) then
         do y = miny to maxy by 1;

            /*---compute Pr[ Y = y ] ---*/
            bign_ = nlot ;
            litn_ = nsample ;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = pprime;
            k_ = z - y;
            n_ = nsample - y;
            link binomial;

            zprob = zprob + hypprob * binprob;

            end;

         end;  /* finish IIb */

      end;  /* finish Case II */


   /*---Case III:  0 < p < 1---*/
   else do;

      /*---IIIa:  pprime = 0 ---*/
      if pprime = 0 then do;

         /* zmin = 0 */
         zmax = min( nsample, d );

         if z <= zmax then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot ;
            litn_ = nsample ;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = p;
            k_ = z;
            n_ = y;
            link binomial;

            /*---increment unconditional probability---*/
            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish IIIa */

      /*---IIIb:  pprime = 1 ---*/
      else if abs( pprime - 1 ) < fuzz then do;

         zmin = nsample - min( nsample, d );
         /* zmax = nsample */

         if z >= zmin then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = p;
            k_ = z - ( nsample - y );
            n_ = y;
            link binomial;

            /*---increment unconditional probability---*/
            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish Case IIIb */

      /*---IIIc:  0 < pprime < 1 ---*/
      else
      do y = miny to maxy by 1;

         /*---obtain Pr[ Y = y ]---*/
         bign_ = nlot;
         litn_ = nsample;
         d_    = d;
         y_    = y;
         link hypergmt;

         /*---obtain Pr[ Z = z | Y = y ]---*/
         condprob = 0.0 ;
         minx     = max( 0, y + z - nsample );
         maxx     = min( y, z );

         do x = minx to maxx by 1;

            p_ = p;
            k_ = x;
            n_ = y;
            link binomial;
            factor1 = binprob;

            p_ = pprime;
            k_ = z - x;
            n_ = nsample - y;
            link binomial;
            factor2 = binprob;

            condprob = condprob + factor1 * factor2;

            end;

         /*---increment unconditional probability---*/
         zprob = zprob + condprob * hypprob ;

         end;  /* finish IIIc */

      end;  /* finish Case III */

   return;  /* finish uncond */

   /*---Compute Binomial Probability---*/
   binomial:

   binprob=0.0;

   if n_ = 0 then do;

      if k_ = 0 then binprob = 1.0 ;

      end;

   else
   if n_ > 0 then do;

      if ( k_ > 0 ) & ( k_ < n_ ) then
         binprob = probbnml( p_, n_, k_ ) -
                   probbnml( p_, n_, k_-1 );

      else
      if k_ = n_ then do;
         if ( p_> 0.0 ) & ( p_ < 1.0 ) then
            binprob = p_**n_;
         else if p_ = 1.0 then
            binprob = 1.0;
         end;

      else
      if k_ = 0 then do;
         if ( p_ > 0.0 ) & ( p_ < 1.0 ) then
            binprob = (1.0 - p_)**n_;
         else if p_ = 0.0 then
            binprob = 1.0;
         end;

      end;

   /*---finish binomial computation---*/
   return;

   /*---Compute Hypergeometric Probability---*/
   hypergmt:

      hypprob = 0 ;
      minarg  = max( 0, litn_ + d_ - bign_ );
      maxarg  = min( litn_, d_ );

      if y_ = minarg then

         hypprob = probhypr( bign_, d_, litn_, y_ );

      else
      if ( minarg < y_ ) & ( y_ <= maxarg ) then

         hypprob = probhypr( bign_, d_, litn_, y_     ) -
                   probhypr( bign_, d_, litn_, y_ - 1 );

   /*---finish hypergeometric computation---*/
   return;

run;

proc sort data=table;
   by nlot d1 d2 d3 nsample c1 c2 c2p;

proc tabulate data=table noseps;
   by nlot d1 d2 d3 nsample c1 c2 c2p;
   class p pprime;
   var accprob;
   table p, pprime*accprob=' '*sum=' '*f=8.4 / rts=7;
run;